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4. Pricing and Valuation of Various Derivatives This chapter discusses the structure, pricing and valuation of various derivative instruments. It is divided into six parts. The first part discusses forward contracts. The second part covers swap pricing and application of interest rate swap and currency swap. The third part covers futures contract including forward rate agreement, interest rate futures, etc. The fourth part covers various options contracts with pricing and their application. The fifth part covers packaged forward contracts. Lastly, Futures Options contracts are discussed. 4.1 Forward Contracts: Forward contract can be defined as an agreement to buy or sell an asset on a specified day for a specified exercise price. The underlying asset can be commodity, currency etc. One of the parties to the contract assumes a long position and therefore he agrees to buy the underlying asset on a certain specified future date for a certain specified price. The other party assumes a short position and therefore agrees to sell the same asset on the same date and for the same price. The forward contracts are normally traded outside the exchange (OTC). Forward contracts are privately negotiated and are not standardized. Further, the two parties must bear each other's credit risk. Since the contracts are not exchange traded, there is no marking to market requirement, which allows a buyer to avoid almost all capital outflows initially (though some counter-parties might set collateral requirements). Thus because of the lack of standardization in these contracts, there is very little scope for a secondary market in forwards. 4.1.1 Salient Features of Forward:

NCFM derivatives module has enumerated the following salient features of forward contracts:

• They are bilateral contracts and hence exposed to counter - party risk. • Each contract is custom designed, and hence is unique in terms of contract size,

expiration date and the asset type and quality. • The contract price is generally not available in public domain. • The contract has to be settled by delivery of the asset on expiration date. • In case, the party wishes to reverse the contract, it has to compulsorily go to the same

counter party, which being in a dominant situation can command the price it wants.

4.1.2 Currency forwards:

Forward contracts in which the underlying asset is currency exchange rate, is called currency forward. It is mainly used by those who have foreign currency inwards or outwards at a future date i.e. importers, exporters and bankers/dealers. An important segment of the forex derivatives market in India is the Rupee forward contracts market. This has been growing rapidly with increasing participation from corporates, exporters, importers, banks and FIIs. Till February 1992, forward contracts were permitted only against trade related exposures and these contracts could not be cancelled except where the underlying transactions failed to materialize. In March

52

1992, in order to provide operational freedom to corporate entities, unrestricted booking and cancellation of forward contracts for all genuine exposures, whether trade related or not, were permitted. Although due to the Asian crisis, freedom to rebook cancelled contracts was suspended, which has been since relaxed for the exporters but the restriction still remains for the importers. Some of the RBI guidelines regarding forward contract are furnished below:

1. Residents: Genuine underlying exposures out of trade/business o Exposures due to foreign currency loans and bonds approved by RBI o Balances in EEFC accounts

2. Foreign Institutional Investors: o They should have exposures in India

3. Non-resident Indians/ Overseas Corporates: o Dividends from holdings in a Indian company o Deposits in FCNR and NRE accounts o Investments under portfolio scheme in accordance with FERA or FEMA

The forward contracts are also allowed to be booked for foreign currencies (other than Dollar) and Rupee subject to similar conditions as mentioned above. The banks are also allowed to enter into forward contracts to manage their assets - liability portfolio. The cancellation and rebooking of the forward contracts is permitted only for genuine exposures out of trade/business up to one year for both exporters and importers, whereas in case of exposures of more than one year, only the exporters are permitted to cancel and rebook the contracts. Also another restriction on booking the forward contracts is that the maturity of the hedge should not exceed the maturity of the underlying transaction.

Cross currency forwards are also used to hedge the foreign currency exposures, especially by some of the big Indian Corporates. The regulations for the cross currency forwards are quite similar to those of Rupee forwards, though with minor differences. For example, a corporate having underlying exposure in Yen, may book forward contract between Dollar and Sterling. Here even though its exposure is in Yen, it is also exposed to the movements in Dollar vis a vis other currencies. The regulations for rebooking and cancellation of these contracts are also relatively relaxed. The activity in this segment is likely to increase with increasing convertibility of the capital account.

4.1.3 Pricing of forward contracts:

For arriving at forward price, the cost of carry method can be applied.

If the investor does not book a forward contract, the alternative for him is to buy at the spot market and hold the underlying asset. In such a contingency he would incur the spot price plus the cost of carry. The cost of carry refers to the difference between the costs and the benefits that accrue while holding an asset. Suppose a groundnut oil producer needs 50000 kg of nuts for processing in two months. To lock in the price of the groundnut today, he can buy it and carry it for two months. One cost of this strategy is the opportunity cost of funds. To come up with the purchase price, he must either borrow money or reduce his earning assets by that amount. Beyond interest cost, however, carry costs vary depending upon the nature of the asset. For a

53

physical asset such as groundnut, he incurs storage costs (e.g., rent and insurance). At the same time, by storing groundnut, he avoids the costs of possibly running out of his regular inventory before two months are up and having to pay extra for emergency deliveries. This benefit is called convenience yield. Convenience yield is a reward for those who keep the advance stock of underlying asset and therefore avoid speculative price in case of emergency purchase. Thus, the cost of carry for a physical asset equals interest cost plus storage costs less convenience yield, that is,

Carry costs = Cost of funds + storage cost - convenience yield.

For a financial asset, such as a stock or a bond (interest bearing security), storage costs are negligible. Moreover, income (yield) accrues in the form of quarterly cash dividends or semi-annual coupon payments. Thus the cost of carry for a financial asset can be modified from the previous formula as follows:

Carry costs = Cost of funds – income accrued.

Carry costs and benefits are modeled either as continuous rates or as discrete flows. Some costs/benefits such as the cost of funds (i.e., the risk-free interest rate(r)) are best modeled continuously. The dividend yield on a broadly based stock portfolio and the interest income on a foreign currency deposit also fall into this category. Other costs/benefits like non-annual cash dividends on individual common stocks, semi-annual coupons on bonds, and warehouse rent payments for holding an inventory of grain are best modeled as discrete cash flows. In the interest of conciseness, only continuous costs are considered here.

Dividend income from holding a broadly based stock index portfolio or interest income from holding a foreign currency (i.e. (i)) is typically considered as a constant, continuous rate. The income, as it accrues, is re-invested in more units of the asset. In this way, buying e(-iT ) units of a stock index portfolio today grows to exactly one unit at time T, and produces a net terminal value of _ST - S e((r - i) T). For a stock index portfolio investment, the cost of carry rate equals the difference between the risk-free rate of interest r and the dividend yield rate i; and equals the difference between the domestic interest rate r and the foreign interest rate i for a foreign currency investment. The total cost of carry paid at time T is

Carry costs = S [e((r - i) T) – 1] . (1)

The value of a forward contract is linked to the cost of carry of the underlying asset. Since a forward contract requires its buyer to accept delivery of the underlying asset at time T, buying a forward contract today is a perfect substitute for buying the asset today and carrying it until time T. The present value of the payment obligation under the forward contract strategy is f e(-rT), and the present value of the latter strategy is S e(-iT). Since both strategies provide exactly one unit of the asset at time T, (i.e., _ST ), their costs must be identical,

f e(-rT) = S e(-iT) . (2a)

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If the relation (2a) does not hold, costless arbitrage profits would be possible by selling the over-priced instrument and simultaneously buying the under-priced one. The relation (2a) is the present value version of the cost of carry relation. A more familiar version is the future value form,

f = S e[(r - i) T]. (2b)

When the prices of the forward and the asset are such that Equation (2a) and/or Equation (2b) hold exactly, the forward market is said to be at full carry. Unless costless arbitrage is somehow impeded, the forward market will always be at full carry. The difference between the forward price and the asset price is frequently referred to as the basis.

4.1.4 Non Deliverable Forward Contracts: An NDF is a short-term committed forward ‘cash settlement’ currency derivative instrument. It is essentially an outright (forward) FX contract whereby on the contracted settlement date, profit or loss is adjusted between the two counterparties based on the difference between the contracted NDF rate and the prevailing spot FX rates on an agreed notional amount. Non-Deliverable Forward (NDF) has become a popular instrument available to corporate treasurers who wish to hedge their exposure to foreign currencies which are not internationally traded and which do not possess a forward market for non-domestic players like Indian Rupee (INR), Philippine Peso (INR), Taiwan Dollars (TWD), Korean Won (KRW), Indonesian rupiah (IDR) and Chinese Renminbi (CNY). NDFs are also distinct from deliverable forwards in that NDFs trade outside the direct jurisdiction of the authorities of the corresponding currencies The NDF markets for some Asian currencies have existed at least since the mid 1990’s. NDFs are commonly quoted for time periods of one month up to one year, and are normally quoted and settled in U.S. dollars. Thus they have become a popular instrument for corporations seeking to hedge exposure to foreign currencies that are not internationally traded.(Singhania K., 2004) The NDF market is typically an offshore market, free from regulatory control of the currency’s home monetary authority. New York, Singapore, and London are major centers with the first two specializing in Latin American and Asian currencies respectively and the third across both sets. Hong Kong is an important trading centre for Asian currency NDFs as well. In 2003, six Asian Currencies – the Korean Won, Chinese Renminbi, New Taiwan Dollar, Indonesian Rupiah, Philippine Peso, and the Indian Rupee – constituted a majority of global NDF markets with the remaining volume coming largely from Latin American currencies and the Russian Rubble. For the Indian Rupee, NDFs are traded primarily in Singapore and Hong Kong with Dubai and Bahrain with comparatively low volume. The NDF market for the Indian Rupee started back in the 1990’s when it provided the foreign investors in India the only avenue of hedging currency risk, in the presence of severe exchange restrictions in a scenario where the Rupee was expected to have a secular decline. Foreign investors would generally sell the NDF Rupee contracts to hedge their underlying positions. The

55

opposite side would typically be taken by Indian trading companies and exporters who could make arbitrage profits as they had access to both the onshore currency markets as well as Dollar flows outside the country. Multinationals also use the Indian Rupee NDF market to hedge their exposure. There is also a demand from arbitrageurs playing the two forward markets. Onshore financial institutions are prohibited from participating in the NDF market. Several major multinational banks like Deutsche Bank, UBS, and Citibank are active traders in the Rupee NDF market. The activity there has risen by over 7.5 times in the last few years while total foreign investment in India has roughly trebled during the period and with easing currency restrictions. Table 4.1: Currency wise access to onshore forward market by non-residents

Access to onshore forward markets by non-residents Chinese Renminbi No offshore entities participate in onshore markets

Indian Rupee Allowed but subject to underlying transactions requirements Indonesian Rupiah Allowed but restricted and limited

Korean won Allowed but subject to underlying transactions requirement Philippine peso Allowed but restricted and limited

New Taiwan dollar Only onshore entities have access to onshore market Source: Ma Guonam et al., HSBC (2003); national data. Generally participants take position in the INR NDF market based on their view on where the INR spot will be, after a certain time period in the on shore market. Entities who have access to both the market take advantage of the arbitrage opportunities, if available between both the markets. Arbitrage opportunity is generally available between the INR NDF market and the on shore market. Rupee dealers in India also keep track of the INR NDF market before taking a view on the INR. Though no authentic data are available, the daily average volume in INR NDF market is estimated to be around USD 150-200 million. Reference: BIS (www.bis.org) Example of NDF Assume on 5 January 2009, Hind Corporation sells INR 48.50 mio. NDF to Cosmic Bank three- months forward for value 6 April 2009, at the NDF rate of USD/INR 48.50. Rate fixing date 5th April 2009. By transacting the above, Hind has locked-in the 3-month forward INR selling rate at USD/INR 48.50, which is equivalent to Hind buying USD 1 mio. (INR 48.50 mio.). On 5th April 2009, the fixing date, both parties will compare the NDF rate with the prevailing USD/INR fixing rate as RBI USD/INR reference spot rate. There are 3 possible scenarios:- a) The prevailing USD/INR rate is equal to the NDF rate or b) The prevailing USD/INR rate is higher than the NDF rate or c) The prevailing USD/INR rate is lower than the NDF rate.

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Now after 3 months the prevailing USD/INR is 51.50 on 5th April 2009. In this instance, the INR has weakened and there is a difference of 3 INR. Hence Cosmic Bank will pay the difference to Hind Corporation on the settlement date (6th April 2009) As the settlement is in USD, Cosmic Bank will have to pay Hind Corporation the difference in USD; this can be worked out by the following method. On 5/1/2009 Hind sells INR 48.50 mio = USD 1,000,000.00 (as USD/INR 48.50) On 5/4/2009 the dollar equivalent will be (for INR 48.50 mio = USD 941,747.5728) (As the exchange rate is USD/INR 51.50) It yields the profit of USD 58,252.4272 to the party (Difference) Hence on 6/4/09, Cosmic Bank pays USD 58,252.4272 to Hind Corporation and the NDF is settled. Assuming the rate other than the realized rate If the USD/INR exchange rate were 43.50 on 5th April 2009. In this instance INR has strengthened and there is a difference of 5 INR. Hence Hind Corporation will pay the difference to Cosmic Bank on the settlement date (6th April 2009). On 5/1/09 Hind sells INR 48.50 mio = USD 1,000,000.00 USD/INR 48.50 On 5/4/09 INR 48.50 mio = USD 11,14,942.5287 (as the rate is USD/INR 43.50) Thus there will be loss to the party of USD 114942.53 (Difference) Hence on 6/4/09, Hind Corporation pays USD 114942.53 to Cosmic Bank and the NDF would have been settled for the given exchange rate. 4.2 SWAPS:

4.2.1 Introduction:

Swap is a derivative instrument. It is a transaction in which two parties agree to pay each other a series of cash flows over a specified period of time. The four popular kinds of swaps are currency swap, interest rate swap, equity swap and commodity swap. Over the years many varieties of swaps have evolved. The common types of swap involve one party making a series of fixed payments and receiving a series of floating payments. In some swap both the parties make floating payment with different bases. Swaps can be used to hedge certain risks such as interest rate risk, or to speculate on changes in the underlying prices. Most swaps are traded Over The Counter (OTC), "tailor-made" for the counter parties. Some types of swaps are also exchanged on futures markets, for instance Chicago Mercantile Exchange Holdings Inc., the largest U.S. futures market, the Chicago Board Options Exchange and Frankfurt-based Eurex AG.

The Bank for International Settlements (BIS) publishes statistics on the notional amounts outstanding in the OTC Derivatives market. At the end of 2006, this was USD 415.2 trillion (that is, more than 8.5 times the 2006 gross world product). The majority of this (USD 292.0 trillion) was due to interest rate swaps. These split by currency is as follows:

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Table: 4.2 Currency wise notional amounts outstanding in the OTC Derivatives market from 2000 to 2006.

Notional outstanding (Year End) in USD trillion

Currency 2000 2001 2002 2003 2004 2005 2006 Euro 16.6 20.9 31.5 44.7 59.3 81.4 112.1 US dollar 13.0 18.9 23.7 33.4 44.8 74.4 97.6 Japanese yen 11.1 10.1 12.8 17.4 21.5 25.6 38.0 Pound sterling 4.0 5.0 6.2 7.9 11.6 15.1 22.3 Swiss franc 1.1 1.2 1.5 2.0 2.7 3.3 3.5

(Source: www.bis.org)

If a swap transaction involves exchange of interest payments then it is known as an interest rate swap. The first interest rate swap took place in London in 1979 and further in 1981, Salomon Brothers negotiated a benchmark currency swap between IBM and the World Bank. Later in1984 banks started developing warehousing whereby a single counter party would approach bank and bank would play a role of counter party. A temporary hedge would be taken in the bond or futures market with opposite exposure until a suitable counter party could be found. Standard terms introduced by ISDA and BBA in 1985 also helped in growth in swap market.

Introduction of swaps in India: OTC rupee derivatives in the form of Forward Rate Agreements (FRAs)/Interest Rate Swaps (IRS) - were introduced by RBI in India in July 1999.

These derivatives enable banks, primary dealers (PDs) and all-India financial institutions (FIs) to hedge interest rate risk for their own balance sheet management and for market-making purposes. Banks/PDs/FIs can undertake different types of plain vanilla FRAs/IRS. Swaps having explicit/implicit option features such as caps/floors/collars are not permitted.

4.2.2 Interest Rate Swap:

There are two parties in a swap transaction, fixed rate payer/ receiver and floating rate receiver/ payer. A fixed rate payer is the provider of floating rate funds and vice a versa.

The following illustration will explain this point:

Suppose two firms A and B have been offered the following rates per annum on a Rs. 1 million five years loan from the market:

Table 4.3: Interest rates combination to two firms

Firm Fixed Rate Floating rate A 12% MIBOR + 1.0 % B 14% MIBOR + 2.0 %

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Firm A requires a floating rate loan and Firm B needs a fixed rate loan. Initially we will assume a direct dealing between these two parties; then after we will consider swap with intermediation of Bank. The first step in designing the swap deal is to calculate the quality differential spread as per absolute advantage or as per comparative advantage.

Table 4.4: Calculation of QSD for the Swap deal

Firm Fixed Rate Floating rate A 12% MIBOR + 1.0 % B 14% MIBOR + 2.0 % QSD 02 % - 01% = 1 %

Here there is comparative advantage to firm B in floating rate compared to fixed rate. Therefore the difference in both rates will be deducted to arrive at the spread differentials. If we pass on the benefits of swap equally between these two parties then each party would get a benefit of 0.5%. Therefore the net cost to firm B should be 14% - 0.5% = 13.5%. Similarly to firm A cost should be (MIBOR + 1%) – 0.5% = MIBOR + 0.5%. The swap can be structured as follows; Diagram 4.1: Swap deal between Firm A and Firm B. MIBOR + 0.5%

12%

12% MIBOR+ 2%

In the diagram Firm A borrows from the market Rs. 1 million @ 12% which is other than their objective i.e. floating rate loan and Firm B borrows @ MIBOR + 2% (floating) even though their objective is in Fixed loan. Now the firm A agrees to pay MIBOR + 0.5% to firm B in exchange of firm B’s acceptance of paying 12% fixed to firm A. which is the rate at which firm A has borrowed from the Market. Thus the net cost to firm A will be [-12% + 12% -(MIBOR + 0.5%)] = MIBOR + 0.5%. Similarly firm B pays to firm A 12% and receives MIBOR + 0.5% which will net their cost to –12% +(MIBOR + 0.5%) – (MIBOR +2%) = 13.5% which is less by 0.5% compared to 14% rate, if they individually go for loan borrowing in the market.

Interest rate swap with intermediation of Bank: Generally the bank’s intermediation is required to design any swap deal. In that case the Bank charges its commission. In the above example suppose Bank charges 0.2% p.a. as commission and both parties will share the

FIRM A FIRM B

Market Market

59

remaining spread equally i.e. 0.4% each, then the swap can be restructured as follows (other things are assumed as it is.)

Diagram 4.2: Swap deal between Firm A and Firm B with intermediary of Bank

MIBOR+ 0.6 % MIBOR+ 2 %

12% 13.6 % 12% MIBOR+ 2%

Table 4.5: Calculation of the net gain to the Bank:

BANK Fixed Floating RECEIPT 13.6% MIBOR+0.6% PAYMENT 12% MIBOR + 2 % NET + 1.6% - (1.4%) = (+0.2%)

Thus Bank is getting 0.2% commission from the spread and each party gets 0.4% of spread each. This swap is known as plain-vanilla swap. Other categories of swap are floating to floating or fixed-to- fixed (in different currencies).

FIRM A BANK FIRM B

Market Market

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4.2.3 Currency Swap: Currency swaps are used for yield enhancement as well as for risk containment. This can be explained from the following hypothetical situation. Suppose an Indian firm establishes one subsidiary in U.S. and one U.S. firm has one subsidiary in India. Now both these subsidiaries want funds for expansion/diversification. If the U.S. parent manages the funds of its subsidiary then it will borrow US$ and converts it into rupees at the prevailing exchange rate. The opposite will be the situation with Indian firm. If we assume that the requirement of funds of both the firms is equivalent, then they can avoid exchange rate risk by entering into swap agreement. This is because every year the subsidiary will generate revenues in the currency, where they are established. But the interest payment will have to be made in the currency of parent country and also the repayment of loan would also be done in parent country’s currency. This exposes both the subsidiary firms to exchange rate risk. Moreover if the subsidiary gets loan from the host country and if it is not very much known in the host country then they have to pay higher rate of interest than what rate is offered to the domestic company. Considering this situation if the Indian parent company borrows rupee loan for the U.S. subsidiary and similarly if the U.S. parent borrows loan for Indian subsidiary in US$ and then they swap the amounts to each other’s subsidiary. Thus both the firms could get benefit of this contract. Firstly they avoid exchange rate risk because there is no transition of funds cross border and secondly the domestic companies are offered loans at cheaper rates. After the exchange of principal, the U.S. subsidiary will pay regular rupee interest to the Indian firm and the Indian subsidiary will pay US$ interest amount regularly to the U.S. parent out of its revenue which is also in US$. At the end there will be re-exchange of principal amounts by both subsidiaries. Thus a currency swap generally involves the following steps:

• Initial exchange of principal • Exchange of interest rate • Re-exchange of the principal at the end of the contract.

A numerical example explaining the above situation is given below: Suppose company A is an Indian manufacturer, wishes to borrow U.S. $ at a fixed rate of interest for its business in California. Company B is a U.S. manufacturer established in Ahmedabad Special Economy Zone (SEZ) wishes to borrow rupees at a fixed rate of interest. Indian firm needs U.S.$ 10,00,000 while U.S. manufacturer wants Rs. 5,00,00,000 when the exchange rate is 1 US$ = INR 50.00. They have been quoted the following rates per annum:

Table 4.6: Interest rates offered to two firms

COMPANY RUPEE (%) DOLLAR (%) A 14.00% 7.00 % B 13.00% 4.80 %

Suppose the bank acting as an intermediary wants 20 basis points as swap charges and each party would share benefit equally. The swap can be designed as below.

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Table 4.7: Calculation of QSD for the Swap deal:

COMPANY RUPEE (%) DOLLAR (%) A 14.00% 7.00 % B 13.00% 4.80 % QSD 1.00 % - 2.20 % = (1.20%)

In this case, the U.S. Company (i.e. Company B) is in a dominant position as they are offered less interest in both the currencies. So comparative advantage theory is applicable and the spread would be considered as the difference (and not the addition) of two differences between the rates. Here the bank is charging 20 basis points i.e. 0.2% of total spread of 1.20% then each party /company would gain 50 basis points i.e. 0.5% each. Therefore the net cost to company A would be 6.50 % $ instead of 7% which is offered to them by U.S. market if they borrow on their own. Similarly company B would have a net cost of 12.5% INR. The swap deal can be explained with the following diagram. Diagram 4.3: Currency Swap deal between Firm A and Firm B with intermediary of Bank 6.50 %($) 4.8%($) 14% (Rs.) 12.5 %(Rs.) 14% (Rs) 4.8 %($) Table 4.8: Calculation of the net gain to the Bank for currency Swap

BANK INR (%) US$(%) RECEIPT 12.50 % 6.50 % PAYMENT 14.00 % 4.80 % NET (-1.5 %) + (1.7%) = 0.2 %

Thus Bank is gaining its agreed charges of 20 basis points while each company is gaining 50 basis points each in terms of cost reduction with swap. Many firms enter into swap agreement with notional principal without any direct hedging purpose but for the yield gain of spread from the swap.

COMPANY A BANK COMPANY B

MARKET MARKET

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4.2.4 Limitations of Swap:

• It is difficult to choose the opposite party to the swap transaction for the swap dealer when one party approaches him.

• The swap deal can not be terminated without the agreement of the parties involved in the transaction.

• The default party risk prevails, as no mark to marking is there in swap. • Less growth of the secondary market for swap, that reduces liquidity. • The swap market is not exchange controlled and it is an OTC market. Therefore

assessment of creditworthiness requires extra time and cost.

4.2.5 Valuation of Swaps:

The value of a swap is the net present value (NPV) of all future cash flows. Initially, the terms of a swap contract are such that the NPV of all future cash flows is equal to zero.

In a plain vanilla fixed-to-floating interest rate swap, where Company A pays a fixed rate and Company B pays a floating rate. In such an agreement the fixed rate would be such that the present value of future fixed rate payments by Company A are equal to the present value of the expected future floating rate payments (i.e. the NPV is zero). Where this is not the case, an Arbitrageur, X, will:

1. Assume the position with the lower present value of payments, and borrow funds equal to this present value

2. Meet the cash flow obligations on the position by using the borrowed funds, and receive the corresponding payments - which have a higher present value

3. Use the received payments to repay the debt on the borrowed funds 4. Earn the difference - where the difference between the present value of the loan and the

present value of the inflows is the arbitrage profit.

Thus a swap can be assumed as the portfolio of borrowing and lending. A firm which converts its fixed rate obligations into floating rate can be compared with a firm that has raised bonds with a fixed coupon rate and then invest in the floating rate deposits.(Chance Don, 2004)

The valuation of a currency swap can be explained with numerical example as follows:

Suppose an American firm has an existing swap agreement with a British firm. The original exchange rate was US$ 1.35 = 1 ₤. Now currency exchange rate is US$ 1.30 = 1₤. The fixed interest rates for the swap are 8% for Sterling and 5% for US$. Interest payments are annual and such payment has just been exchanged. The swap has a remaining life of four years. The American firm is a recipient of sterling (₤) and a payer of dollars. The original amounts agreed were US$ 13.5 million and ₤ 10 million. At present the interest rate in US$ is 4% and in sterling is 6%.

From this information we can calculate the value of the swap to both the firms by calculating the present values of their cash inflows and cash outflows. Then by calculating their difference in

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one currency we can comment on the profit or loss to both these parties. As it is a zero sum deal, the profit of one firm will become loss to the other firm.

We consider the payoff from the U.S. firm’s point of view. As they are recipient of sterling (₤), their cash inflows are in ₤ terms while outflows are in US$ terms. The following table shows these calculations:

Table 4.9: Inflows and outflows from Currency Swap

YEAR INFLOWS (₤) In millions

OUTFLOWS ($) In millions

1 0.8 0.675 2 0.8 0.675 3 0.8 0.675 4 0.8 0.675

Calculations of Present value of inflows (Discount factor will be 6% which is the current rate of interest in ₤):

P.V. of inflow = [(0.8)/ (1.06) + (0.8/(1.06)2 + (0.8)/ (1.06)3 + (0.8)/ (1.06)4 + (10)/ (1.08)4 ] = ₤ 10.693021225 millions

P.V. of outflow = [0.675 * PVIFA (4%, 4 yrs.) + 13.50 * PVIF (4% , 4 yrs.) ] = US $ 13.990035855 millions.

If we convert ₤ 10.693021225 millions in $ at the current exchange rate i.e. 1.3 $ per pound sterling then we would get $ 13.9009274593 millions.

Now the value of swap to the American firm would be difference between the P.V. of inflow and outflows. i.e. US$ (-0.08910840 millions) = -89108 $ approximately. The same will be the profit for British firm in the swap. Thus the swap valuation depends on the present rates of interest in two underlying currencies as well as the prevailing currency exchange rate.

4.2.6 Pricing of Interest Rate Swap: The pricing of swap means derivation of fixed rate (agreed upon rate) for the one leg of swap. Here we have to use the term structure of interest rate at the contract initiation date. For pricing a swap firstly the present value of both future fixed and floating rate payments are found out. To avoid arbitrage, the present value of fixed and floating rate cash flow must be same. In order to find out the present value of fixed rate payments, term structure of interest rates is considered for appropriate period. The cash flow for floating rate payment is known for one period only. Therefore the present value of single cash flow is only calculated in case of floating rate payment. It is considered as 1. Because the LIBOR for the interest calculation and LIBOR used for discounting are same at the beginning of payment period and also at the end of payment period. (Chance Don, 2008)

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For simplicity of calculations, portfolio of two bonds is considered. The investor has short sold the fixed rate paying Bond and invested the proceeds in Floating rate paying Bond. The following formula shows the stepwise calculations for swap pricing. The assumed agreed rate for fixed leg of swap is assumed as R, which is to be derived. Assume the principal of Re. 1 for interest calculations.

1. Value of Floating Rate Bond can be worked out by the following formula. 2. At the beginning or at end of any payment period V (FLRTB) =1.

(Between payment dates 0 and 1: i.e. at time t)

Here V (FLRTB) = Present Value of floating rate payments of Bond.

Value of fixed rate bond:

3. ( ) ( )1

0 01 360 i n

ni i

FXRTB t ti

t tV R B B−

=

− = +

∑

Where ( )

( )

1

1

01

0

1

1360

t

t

BtL

= +

And ( )

( )

01

0

1

1360

n

n

t

t

BtL

= +

Formula for calculating fixed rate for swap can be written as follows:

( ) ( ) ( )0 01

i n

n

FXRTB t ti

V R q B B=

= +∑

Where ( ) ( )( )1

360i it t

q − − =

Which remains constant e.g. (360-180) = (540 – 360)

( )

( )

10 1

( )1

1

1360

1360

t

FLRTB

t t

tLV

t tL

+ =

− +

65

Now, the (VFLRTB ) and (VFXRTB ) should be same for no arbitrage.

⇒ ( ) ( ) ( )0 01

1i n

n

t ti

R q B B=

= +∑

R and q are constant over here, therefore the formula can be re-written as follows:

( ) ( ) ( )0 01

1 *i n

n

t ti

R q B B=

= + ∑

This can be re-arranged as follows:

⇒ ( ) ( ) ( )0 01

1 *n i

n

t ti

B R q B=

− = ∑

⇒ ( )( )( ) ( )

0

01

1n

i

t

n

ti

BR

q B=

−=

∑

⇒ ( )( )( )

0

01

11 n

i

t

n

ti

BR

q B=

− =

∑

This is the formula for calculating the Fixed agreed upon rate for a interest rate swap contract. The following example explains the application of the derived formula for interest swap pricing. Example: A company enters into a two year Rs. 50,00,000 notional principal interest rate swap. It promises to pay a fixed rate and receive LIBOR. The payments are made every six months. Assuming 180 days period for each interest payment and considering 360 days’ basis, the term structure of interest rate is given below: Table 4.10: Interest rates offered at different maturity periods

Time period Rates annualized (%)

180 days 10 360 days 11 540 days 11.50 720 days 12

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By applying the formula for R we will get the following answer

( )( )( )

0

01

11 n

i

t

n

ti

BR

q B=

− =

∑

=» 360 1 0.806452 *(100) 11.02180 0.95+0.90 +0.85 + 0.81

R − = =

The excel calculations give the following results: Table 4.11: Calculation of the Fixed agreed upon rate for interest rate swap contract

Time period Annualized Rates

Discounted Bond price (Bo) values of Principal Re.1

180 0.1 0.952381 360 0.11 0.900901 540 0.115 0.852878 720 0.12 0.806452

fixed rate of swap R = 11.0202 %

By solving with Excel worksheet we get R = 11.02 %. This rate is the fixed rate for plain vanilla swap.

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4.3 FUTURES: 4.3.1 Introduction: A futures contract is an agreement between two parties to buy or sell an asset at a certain time in the future at a certain price. Future contracts are standardized and exchange traded. To provide enough liquidity in the futures contract, the exchange specifies certain standard features of the contract. It is a standardized contract with standard underlying instrument, standard quantity and quality of the underlying instrument that can be delivered and a standard timing of such settlement. The futures contracts are generally offset prior to maturity by entering into an equal and opposite transaction. As per NSE data more than 99% of futures transactions are offset this way. Distinction between futures and forwards can be summarized as follows: Table 4.12: Distinction between futures and forwards

Futures Forwards Trading on an organized exchange Over the counter in nature Standardized contract terms Customized contract terms More liquid Less liquid Requires margin payments No margin payment Follows daily settlement Settlement happens only at end of period

(Source: NCFM Derivatives Market (Dealers) module) 4.3.2 Pricing of Futures: The relationship between futures price and spot price can be written in terms of cost of carry. This measures the storage cost plus the interest paid to finance the asset less the income earned on the asset.

• For non-dividend paying stock, the cost of carry is r (rate of interest) because there are no

storage costs and no income is earned. • For an equity index, it is r-q (where q is dividend earned on stock portfolio of underlying

index) as income is earned at rate q on the asset.

• For a currency, it is r- rf , where rf is interest on foreign currency, • For a commodity with storage cost that are a proportion u of price, it is r + u; and

likewise.

Defining cost of carry as “c” for an investment asset, the futures price is F0 = S0 e

cT

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For a consumption asset, it is

F0 = S0 e (c-y) T

Where y is the convenience yield. (Source: “Futures and options markets” by John C. Hull) The effective price of the underlying asset can be found out by using futures as follows: Suppose Sp1 is the spot price at time T1

Sp2 is the spot price at time T2

Ft1 is the futures price at time T1

Ft2 is the futures price at time T2

Sp1 - Ft1 = Basis at time T1 = b1

Sp2 - Ft2 = Basis at time T2 = b2

Assume that an exporter is going to receive foreign exchange at time T2.

Therefore to hedge his position (to avoid the unfavorable movement of exchange rate i.e. expecting the appreciation of home currency at that time) he enters into short currency futures contract at time T1 for price Ft1

He closes his position at time T2 by going long in currency futures. The payoff will be calculated as Ft1 - Ft2 (i.e. the difference between the selling and purchase price of futures.) He will actually sell the foreign exchange in the market at ongoing exchange rate at time T2 i.e. Sp2 . The effective price at which the foreign exchange sold is; = Sp2 + ( Ft1 - Ft2 ) = Ft1 + ( Sp2 - Ft2 ) = Ft1 + b2 Where b2 represents basis at time T2 as defined earlier. As b2 is unknown, the futures transaction is exposed to risk. If basis remains constant i.e. b1 = b2 , then the effective price will be Ft1 + ( Sp1 - Ft1 ) = Sp2 .

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Here the risk is completely eliminated and the home currency inflow will be exactly as price of underling at T1. It means you have freezed the spot rate for future execution. 4.3.3 Concept of Hedge ratio: The market is exposed to basis risk, as the abovementioned situation does not always work; the hedge ratio is to be found out because the exposure in futures has to be different than the exposure in the underlying asset. A hedger has to determine the number of futures contracts that provide best hedge for his exposure. It helps the hedger to determine the number of contracts that must be entered into for minimizing the risk of the combined cash-futures position. The hedge ratio is defined as the number of futures contracts to hold for a given position in the underlying asset.

Futures PositionHedge Ratio =Underlying Asset Position

The Minimum Variance Hedge Ratio: A portfolio theory is used to derive the mathematical model for defining minimum variance hedge ratio as the proportion of the futures to the cash position that minimizes the net price change risk. If h stands for minimum variance hedge ratio then:

p p

t

F * σSh =

σ F

Fp = Coefficient of correlation between Sp and Ft σSp = Standard deviation of ∆ Sp σFt = Standard deviation of ∆ Ft ∆ Sp = Change in the spot price during the period of hedging

∆ Ft = Change in the futures price during the period of hedging

The following example explains the concept of hedging using currency futures. Assume a U.S. based exporter is exporting goods to his U.K. based client. On November3 2008, the exporter got the confirmation from the U.K. importer that the payment of Pound sterling 6,25,000 will be made on January 30 2009. Thus the U. S. exporter is exposed to the risk due to currency fluctuations. The depreciation of pound sterling will create loss on the dollar receivables. To hedge this risk the exporter can sell GBP future contracts on the exchange. It can be performed as follows:

• Spot market rate on November 3 is (US$/ per GBP) is 1.6370 (as per cross rate on RBI reference rates). Expected cash inflows are (6, 25,000* 1.6370 = US$ 10,23,125 US$.

• If he were able to convert GBP into US$. But he cannot do so as he did not receive the GBP. Therefore he decided to go short on futures on GBP.

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• Futures market: Assuming one contract is of GBP 1, 25,000, he has to enter into five sell contracts on GBP. Suppose the rate is 1.6400 for expiry February 2009. So the equivalent notional amount in US$ will be 10,25,000 ( i.e. 6,25,000 *1.6400 )

On January 30, 2009 SPOT MARKET • Dollar has appreciated and the spot rate is 1.4243 (US$ / per GBP). The dollar value of

GBP 6,25,000 now is US$ 8,90,187. Thus the loss on spot position is (10, 23,125 – 8,90,187 = US$ 1,32,938)

On January 30, 2009 FUTURES MARKET • Buy five February GBP futures contracts. Let the futures rate be 1.4273. It gives the

exporter the notional right to buy GBP 6,25,000 by paying US$ 8,92,063.( i.e. 6,25,000* 1.4273)

• Thus profit on the futures contract is (US$ 10,25,000 – 8,92,063) = US$ 1,32,938. • The loss in the spot market arising from the appreciation of US$, is set off by the profit in

the futures contract. Here in this example the exporter finally receives the payment as per the rate on November 3, 2008. This has happened because of the same basis change in spot and futures market, i.e. No basis risk.

• But generally, because of basis risk, the exposure in the futures contract should be different than the exposure in the spot market. For that the concept of hedge ratio is applied as explained earlier.

Thus we can summarize the hedging procedure as follows: • Hedging in the futures market involves a two step process. • Depending upon the cash/spot market position, a hedger initially either buys or sells

futures. For example, an investor who owns or plans to purchase equity stock will sell futures to hedge this cash position. A long hedge involves purchasing futures to protect itself a price increase in the underlying, prior to purchasing it either in the spot or forward market.

• In the second step, once cash market transaction materializes, the futures position is closed out i.e. if the person has gone short a futures contract will now go long on the same contract. It is worth noting that both futures transactions should have same size and expiry/ delivery month.

Application of Futures as Speculation: For this if you are bullish on the security then you buy futures.

• Suppose a speculator who has a view of bullish trend of the market. He believes that a particular security that trades at Rs. 500 is undervalued and expects the price to go up in the next two months.

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• If he buys 200 shares then his investment would be Rs. 1,00,000. Suppose after two months if the price moves up to Rs. 510 in two months, then his portfolio value enhances to Rs. (510*200 = Rs. 1,02,000) i.e. a 12% annual basis return.

• If he purchases stock futures instead of stock purchase then the return can be calculated as follows:

• Suppose the futures, currently trades at Rs. 506. If the minimum contract value is Rs. 1,00,000 then he buys 200 security futures (lot size) for which he pays margin of Rs. 20,000. After two months, the price of futures increases to Rs. 510 in two months period. On the day of expiration, the futures price converges to the spot price and he earns a profit of Rs. ((510-506) * 200) Rs. 800. This will be equal to annual return of 24%.

• Thus because of the leverage on futures the security futures form an attractive means for speculation. If the view of investor becomes wrong and the price of underlying stock declines then the losses will be manifold compare to cash position in stock.

• Similarly for speculation on bearish stock, the opposite strategy can be adopted. If the investor can’t do short selling in a particular stock then he cannot take advantage of his bearish view of the market price. He can at most go for intra-day short selling. But with futures, he can always go short at the on going futures price and after a month or two, he can square up the position by going long, or can exercise it on expiry.

4.3.4 Stock index futures: a) Meaning: When the underlying is any stock composite index, i.e. sensex or nifty in India, then it is called Stock Index Futures. Index futures offer ease of use for hedging any portfolio of its composition. They are cash settled and hence do not suffer from any settlement delays and problems related to delivery. The contract specifications for S & P Nifty Futures are given below: Table 4.13: The contract specifications for S & P Nifty Futures Underlying index S & P CNX Nifty Exchange of Trading NSE India Ltd. Contract Size Permitted lot size is 100

(Minimum value Rs. 2,00,000) Price steps Re. 0.05 Price bands Not applicable Trading Cycle Maximum 3 months trading cycle Expiry day Last Thursday of the month Settlement basis Mark to market and final settlement will be cash settled on T+1 basis Settlement price Daily settlement price will be the closing price of futures for the

trading day and the final settlement price shall be the closing value of the underlying index on the last trading day.

(Source: nseindia)

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b) Pricing Of Index Futures: The price can be decided as follows:

• Consider an investor who wants to hold a portfolio of market index for a period of one year. During the period of his holding in one year, he receives dividends, and at the end of the year, he will have capital gain / loss by selling the portfolio.

• Suppose the current market index is Io and at expiration its value is It and the dividends received are Dt then, the rupee return earned by the investor can be written as

( It – Io) + Dt • If an alternative of index futures is considered then, he will buy the index futures contract

and invest all his money into risk free securities. (Margin money is ignored here). If the current futures price is Fo , the expiration day price as Ft and the interest earned in rupees as Rf , the rupee return from index futures to the investor will be

( Ft – Fo) + Rf • If the investor has to be neutral between the two alternatives then both these rupee returns

should be same i.e. ( It – Io) + Dt = ( Ft – Fo) + Rf

• As on expiry the spot and futures prices converge, Ft = It • Thus the equation will be reduced to

Fo = Io + (Rf - Dt ) • Therefore the current index futures price must be equal to the index value plus the

difference between the risk free interest and dividends expected over the life of the contract. The difference of interest and dividend is called the cost of carry. This cost of carry is generally positive.

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c) Application of Index Futures: Index futures can be used for arbitrage and hedging purposes. • In the stock market, an arbitrage opportunity arises when the same scrip trades at

different prices in different markets. In such a situation, investors buy the stock in one market at a lower price and sell it in the other for more, cashing in on the difference, net of the transaction cost. However, such an opportunity vanishes quickly as investors rush in to take advantage of this price difference. Thus, the arbitrage process helps correct the discrepancies in pricing.

• The same principle can be applied to index futures. Being a derivative product, index futures derive their value from the stocks that constitute the index (either the Sensex or the Nifty). At the same time, the index futures value is linked to the stock index value.

Application Of Stock Index Futures For Arbitrage:

On 30th April the Nifty is traded at 3330. The risk free rate is 6.5% p.a., the annual dividend yield is suppose 3.5% p.a., and the July Nifty futures contract i.e. 90 days from here onwards is at 3420. The lot size for Nifty futures is 50.

The theoretical index futures pricing can be arrived at as follows:

Fo = 3330 + [(3330* 0.065* 0.25)] – [3330*0.035*0.25]

= 3330 + 24.975

= 3354. 975

Here the index future is overpriced than the calculated theoretical price.

Arbitrage strategy:

1. Short Nifty futures at 3420 2. Long cash Nifty at 3330.

Now the investor will earn risk less profit of Rs. [ (3420-3354.975 * 50 = Rs 3251.25], whether the bullish or bearish trend prevails.

We can consider two extreme situations on expiry of July futures.

1. Suppose the Nifty closes at 3030,then arbitrageur’s profit can be calculated as follows:

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Table 4.14: Calculation of arbitrage profit from nifty futures in bearish trend

Particulars Rs Profit from short sell of Nifty futures [3420 – 3030] *50 19,500

Cash dividend received @ 3.5% [( 3330 * 0.035 * 0.25)] * 50 1456.875

Less: Loss on sale of stock portfolio index [ 3030 – 3330] *50 -15,000

Interest foregone [3330*0.065*50*0.25] -2705.625

Total pay-off (ignoring transaction cost) 3251.25

2. Suppose Nifty closes at 3663 on expiry of futures contract:

Table 4.15: Calculation of arbitrage profit form nifty futures in bullish trend

Particulars Rs. Loss from short sell of Nifty futures [3420 – 3663] *50 -12150

Cash dividend received @ 3.5% [( 3330 * 0.035 * 0.25)] * 50 1456.875

Profit on sale of stock portfolio index [ 3663 – 3330] *50 16,650

Interest foregone [3330*0.065*50*0.25] -2705.625

Total pay-off (ignoring transaction cost) 3251.25

The annual return is [3251.25 /(50* 3330)] *4 = 0.0781 = 7.81%

This return exceeds the risk free rate of 6.5%.

Application of Stock Index Futures for Hedging:

The basic Principle is:

"If long in cash underlying: Short Future; If short in cash underlying: Long Future"

This is illustrated in the following example.

Example: If you have bought 500 shares of company A and want to hedge against market movements, you should short an appropriate amount of Index Futures. This will reduce your overall exposure to events affecting the whole market (systematic risk). Given this situation if

75

the entire market falls (most likely including Company A), then your loss in company A would be offset by the gains in your short position in Index Futures.

Some instances where hedging strategies are useful include:

• Reducing the equity exposure of a Mutual Fund by selling Index Futures; • Investing funds raised by new schemes in Index Futures so that market exposure is

immediately taken.

Another usage of futures is for portfolio beta management as explained below with numerical example.

d) Beta Management:

Stock index futures can be used to hedge an equity portfolio

If P = Current value of the portfolio

A = Current value of the stocks underlying one futures contract

• If the portfolio is identical to the index then a hedge ratio of 1 will be appropriate. • The formula for number of futures contract is N* = (P / A) • Suppose the index is representative of the market as a whole then appropriate hedge ratio

can be written as

*N PA

β=

Here β measures systematic risk.

Hedging the exposure to the Price of an individual stock:

• An investor holds 1400 Infosys Technology equity shares. He is concerned about the volatility of the market during next month due to Parliamentary election results to be announced in May 2009. The current price( 14-04-2009) is Rs. 1400. The current level of Nifty (Market Index) is 3380, and the June future price of Nifty is 3400.The beta of Infosys is 1.2. The lot size of Nifty is 50.

• Strategy:

1. Short 14 June futures contracts on Nifty

(i.e. *N PA

β= = [1.2 * ((19,60,000)/ (3400*50)) ] = 14 approx.)

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2. Closes out position one month later.

After one month if the price of Infosys declines to Rs.1200 and the June Nifty futures price is 3000. Then the profit / loss will be as under

1. In Infosys the Investor losses [(1200 – 1400) * 1400] = Rs. –2,80,000 2. In Nifty Futures the gain is [(3400 – 3000) * 50* 14] = Rs. +2,80,000 3. Thus the losses are recovered by entering into hedging contract in Nifty futures;

otherwise the Investor would have lost lacs of rupees.

Hedging the exposure to the Price of portfolio of equity stocks:

• A hedge using index futures removes the risk arising from the market moves and leaves the hedger exposed only to the performance of the portfolio relative to the market.

• The other reason for hedging is that the hedger plans to hold a portfolio for a long period of time and requires short term protection in an uncertain market situation. The alternative strategy of selling the portfolio and buying it back later might involve high transaction costs.

• The following example discusses the use of Nifty futures during the times of unusual volatility in the market to adjust or eliminate systematic risk.

A portfolio manager who on January 30, 2009, is concerned about the market over the period ending March 17, 2009. The portfolio has generated good profit till date and the portfolio manager would like to protect the portfolio value over this time period. The manager knows that the portfolio is exposed to a loss in value resulting from a decline in the market as a whole, the systematic risk effect, as well as losses resulting from the unsystematic risk of the individual stock. The portfolio consists of six stocks (given below) and he is not much worried about unsystematic risk. Therefore he tries to eliminate/ reduce systematic risk by entering into Nifty futures selling contract i.e. the opposite of stock position. Assuming the Nifty futures have beta value of 1, he firstly calculates the portfolio beta as the weighted average beta, where the weights are the respective proportionate market values of each stock.

The prices and beta values of each stock are given below:

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Table 4.16: Calculations for Weighted beta of portfolio

Scrip Price

(in Rs.) (30-1-09)

Number of shares

Market value (Rs.)

Weight Beta Weighted (Portfolio) beta

SBI 1150 100 115000 0.15970 1.15 0.183658619 RPL 86.35 400 34540 0.04797 0.35 0.016788273

ONGC 654.95 200 130990 0.18191 1.07 0.194642445 TCS 511.6 250 127900 0.17762 0.52 0.092361190

HDFC 1531.6 100 153160 0.21270 1.05 0.223331658 BHEL 1320.8 120 158496 0.22011 0.72 0.158477071

720086 Total = 1 Total = 0.869259255 Price of Nifty futures on 30-1-09 Rs. 2869.35 Multiplier 50 Price of one contract Rs. 143467.5 Optimal number of contracts 4.362949

Strategy: Sell 4 nifty futures (Rounded off number of contracts.) It is arrived at by applying

hedge ratio formula. i.e. number of contracts will be equal to PA

β

Now on 17-3-2009, the following is the situation of stock prices and Nifty futures.

Table 4.17: Calculation of Total Market Value

Scrip PRICE (17-03-09) Number of shares Market Value(Rs.)

SBI 950 100 95000 RPL 80.9 400 32360

ONGC 715.9 200 143180 TCS 497.55 250 124387.5

HDFC 1364.6 100 136460 BHEL 1401.65 120 168198

Total market value 699585.5 Price of Nifty Futures on 17-03-09 = 2744.9

Multiplier = 50 Value of one contract Rs. 137245 Decline in portfolio Rs. 20500.5 = 2.84695 (% decline)

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Strategy: Buy 4 Nifty futures contract @ 2744.90

Table 4.18: Calculations of net gain from beta hedging

Change in futures 6222.5 (% Change = 4.337219) Number of contracts 4 Profit Rs.24890 Net gain Rs.4389.5 24890 – 20,500.50

• The hedge certainly helped them but was far from perfect. There are several possible explanations for this. One is that, the betas are estimation, derived from the historical data (ex post beta). Beta estimates over the recent past have not necessarily been stable.

• It is also possible that the portfolio was not sufficiently diversified and some unsystematic risk (company or industry specific) contributed to the loss.

• Some of the stocks may have paid dividends during the hedge period. These factors are ignored in this example.

• If the market had moved up, then the portfolio would have shown a profit, but this would have been offset by losses on the Nifty futures transaction.

Changing Beta:

Sometimes futures contracts are used to change the beta of a portfolio to some value other than zero.

• In general, to change the beta of the portfolio from β to β*, where β > β*, the following change is required in contract’s position:

( *) *

( * ) *

Pshort contracts whenA

SimilarlyPlong contracts whenA

β β β β

β β β β

− >

− <

• Where denotes the value of the portfolio and is the value of assets in one futures contract.

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4.3.5 Forward Rate Agreement:

a) Introduction:

A Forward rate Agreement is a type of forward contract. It is used for hedging the interest rate risk. For a borrower the purchase of an FRA is appropriate while for an investor selling of an FRA can be suggested.

Suppose a manager believes that the interest rates are going up and wants to lock in a specific rate to be paid on a future loan, he could do so by entering into selling a forward contract on a fixed income security like Treasury notes or bills. So if the rates rise, the security price will fall and the short position will be profitable, which helps in offsetting the effect of higher borrowing rate.

He can hedge the same risk by entering into FRA. Here he would take a long position in FRA to protect his borrowing rate in future. This can be understood by studying the formula of pay-off for FRA holder (buyer) and writer (seller), which is given in the following Para:

b) Structure and Use of FRAs:

A forward rate agreement (FRA) is a cash-settled forward contract on a short-term loan.

As a hedging vehicle, FRAs are similar to Eurodollar futures, but because they trade OTC, they have the advantage that they can be customized for the needs of the counter parties.

FRAs settle on the first day of the underlying loan, which is called the settlement date.

A 3×9 FRA is a 3-month forward on a 6-month loan—the loan commences in 3 months and matures in 9 months. The interest rate on the loan—called the FRA rate—is set when the contract is first entered into. Because they are cash settled, no loan is ever extended. Instead, the contracts settle with a single cash payment linked to LIBOR (or EURIBOR).

Most transactions are fairly standardized. The underlying loan is typically for 3 or 6 months, and quotes are generally available for 1×4, 1×7, 3×6, 3×9, 6×9 and 6×12 deals.

Consider an FRA based on 90-day LIBOR and suppose the FRA is expiring today. If the rate is 8%, the FRA pays off at the 8% rate today. The Eurodollar deposit (the underlying interest bearing instrument) itself, which is the source of 8 % rate, pays off 90 days later. Therefore, to use LIBOR to determine FRA pay off requires an adjustment, as the payoff is settled today itself. This adjustment is to discount the FRA payment for 90 days at 90 days rate. As this pay offs are deferred for 90 days in case of interest rate swaps and options, they do not require this adjustment in such cases. This system of payment is called “advanced set, settled in arrears”.

In general an FRA on m-day interest rate, pays off at expiration but the pay off is discounted for m days at the m day rate. Such system of payment is referred to as “advanced set,

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advanced settled” (Source: “Derivatives and Risk Management Basics”- by Don M. Chance and Robert Brooks)

The formula for payoff for long FRA is as follows:

Where

notional = the notional amount of the loan,

The reference rate = typically a LIBOR or EURIBOR,

Days = the number of days of the loan, calculated using the actual number of days in each month, and

Basis = the day count basis, applicable to money market transactions in the currency of the loan—360 days for USD or EUR; 365 days for GBP

If on the agreed date (fixing date) the FRA-contract-rate differs from the agreed reference rate (suppose LIBOR), a settlement payment depending on the difference must be paid by one of the contractors. The principal is not exchanged and there is no obligation by either party to borrow or lend capital.

The FRA is not an obligation to borrow or lend any capital in the future. At settlement date, the principal just serves as the basis to calculate the difference between the two interest rates, or rather the settlement payment that results from this difference.

Considering the above setup of FRA, it can be used

1. By market participants who wish to hedge against future interest rate risks by setting the future interest rate today (at trading date)

2. By market participants who want to make profits based on their expectations of the future development of interest rates

3. By market participants who try to take advantage of the different prices of FRAs and other financial instruments, e.g. futures, by means of arbitrage.

The payoff depends on the rate of LIBOR on expiry of the term as explained below: Pay-off Formula of FRA =

81

The abovementioned formula can be applied as follows:

c) Example of application of FRA

Suppose a firm plans to borrow Rs. 20,00,000 in 30 days at 90-day LIBOR plus 100 basis points. The loan will be repaid with interest 90 days later. It is worried about the possibility of increase in rate of interest. The firm would like to lock in the rate of interest. The firm decides to go long on FRA. The rate on 30- day FRAs based on 90-day LIBOR is 8 %( which is called the agreed upon rate). Interest on the loan and the FRA is based on the factor (90/360). The possible results for a range of LIBOR are shown as follows:

Notional = 20,00,000 Rs Agreed rate = 0.08

In the excel worksheet the following are the calculations.

Table 4.19: Pay off and interest rate calculations for with and without FRA contract

LIBOR on Day 30

(%)

FRA payoff (Rs.)

Payoff on day 120

Amount of loan on Day 120

Total amount paid on day

120

Effective rate on Loan

(in %)

Rate without FRA(%)

6.0 -9852.22 -10000 2035000 2045000 9.44 7.29 7.0 -4914.00 -5000 2040000 2045000 9.44 8.36 7.5 -2453.99 -2500 2042500 2045000 9.44 8.90 8.0 0.00 0 2045000 2045000 9.44 9.44 9.0 4889.98 5000 2050000 2045000 9.44 10.53

10.0 9756.10 10000 2055000 2045000 9.44 11.63 10.5 33734.94 35000 2080000 2045000 9.44 12.18 11.0 14598.54 15000 2060000 2045000 9.44 12.74 12.0 19417.48 20000 2065000 2045000 9.44 13.85 13.0 24213.08 25000 2070000 2045000 9.44 14.97

The following formula is applied to calculate the FRA pay off

• For calculating loan repayment amount, the following formula is applied: Loan Payment = {Notional Principal * (1+ ((LIBOR +1%) * (90/360)))}

• For calculating effective rate, the following formula is applied: Effective rate = [(Total amount paid/ 20,00,000)(365/ 90 ) ) –1] * 100

• We can say that by entering into FRA, the firm has freezed the borrowing rate as 9.44%. The graph of the above data is presented below:

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Graph 4.1: Cost of loan with and without FRA contract.

Cost of Loan With & Without FRA

0.002.004.006.008.00

10.0012.0014.0016.00

0.06

0.075 0.0

90.1

05 0.12

LIBOR On Expiry

Cos

t of L

oan

(%)

Effective rate onloanRate withoutFRA

d) Pricing of FRA:

The pricing of FRA can be derived, by considering a hypothetical portfolio of borrowing and lending.

A person borrows for (h + m) days so that it will be (1+ F (m/360) on maturity, in particular, for 120 (i.e. 30+ 90) days. He also lends so that Re. 1 is to be paid back in h days (30 days).

Now after period h (30 days), the loan owed is not due but the market value of the same can be calculated by discounting at appropriate existing LIBOR rate as follows:

( )( ) ( )

1 3601 360h m

mF

mL

+

+

We want to repay the loan early. Therefore the amount to be repaid is the value calculated above. The lending portfolio returns Re. 1 on expiry of 30 days (i.e. now). The net cash flow at this juncture can be written as follows:

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( )

( ) ( )1 3601

1 360h m

mF

mL

+−

+

By taking L.C.F. we get the following form of pay off:

( ) ( ) ( )( )( )( ) ( )

( ) ( )( )( )

1 1360 360 3601 1360 360

h m h m

h m h m

m m mL F L F

mmL L

+ − + −=

+ +

But the last version of the formula is exactly equal to the formula of Forward Rate Agreement. Thus this strategy is the replication of FRA. Therefore the value of FRA at start is exactly equals to the difference between market values of lending and borrowing portfolio. This can be written as follows:

( ) ( )0 0

11 360

1 1360 360h h m

mF

h h mL L +

+ −+ + +

But in order to remove arbitrage possibilities, the value of the difference must be equal to zero. Therefore by equating this difference to zero and putting the formula in terms of F (i.e. the forward rate after h days for m days.) we get the following formula for Forward Rate:

F ( )

( )

0

0

1360 3601

1360

h m

h

h mL

h mL

+

+ + = − +

This rate can be used for fixed leg of FRA contract.

The same procedure can be explained in simple way by using term structure as follows:

For pricing FRA contract, the concept of term structure of interest rate is used.

• i.e. (1+ y2 ) 2 = (1+ y1 ) * ( 1 + f2 ) • Here f2 will be the agreed upon rate for the FRA contract for period 2.

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• If y2 is LIBOR applicable for 120 day underlying, and y1 is LIBOR applicable for 30 day underlying maturity, then f2 will be equal to the implied FRA agreed upon rate at the time of entering into 90- day underlying maturity contract.

• Suppose the following LIBOR rates are available for various terms of maturity:

Table 4.20: LIBOR rates are available for various terms of maturity:

Maturity Rate 30 days 10% 120 days 11.50%

Then the forward rate for 90- days underlying after 30 days can be calculated as follows:

• (1+ (0.1150 (120/360) )) = (1+ (0.10(30/360)) ) * ( 1 + f2 ) • Solving for f2 gives the value = 11.901 % which can be considered as agreed upon

rate for FRA contract.

4.3.6 Interest Rate Futures: a) Introduction: Interest rate futures can be defined as follows:

“An interest rate futures contract is an agreement to buy or sell a standard quantity of specific interest bearing instrument, at a pre-determined future time and at agreed price between the parties.” • If the money is lent on a floating rate basis then they lose if the interest rates go down in

future. • If the money is borrowed on floating rate basis then they lose if the interest rate goes up

in future. Both the parties can hedge this interest rate risk by entering into the interest rate futures contract.

• Buying an interest rate futures contract allows the buyer of the contract to lock in a future investment rate. Interest rate futures are based on an underlying security which is a debt obligation and moves in value as interest rates change.

• When interest rates move higher, the buyer of the futures contract will pay the seller an amount equal to that of the benefit received by investing at a higher rate versus that of the rate specified in the futures contract. Conversely, when interest rates move lower, the seller of the futures contract will compensate the buyer for the lower interest rate at the time of expiration.

• To determine the gain or loss of an interest rate futures contract, an interest rate futures price index is created. In case of buying, the index can be calculated by subtracting the futures interest rate from 100, or (100 - Futures Interest Rate). As rates fluctuate, price index also fluctuates.

• Interest rate futures contracts were first traded on October 20, 1975, in the Chicago Board Of Trade. Such contracts can have short-term (less than one year) and long-term (more than one year) interest bearing instruments as the underlying asset.

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• The underlying assets of an interest rate futures contract are different interest bearing instruments like Treasury Notes, Treasury Bills, Treasury Bonds, deposits and so on. According to Bank for International Settlements (BIS), the size of such instruments is globally set at more than $45 trillion.( Roy, S and Morvelkan, 2003)

• Fixed income derivatives existed in the Indian markets in the form of interest rate swaps, which are OTC products. RBI had given permission to introduce interest rate futures, partially. As in the first phase only 10-year notional bond futures and futures on notional T-bills with 91 day underlying were introduced. The trading started on NSE on June 23, 2003.

Table 4.21: Year wise trading of interest rate futures at NSE

Interest Rate Futures Year No. of

contracts Turnover (Rs. Cr.)

2008-09 0 0 2007-08 0 0 2006-07 0 0 2005-06 0 0 2004-05 0 0 2003-04 10781 202 2002-03 - - 2001-02 - - 2000-01 - -

(Source: www.nseindia.com)

• RBI had earlier allowed banks to take trading positions in IRFs to add depth and liquidity to the market. The guidelines were also applicable to overseas branches of the domestic banks.

• SEBI and RBI had worked hard to re-introduce interest rate futures by January 2009. But till date (April, 2009) they have not come in the financial market in India. A panel comprising officials of RBI and Securities and Exchange Board of India, the stock market regulator, is working on specifications, margins and regulatory issues for the banks to deal in contracts.

• A few years ago (i.e. 2003-04) also such instruments were launched but there were some pricing and tenure related problems that arose at that time and therefore this time they have been handled with special attention. (In India, in year 2003, RBI had started these instruments in the form of a notional 10-year zero coupon bonds and a notional 90-day treasury bill. However they did not prove successful because of the huge price difference between spot and futures market.)

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• According to market sources, the interest rate futures would help to spot the market scenario more accurately as these contracts will detect the interest rate in a better manner for corporates that have interest rate risk. Currently there are interest rate derivatives on notional T-bills and notional 10-year bonds floating in the market. These instruments are offered by National Stock Exchange (NSE) but, there has been absolutely no activity associated with them.

Pricing of Interest Rate Futures:

• In futures contract, one tick means one basis point or 0.01%. The tick means a change in the value of Rs. 2,00,000 invested in a notional T-bill or bond for a one basis point change in the yield. For a short futures contract at a yield of 5%, the price of the notional T-bill can be worked out as follows:

{ [ 100 / ( 1 + (0.05* (91/ 365))]} = Rs. 98.7688 (Here face value is Rs. 100) • Suppose the yield increases to 5.01 %, then the revised price can be calculated as

follows: {[100 / (1 + (0.051 * (91 / 365))]} = Rs. 98.7663.

The absolute change in price is (Rs. 98.7688 - Rs. 98.7663. = Rs. 0.0025 ) Thus the change in the value for Rs. 2,00,000 investment is Rs. 5. • The same way for bond futures, the tick value for one basis point change for notional

10 –year zero-coupon bond will be Rs. 120. Futures pricing formula for underlying T-bills is as follows:

• ( ) %* 1 *100 360

discount Days to maturityP Facevalue = −

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In the next Para, application of IRF is discussed: b) Hedging using Interest Rate Futures:

Hedging a Rise in the interest rate: • This can be considered in other way as Borrower’s decision, who is worried about the

high rate to be applicable at the time of borrowing. This could be done by, going short in interest rate futures. The profit maxim is “Buy Low and Sell High”

• Because the rate of interest and the bond price are inversely related. If someone expects the rate to go up in future, then the expected futures price will be lesser than the currently offered futures price. To get benefit of this situation, one should sell the futures today (as the price is high) and buy it at later when the price goes down because of increase in rate of interest.

Hedging a Fall in the interest rate:

• This can be considered as Investors’ decision. Investor, who is worried about the

expected decrease in rate of interest at the time of investment, can use this for hedge.

• Hedging can be done, by going long in interest rate futures today. If the rate falls at the time of investment, value rate of interest futures must have gone up. This will provide cushion to the investor, as he earns profit in futures.

4.4 Options: 4.4.1 Introduction: An option is the right to buy or sell an asset for a particular time period and at a pre-decided rate. It gives the buyer a right and not an obligation to buy or sell. In India, options were traditionally traded on the Over The Counter market with the names teji-mandi. Commodity options were banned as per the forward Contract Regulation Act, 1952. Similarly options on securities were also banned in the Securities Contract (Regulation) Act in 1969. But after liberalization era, government realized the value of options in financial market for hedging and liquidity, and therefore trading in options in securities was legally allowed in 1995. BSE and NSE both have started trading in options and futures for individual stock and index. This is a landmark for the beginning of options in organized market in India. In the foreign exchange market, the Reserve Bank Of India has allowed all authorized forex dealers and corporates that have forex exposure to deal in certain options, which are generally traded in the dollar / rupee rate. They are Over the Counter Options. As explained earlier in chapter one, there are two types of option contracts; call option and put option.

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Call option: It is a contract that gives its owners the right and not the obligation to buy stock or any financial asset at a specified rate on or before a specified time period. The application of a call option can be understood with the following example: Suppose an investor owned a call to buy one lot of contract i.e. 264 shares at a strike price of Rs. 1200 per share. The price rises subsequently to Rs. 1250; the option owner may take any of the following three actions on or before expiration: Do Nothing: If the price of the underlying asset is less than the strike price, then the call option is out of the money and it expires worthless. But it is not applicable here. Close out his position in the options market: Here the buyer of the call option can sell the option at the ongoing premium rate, which will be obviously higher than the premium that he has paid for buying it earlier. This is generally opted for by the option holder in case of in-the-money option position i.e. when the option is worth exercising. Exercise call on or before expiration: If the price of the share exceeds the strike price of the option then it can be exercised and thereby earning the difference between the settlement price and strike price of the option. Here on instruction of exercise to the Exchange, the clearing-house of the exchange will randomly ask the writer of the option to oblige the buyer. In the abovementioned example, the buyer will get the difference of Rs. (1250 – 1200)* 264 = Rs. 13,200. The following section discusses various models for pricing of option contracts. 4.4.2 The Binomial Model for option pricing: a) Single Period Valuation Model: The discounted cash flow method is not applicable for option valuation, as the prices of a stock are not predictable. It is impossible to determine the opportunity cost of capital because the risk of an option is virtually indeterminate, as it changes every time with the changing stock price. The basic idea underlying this Binomial Pricing Model is to set up a portfolio, which imitates the call option in its payoff. The researchers finally derived one such portfolio, which is discussed below. The cost of this portfolio is nothing, but the cost of purchasing the option i.e. call premium. The portfolio consists of the following combination of stock and debt borrowing or lending: 1) The stock currently selling for S, can take two possible values next year, uS or dS Where (uS > dS ) 2) An amount of B can be borrowed or lent at a rate of r. The interest factor (1+r) may be represented, for the sake of simplicity as R. (i.e. 1+ r = R) The value of R is greater than d but smaller than u to avoid any arbitrage. Otherwise stock investment dominates the borrowing option. The exercise price is assumed to be X.

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The value of the call option, just before expiration, if the stock price goes up to uS, will be: Cu = Max [0, uS – X] Similarly, the value of the call option, just before expiration, if the price goes down to dS then; Cd = Max [0, dS – X] Now we construct a portfolio consisting of Δ shares of the stock and B rupees of debt. As the portfolio is assumed to give identical returns to call option return, we can derive the following equalities: When stock price rises: Δ uS+ RB = Cu When the stock price falls: Δ dS + RB = Cd By solving these two equations for Δ and for B , we get the following formula:

( )u dC C

S u d−

=−

Δ

( )

d uu C d CBu d R

−=

−

Since the portfolio has the same payoff as that of a call option, the value of the call option will be: C = Δ S + B If B is positive then it will be investment or else if B is negative then, it indicates borrowing of funds. (Chandra P., 2002) Thus the following input values are requited to find out call value:

• Stock price (S) • Price increase (u) • Price decrease (d) • Exercise price (X) • Rate of interest (R)

From this data, firstly, values of Cu and Cd are found out. Then Δ and B can be derived. Finally from the equation C = Δ S + B, the value of C is derived which is the option pricing as per the binomial model. The following example discusses the application of this concept for option calculation: Suppose the stock is currently selling for Rs. 100. It may take on the values of either Rs. 140 (i.e. Su) or Rs. Rs. 90(i.e. Sd) after one year. The exercise price (X) is Rs. 110. The rate of interest is 10%. i.e. R = 1.10. From this data, we can first derive values of u and d as 1.4 and 0.9 respectively. Cu = Max [0, 140 – 110] = Rs. 30 Cd = Max [0, 90 – 110] = 0 Now the portfolio components can be worked out as follows:

90

( )u dC C

S u d−

=−

Δ

( )30 0

100 1.4 0.9−

−Δ = = (30 / 50) = 0.6

( )d uu C d CBu d R

−=

−

( ) ( )( ) ( )1.4*0 0.9*301.4 0.9 * 1.10

B−

=−

= (-27 / 0.55) = - 49.1

Thus the portfolio consists of 0.6 equity shares and borrowing of 49.1. The value of call option is: C = Δ S + B C = (0.6) 100 + (-49.1) = Rs. 10.9 Thus the call option should be priced at Rs. 10.9. The binomial option pricing model starts by evaluating what a call premium should be if the underlying asset could only be 1 of 2 prices by expiration. A variable that can only be 1(decrease in price) of 2 (increase in price) values, is known as a binomial random variable. By subdividing the time into smaller time intervals with 2 possible prices that are closer together, a more accurate option premium can be calculated. As the number of time periods increases, the distribution of possible stock prices approaches a normal distribution—bell shape curve. The binomial model breaks down the time to expiration into potentially a very large number of time intervals, or steps. A tree of stock prices is initially produced working forward from the present to expiration. At each step, it is assumed that the stock price will move up or down by an amount calculated using volatility and time to expiration. This produces a binomial distribution, or recombining tree, of underlying stock prices. The tree represents all the possible paths that the stock price could take during the life of the option. At the end of the tree – i.e. at expiration of the option - all the terminal option prices for each of the final possible stock prices are known, as they simply equal their intrinsic values. Next the option prices at each step of the tree are calculated working back from expiration to the present. The option prices at each step are used to derive the option prices at the next step of the tree, using risk neutral valuation, based on the probabilities of the stock prices moving up or down, the risk free rate and the time interval of each step. Any adjustments to stock prices (at an ex-dividend date) or option prices (as a result of early exercise of American options) are worked into the calculations at the required point in time. At the top of the tree, we are left with one option price.

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This process of pricing of option is discussed below:

Diagram 4.4: Tree showing Single Period Binomial option pricing process

p p

(1-p) (1-p)

Here R dpu d

−= −

Cu = Max [0, Su – X )] Cd = Max [0, Sd - X )]

(1 )1

u dpC p CC

r+ −

= +

Similarly for put option we can write as follows:

( )0,u uP Max X S= −

( )0,d dP Max X S= −

(1 )1

u dp P p PP

r+ −

= +

Example: S = Rs. 100 Su = Rs. 140 Sd = Rs. 90 u = 1.4 d = 0.9 r = 7.5 % R = 1.075

S

Su

Sd

C

Cu

Cd

92

Given this information, we can find out the value of call option as follows: Cu = Max [0, Su – X )] ( )( )0, 140 100Max= − = Rs. 40 Cd = Max [0, Sd - X )] ( )( )0, 90 100Max= − = 0

R dpu d

−=

−

1.075 0.90 0.351.40 0.9

p −= =

−

(1 )

1u dpC p C

Cr

+ − = +

= ( )( ) ( )0.35 40 ( 1 0.35 0)13.023

1.075− −

= Rs.

For Put Option

( )0,u uP Max X S= − = ( )( )0, 100 140 0Max − =

( )0,d dP Max X S= − = ( )( )0, 100 90 10Max − =

(1 )1

u dp P p PP

r+ −

= + = ( )( ) ( )( )0.35 0 1 0.35 10

6.04651.075− −

= Rs.

b) Two Period Binomial Model

The two period binomial option pricing formula provides the option price as a weighted average of the two possible option prices the next period, discounted at the risk free rate. The two futures option prices are derived from the one period binomial model.

93

Diagram 4.5: Tree showing Two Period Binomial option pricing process

Su2

p

p Su

(1-p) Sud

(1-p) p

Sd

(1-p)

Sd2

Diagram 4.6: Tree showing Two Period Binomial Call Option valuation

Cu2

p

p Cu (1- p)

Cud

(1- p) p

Cd

(1 – p)

Cd2

( )2 20,u u

C Max S X= −

( )2 20,d d

C Max S X= −

( )0, ududC Max S X= −

( )0,u uC Max S X= −

( )0,d dC Max S X= −

S

C

94

Similarly for two step binomial put option we can write as follows: ( )2 20,

u uP Max X S= −

( )0,ud udP Max X S= −

( )2 20,d d

P Max X S= −

Now we will consider an example of Two Period Binomial Model for Call and Put Option Valuation: Continuing with the one period Binomial Model S = Rs. 100 Su = Rs. 140 Sd = Rs. 90 u = 1.4 d = 0.9 r = 7.5 % R = 1.075

2 196u

S = 126udS = 2 81d

S =

Therefore ( )( )2 0, 196 100 .96u

C Max Rs= − =

( )( )0, 126 100 .26udC Max Rs= − =

( )( )2 0, 81 100 .0d

C Max Rs= − =

( )2 11

uduu

pC p CC

r+ −

=+

( )( ) ( )( )0.35 96 0.65 26

46.9771.075uC

+= =

( ) 211

ud dd

pC p CC

r+ −

=+

( )( ) ( )( )0.35 26 0.65 08.465

1.075dC+

= =

(1 )1

u dpC p CC

r+ −

= +

( )( ) ( )( )0.35 46.977 0.65 8.4651.075

C+

= = Rs. 20.413

Another formula for directly calculating Call option price is given below:

( ) ( )( )( )

2 2

22

2

2 (1 ) 1

1

udu dp C p p C p C

Cr

+ − + −=

+

( ) ( ) ( )( )( )

22

2

0.35) *96 2(0.35)(1 0.35) * 26 1 0.35 * 0

1.075C

+ − + −= = Rs. 20.413

95

TWO PERIOD BINOMIAL MODEL FOR AMERICAN OPTION

The procedure for American style option is to work back through the tree from the end to the beginning, testing at each node to see whether early exercise is optimum. The value of the option at the final nodes is the same as for the European option. At earlier nodes the value of the option is the greater of 1) The value as per the formula for C; 2) The pay-off from early exercise. This can be better explained with the help of put option pricing example as follows: Example: Suppose the following information is available for an equity stock: Current price = S = Rs. 100. Exercise price = X = Rs. 100 u = 1.4 d = 0.80 r = 7.5% R = 1.075 We can derive the following:

2uS = 196 udS = 112 2d

S = 64 1.075 0.80 0.461.40 0.80

p −= =

−

( )( )0, 100 140 0uP Max= − = ( )( )0, 100 80 20dP Max= − =

( )( ) ( )( )0.46 0 0.54 * 20.10.05

1.075P Rs

+= =

Now for two Periods put option: ( )( )0, 100 112 0udP Max= − = ( )( )2 0, 100 196 0

uP Max= − =

( )( )2 0, 100 64 36d

P Max= − =

( )0.46 (0) (0.54)(0)0

1.075uP+

= = ( )0.46 (0) (0.54)(36) 19.44 18.0841.075 1.075dP

+= = =

( ) ( ) ( )( )0.46 0 0.54 18.089.084

1.075P

+= =

As per American style put option, early exercise gives higher value when price goes down, then the two-step calculated Pd value of 18.08 would be replaced by value of Rs. 20. Therefore we get

( )( ) ( )( )0.46 0 0.54 20 10.80 .10.051.075 1.075

P Rs+

= = =

Which is higher than, the earlier calculated value of Rs. 9.084

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c) Binomial Valuation for Dividend Paying Stock: Let the dividend be paid and stock go ex-dividend at time 1. Suppose the company pays 10% dividend of the price at period 1. The effect of dividend is seen, as the reduction in price of shares. Continuing with the call option two period valuation model; Current price = S = Rs. 100. Exercise price = X = Rs. 100 u = 1.4 d = 0.90 r = 7.5% R = 1.075

140 14 126uS = − =

2 126(1.4) 176.40u

S = = 126(0.9) 113.40udS = = 90 9 81dS = − =

2 81* (0.9) 72.9dS = =

( )2 0,176.40 100 76.40u

C Max= − =

( )0,113.40 100 13.40udC Max= − =

( )2 0,81 100 0d

C Max= − =

2 (1 )1

uduu

p C p CC

r+ −

=+

= ( )( ) ( )( )0.35 76.40 0.65 13.4032.977

1.075+

=

( ) ( )( ) ( )( )21 0.35 13.40 0.65 04.363

1 1.075ud d

d

pC p CC

r+ − +

= = =+

European call option at (time 0) = ( )( ) ( )( )0.35 32.977 0.65 4.36313.375

1.075C

+= =

But in case of American style for early exercise Cu = 40 (i.e. cum dividend price realization) Therefore replacing this value in place of Rs. 32.977 as calculated earlier;

We get ( )( ) ( )( )0.35 40 0.65 4.36315.66

1.075C Rs

+= =

This is more than Rs. 13.375 calculated earlier. d) Advantages & Limitations of Binomial Model of option pricing. Advantage: The most valuable feature of the model is that it enables the construction of a dynamic risk free hedge, which leads to a formula for option price. Therefore if the option price is not in conformity with the formula, then there is a possibility of risk less arbitrage profit, which is more than risk free market rate.

97

The big advantage the binomial model has over the Black-Scholes model (which is discussed later on) is that it can be used to accurately price American options. This is because with the binomial model, it is possible to check at every point in an option's life (i.e. at every step of the binomial tree) for the possibility of early exercise (e.g. where, due to e.g., a dividend, or a put being deeply in the money the option price at that point is less than its intrinsic value). Where an early exercise point is found, it is assumed that the option holder would elect to exercise, and the option price can be adjusted to equal the intrinsic value at that point. Then it flows into the calculations higher up the tree and so on. Limitation: The main limitation of the binomial model is its relatively slow speed. It is great for half a dozen calculations at a time but even with today's fastest PCs; it is not a practical solution for the calculations of thousands of prices in a few seconds. e) Relationship to the Black-Scholes model The same underlying assumptions regarding stock prices underpin both the binomial and Black-Scholes models: that stock prices follow a stochastic process described by geometric Brownian motion. As a result, for European options, the binomial model converges on the Black-Scholes formula as the number of binomial calculation steps increases. In fact the Black-Scholes model for European options is really a special case of the binomial model where the number of binomial steps is infinite. In other words, the binomial model provides discrete approximations to the continuous process underlying the Black-Scholes model. Whilst the Cox, Ross & Rubinstein binomial model and the Black-Scholes model ultimately converge as the number of time steps gets infinitely large and the length of each step gets infinitesimally small. This convergence, except for at-the-money options, is anything but smooth or uniform. 4.4.3 Black –Scholes –Merton Model of Option Pricing: a) Historical Background: Attempts to value derivatives have a long history. As far back as 1900, the French mathematician Louis Bachelier reported one of the earliest attempts in his doctoral dissertation, although the formula he derived was flawed in several ways. Subsequent researchers handled the movements of stock prices and interest rates more successfully. But all of these attempts suffered from the same fundamental shortcoming: risk premia were not dealt with in a correct way. The value of an option to buy or sell a share depends on the uncertain development of the share price to the date of maturity. It is therefore natural to suppose - as did earlier researchers - that valuation of an option requires taking a stance, on which risk premium to use, in the same way as one has to determine which risk premium to use when calculating present values in the evaluation of a future physical investment project with uncertain returns. Assigning a risk premium is difficult, however, in that the correct risk premium depends on the investor's attitude

98

towards risk. Whereas the attitude towards risk can be strictly defined in theory, it is hard or impossible to observe in reality. In the late 1960s, Fisher Black finished his doctorate in mathematics at Harvard. Passing up a career as mathematician, he went to work for Arthur Little, a management-consulting firm in Boston. There Black met a young MIT finance professor Mr. Myron Scholes, and the duo began an interchange of ideas on how financial markets worked. Afterwards Black joined the MIT finance faculty, where he made many contributions in the field of assets’ pricing; of course other than options. Black and Scholes then began studying options, which at that time were only traded on the over-the-counter market. They first reviewed the attempts of previous researchers to find the option pricing formula. Black and Scholes took two approaches to finding the price. One approach assumed that all assets were priced according to Capital Assets Pricing Model. The other approach used the stochastic calculus. They obtained an equation using the first approach, but the second method left them with a differential equation they were unable to solve. But they gave priority to this mathematical model; therefore they continued to work on this unsolved formula, which they had not thought would create a history. Fisher Black started out working to create a valuation model for stock warrants. This work involved calculating a derivative to measure how the discount rate of a warrant varies with time and stock price. The result of this calculation held a striking resemblance to a well-known heat transfer equation. Black eventually found that the differential equation could be transformed into the same one that described the movement of heat as it travels across the object. There was already a known solution, and Black and Scholes applied it to their problem and finally obtained the correct formula, which they had obtained by the first method. Black and Scholes' improvements on the Boness model come in the form of a proof that the risk-free interest rate is the correct discount factor, and with the absence of assumptions regarding investor's risk preferences. Their paper reporting their findings was rejected by two academic journals before finally being published in the “Journal of Political Economy” which reconsidered an earlier decision to reject the paper. During this time, another young financial economist at MIT, Robert Merton, was also working on option pricing. Merton discovered many of the arbitrage rules. Merton also co-incidentally simultaneously derived the formula. Merton’s modesty compelled him to ask a journal editor that his paper not to be published before of Black and Scholes. Both papers were published, with Merton’s paper appearing in the “Bell Journal of Economics and Management Science” at about the same time. Merton, however, did not receive as much credit as Black and Scholes at the initial stage. Black and Scholes names become permanently associated with the model. Fisher Black left academia in 1983 and went to work for the Wall Street firm of Goldman Sachs. Black died in 1995 at the age of 57 only. Scholes and Merton remained in the research and extensively involved in real-world derivatives applications. In 1997 the Nobel Committee awarded the Nobel Prize for Economic Sciences to Myron Scholes and Robert Merton, while recognizing the contributions of Fisher Black. Thus even if one does not agree with everything about the model, knowing something about it, is important for surviving in the options market. In India, for exchange traded equity and index options, the starting option prices are offered as per the Black-Scholes Model formula at the Equity Stock exchanges (which is mandatory). Thousands of traders and investors now use this formula every day to value stock options in markets

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throughout the world. (SOURCE: “Derivatives and Risk Management Basics” by Don M. Chance and Robert Brooks) The following section discusses assumptions for the Black- Scholes Option Pricing Model: b) Assumptions Underlying The Black- Scholes Option Pricing Model: Returns are log normally distributed: This assumption suggests that the returns on the underlying stock are normally distributed, which is reasonable for most assets that offer options. There remains theoretical objection to the assumption that individual stock returns are normally distributed. Given that a stock price can not be negative, the normal distribution can not be truly representative of the behaviour of a holding period rate of return because it allows for any outcome, including the whole range of negative prices. Specifically rates of return lower than –100 % are theoretically impossible because they imply the possibility of negative security prices. The failure of the normal distribution to rule out such outcomes must be viewed as shortcoming. An alternative assumption is that the continuously compounded annual rate of return is normally distributed. If we call this rate “r” and we call the effective annual rate “re.” then re =(er - 1). And because er can never be negative, the smallest possible value for re is –1 implying –100% return. Thus this assumption nicely ruled out the possibility of negative prices while still conveying the advantages of working with normal distributions. Under this assumption the distribution of re will be lognormal. As, if the log of a variable is normally distributed, then the variable is said to be log normally distributed. Interest rates remain constant and known and the Volatility of the stock returns on the stock is constant The Black and Scholes model uses the risk-free rate to represent this constant and known rate: In reality there is no such thing as the risk-free rate, but the discount rate on U.S. Government Treasury Bills with 30 days left until maturity is usually used to represent it. During periods of rapidly changing interest rates, these 30-day rates are often subject to change, thereby violating one of the assumptions of the model. The assumption that the volatility, which is the standard deviation, is constant is an important one. In fact it is virtually inconceivable that any risky asset will have the same level of volatility over a period of time such as a year. Actually the models with changing volatility are very complicated. Also it is not clear that such models are any better. No commissions are charged. (No Taxes or Transaction Costs). Usually market participants do have to pay a commission to buy or sell options. Even floor traders pay some kind of fee, but it is usually very small. The fees that Individual investors pay is more substantial and can often distort the output of the model. This Black- Scholes-Merton

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Option Pricing Model is based on the ability of actively trade stocks, risk free instruments and options. This activity will trigger tax consequences and transaction costs. The stock pays no dividends during the option's life Most companies pay dividends to their shareholders, so this might seem a serious limitation to the model considering the observation that higher dividend yields elicit lower call premiums. A common way of adjusting the model for this situation is to subtract the discounted value of a future dividend from the stock price. As such they have relaxed this assumption afterwards and incorporated dividend paying stocks for option pricing. European exercise terms are used European exercise terms dictate that the option can only be exercised on the expiration date. American exercise term allow the option to be exercised at any time during the life of the option, making American options more valuable due to their greater flexibility. This limitation is not a major concern because very few calls are ever exercised before the last few days of their life. This is true because when you exercise a call early, you forfeit the remaining time value on the call and collect the intrinsic value. Towards the end of the life of a call, the remaining time value is very small, but the intrinsic value is the same. Markets are efficient This assumption suggests that people cannot consistently predict the direction of the market or an individual stock. If you were to draw a continuous process you would do so without picking the pen up from the piece of paper. c) Option Pricing Formula:

( ) ( )0 1 2

rtC S N d Xe N d−= −

( )2

0

1

ln *2S r tX

dt

σ

σ

+ + =

2 1d d tσ= − Here C = Current call premium. S0 = Current stock price. N(d) = the probability that a value in a normal distribution will be less than d. X = strike price. t = time (in years) until expiration. r = risk-free interest rate. e = 2.71828, the natural log base.

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ln = natural logarithm function. σ = standard deviation of the stock's annualized continuously compounded rate of return. The following example explains the use of this formula: Suppose the following details are available for the equity of a Company:

0 . 150 . 145S Rs X Rs= = Risk free rate = 6% p.a.

0.80 0.0822t yearsσ = = Then firstly we can find out value of 1d as follows:

( )20

1

ln *2S r tX

dt

σ

σ

+ + =

( )

( )

2

1

0.80150ln 0.06 0.0822145 2 0.06514

0.22940.80 0.0822d

+ + ⇒ = = = 0.2840

And 2 1d d tσ= −

( )2 0.284 (0.80) 0.0822 0.0546d⇒ = − =

From the Normal distribution table we can derive the values of probabilities related to 1 2d and d as follows :

1

2

( ) 0.61182( ) 0.52174

N dN d

=

=

Finally we would apply the formula for call option as below:

( ) ( )0 1 2

rtC S N d Xe N d−= −

( ) ( )( 0.06*0.0822)150*0.61182 145* 0.52174C e −⇒ = − C = 16.49 For pricing put option, the concept of put-call parity can be applied as under: d) Put-Call Parity A portfolio consisting of stock and a protective put on the stock establishes a minimum amount of value for the portfolio that also has an unlimited upside potential. If the stock declines below the strike of the put, the put increases in value by a dollar for every dollar decline of the stock

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below the strike price. If the stock climbs above the strike price, the put expires worthless, leaving only the stock. An equivalent portfolio can be established by buying a call and a risk-free T-bill that matures on the expiration day of the call. Because a portfolio that has a minimum value but unlimited upside potential can be established using either a put or a call, then they must be equivalent, because if they weren’t, then arbitrageurs could take advantage of the price discrepancy, writing one option and buying the other until equivalence was achieved. Consider two portfolios as follows: A portfolio consisting of a European call option (C) and cash equal to present value of exercise price of stock under option. (i.e. Xe –rt ). Another portfolio consisting of a put option (P) with the same strike price X and hold a stock. (The underlying asset for the options is the same share, which is a part of portfolio 1) The value of both the portfolios at expiration of the options will be the same i.e. Max (ST , X)

Table 4.22: Value of portfolio at expiration time T

Portfolio ST > X ST < X Cash + Call (ST – X) + X = ST 0 + X = X Put + Stock 0 + ST = ST (X -ST ) + ST = X

Thus both portfolios give the identical return. This is called put-call parity strategy or relationship. Now by comparing both the portfolios we get the following equation. C + Xe –rt = P + S0 From this we can derive the following version in terms of P P = C - S0 + X e –rt

But ( ) ( )

0 1 2rtC S N d Xe N d−= −

Then the price of a put option is: P = X e-rt N(-d2) - S0 N(-d1) d1 and d2 are derived the same way as found in call option formula. e) Option pricing for stock paying dividend, stock indices and currency exchange rate: In case of the stock which pays dividend before the expiry of the option maturity, the price of the underlying is affected (i.e. reduces) because of return cash drain, at the same time the underlying asset holder is eligible to receive this return, the option holder is not entitled to it. The same concept is applicable to any asset in general, which offers return in between to date of initiation

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and date of expiry. In such cases the price of the option has to be adjusted accordingly as follows:

( ) ( )

( )

( )

0 1 2

2 0 1

20

1

20

2 1

( ) ( )

ln 2

ln 2

qt rt

rt qt

C S e N d Xe N d

P Xe N d S e N dWhere

S r q tXd

tS r q tX

d d tt

− −

− −

= −

= − − −

σ+ − + =

σ

σ+ − − = = −σ

σ

Where q = the return from the underlying asset, For example in case of equity stock paying dividend, it is dividend yield; in case of stock indices option calculation, it is the total dividend yield during the option’s life; in case of currency future, q is the foreign currency interest rate. In case of currency options, the formula for option pricing can be written in terms of forward exchange rate as follows: If the forward rate is denoted by 0F and spot exchange rate by 0S , then the formula for forward rate will be as follows:

( )

( ) ( )( ) ( )

0 0

0 1 2

2 0 1

20

1

20

2 2 1

ln 2

ln 2

fr r t

r t

r t

F S e

ThenC e F N d X N d

p e X N d F N d

WhereF t

Xd

t

F tX

d OR d d tt

−

−

−

=

= − = − − −

σ + =

σ

σ − = = − σ

σ

Verification of Put-Call parity: Consider a portfolio that has one call with a strike price of Rs.100 that expires in 6 months, and a T-bill with a face value of Rs.10,000, bought at a discount, that matures on the call's expiration day. If the call is in the money, then the Rs.10,000 from the T-bill can be used to purchase the 100 shares of stock at the strike price, resulting in a portfolio value equal to 100 shares of stock.

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If the call expires worthless, then the portfolio is worth Rs.10,000 from the T-bill. Thus, the minimum portfolio value is Rs.10,000. Similarly, the portfolio consisting of stock and a put option would have as its minimum value the strike price of the put. Thus, if the two portfolios provide equal values, then they should cost the same to establish—otherwise, arbitrageurs would profit from the difference until the difference fell below trading costs, buying one option and selling the other. Arbitrageurs would not have much effect on the stock price or the interest rate of the T-bill, but they would have an effect on the prices of the options, and it is this effect that would equilibrate the prices. Thus, the stock plus put must equal the T-bill plus call. These equations assume that the options are not exercised before expiration. If they are, then the payoff of the portfolios will be different. Example: Verifying the Put-Call Parity with Real Prices On November 18, 2006, market data yielded the following information on Microsoft (MSFT), with the two options having a strike price of $30 and that expired two months later, in January, 2007: Stock Price = $29.40 MSFT Put bid/ask = $0.90/$1.00 = $0.95 average price. MSFT Call bid/ask: $0.80/$0.85 = $0.825average price. Lowest Margin Interest: 8% per annum Substituting the above numbers into the put-call parity equation and using the average prices of the put and call, and using 1/6 of a year = 2 months, we get: .0825 + 30/(1.08)1/6 = 29.40 + .95 30.44 ≈ 30.35 As the two sides of the equation are well within even an arbitrageur's trading costs. Example 1—Conversion Arbitrage - Profiting from an Overpriced Call Let us suppose that, in the above example, the call was selling for $1.80 instead of $0.80. We can then profit from what is known as conversion arbitrage. We sell the left hand of the equation and buy the right for an immediate profit. Table 4.23: Example-1 of (Conversion) arbitrage for non- put-call parity situation

Conversion Arbitrage Profits per Share

Position Immediate Payoff

Cash Flow in two months

St = Stock Price at expiration. St ≤ 30 St > 30 Buy Stock -29.40 St St Borrow $30/(1.08)1/6 = 29.62 +29.62 -30 -30 Write (sell) call +1.80 0 30 - St Buy put -1.00 30 - St 0 Total = $1.02/share 0 0

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Example 2—Reverse Conversion Arbitrage - Profiting from an Overpriced Put The put value is changed to 1.85, so that it is now overpriced. This arbitrage is called a reverse conversion, because it is basically the reverse of a conversion. Now we want to buy the left side of the put-call parity equation and sell the right side. Note that the call covers the shorted stock if the stock rises above the strike, and the put is covered by the shorted stock, if the stock price is less than the strike, which explains St - 30. For instance, if the stock price is $20 on expiration day, the liability from the put is $10, which is mostly covered by the profit of $9.40 from the shorted stock. Table 4.24: Example-2 of (Reverse Conversion) arbitrage for non- put-call parity situation

Reverse Conversion Arbitrage Profits per Share

Position Immediate Payoff

Cash Flow in two months

St = Stock Price at expiration. St ≤ 30 St > 30 Sell Short Stock + 29.40 - St - St Buy Discounted Note for PV(30) - 29.62 + 30 + 30 Buy call - 0.85 0 St - 30 Write (sell) put + 1.85 St - 30 0 Total = $0.78/share 0 0 (Source: www.thismatter.com) 4.4.4 Factors Influencing Option Prices considered in Black –Scholes Model: The various factors affecting the option premium which have been covered in the formula, in earlier sections, are discussed below: 1. Current Stock Price: The value of the call option increases as the underlying stock price increases, while its value decreases whenever stock price declines. For put option, the net flow on exercise is the amount P that is strike price less the stock price. So the value of a put option decreases with increase in the stock price, and its value increases with decline in stock price. This can be explained with following relationship: C = Max [0, ST – X] P = Max [0, X – ST] Where ST is stock price, and X is strike price. 2. Strike price: The value of the call option increases with decline in strike price and value of call option decreases with the increase in stock price. This happens because the value of call option depends on the difference between the stock price and the strike price. The value of put option increases with the increase in the strike rice, while it decreases with decrease in strike price. 3. Time to Expiration: The more the time remaining for an option to expire, the higher the premium. This is because a longer time period increases the possibility of the price of underlying asset moving to the in-the-money range, where the purchase and sale of the asset at strike price

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will be profitable. Thus, an option with two months expiry will have a higher premium compare to one-month option. The time value is what makes the difference to the option premium and its intrinsic value. A deep out of the money option has less potential to gain intrinsic value because asset price has to change substantially to do so. So it will have a less time value of money. Similarly, a deep in-the-money option is more likely to lose intrinsic value than to gain value. As a consequence it also has a little time value. Time value is the maximum when an option is at-the-money at which there is no intrinsic value but there is a chance to gain it. 4. Volatility of asset: The volatility of underlying asset price is a measure of uncertainty about the movement of future asset price. With the increase in volatility, chances are that the price movement in underlying asset price will increase. The purchaser of a call or a put option, benefit from the price movement, since he has only one side risk. Therefore as the volatility increases, the risk of the option writer increases, as the chances of option going in the money rises. The effect of volatility on option is always the positive, which means the value of both call and put option increases as volatility increases. 5. Risk Free Interest Rate: The impact of risk free interest rate on the price of an option cannot be precisely defined. Whenever the interest rate rises, the expected growth rate of the stock price increases but the present value of all future cash flows to be received by the owner of the option declines. Therefore because of these effects the value of a put option decreases as the risk free interest rate increases. The increase in the growth rate of stock price enhances the value of call option, but the present value effect tends to decrease it. But the effect of former dominates the latter effect. Thus the price of call always increases, as the interest rate increases. Other variables are assumed to remain constant here. 6. Dividends: The value of stock increases, in anticipation of dividend declaration and it declines after the record date as there is a huge cash outflows for the dividends paying firm. Hence the price of European call option whose expiry is beyond the record date tends to decline whereas that of put option tends to increase. The impact of dividend on the price of option is essentially a consequence of the impact of dividend on stock price. The following Para discusses Upper and Lower Bounds for call and put options: For arbitrage restrictions there are lower and upper boundaries for call and put options: Upper Bound: The highest amount that a call option can be sold for is the market price of the underlying asset, which is called the upper bound of call option. The highest amount that a put option can be sold for is its strike price itself. Lower Bound: For call option the lower bound for can be explained as under: C > Max [ 0, S – X e-rt ] For put option it can be termed as;

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P > Max [0, X e-rt - S] Where S = Stock Price

X = Exercise Price of underlying r = risk free rate of interest t = time remaining till expiry of option contract.

Advantages & Limitations of Black – Scholes Option Pricing Model Advantage: The main advantage of the Black-Scholes model is speed. It enables to calculate a very large number of option prices in a very short time. Limitation: The Black-Scholes model has one major limitation: it cannot be used to accurately price options with an American-style exercise as it only calculates the option price at one point in time at expiration. It does not consider the steps along the way where there could be the possibility of early exercise of an American option. As all exchange traded equity options have American-style exercise (i.e., they can be exercised at any time as opposed to European options which can only be exercised at expiration), this is a significant limitation. The exception to this is an American call on a non-dividend paying asset. In this case the call is always worth the same as its European equivalent as there is never any advantage in exercising early.

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4.4.5 Exotic Options: Exotic options can be broadly classified into path-dependent and path independent or free- range options. They are over the counter traded options. They are flexible in nature for pricing and option alternatives. 1. Digital options (also known as: binary options, all-or-nothing options, cash-or-nothing options, asset-or-nothing options): It pays a set amount if the underlying is above or below the strike price, or nothing at all—hence, the names for these options, because they pay 1 of 2 values. A digital call pays off if the underlying asset is above a certain value at expiration—the strike price; a digital put pays off if the underlying asset is below the strike price at expiration. Graph 4.2: Payoff of Digital Call Option

At expiration, a digital call option is worth Re. 1 if TS X> and zero otherwise. For Binary put option the expiration value is Re. 1 if X > ST and zero otherwise. Before expiration, the values of digital options are given by following formula:

( )2rt

digitalC e N d−= And ( )2r t

digitalP e N d−= − If the payoff of digital option is multiplied by the strike price, the digital option is known as a Strike or nothing option. Before expiration the valuation will be as follows for strike or nothing option:

( )2rt

sonC X e N d−= And ( )2r t

sonP X e N d−= − Similarly in case of asset or nothing option the valuation of the option before expiry will be as follows:

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( )1AoNC S N d= And for put option ( )1AoNP S N d= −

Here we can visualize that the formula for asset or nothing and strike-or-nothing are parts of Black Scholes Model. In other words a long position in call option is equal to a long position in asset-or-nothing call option and a short position in strike-or-nothing call option.

2. European Gap Option: It also provides discontinuous payoff profile. The payoff of a gap call option is ST - G, if ST > X and zero otherwise. To make a call option different from simple call option, value of G must be different than X.

Before expiration the value of a gap option is the value of plain call option plus adjustment for difference between X and G, which is given below:

For Gap call option ( ) ( )( ) ( ) ( )( )1 2 2rt rt

GAPC S N d X e N d X G e N d− −= − + −

For Gap put option ( ) ( )( ) ( ) ( )( )2 1 2rt rt

GAPP X e N d S N d G X e N d− −= − − − + − −

3. European Pay later Options: In this option the initial cost is zero. The premium will be paid if the option expires in the money. If premium is denoted by c then the payoff at expiration is for a pay later call option = ST –X- c; if ST > X or else it will be zero. This option is at par with a combination of plain call option less a payment of premium c times the value of a Digital call option. Thus we can write the valuation formula for pay later call option as follows:

At time o the value of option: PL( ) ( )( )

( )( )1 2 0

2 0

r t

r t

S N d X e N dc

e N d

−

−

−=

At any time after initiation, the value will be

( ) ( )( ) ( )( )1 2 2r t r t

paylater ct tC S N d X e N d PL e N d− −= − −

The holder of a pay later option does not have to pay for the option if it expires out of money. But if the option expires in the money with a very thin margin i.e. (low ST –X) then there might be negative payoff if the value of c is larger than this difference.

For example if S0 = Rs. 100 X = Rs. 100 r = 6% p.a. t = 0.25 σ = 0.30

Then the plain call value will be Rs. 6.67, and the value of pay later option will be Rs. 13.29. If stock increases thinly to say Rs. 103 (ST ), then the option is in the money by Rs. 3 , and call pay later holder will lose Rs. (13.29 – 3 ) 10.29 compared to plain call option loss of Rs. (6.67 – 3) 3.67. (SOURCE: “DERIVATIVES- Valuation and Risk Management” By David A. Dubofsky and Thomas W. Miller, Jr.)

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4. European Chooser Option: They are also known as “as you like it” option or “option for the undecided”. At the time of purchase, the option is neither a call option nor a put option. The buyer of a chooser option has a right, at a pre specified time to choose whether the option finishes as a call option or put option. They are lower cost alternative to purchasing a Straddle, which is explained in the chapter option strategy.

At the beginning of the option period, the payoff of chooser option is similar to the following two transactions’ payoff.

• A long position in a call option with strike price X and time to maturity T • A long position in put option with strike price equal to X e-r (T – t) and a time to maturity t.

Where t is the choice date for the option.

The holder of a chooser option will choose call option at time t if the stock price is more than the discounted strike price.

5. Compound options are options on options that give the holder the right to acquire another option by a specific date and for a specific premium. There are calls on calls, puts on puts, calls on puts, and puts on calls. Compound options have two exercise dates and two strike prices. Black Scholes pointed out that plain call options are also compound options. As the holder of the call option has an option on the firm’s stock, which represents the residual claim over the firm’s assets. Compound options are used by corporations to hedge foreign exchange risk for a business venture that may or may not occur.

The meaning of this option can be explained with the following example:

Suppose an investor purchases a call on call. On the first exercise date, T1, the holder of the compound option is entitled to pay the first strike price, X1 , and receives a call option. The call option gives the holder the right buy underlying asset for the second strike price, X2, on the second exercise date,T2. The compound option will be exercised on the first exercise date only if the value of the second option on that date is greater than the first strike price. (Source: “Fundamentals of Futures and Options Markets.” - By John C. Hull,2005).

6. Option on the Maximum or Minimum of more than one asset:

These are exotic option in the sense that there are two underlying assets. These are call options on the maximum of two stochastic values, put options on the minimum of the two stochastic values; call options on the minimum of two stochastic values and put on maximum of two stochastic values. They are sometimes called Rainbow options.

Thus multi asset options consist of a family of options whose payoff depends on the prices of more than one asset.

Option to exchange one asset for another: They are also referred to as exchange options. e.g. option to exchange one foreign currency asset for another foreign currency. An option to obtain

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the better or worse of two assets is closely related to an exchange option. It is a position in one of the assets combined with an option to exchange it with the other asset.

( ) ( )( ) ( )

, ,0

, ,0T T T T T

T T T T T

Min U V V Max V U

Max U V U Max V U

= − −

= − −

Path Dependent Options:

In path dependent options, the option has a value that depends on the path of the price of the underlying asset before option expiration. Such exotic options are known as path dependent options. They are discussed below:

1. Barrier Option: Barrier options pay off if an asset reaches a certain price. Knock-in options are created with predetermined characteristics when the underlying reaches a certain price. Knockout options are options that terminate if the underlying reaches a certain price. Since the option ceases to exist, there is no payoff even if the price moves back within the knockout barrier before the original expiration. Thus, an option with a knockout barrier has a maximum specified value and payoff. The graph of a call option and put option with knockout barrier are given below: Graph 4.3: Payoff for Call Option with a knockout barrier

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Put option with knock-out barrier is shown below: Graph 4.4: Payoff for Put Option with a knockout barrier

Thus Call and Put barrier options are divided into two groups each. They are “out barrier options (also referred to as knock outs) and in barriers (also referred to as knock-ins). A knock out option ceases to exist when value of the underlying touches the barrier level. While a knock-in option comes into existence when the underlying asset price reaches the barrier level. The name of the barrier option also depends on the price path of the underlying asset.

They are “up and in” and “up and out” as well as “down and in” and “down and out” options. This is how eight different combinations are possible for call and put barrier options.

Barrier options sometimes provide rebate feature meaning thereby the holder of an “out” option receives part of premium back if the option is knocked out. Similarly the holder of an “in” option would receive a part of the premium back if the option expires without being “knocked in”.

The pricing of barrier option is not very complex. The value of European down-and-in barrier option plus the value of a European down-and-out barrier option equals the value of call option as per Black Scholes Model.

Black Scholes down out down inC C C− − −= +

Black Scholes up out up inC C C− − −= + Where C equals the call price .

When the barrier level is less than or equal to strike price then the value of an up-and-out call will be zero. Therefore such combination is meaningless as positive intrinsic value is never possible. When the barrier level is less than or equal to the strike price, the value of an up-and-in call equals the value of Black-Scholes call.

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3. Asian option (average rate option, average price option): It pays according to the average value of the underlying during the contract period. A business would use this option to hedge against price increases or decreases over a certain period, but must buy and sell the underlying asset every day or more frequently than the available expiration dates for options or futures.

There are two basic types of average options; average price and average strike. At expiration an average price call option pays off according to the following formula: Average Price = Maximum (Average Spot Price – Strike Price, or 0). The payoff to an average price put option is: Maximum (Strike price – Average price, 0) For average call strike (same as floating strike price and not fixed as usual) payoff is: Maximum (ST – SAVG, 0) Similarly the expiration payoff of an average strike put = Max (SAVG - ST, 0) Asian options are cheaper than ordinary call or put options as volatility is reduced when averaging of prices is done. The pricing of these options are done with simulations. 4. Look back option: It generally pays according to the highest value reached by the underlying during the contract period. In lookback options two possibilities are considered; one in which the strike price is fixed and another in which the strike price is floating. Some lookback options use the highest value reached by the underlying during the contract period to determine the amount of settlement. Sometimes these options are called “no regret option” as the holder of a lookback option can buy the underlying asset at its lowest price between option initiation and option expiration. The formula for this type of lookback is: Lookback pay off = Maximum (Spot Price at Expiration – Minimum Spot Price over Term of Contract, or 0). These standard lookback options are known as “floating-strike” options. Another payoff formula is given below:

Lookback = Maximum (Maximum Spot Price – Strike Price, or 0). Such options are known as “extreme” lookback options. They are also known as fixed-strike lookback option. 5. Bermuda options have an exercise option that is somewhat between that of American- and European-style options. (Supposedly, the name Bermuda derives from the island’s location between the United States and Europe.) Whereas an American-style option can be exercised any time before expiration and a European option can only be exercised on or near the expiration date, a Bermuda option can only be exercised on specific days before expiration or on the expiration date. For instance, a Bermuda option may allow exercise only on the 1st day of each month before expiration, or on expiration. 6. Shouts and Ladders: A shout option gives the option holder the right to capture the intrinsic value portion of the option premium before expiration and also retain the time value. Many a times in American Style option, the option is in the money before expiry and high volatility is there in the market. In such a case the confidence of the option holder is shaken, and he exercises

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the option for profit booking with a guilty feeling of not waiting further for higher profit opportunity. For such situations shout strategy helps as a stop loss strategy, in over- the- counter. Shout option thus permits the holder at any time during the life of the option to establish a minimum pay off that will occur at expiration. A slight variation to this is called cliquet option, in which the exercise price can periodically increase as the stock price rises. If a shout option has numerous shouting opportunities then it is known as a ladder. 4.4.6 Option Greeks: The Greeks refer to the analysis of how an option’s value changes if there is a change in one of the factors that determines option’s value. As per Black Scholes Model they are Volatility (standard deviation), Risk less rate of interest, Stock Price, Time to expiration, Strike price. ( ), , , ,r S t Xσ . These sensitivities are determined by taking the partial derivative of the call pricing formula of Black –Scholes with respect to one of these determinants of value. Each result has been assigned a Greek letter. These options Greeks are very useful for risk management for those who deal with options. The theoretical call value formula can be partially differentiated with respect to each of its five parameters mentioned above. The outcomes are formulas that predict how much a call option value will change if only one input parameter changes by a small amount; all other remaining same. It means no other parameter values are changed. (This is known as sensitivity analysis) As every sensitivity is defined by a Greek letter, this statistics is known as option Greeks. Delta A by-product of the Black-Scholes model is the calculation of the delta: the degree to which an option price will move given a small change in the underlying stock price. The formula for a the delta of a European call on a non-dividend paying stock is: Delta = N (d1) The graph of call price with stock price is shown below: Graph 4.5: Call premium at different stock prices

Here the slope of call price curve is called Delta. For example, an option with a delta of 0.5 will move half a rupee for every one rupee movement in the underlying stock.

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A deeply out-of-the-money call will have a delta very close to zero; a deeply in-the-money call will have a delta very close to 1. Thus, if a call is deep out of money then its delta is about 0. The delta of a call increases as the stock price increases. When the call is deep in the money, it sells about its intrinsic value and the delta of the call approaches one. The delta of an out of the money call will be below 0.5, and it falls toward 0 as time passes. At expiration the delta of an out of the money call is 0. A deep in the money put option will have delta of –1. At the money put deltas are about –0.5. Call deltas are positive; put deltas are negative, reflecting the fact that the put option price and the underlying stock price are inversely related. The put delta equals the call delta - 1. The delta is often called the hedge ratio: If you have a portfolio short “n” options (e.g. you have written n calls) then n multiplied by the delta gives you the number of shares (i.e. units of the underlying) you would need to create a risk less position – i.e. a portfolio which would be worth the same whether the stock price rose by a very small amount or fell by a very small amount. In such a "delta neutral" portfolio any gain in the value of the shares held due to a rise in the share price would be exactly offset by a loss on the value of the calls written, and vice-a-versa. As the delta changes with the stock price and time to expiration, the number of shares would need to be continually adjusted to maintain the hedge. How quickly the delta change with the stock price is given by gamma. Gamma A question may arise as to, why to adopt a delta neutral portfolio when your objective is to make a profit? The Answer is that the above strategy would protect the downside risk while still allow earning profit from most of the upside. A delta neutral portfolio is only delta neutral within a narrow price range of the underlying. Delta itself changes as the price of the underlying changes. Gamma is the change in delta for each unit change in the price of the underlying. The absolute magnitude of delta increases as the time to expiration of the option decreases, and as its intrinsic value increases. Gamma is a derivative of delta. It measures how delta changes as the stock price changes. The

Gamma for call and put options are same. Gamma ( Γ ) = 2

2

CS

∂∂

.The Gamma changes in

predictable ways. Delta can never be greater than 1, or, in the case of a put, less than -1. When delta is close to 1 or -1, then gamma is near zero, because delta doesn’t change much with the price of the underlying. Gamma and delta are greatest when an option is at-the- money—when the strike price is equal to the price of the underlying. The change in delta is greatest for options at the money, and decreases as the option goes more into the money or out of the money. Both gamma and delta tend to zero as the option moves further out of the money. The total gamma of a portfolio is called the “position gamma.”

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Theta (θ ) The option premium consists of a time value that continuously declines as time to expiration nears, with most of the decline occurring near expiration. Theta is a measure of this time decay, and is expressed as the loss of time value per day. Thus, a theta of - 0.1 indicates that the option is losing Re. 0.10 per day. Theta is very little for a long-term option, and increases as expiration nears. Theta is also greatest when the option is at the money, because this is the price where the time value is greatest, and, thus, has a greater potential to decay. For the same reason, theta is greater for more volatile assets, because volatility increases the option premium by increasing the time value of the premium. The call option theta is a U- shaped function of the stock price. That means it is directly related for out of the money calls and inversely related for in the money calls. The rate of time value decay is highest for at the money calls. Although theta is not very useful for individual options, it is sometimes used to assess the changes in value of a portfolio. The holding of options has a negative position theta because the value of options continuously declines with time. However, because time decay is generally considered to favor the option writer, a short position in options is said to have positive position theta. The net of the positive and negative position thetas is the total “position theta” of the portfolio. Vega Vega measures the change in the option premium due to changes in the volatility of the underlying, and is always expressed as a positive number. Because volatility only affects the time value of an option premium, vega tends to vary like the time value of an option—greatest when the option is at the money and least when the option is far out of the money or in the money. Because volatility is difficult to measure, some traders sometimes use implied volatility, and Thus, calculate vega as the change in the option premium per unit change in the implied volatility. Like theta, vega is not very useful for forecasting the value of individual options, but it is sometimes used as a measure of the change in value of a portfolio in response to changing volatility—the position vega. For any given time until expiration, the time value of an option is greatest when the option is at the money, and diminishes as it moves farther either out of the money or in the money. Because theta and vega only measure the effect of time passage and volatility on the time value of an option, both theta and vega are greatest when the time value is greatest, and declines with time value when the price of the underlying moves away from the strike price

117

Rho (ρ ): The sensitivity of the call price to the risk free rate is called its rho and is given by the formula: ρ = ( )2

r tX t e N d− . Black- Scholes Model assumes that the interest rate does not change during the life of the option. Rho is directly related to stock price. Prevailing Interest Rates, Call Premiums

Put Premiums

Historically, higher interest rates generally result in higher call premiums and lower put premiums, and interest rates are a factor in option pricing models. Rho is the amount of change in premiums due to a 1% change in the prevailing risk-free interest rate. Thus, a rho of 0.05 means that the theoretical value of call premiums will increase by 5%, whereas the theoretical value of put premiums will decrease by 5%, because put premiums move opposite to interest rates. The values are theoretical because it is market supply and demand that ultimately determines prices, but interest rates do have some effect. We can summarize the Option Greek Formula in the table form as follows: Here ( )1N d is the standard normal cumulative distribution function, For N(x) =

.

And ( )'1N d is the standard normal probability density function:=

21

2

2

d

eπ

−

Table 4.25: Call and Put Option Greek valuation formula

Greeks

Meaning Calls Puts

Delta ( ∆ )

( )1N d ( ) ( )1 1 1N d N d− − = −

Gamma

( Γ ) ( ) '

1

0

NS

dT σ

Vega (V)

( ) '0 1S N d T

Theta (θ )

Ct

∂∂

( ) ( )( ) '

0 12

S N2

r tdr X e N d

T− σ

− −

( ) ( )( ) '

0 12

S N-

2r td

r X e N dT

− σ+ −

Rho (ρ )

( )2r tX t e N d− ( )2

r tX t e N d−− −

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4.4.7 Interest Rate Options: Interest rate options are like other derivatives namely, Forward Rate Agreement and Interest rate Futures. Options differ in the sense that they do not impose firm commitment to make fixed interest payment but represent a right to make a fixed payment and receive a floating interest payment and vice-a-versa. Interest rate options have an exercise rate or strike rate instead of exercise price or strike price. Calculation of Pay-off in interest rate call option:

• An interest rate call option gives the holder the right to make a known interest payment, based on the exercise rate, and receives unknown interest payment (i.e. LIBOR, MIBOR etc.)

• To purchase the option, the buyer pays a premium, which is called the option price. It has to be paid upfront.

• The following is the formula for calculating pay-off of call option holder. ( ) ( )( )( Pr ) 0, 360

mNotional incipal Max LIBOR X−

• Here notional principal is the amount on which the interest is calculated. • X is the agreed upon rate for payment by the call option holder, if he wishes to exercise

the option. • m is number of days for the loan/option period, for which hedging is required. • In contrast to the payment of Forward Rate Agreement, the pay-off of an interest rate

option does not occur at the expiration. If the underlying is m-day MIBOR (Mumbai Inter Bank Offered Rate), the pay off occurs m-day after the expiration of the option. As payment of pay off is deferred till maturity (i.e. completion of m days), no discounting is required unlike FRA pay off. The following example discusses the above-mentioned points:

• A company is planning to borrow Rs. 50,00,000 in 30 days at 90-day LIBOR plus100 basis points. Loan would be repaid 90 days later. He is worried about the rate applicable after 30 days. To hedge the position he enters into interest rate call option contract. To hedge future borrowing, he has to buy the interest rate call option. Here he preserves the flexibility to benefit from the decrease in the interest rate. Suppose the interest on the loan and the call is based on the factor (90/360), the call has an exercise rate of 12% and the cost (premium) is Rs. 6250.The call premium is compounded for 30 days at the 30-day rate of 13%. It gives the amount of Rs. 6320, which is the cost of call at the time of actual loan borrowing.

• The net proceeds of the loan will be (Rs. 50,00,000 – Rs. 6320) Rs. 49,93,680. • The following table shows the effective cost of borrowing with and without opting for

call option, by assuming different LIBOR rates for borrowing.

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Table 4.26: The effective cost of borrowing with and without opting for call option for different LIBOR

LIBOR on Day 30

Call pay-off (Day 120) Rs.

Amount due on loan (on day 120)Rs.

Total Amount

Paid

Effective cost of borrowing with

call (%)

Effective cost Without call

(%) 6% 0 5087500 5087500 7.84101 7.28926

6.5% 0 5093750 5093750 8.37931 7.824807 7% 0 5100000 5100000 8.91963 8.362365 8% 0 5112500 5112500 10.0064 9.443536 9% 0 5125000 5125000 11.1012 10.53282

10% 0 5137500 5137500 12.2043 11.63024 10.5% 0 5143750 5143750 12.7589 12.18202 11% 0 5150000 5150000 13.3156 12.73586 12% 0 5162500 5162500 14.4352 13.84971 13% 12500 5175000 5162500 14.4352 14.97182 14% 25000 5187500 5162500 14.4352 16.10225 15% 37500 5200000 5162500 14.4352 17.24104

15.5% 43750 5206250 5162500 14.4352 17.81358 16% 50000 5212500 5162500 14.4352 18.38822

Formula list, which are used for various calculations is given below:

1. The amount due on loan is calculated as follows:

( ) ( )( )905000000 1 0.01 360LIBOR+ +

2. The Effective cost on loan without call option is calculated as follows:

( )

( )

36590

1 * 1005000000

AmountDueOnLoan

−

3. The effective cost of Loan with call option is calculated by the following formula:

( )

( )365

901 * 100

4993680AmountDue OptionPayoff − −

120

The following graph shows the comparison of cost of loan with and without Call option.

Graph 4.6: Cost of loan with and without Call option for various rates of interests

Cost of Loan with & without Call

0.00000

5.00000

10.00000

15.00000

20.00000

6

7.8 9

10.5 12 14

15.5

LIBOR on Expiry (%)

Cos

t of L

oan

(%)

cost of Loanwith callCost of LoanWithiout call

Interest rate put option

An interest rate put option gives the holder a right to pay floating rate of interest and receive fixed rate of interest. Therefore it is helpful for investors (e.g. Banks, Financial Institutions and other money lending institutions) who have risk of decrease in the rate of interest in the market, on floating rate investment. By paying a premium, the holder receives a right and not the obligation to pay floating and receive fixed rate payment. So if there is an increase in rate of interest, then the holder will not exercise the put option, thereby still is eligible to get benefit of increase in the interest rate in market.

The formula for pay off of a put buying can be written as follows:

( ) ( )Pr 0,360mNotional incipal Max X LIBOR

−

Here X = the agreed upon fixed rate to be received. m = Days remaining till expiry of investment

121

The following example explains the application of interest rate put option:

• A Financial Institution plans to lend Rs. 50,00,000 in 60 days at 120-day LIBOR plus 100 basis points. The loan will be paid back with principal and interest 120 days later. It is worried about the fall in the market interest rate at the time of lending after 60 days. For protecting against the decrease in the rate of interest, while preserving the flexibility to benefit from an increase in the interest rates; Financial Advisor suggests them to go for interest rate put option. Suppose interest on the loan and the put option calculations are based on the factor (120/360). The put will have an exercise rate of 10% and the premium charged is Rs.2500. The cost of the put is compounded for 60 days at the current 60 day-LIBOR of 11.50%, which amounts to Rs. 2550 approximately. This is the effective cost of the put at the time loan is given. Therefore the effective amount lent is Rs. 50,02,550. (i.e. Rs. 50,00,000 + Rs. 2550)

• The following table shows the effective return on lending with and without opting for put option, by assuming different LIBOR rates for lending.

Table 4.27: The effective return on lending with and without opting for Put option for different LIBOR

LIBOR on Day 60

(%)

Put Option Pay off (Rs.)

Amount received on Loan

repayment (Rs.)

Total amount received with option pay off

Effective return

with put (%)

Effective return

without put (%)

5 % 83333 5100000 5183333 11.40 6.215.5 % 75000 5108333 5183333 11.40 6.74

6 % 66667 5116667 5183333 11.40 7.276.5 % 58333 5125000 5183333 11.40 7.80

7 % 50000 5133333 5183333 11.40 8.338 % 33333 5150000 5183333 11.40 9.419 % 16667 5166667 5183333 11.40 10.49

10 % 0 5183333 5183333 11.40 11.5810.5 % 0 5191667 5191667 11.95 12.12

11 % 0 5200000 5200000 12.50 12.6712 % 0 5216667 5216667 13.60 13.7713 % 0 5233333 5233333 14.70 14.88

14 % 0 5250000 5250000 15.82 16.0015 % 0 5266667 5266667 16.94 17.12

Formula list, which are used for various calculations:

1. The amount due on loan is calculated as follows:

( ) 1205000000 1 0.01360

LIBOR + +

122

2. The Effective Return on investment without put option is calculated as follows:

( )365120Re 1 * 100

50,00,000Amount ceived On Loan

−

3. The effective return of Loan with Put option is calculated by the following formula:

( )365120Re 1 * 100

50,02,550Amount ceived On Loan Option pay off

+ −

The following graph shows the comparison of return on loan with and without Put Option.

Graph 4.7: The effective return on lending with and without opting for Put option for different LIBOR

Return On Loan With & Without PUT

0

5

10

15

20

5 6 7 910

.5 12 14

LIBOR On Expiry (%)

Ret

urn

On

Loan

(%)

EffectiveReturm with put Effective returnwithout put

Thus by purchasing interest rate put option, the investor ensures minimum return of 11.40 %, irrespective of lower rate of LIBOR.

123

Interest rate Caps:

Interest rate calls and puts are not generally useful for long term loans i.e. more than one payment. Therefore they are combined to form a series of receipts and payments. Thus it becomes a combination of Forward Rate Agreements (FRAs).

The combination of interest rate calls for protecting the increase in the rate of interest rate is called “interest rate cap”, while each component individually is referred to as caplet. The following example explains the concept of interest rate cap in detail.

Suppose a company borrows loan in January beginning of Rs. 50,00,000 over one year with quarterly payments. The payments are due in April, July, October and January next year. On each of these dates, starting with January, LIBOR in effect will be the rate to be paid over next quarter. The current LIBOR is 11%. The firm wants to fix the rate on each payment; therefore it buys a cap for an upfront premium payment of Rs. 15,000 with an exercise rate of 11%. The pay off will be calculated as follows:

( )Pr 0, 0.11360daysincipal Max LIBOR

−

Here LIBOR is understood to be set on the previous settlement date and days mean actual number of days in the quarter. LIBOR for the first payment is set when the loan is initiated. Here also the decision for exercise is made at the beginning of the settlement period, which is the date on which the rate is set, but the payoff occurs at the end of the settlement period. The rates of LIBOR are given below: Table 4.28: LIBOR for different value dates maturity.

Date Day count LIBOR (%) 1st Jan 11 1st Apr 90 10 1st Jul 91 13.5 1st Oct 92 12.75 1st Jan 92

The calculations of cash flows are given below: Table 4.29: Calculations of net cash flows with and without interest rate cap

Date Day count

LIBOR Interest due (Rs.)

cap payment

(Rs.)

Principal repayment

(Rs.)

Net cash flow with

cap

Net cash flow without cap

1st Jan 0.11 -15,000 0 4985000 5000000 1st Apr 90 0.1 137500 0 0 -137500 -137500 1st Jul 91 0.135 126388.9 0 0 -126389 -126389 1st Oct 92 0.1275 172500 31944.44 0 -140556 -172500 1st Jan 92 162916.7 22361.11 5000000 -5140556 -5162917

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Interest rate Cap At each interest payment date, the cap will be worth:

( )( )Pr 0, 360Daysincipal Max LIBOR AgreedRate −

The calculation for annualized rate the firm actually paid is given below:

( ) ( ) ( ) ( )1 2 3 4

137500 126389 140556 51405564985000= 1+y 1 1 1y y y

+ + ++ + +

Solving for y gives value of 2.8037 %, annualizing it gives the value of 11.70%. i.e., ( )4(1 ) 1 *100y+ − formula is used to calculate annualized cost. For loan without caps gives the rate of y = 2.9875% annualized rate = 12.50 %. Thus we can say that the cap has proved to be helpful to the borrower. Interest rate floors A combination of interest rate put options is called floors. It gives the investor hedging over a series of floating rate receipts. As the rate for receipt is pre-decided, the investor is assured of a minimum fixed rate of return for a series of cash flows. The pay off for floors can be calculated similar to interest rate caps. Each component of put is referred to as a floorlet. At each interest payment date, the floor will be worth:

( ) ( )Pr * 0, 360Daysincipal Max AgreedUponRate LIBOR −

Interest rate Collars: A combination of a long cap and short floor is called an interest rate collar. It is used by a borrower and consists of a long position in a cap, which is financed by selling a short position in a floor. It is not necessary that the premium from selling the floor be exactly equal to the premium from buying the cap but if they are equal, they are referred to as zero-cost collar.(However it is somewhat misleading as the cost of borrower is in its willingness to give up the benefits from a decrease in interest rates below the exercise price of the floor. It establishes a range of rates within which there is an uncertainty, but the maximum and minimum rates are locked-in.(Source: “Derivatives and Risk Management Basics” by Don Chance and Robert Brooks)

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Example: On 1st March, a company takes a loan of Rs. 6000000 for two years with quarterly interest payments at LIBOR. To protect against rising interest rates, the company buys an interest rate cap with an exercise rate 11% p.a. for a premium of Rs. 30,000. To recover the premium paid, he enters into interest rate floor with an exercise rate of 9% for Rs. 30,000. The interest payments and the payoffs of caps and floor are based on exact days and 360-day basis. Table 4.30: Calculation of net cash flows with and without collar

Date Day count LIBOR Interest Cap

receipt Floor

payment Principal

repayment

Net cash flow with

collar

Net cash flow

Without collar

1-Mar 0.11 -30000 30000 0 6000000 6000000 1-Jun 92 0.125 168667 0 0 0 -168667 -168667 1-Sep 91 0.1275 189583 22750 0 0 -166833 -189583 1-Dec 91 0.1 193375 26542 0 0 -166833 -193375 1-Mar 91 0.105 151667 0 0 0 -151667 -151667 1-Jun 91 0.1 159250 0 0 0 -159250 -159250 1-Sep 92 0.09 153333 0 0 0 -153333 -153333 1-Dec 92 0.08 138000 0 0 0 -138000 -138000 1-Mar 89 118667 0 -14833 -6000000 -6133500 -6118667

After getting cash flow figures we can work out effective cost by applying IRR function in Excel work- sheet as shown earlier. Table 4.31: Effective cost of loan with and without collar (annualized)

Particulars Per Quarter Annualized (%) Effective cost of loan with collar 2.590544 % 10.77183 Effective cost of loan without collar 2.670707 % 11.11846

By comparison we can conclude that the collar has helped the buyer in controlling the borrowing cost to 10.77 % instead of 11.12 % (without hedging).

126

4.4.8 Interest Rate Swaption: A swaption is a variation of an interest rate option. It is based on an underlying swap and has a fixed maturity. A swaption is an option in which the buyer pays a premium up front and acquires a right to enter into a swap. Now like simple call and put option, in swaption also the party may choose payer option or receiver option. In payer swaption the buyer has a right to pay a fixed rate of interest and receive floating rate of interest, while in receiver swaption the buyer has a right to enter as a floating ratepayer and fixed rate receiver. The parties agree up front that exercise will be accomplished by entering into the underlying swap or by an equivalent cash settlement. The pay off of a payer swaption at expiration is:

( ) 0( )1

Pr *( [0, )*360 i

n

ti

daysPayoff Notional incipal Max R X B=

= −

∑

R is the swap rate at expiration of the swaption, X is the agreed rate of swaption.

0( )1

i

n

ti

B=∑ is the present value factors over the life of the swap.

The pay off of a receiver swaption (i.e. floating rate payer and fixed rate receiver) is as follows:

( ) 0( )1

Pr *( [0, )* *360 i

n

ti

daysPayoff Notional incipal Max X R B=

= − ∑

The following example discusses the concept of swaption: A company projects that they will have to enter into swap after two years from today for three year duration, as they will need funds for three years after two years. The amount is Rs. 50,00,000. They are concerned about the rising interest rates during this period of two years’ buffer, before the actual borrowings of Rs.50,00,000. Therefore to hedge this, they would enter into swaption, as a purchaser of two-year swaption where the underlying is a three-year, pay fixed- receive floating swap (i.e. payer swaption). The company contracts the fixed rate of interest of 12.5% The Company pays the premium upfront, suppose, Rs. 20,000. The interest payments will be on annual basis. Suppose after passing of two years, the following term structure of interest rate is available: Table 4.32: Term structure of interest rates

Term (Days) Rate (%) 360 13.00 720 14.00

1080 15.50 From this information we can find out the fixed rate for the swap for three years.

127

Table 4.33: Calculation of fixed rate for three-year swap

Term (Days) Rate Discount for bond

price 360 0.13 0.885 720 0.14 0.781

1080 0.155 0.683 R = 0.1351

Where R is calculated as follows:

( )1 0.6826 360* 0.13510.8850 0.7813 0.6826 360

R− = = + +

If the company exercises the swaption, then it would pay the previously agreed 12.5% rate and receive LIBOR. Thus the swap would yield him positive returns. Suppose the company entered into another swap contract at the ongoing rate of 13 % as receiver in swaption. Thus they have the following positions:

• A swap to pay 12.5% and receive LIBOR • A swap to pay LIBOR and receive 13.51 %.

Ultimately they receive @13.514% and pay @ 12.50 %.

The payment will be Rs. 50,667. i.e. ( ) ( ) 36050,00,000 * 0.1351 0.1250360

−

The pay off of a payer swaption at expiration is as follows:

( ) 0( )1

Pr *( [0, )*360 i

n

ti

daysPayoff Notional incipal Max R X B=

= −

∑

R is the swap rate at expiration of the swaption i.e. 13.514 %, X is the agreed rate of swaption .i.e. 12.50 %.

0( )1

i

n

ti

B=∑ is the present value factors over the life of the swap.

=» ( ) ( )360(50,00,000) 0, 0.1351 0.1250 * 0.8850 0.7813 0.6826360

Max − + +

= Rs. 1,19,030 Which is the present value of future cash flows of pay offs.

128

Similarly the formula for receiver swaption is as follows: The pay off of a receiver swaption (i.e. floating rate payer and fixed rate receiver) is as follows:

( ) 0( )1

Pr *( [0, )* *360 i

n

ti

daysPayoff Notional incipal Max X R B=

= − ∑ as discussed earlier. Thus

depending on the expectations of the future interest rates a party can enter into swaption as either payer or receiver swaption. 4.4.9 Forward Swaps: The forward contracts on swaps are called forward swaps. A forward swap commits the two parties to enter into a swap at a pre-decided fixed rate. The party, who has gone long, commits to enter into swap to pay the fixed rate. The other party commits to enter into swap to receive the fixed rate. Unlike swaption, in forward swap no premium is paid upfront, but at the same time there is an obligation in forward swap.

4.5 Packaged forward contracts: These contracts are regarded as second-generation risk management products. It includes strategies that cover options and forwards. The broad objectives can be written as follows:

• Customized risk profiles can be used to overcome the difficulty of traditional forward contract that has obligation to exercise the performance on maturity, which may be adverse for the user.

• Combinations of bought and sold options are used to compensate for the premium paid for purchasing the option.

• Packaged forwards include a number of special hybrid structures which are designed to assist in managing the problem of uncertainty or contingent hedging.(Source: “swaps/financial derivatives” by Satyajit Das)

The various types of packaged forwards are explained below with examples:

(1) Range forwards: This is an agreement to buy or sell an asset of a specified future date at a price that lies within a particular range of prices. The actual exercise price depends on the actual spot price at the time of exercise of contract. The main features of this contract can be summarized as under:

• Absence of an up-front cost as a result of embedded zero cost premium structure (i.e. it is a combination of bought and sold puts and calls (collar) in which the strike prices are set so that the net premium is zero.)

• Protection against adverse changes in the asset price through the embedded purchase option

• Opportunity for the user to benefit from favorable movements in asset price, but only within a range.

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The following example explains the above-mentioned points: Suppose a UK based company needs to buy 10, 00,000 dollars for value nine months from today, the current market rates are (Pounds/ per dollar) are as follows:

Spot rate 1 US$ = ₤ 0.75 9 months forward rate 1 US$ = ₤ 0.74 The company enters into range forward contract to purchase dollars with a range of ₤ 0.71 and ₤ 0.76. The range forward can be explained as follows:

• If the spot exchange rate at expiration is 1 US$ = ₤ 0.71 or lower, then the company will buy dollars at ₤ 0.71.

• If the spot exchange rate at expiration is between 1 US$ = 0.71 ₤ and 0.76 ₤, then the company will buy dollars at the prevailing spot rate for pounds.

• If the rate turns out to be 1 US$ = ₤ 0.76 or more, then the company will buy at 0.76 ₤ only.

The payoff for the different expected rates are written in the following table:

Table 4.34: The payoff from FRC and Range forward contract at different exchange rates:

Exchange rate(₤) Forward rate contract

Range forward contract

1 US$ = 0.9000 7,40,000 ₤ 7,60,000 ₤ 1 US$ = 0.8500 7,40,000 ₤ 7,60,000 ₤ 1 US$ = 0.8000 7,40,000 ₤ 7,60,000 ₤ 1 US$ = 0.7800 7,40,000 ₤ 7,60,000 ₤ 1 US$ = 0.7600 7,40,000 ₤ 7,60,000 ₤ 1 US$ = 0.7500 7,40,000 ₤ 7,50,000 ₤ 1 US$ = 0.7300 7,40,000 ₤ 7,30,000 ₤ 1 US$ = 0.7100 7,40,000 ₤ 7,10,000 ₤ 1 US$ = 0.7000 7,40,000 ₤ 7,10,000 ₤ 1 US$ = 0.6800 7,40,000 ₤ 7,10,000 ₤ 1 US$ = 0.6500 7,40,000 ₤ 7, 10,000 ₤ 1 US$ = 0.6000 7,40,000 ₤ 7,10,000 ₤

The graph of the above data is given below:

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Graph 4.8: The payoff from FRC and Range forward contract at different exchange rates

Forward Rate Contract & Range Forward Contract

680000700000720000740000760000780000

0.9 0.78

0.73

0.68

Exchange Rates (GBP /US$)

Con

trac

t Val

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Forward ratecontractRange forwardcontract

(2) Break forwards: This is an agreement to purchase or sell an asset at an agreed price, with

the ability to close out the contract, if the price of the asset goes in the purchaser’s or seller’s favour. It is also known as “ a forward with optional exit”

The main features of this contract can be summarized as under:

• Absence of an up-front cost • Protection against adverse changes in the asset price through the embedded purchase

option. • Opportunity to benefit from the favorable asset price movement. (i.e. above or below

the agreed break rate.) • Break loading is added to the available plain forward rate to derive adjusted forward

rate. • The contract works as a embedded option contract (i.e. buying call of dollars and

selling put of dollars at the same rate) and buying a put at break rate. (The cost of this option is loaded in the prevailing plain forward rate, which is known as Adjusted forward rate.)

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Continuing with the above U.K. based company in case of break forward it works as under: Consider the following rates: Spot rate 1 US$ = ₤ 0.75 9 months forward rate 1 US$ = ₤ 0.74 Break loading 2.5% * ₤ 0.74 = ₤ 0.0185 Adjusted forward rate 1 US$ = ₤ 0.7585 Break rate 1 US$ = ₤ 0.7300 The strategy is as follows: The forward purchase of Dollar at 1 US$ = ₤ 0.7585 is equivalent to a purchase of Dollar call and a sale of Dollar put at a strike price of ₤ 0.7585; Long a Dollar put with a strike price of break rate i.e. ₤ 0.7300

• If dollar strengthens above 0.73₤ then the company purchases Dollars at agreed fixed price of ₤ 0.7585 per dollar.

• If the rate is equal to break rate (i.e. 1 US$ = ₤ 0.73) then the break facility is activated. Buying at 0.7585 and selling at 0.73 will lock the loss of 0.0285 per dollar. It means total loss of ₤ 28,500.(0.0285* 10,00,000)

• If the dollar weakens below break rate ₤ 0.7300, then also the break facility is activated. In addition the company would sale dollars at the ongoing lower market rate than the break rate. Thus the net rate applicable would be ongoing rate plus 0.0285.

The payoff for the different expected rates are written in the following table:

Table 4.35: The contract value for FRC and Break Forward Contract for different Exchange Rates

Exchange rate (₤)

Forward rate

contract Break forward

contract 1 US$ = 0.9000 7,40,000 ₤ 7,58,500 ₤ 1 US$ = 0.8500 7,40,000 ₤ 7,58,500 ₤ 1 US$ = 0.8000 7,40,000 ₤ 7,58,500 ₤ 1 US$ = 0.7800 7,40,000 ₤ 7,58,500 ₤ 1 US$ = 0.7600 7,40,000 ₤ 7,58,500 ₤ 1 US$ = 0.7500 7,40,000 ₤ 7,58,500 ₤ 1 US$ = 0.7300 7,40,000 ₤ 7,58.500 ₤ 1 US$ = 0.7100 7,40,000 ₤ 7,38,500 ₤ 1 US$ = 0.7000 7,40,000 ₤ 7,28,500 ₤ 1 US$ = 0.6800 7,40,000 ₤ 7,08,500 ₤ 1 US$ = 0.6500 7,40,000 ₤ 6,78,500 ₤ 1 US$ = 0.6000 7,40,000 ₤ 6,28,500 ₤

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The graph of the above table is shown below: Graph 4.9: The contract value for FRC and Break Forward Contract for different Exchange Rates

Break Forward Graph

600000650000700000750000800000

0.9 0.78

0.73

0.68

Exchange Rate (GBP/US$)

Con

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Forward ratecontractBreak forwardcontract

(3) Participating forwards: It is an agreement to buy or sell an asset at an agreed price. The contract specifies the worst-case price at which the hedger can buy or sell the asset. The contract also specifies an agreed participation rate. The party receives the portion of any favorable movement in the asset price. The portion is equal to the agreed participation rate. Functionally it is a combination of bought and sold puts and calls. The face value of the two options is different. Premium neutrality is obtained by adjusting the strike rate of the two options.

The main features of this contract can be summarized as under: • Absence of an up-front cost as a result of the embedded zero cost premium structure

of collar. • Protection against adverse changes in the asset price through the embedded purchased

option. • Flexibility in terms of desirable price level to hedger for protection.

Continuing with above example of U.K. based company wanting dollars 10,00,000. It enters into participating forwards for nine months to buy dollars, wherein it is protected from weaker UK₤ below 1 US$ = ₤ 0.76. However the company will participate in any appreciation in the UK₤ above 1 US$ = ₤ 0.74. The agreed participation ratio is 50%.

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This works as under: • If the rate on expiry is 1 US$ = ₤ 0.76, the company buys at ₤ 0.76 • If the rate is 1 US$ = ₤ 0.74 or less; then the corporation buys at ₤ 0.74 for 50 % of the

total amount of the face value and remaining 50% at ongoing spot value. • Thus effective rate = {[(participation ratio * Spot rate) + ((1 – participation ratio) *

agreed price))] * Notional amount} The payoff for the different expected rates are written in the following table:

Table 4.36: The contract value of FRC and Participating Forward Contract for different exchange rates

Exchange rate(₤) Forward rate contract Participating forward contract 1 US$ = 0.9000 7,40,000 ₤ 7,60,000 ₤ 1 US$ = 0.8500 7,40,000 ₤ 7,60,000 ₤ 1 US$ = 0.8000 7,40,000 ₤ 7,60,000 ₤ 1 US$ = 0.7800 7,40,000 ₤ 7,60,000 ₤ 1 US$ = 0.7600 7,40,000 ₤ 7,60,000 ₤ 1 US$ = 0.7500 7,40,000 ₤ 7,50,000 ₤ 1 US$ = 0.7400 7,40,000 ₤ 7,40,000 ₤ 1 US$ = 0.7300 7,40,000 ₤ 7,35,000 ₤ 1 US$ = 0.7100 7,40,000 ₤ 7,25,000 ₤ 1 US$ = 0.7000 7,40,000 ₤ 7,20,000 ₤ 1 US$ = 0.6800 7,40,000 ₤ 7,10,000 ₤ 1 US$ = 0.6500 7,40,000 ₤ 6,95,000 ₤ 1 US$ = 0.6000 7,40,000 ₤ 6,70,000 ₤

The graph of the above data is given on next page:

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Graph 4.10: The contract value of FRC and Participating Forward Contract for different exchange rates

Participating Forward Graph

600000650000

700000750000

800000

0.90.78 0.74 0.7 0.6

Exchange Rate ( GBP / US$ )

Con

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Val

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Forward ratecontractParticipatingForwards

4.6 FUTURE s OPTION (Future with Options) 4.6.1 Introduction: A futures option is the right and not the obligation, to enter into a futures contract at a certain futures price by a certain date. A call futures option contract is the right to enter into a long futures contract at a certain (pre decided price). A put futures option is the right to enter into a short futures contract at a pre fixed price. When a call option is exercised, the holder acquires from the writer a long position in the underlying futures contract plus a cash amount equal to the excess of the futures (last traded price) over the strike price. When a put option is exercised, the holder gets a short position in the underlying futures contract plus a cash amount equal to the excess of the strike price over the futures price (last traded). The futures option contract is more liquid than the underlying asset.

• A futures price is known immediately from the trading on the futures exchange. Whereas the spot price of the underlying asset may not be so readily available.

• Exercising the contract does not lead to delivery of the underlying. As in most of the cases the futures contract is closed out prior to delivery. It helps the investor with limited capital that may find it difficult to come up with the funds to buy the underlying asset when the option is exercised.

• It may be used for hedging, arbitraging and speculation.

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4.6.2 Put call parity in futures option: This can be explained with the following two portfolios Portfolio1: A European call futures option with an amount of cash equal to Xe-rt

Portfolio 2: A European put option with long futures contract plus an amount of cash equal to F0e-rt

If Ft > X (Bullish market) For portfolio 1: Then call option in portfolio 1 is exercised and the gain will be = Ft -X Cash is invested at risk free rate r, so it will grow to X on expiry. Thus total value will be Ft.

For portfolio 2 The put option is out of the money and value will be nil. Cash will grow to F0 at time t The long futures will give value equal to (Ft - F0) Thus the total value will be Ft If Ft < X (Bearish market) For portfolio 1: The call option in portfolio 1 is out of the money and value will be nil. Cash is invested at risk free rate r, so it will grow to X on expiry. Thus total value will be X.

For portfolio 2 The put option in portfolio 2 is exercised and the gain will be ( X - Ft ) The long future will give value equal to (Ft - F0 ) Cash will grow to F0 at time t. Thus total value will be X. As both the portfolios have the same value at time t and there are no early exercise opportunities, it is inferred that they are worth the same today.

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Value of portfolio 1 today is

c + Xe-rt = p + F0e-rt

c = call future’s premium p = put future’s premium X = strike price of option call & put F0 = The price of futures at time 0(when the futures contract is entered). Ft = The futures price at time t T = Expiry time of futures r = risk free rate of interest This equation is called put-call parity for European style options. 4.6.3 Examples of Futures Options: Call futures option: (Source: “Futures and Options market” by John C. Hull) A person buys an April call futures option contract on Gold. The contract size is 100 ounces. The strike price is 310. During he life of the contract, the price of gold futures reaches to 350 and the previous / most recent settlement price is 348. Thus the investor decides to exercise this futures option contract. The investor receives a long futures contract in addition to a cash equal to [(348 – 310) * 100] = $ 3800. If the investor decides to close its futures position today itself then he further will have a pay off of [(350 – 348) * 100] = $ 200. Thus the total pay off from the exercise is 3800 + 200 = $ 4,000. Put futures option: A person buys a July put futures option contract on a particular commodity. The contract size is 5,000 units of a measure the strike / exercise price is 250 cents. Suppose the price goes down to 220 cents during the life of the July contract. The previous settlement price was 218. The person decides to exercise the contract as it is in the money. The investor receives a short futures contract in addition to a cash pay off of $ [(2.50 – 2.18) * (5,000)] = $ 1,600. If the person decides to close out his short position right now, then the pay off will be [{2.18 –2.20)* 5,000] = $ - 100 (loss) Thus the net pay-off from the exercise of contract equal to [1600 – 100] equal to $ 1500.