Put-Call Parity and Arbitrage

7/27/2019 Put-Call Parity and Arbitrage

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Value & Price

Relative Valuation & Absolute Valuation

Trading Desk & Broker & Dealer & Market-Maker

1. Underwriting :

2. Brokers :

3. Dealers :

4. Trading Desk :

5. Market-Maker :

What is the difference between a broker and a market maker?

Abrokeris an intermediary who has a license to buy and sell securities on a client's behalf.

Stockbrokers coordinate contracts between buyers and sellers, usually for acommission. Amarket maker, on the other hand, is an intermediary that is willing and ready to buy and sell

securities for a profitable price.

A broker makes money by bringing together securities' buyers and sellers. Brokers have theauthorization and expertise to buy securities on an investor's behalf - not just anyone is

allowed to walk into theNew York Stock Exchange and purchase stocks; therefore,investors must hire licensed brokers to do this for them. A flat fee or percentage-based

commission is given to the broker for carrying out a trade and finding the best price for a

security. Because brokers are regulated and licensed, they have an obligation to act in the

best interests of their clients. Many brokers can also offer advice on what stocks, mutualfunds and other securities to buy. Due to the availability of internet-based automated stock

brokering systems, clients often do not have any personal contact with their brokerage

firms.

A market maker makes a profit by attempting to sell high and buy low. Market makersestablish quoteswhereby thebid price is set slightly lower than listed prices and theaskprice is set slightly higher in order to earn a small margin. Market makers are useful

because they are always ready to buy and sell as long as the investor is willing to pay a

specific price. This helps to createliquidityand efficiency in the market. Market makers

essentially act as wholesalers by buying and selling securities to satisfy the market; theprices they set reflect market supply and demand. When the demand for a security is low

and supply is high, the price of the security will be low. If the demand is high and supply is

low, the price of the security will be high. Market makers are obligated to sell and buy at theprice and size they have quoted. (For further reading, see Why The Bid-Ask Spread Is So

Important.)

:You've probably heard the termsspread orbid and ask spread before, but you may not knowwhat they mean or how they relate to the stock market. The bid-ask spread can affect the

price at which a purchase or sale is made - and an investor's overall portfolio return. What

this means is that if you want to dabble in the equities markets, you need to become familiarwith this concept.
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Supply and Demand

Investors must firstunderstand the concept of supply and demand before learning the ins

and outs of the spread. Supply refers to the volume or abundance of a particular item in themarketplace, such as the supply of stock for sale. Demand refers to an individual's

willingness to pay a particular price for an item or stock.

There are two main lines of business in investment banking. Trading securities for cash orfor other securities (i.e. facilitating transactions, market-making), or the promotion of

securities (i.e. underwriting, research, etc.) is the "sell side", whilebuy side is a term used

to refer to advising institutions concerned with buying investment services. Private equityfunds, mutual funds, life insurance companies, unit trusts, and hedge funds are the most

common types of buy side entities.

It is often the case that a market maker is also a broker. This can sometimes create theincentive for the broker to recommend securities for which he or she also makes a market.

Therefore, investors should make sure that there is a clear separation between a broker and

a market maker.

A broker-dealer firm that accepts the risk of holding a certain number of shares of a

particular security in order to facilitate trading in that security. Each market makercompetes for customer order flow by displaying buy and sell quotations for a guaranteed

number of shares. Once an order is received, the market maker immediately sells from its

own inventory or seeks an offsetting order. This process takes place in mere seconds.

Trading desks can be either large or small depending on the organization and are

occupied by licensed traders, usually specializing in trading one particular type ofinvestment product (e.g. forex traders, commodities traders, stock traders, etc.). The

instantaneous trade executions can be particularly important for day traders looking for

arbitrage opportunities that usually last only minutes or even seconds.

The Nasdaq is the prime example of an operation of market makers. There are more than

500 member firms that act as Nasdaq market makers, keeping the financial marketsrunning efficiently because they are willing to quote both bid and offer prices for an asset.

A trading desk is a desk within an organization where the buying and selling of securities

takes place. Companies that manage these desks include banks and other financial

institutions. Examples of investments exchanged at trading desks are fixed-incomesecurities, equities, futures, commodities and foreign exchange. Depending on the type of

business where it is housed and the variety of exchanges that occur, a trade desk can be

large or small.

Licensed traders who specialize in one type of investment, such as a commodity trader orstock trader, man the desks. The trades that occur take place instantly, which is important

for day traders who seek financial opportunities that often do not last more than a few

minutes. The day tradingdesk can help day traders make their exchange quickly and take
http://www.investopedia.com/articles/economics/11/intro-supply-demand.asphttp://www.investopedia.com/articles/economics/11/intro-supply-demand.asphttp://en.wikipedia.org/wiki/Sell_sidehttp://en.wikipedia.org/wiki/Buy_sidehttp://www.wisegeek.com/what-are-futures.htmhttp://www.wisegeek.com/what-are-futures.htmhttp://www.wisegeek.com/what-does-a-stock-trader-do.htmhttp://www.wisegeek.com/what-is-day-trading.htmhttp://www.investopedia.com/articles/economics/11/intro-supply-demand.asphttp://en.wikipedia.org/wiki/Sell_sidehttp://en.wikipedia.org/wiki/Buy_sidehttp://www.wisegeek.com/what-are-futures.htmhttp://www.wisegeek.com/what-does-a-stock-trader-do.htmhttp://www.wisegeek.com/what-is-day-trading.htm


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advantage of the opportunity before it disappears.

Also known as a dealing desk, the primary function of a trade desk is to promote

progressive and reactive pricing functions. Every exchange made in the stock market has

a ripple effect. Trade desks are a good way to manage the transactions and their

consequences. Above all, a trading desk provides a more solid, market-oriented methodof making deals.

2007/11/06

Emanuel Derman Lecture said :

Why do practitioners concentrate on relative valuation for derivatives pricing?

Because most of the firms youll end up working at are not interested in simple

speculation or consumption of financial products.

You can make money by using derivatives like lottery tickets, to speculate.

Then youre interested only in the price vs. the probability of payoff.
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But many financial firms are more interested in more complex bets, by trying to be

arbitrageurs.

An arbitrageur tries to make money by identifying securities that are mispriced relative to

another security, and then exchanging them.

More generally, an arbitrageur is a kind of manufacturer that manufactures one security out

of others more cheaply than it can be bought.

The price of the ingredients and the labor have to be less than the price of the finished

product, if not with 100% confidence then at least with a high probability.

Therefore, were interested in relative value.

In this course on derivatives we are going to take the viewpoint of a manufacturer, because

the great insight of Black-Scholes is that derivatives can be manufacture our of stock and

bonds.

Options trading desks are in many ways like manufacturers, or inverse manufacturers.

They acquire simple ingredients stocks and Treasury bonds, for example and

manufacture options out of them by replication or hedging.

The more sophisticated ones do the reverse: they acquire relatively simple options and

construct exotic ones out of them, or acquire exotic ones and deconstruct into their simpler

parts and sell them off.

From their point of view, relative value is very important, because you are buying

ingredients and selling a finished product, or vice versa.

Similarly, exotic options valuation is often a question of replicating exotics out of more

liquid vanilla options.

Well cover this later in the course.

In this course well mostly take the viewpoint of a trading desk or a market-maker who

buys what people want to sell and sells what people want to buy, willing to go either way,always seeking to make a profit.

For desks like that, valuation is always a relative concept.

Much of relative value modeling is then more or less sophisticated versions of fruit salad

problems: given the price of apples, oranges and pears, what should you charge for fruitsalad? Or, the inverse problem: given the price of fruit salad, apples and oranges, what is

the implied price of pears?


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But dont think that you can escape all sentiment in financial theory; the models of

quantitative finance involve expectations and estimates of future behavior, and those

estimates and expectations are peoples estimates.

Financial economists refer to their essential principle as the law of one price, or the

principle of no riskless arbitrage, which states that .........Any two securities with identical future payoffs, no matter how the future turns out, should

have identical current prices.

The law of one price is not a law of nature.

Its a general reflection on the practices of human beings, who, when they have enough time

and enough information, will grab a bargain when they see one.

The law usually holds in the long run, in well-oiled markets with enough savvy participants,

but there are always short- or even longer-term exceptions that persist.

Dynamic replication is very elegant, and all the advances in the field of deriva-tives over the past 30 years have been connected with extending the fundamen-

tal insight that you can sometimes replicate complex securities dynamically.

But, and practitioners who work with trading desks know this from their ownexperience, its not a walk in the park.

Neftcis Principles of Financial Engineering, pp. 188-89, is one of the few

books Ive seen that points out the actual problems that arise when you try to

replicate dynamically:Real life complications make dynamic replication a much more fragile

exercise than static replication. The problems that are encountered in static

replication are well known. There are operational problems, counterpartyrisk, and so the theoretically exact synthetics may not be identical to the

original asset. There are liquidity problems and other transactions costs.

But all these are relatively minor and in the end, static replicating portfoliosused in practice generally provide good synthetics.

With dynamic replication, these problems are magnified, because the

underlying positions needs to be readjusted many times. For example, the

effect of transaction costs is much more serious if dynamic adjustments arerequired frequently. Similarly, the implications of liquidity problems will

also be more serious. But more importantly, the real-life use of dynamic

replication methods brings forth new problems that would not exist withstatic synthetics.

We have to worry not just about current liquidity and bid-ask spreads, but

about how they vary in the future.Dynamic replication is imperfect; it depends upon models, which imply

assumptions and the approximations involved in working in discrete time

steps.


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Even if the theory is easy, the strategy needs to be implemented using

appropriate position-keeping and risk-management tools. The necessary

software and human skills required for these tasks may lead to significantnew costs, but also to many jobs producing and taking care of these tools.

Finally, dynamic replication is often used to replicate securities with non-

linear payoffs. This leads to exposure to the level of volatility, and whoknows what the future level of volatility will be. Managing exposure to

volatility can be much more difficult than managing exposure to interest

rates or currencies, because there are (almost) no underlyers to trade.So, in this course, wherever we can, we will first try use static replication for

valuing new securities. If we cannot, then we will use dynamic replication.

And finally, if we cant replicate exactly, then you have to speculate on risk and

return, which is less mathematical but more difficult.

Lecture_02

..... because we are taking the viewpoint of a ......dealer or

market-maker

who, for practical reasons, wants to use a single average discount rate for all securities inhis portfolio with the same expiration.

A dealer or market-maker in options, however, has additional consistency con-

straints. As a manufacturer rather than a consumer of options, the market-maker must make prices consistent with the value of his raw supplies. He must

notice that a portfolio F=C-K consisting of a long position in a call and a short position in a

put with the same strike K has exactly the same payoff as a forward contract with expirationtime T and delivery price K whose fair current value is :

Put-Call Parity

where r is the true riskless discount rate for time to expiration (T-t).

The individual formulas of Equation 2.18 and Equation 2.19 must be consistent with the

constraint that when they are combined to value the portfolio F which statically replicates aforward contract, they should produce the same value.

If they are not calibrated to satisfy this, the market-maker will be valuing his options, stockand forwards inconsistently.

What is necessary to satisfy this ??


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Put-Call Parity & Arbitrage

Anarbitrageoperation may be represented as a sequence which begins with zero balancein an account, initiates transactions at time t = 0, and unwinds transactions at time t = T sothat all that remains at the end is a balance whose value B will be known for certain at the

beginning of the sequence.

If there were no transaction costs then a non-zero value for B would allow an arbitrageur

to profit by following the sequence either as it stands if thepresent valueof B is positive,or with all transactions reversed if the present value of B is negative.

However, market forces tend to close any arbitrage windows which might open; hencethe present value of B is usually insufficiently different from zero for transaction costs to

be covered.

This is considered typically to be a "Market Maker/ Floor trader" strategy only, due toextreme commission costs of the multiple-leg spread.

If the box is for example 20 dollars as per lower example getting short the box anythingunder 20 is profit and long anything over, has hedged all risk .

A present value of zero for B leads to a parity relation.
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Two well-known parity relationsare:- Spot futures parity. The current

price of a stock equals the current price of a futures contract discounted by the time

remaining until settlement:

Put call parity. A longEuropean call c together with a short European put p at the same strike price K is

equivalent to borrowing and buying the stock at price S.

In other words, we can combine options with cash to construct a synthetic stock:

Note that directly exploiting deviations from either of these two parity relations involves

purchasing or selling the underlying stock.

Documents

Put-Call Parity and Arbitrage