Unit 5 DFA 4212options

7/29/2019 Unit 5 DFA 4212options

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OPTION CONTRACTS

5.0 Overview

5.1 Learning Outcomes

5.2 Options: Definition

5.3 Differences between Options and Forward/Futures Contracts

5.4 The Origin and Development of Options Markets

5.5 Payoff Profiles

5.6 The Pricing of Options

5.7 Summary

5.8 Activities

5.9 References & Readings

5.0 OVERVIEW

This unit discusses on the use of option contract to manage financial risk pertaining to

market, interest and exchange rate mainly after having dwelled into the definition and

different kinds of options. Emphasis is put on the pricing of options and on the Black

Scholes options pricing model.

5.1 LEARNING OUTCOMES

By the end of this Unit, you should be able to do the following:

1. Define an option contract.

2. Discuss the different type of options.

3. Explain how to use currency and interest options to hedge exchange and interest rate

risk.

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4. Differentiate between an American and a European option; between a call and a put

option.

5. Draw the pay-off profile of a long call, along put, a short call and a short put.

6. Outline the factors that explain the price of an option(both call and put).

7. Analyse the intuition behind the Black-Scholes formula.

8. Explain the Black-Scholes formula in the following form:

C= S N (d1) - Xe-rT N (d2)

5.2 OPTIONS: DEFINITION

Options are more complex derivatives that have a number of different varieties. Options

can be European or American and can be calls or puts. Below we shall give a brief

outline of the nature of the option contract for the aforementioned classes.

An optionis a contract that gives the owner the right, but not the obligation, to buy or sell

a specified asset at a specified price, on or by a specified date.

Aput optionconfers to the owner the right to sell the asset.

A call optionconfers to the owner the right to buy the asset.

Put and call options are traded for many assets, including corporate stocks, major

currencies, and major stock indices. The majority of options traded on exchanges,

however, are options on futures contracts traded on the exchanges. An option involves

two parties. The ownerorpurchaserof a put (call) option obtains the right to sell (buy)

an asset at a specified price by paying apremium to the writerorsellerof the option, who

assumes the collateral obligation to buy (sell) the asset, should the owner of the option

choose to exercise it. An individual who takes on this option must additionally pay a

given amount of money (the option price) at the outset. The agent who provides this

option is said to have written the option and receives the option price while promising to

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deliver the asset specified at the future date if the buyer exercises his right to purchase.

The owner is said to take a longposition in the option; the writer is said to take ashort

position in the option.

When discussing option contracts, several other technical terms must be clearly

understood. The strike price or exercise price is the price at which the commodity or

asset may be bought or sold by the owner of the option under the terms of the option

contract.

The expiration date, exercise date, or maturity refer to the date at which the option

expires or, equivalently, the last date on which the owner may exercise his option.

AnAmerican option may be exercised at any time up to expiration date.

AEuropean optionon the other hand, may be exercised only on the expiration date.

An option is said to be in the money if its immediate exercise would produce positive

cash flow. Thus, a put option is in the money if the strike price exceeds the spot price of

the underlying asset and a call option is in the money if the spot price of the underlying

asset exceeds the strike price. An option that is not in the money is said to be out of the

money.

Example of a call option

A call option on 100 IMB shares at a strike price of $90 that expires on 31 January 2009

gives the call option holder the right to buy 100 IBM shares for $90 a share prior to 31

January 2009. The writer of the option is obligated to sell the share if and when the

holder decides to exercise the option.

Suppose the IBM stock is trading at $95 per share. Is it beneficial for the holder to

exercise the right?

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Yes it is. The option holder can buy the IBM shares at $90 instead of the actual trading

price which is currently at $95. So if the option holder exercises the option he will be

buying the shares at a discount of $5 per share.

Propositions concerning a call option

1. If the value of the underlying asset is greater than the strike price, the option is

exercised. Why?

Because the holder of the option will be having the right to purchase the asset at a

discount. Other market participants who do not have that right have to buy at the

higher price.

2. If the value of the asset is smaller than the strike price, the option is not exercised.

In the case of the IBM example, the price of the shares may be trading at say $80

per unit. Now the holder of the call option has the right to buy at $90 but the

shares are actually trading at below that price so it is pointless for him to exercise

that right because he will be buying the shares at $90 instead of the current $80

You can take one of four positions on an options contract:

1. Buying a call option-known as being long on a call.

2. Selling a call option-known as being short on call.

3. Buying a put option-know as being long on a put

4. Selling a put option- known as being short on a put.

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The table below give a summary for all option positions possible:

London (holder) Short (writer)

Call Right, but no obligation to

buy

Obligation to sell

Pot Right, but no obligation to

sell

Obligation to buy

Table 1: A summary of option positions

5.3 DIFFERENCES BETWEEN OPTIONS &

FORWARDS/FUTURES CONTRACTS

Options differ from forward/futures contracts as shown in Table 2.

Options contracts Forward contract/futures contract

The buyer of an option(long position) is not

obliged to transact

The buyer of a forward/futures contract

(long position) is obliged to transact; the

buyer of a futures contract can establish a

reversing trade prior to the maturity of the

contract, but has to trade at maturity.

Asymmetric pay-off profile Symmetric pay-off profile

Premium paid for option Margin requirement (futures contract), bid-

ask spread(forward contract)

Table 2: Differences between options and forwards/futures contracts

5.4 THE ORIGIN AND DEVELOPMENT OF THE OPTIONS

MARKET

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Option contracts on foreign exchange (currency options) and interest rate (interest rate

options) are among the most recent innovations in international financial markets. In

April 1973, organised trading in call options began on the Chicago Board Options

Exchange (CBOE), followed by call option trading on the American Stock Exchange

(AMEX). Both American- and European- style options on spot exchange are traded over-

the-counter at the Philadelphia Stock Exchange (PHLX). Currency options are traded

either between banks or between banks and their customers.

International Monetary Market (IMM) foreign currency futures are traded at the Chicago

Mercantile Exchange (CME). Interest rate options are typically options on interest rate

futures.

Since options can be written on any asset or commodities whose price is determined by

market forces, that is the forces of demand and supply, option instruments, with the

increasing level of risks, have quickly become an interesting instrument. This is fostered

by the relatively liquid market it represents coupled with the ever increasing level of risk

in the world due to the process of liberalisation.

Due to the increased level of risks and also to the attractiveness of options contracts,

nowadays, a number of options market have emerged, among which features the Korea

exchanged and the Singapore exchange. Their growth rate has been phenomenon and

they offer a wide variety of options related to commodities and financial instruments.

5.5 PAY-OFF PROFILES

As seen in the following Figure, the payoff from a put option at the expiration date is a

function of the strike priceKand the spot pricesTof the underlying asset on the delivery

date. From this diagram, we conclude that if a put option is held until expiration (which

must be so for a European option, but not an American option), then the option will be

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exercised if, and only if,sT < K, in which case the owner of the option will realise a net

payoffK - sT > 0 and the writer of the option will realise a net payoffsT - K < 0.

.

In interpreting the payoff diagrams for put and call options, one must keep in mind that

options are bought and sold at a premium. Although the cashflow at expiration for an

owner of an option is non-negative, the owner paid a premium to acquire the option

initially. Similarly, although the cashflow at expiration for the writer of an option is non-

positive, the writer received a premium to write the option initially. Thus, the net

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cashflow from an option over time can be either positive or negative for the owner or

writer alike.

5.6 PRICING OF OPTIONS

The Factors Affecting the Price of a European Call and Put Option

The higher the likelihood that the option is exercised the higher should be the option

premium as the writer has more probability of default. There are five factors influence the

likelihood of a call (and put) option being exercised and thus the price to be paid for a

European call (and put) option. Payment of the option premium gives the buyer the right

to buy the underlying asset at a predetermined price at a given date in the future. The

factors are:

1. The current price of the underlying asset. The higher the current price of the asset the

more likely the option is to be exercised for any given strike price and consequently, the

higher the price of a call option.

2. The strike price. The higher the strike price of a call option, the less likely it is that it

will be exercised and consequently the lower the price.

3. The time left to expiration. The longer the time left to expiration, the higher the chance

of the option being exercised and hence the higher its price.

4. The volatility. The more volatile an option is, the more likely that its price at the time of

the expiration will exceed the strike price and consequently, the higher the price of the

option.

5. The risk-free rate of interest. The buyer of a call option is paying the issuer cash for an

option that can be exercised to buy an underlying security at a future date. The option-

holder is thus benefiting from the fact that the difference between the option premium

and actually buying the underlying security can be invested at a risk-free rate of interest

until the option expires. A rise in the risk-free rate of interest makes it more attractive to

buy the option rather than the underlying security. For this reason, other things being

equal, a call option needs to be priced more highly when interest rates are high than when

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interest rates are low. The higher the risk-free rate of interest, the higher a call option

price. Nonetheless, changes in the risk-free rate of interest are usually only a marginal

factor in the pricing of options.

The binomial option pricing framework was originally used to price call options. It was

based on the notion of no arbitrage and also assumed a binomial probability function.

This framework was very simple but the model was not sufficient for pricing options in

the real world as it failed to account for the dynamics of the underlying asset. Obviously,

the single branch binomial process is far from the frequent, random movements we see in

asset prices and in real life.

5.6.1 The Black-Scholes Option Price

The most famous formula for pricing European call options was the Black-Scholes

Option Price and was derived by Fischer Black and Myron Scholes in a seminal 1974

paper. In this small section, we present their option pricing formula. This framework is an

extension of the binomial model and assumes normal distribution. The arguments

employed in the Black-Scholes analysis are precisely those we came across in the

binomial option pricing framework (i.e. replication followed by arbitrage piercing). Thedifference between their model and that shown above is that the process for the dynamics

of the underlying asset price is specified in continuous time (although our binomial

model has very similar features).

The underlying assumptions of the Black-Scholes options pricing formula for European

call options on equity are:

The underlying asset being analysed pays no dividends or interest during its lifetime.

The option is a European option; that is, it cannot be exercised prior to maturity.

The risk-free rate of interest is fixed during the life of the option.

The financial markets are perfectly efficient with zero transaction costs(no bid-ask

spreads) and no taxes.

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The price of the underlying is log-normally distributed, with a constant mean and

standard deviation.

It is possible to short-sell the underlying asset and use the proceeds

obtained without restriction.

The price of the underlying asset moves in a continuous fashion.

The Black-Scholes option price is

C = SN (d1) - Xe-rt N ( d2)

c = option premium

s = spot price

x = exercise price

r = rate of interest

t = time period

Where N( ) is the standard normal distribution function and

Similarly with the same set of arguments, the premium for a European put option expires

in time Tand has strike priceKis given by

where d1 and d2 are defined as above andNis the cumulative distribution function for a

standard normal variate.

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Example

A call option with an exercise price of $50 has three months to expiration. The

continuously compounded annual risk-free rate is 4%; the stock currently trades for $45;

and the volatility of the asset price is 0.4. Then T = 0:25, K = 50, S = 45, risk = 0:4, and r

= 0.04. According to the Black-Scholes formula,

and the price of the European Call option is

Valuing American Options

As mentioned above, the main difference between European and American options is that

American options are usually priced slightly higher than European options because of the

extra advantage that they give to the holder of being able to exercise the option at any

date prior to maturity. Because American options have an early exercise feature, they

cannot be priced using the Black-Scholes formula. The price of an American option,

however, can be derived numerically using a recursive procedure in which the time to

expiration of the option is divided into many smaller subintervals.

The price of a Put Option, P is given by

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Properties of the Black-Scholes Option Price

The above equation may look daunting. However, we can draw some very intuitive

arguments about the determinants of option price from it. These are:

call option are greater if the underlying asset price is greater (S).

call prices increase with decreases in the exercise price (X).

option prices increase with volatility of the underlying (()

option prices increase as time to maturity (T-t) is increased.

The Drawbacks of the Black-Scholes Option Pricing Formula

The Black-Scholes option pricing formula is only applicable to European options.

The Black-Scholes option pricing formula assumes that the log of the underlying

security follows a log-normal distribution. In the real world, distributions tend to be

leptokurtic, that is, they have fatter tails than the normal distribution.

5.7 SUMMARY

An option is a contract that gives the owner the right, but not the obligation, to

buy or sell a specified asset at a specified price, on or by a specified date.

A put option confers to the owner the right to sell the asset.

A call option confers to the owner the right to buy the asset.

The owner or purchaser of a put (call) option obtains the right to sell (buy) an

asset at a specified price by paying a premium to the writer or seller of the option,

who assumes the collateral obligation to buy (sell) the asset, should the owner of the

option choose to exercise it.

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An American option may be exercised at any time up to expiration date.

A European option on the other hand, may be exercised only on the expiration

date.

Option contracts on foreign exchange (currency options) and interest rate (interest

rate options) are among the most recent innovations in international financial markets.

The higher the likelihood that the option is exercised the higher should be the

option premium as the writer has more probability of default.

There are five factors influence the likelihood of a call (and put) option being

exercised and thus the price to be paid for a European call (and put) option.

1. The current price of the underlying asset.

2. The strike price

3. The time left to expiration

4. The volatility

5. The risk-free rate of interest

The binomial option pricing framework was originally used to price call options

The most famous formula for pricing European call options was the Black-

Scholes Option and was derived by Fischer Black and Myron Scholes in a seminal

1974 paper.

This framework is an extension of the binomial model and assumes normal

distribution.

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American options are usually priced slightly higher than European options

because of the extra advantage that they give to the holder of being able to exercise

the option at any date prior to maturity.

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Unit 5 DFA 4212options