Derivatives and Options

This chapter presents a brief introduction to the subject of options. The material is extremely brief and elementary. More detailed treatment of the theory of options can be found in Chriss[1] (which uses minimal mathematics) and Hull[2] (which has a deeper and more rigorous approach). The efficient market hypothesis and the principle of no arbitrage are treated in some detail in Jacob and Pettit[3].

Securities

Before we can discuss derivatives, we need to understand what securities are. All the financial instruments we see around us such as shares, bonds, futures, options, etc. are considered securities. More precisely, a securities contract (security, for short) is a contract issued by a government or company in order to acquire financial capital which is used to acquire real capital, labour and management talent. Examples of securities are common stocks which are issued by companies to acquire capital (which make the buyer a partial owner of the company), bonds and fixed income securities which are issued by governments and companies which need to borrow money for both short and long-term use, debt/equity combinations (which are basically combinations of stocks and bonds) and third-party financial contracts (or derivatives, which are the most important from our point of view).

Common Stocks

The precise nature of the common stock contract is normally defined in accordance with the laws of the country. However, almost all common stock contracts have the following features :

1. They give the owner the right to receive his share of the remaining value of the company after all the outstanding loans, fines etc. have been cleared.
2. They give the owner the right to a share of the earnings (in proportion to the number of shares owned by him) of the firm paid out in the form of dividends (which normally take the form of the company giving all shareholders a certain amount of money for each share that they own).
3. The owner has the right to sell the stock to another investor in the secondary markets (more commonly known as stock exchanges).

While most common stock contracts have the above features, there are several varieties of them which are distinguished by other features of the contracts. For example they may be voting or non-voting (i.e., the owners may or may not have the right to vote about major decisions that affect the company), they may have restrictions on the proportion of the stock that can be held by one investor and so on and so forth.

Bonds and Fixed Income Securities

Bonds and fixed income security contracts are issued by governments and companies essentially for the purpose of borrowing money. They ordinarily specify the financial obligations the issuer has to the owner. They generally require the issuer to pay interest and principal in specified amounts at specified future dates. The number of such dates (excluding the maturity) is called the number of coupons of the bond, the terminology arising from the fact that the bond-holder has to tear off coupons from the contract to claim the payments before the final date.

Debt/equity combinations are financial contracts which are combinations of debt and equity securities (in other words, a combination of stocks and bonds). One common type of such a security are the warrants issued with some stocks. These warrants are securities entitling the bond-holder to purchase a specified number of shares of the firm at a fixed price. These are similar in nature to the options we will consider later.

Derivatives

Third-party financial contracts or derivatives constitute one of the fastest growing markets in recent history and their importance today for companies and financial institutions is difficult to overestimate. Hence, the valuation of derivatives is an important field of financial research.

A derivative is an instrument whose value is dependent on another security (called the underlying security). The derivative value is therefore a function of the value of the underlying security. The two most commonly traded derivatives are futures and options. A futures contract is an agreement between two parties to buy or sell an asset (the underlying security) at a certain time in the future for a certain price. They are traded on exchanges.

Options

Figure 1.1: The profit of a call option at maturity as a function of the stock price at maturity. The strike price is taken to be $100 and the original price of the option is assumed to be $10.

Figure 1.2: The profit of a put option at maturity as a function of the stock price at maturity. The strike price is taken to be $100 and the original price of the option is assumed to be $10.

There are two basic types of options that are traded in the market. A call option gives the holder the right to buy the underlying asset by a certain date for a certain price. A put option gives the holder the right to sell the underlying asset by a certain date for a certain price. This price is called the strike price and the date is called the exercise date or maturity of the contract.

There is a further classification of options according to when they can be exercised. An European option can only be exercised at maturity while an American option can be exercised at any time up to the maturity. The value of an Asian option is dependent on the average value of the underlying security during the term of the contract while a Bermudan option can only be exercised on certain days between the present time and the maturity of the contract.

In this thesis, I will only consider European call and put options. For call options this is not a significant liability as European and American call options with the same maturity and strike prices on a non-dividend paying stock have an identical value. However, the value of an American put option is always higher than that of an European put as early exercise can be optimal under certain conditions.

From the definition of a call option, we can see that the value of an European call option at maturity is given by

$$C = (S-K)\theta(S-K)$$ (1)

(if $S<K$ then the option will not be exercised and if $S>K$, the profit on the option will be $S-K$) where $C$ is the value of the call option at maturity, $S$ is the value of the underlying security at maturity and $K$ is the strike price of the option. ($\theta$ represents the Heaviside function defined by $\theta(x) = 0$ if $x<0$, $\theta(0) = 0.5$ and $\theta(x) = 1$ if $x>0$).

We see that the payoff of a call option at maturity is either positive or zero. Hence, the call option must have a positive value before maturity, which of course is the price of the option. If we also consider the original price of the option, the profit is then given by

$$Y = (S-K)\theta(S-K)-C_0$$ (2)

where $Y$ is the profit and $C_0$ is the initial price of the call option. For the rest of this section (in fact for most of the rest of the thesis), we will assume that the present value of the underlying security is $100. This does not change anything as all the prices can be rescaled by a constant factor without affecting the theory. The profit of the call option (assuming a strike price of $100 and an initial option price of $10) as a function of the value of the underlying security at maturity is shown in figure 1.1.

Similarly, the payoff of a put option at maturity is given by

$$Y = (K-S)\theta(K-S)-P_0$$ (3)

(if $K<S$, the option will not be exercised and if $K>S$, the profit is $K-S$) where $P_0$ is the initial price of the put option. The profit (with the same assumed values as for the call option) as a function of the value of the underlying security is shown in figure 1.2.

It can be shown that if, for a given underlying security, call and put options were available for all strike prices, then any payoff (excluding the original price of the options) whatsoever can be duplicated. Hence, if we can calculate the value of all possible call and put options on an underlying security, we can calculate the value of any derivative based on that underlying security.

The Practical Uses of Options

Speculation

Options give a person using them very high leverage for speculation. For example, if one is convinced that the price of Microsoft stock is going to increase over the next three months, one can either buy the stock or call options on the stock. If we assume that the stock's present price is $100, that the price of a call option maturing in three months for a strike price of $100 is $10 (we assume that one call option contract gives us the right to buy 1 share) and that we have $10,000 to invest, we can either buy 100 shares or 1,000 call options on the stock. If the price rises to $120 in three months, we would make a profit of $2,000 if we had bought the stock or a profit of $10,000 if we had bought the options. Of course, if the price drops, the loss on the options will be greater than the loss on the stock.

Hedging

While speculators want to increase their exposure in the market, hedgers are trying to do exactly the reverse. For example, if a fund manager wants to insure his holding of Microsoft stock, he can sell call options on it. If the price decreases below the strike, he will make a profit equal to the price of the options when he sold them while if the price increases, he incurs a loss as the options are called. This compensates for the decrease and increase respectively in the value of the Microsoft stock he holds. The hedger's aim is to prevent price changes in the price of the underlying security from affecting the value of his portfolio. Hedging can also be performed (and is, in fact, usually done) using futures.

The Principle of no Arbitrage

This principle is also called the law of one price. It states that two equivalent goods in the same competitive market must have the same price. Anyone who has ever gone shopping knows that this is not exactly true. The reason this is so is that the shopping malls are quite far from being efficient markets. Hence, before we apply this principle, we must understand what makes markets efficient.

There are four conditions that efficient markets must satisfy. These are

1. Liquidity - this means that all players are small relative to the size of the market and that there are no transaction costs.
2. Price Continuity - this means that the price is set only by the underlying supply-demand situation and that temporary imbalances do not affect the price.
3. Fairness - this means that there is no discrimination between different players in the market.
4. Equal access to information - this means that all market players have equal access to all available information.

We know that at least part of the first condition is never satisfied - there is no market where the transaction costs are zero. This does limit the application of the no arbitrage principle as we will see below. Most stock exchanges put in a lot of effort to ensure that the other conditions are satisfied. The structure of the stock market (specifically, the presence of market makers) ensures that price continuity is more or less satisfied. Fairness is easily ensured as only the price and time of the orders are relayed to the floor of the exchange. While it is obviously true that some people have more information than others, the presence of insider trading regulations minimize the effect of the inequality in access to information.

Let us now consider an imaginary market where all the above conditions are satisfied. In this market, participants, acting in their own self-interest, use available information to attempt to secure a higher return for their investments. In doing so they collectively ensure that price movements in response to new information are instantaneous and unbiased and will fully reflect all relevant information. Competition will drive security prices from one equilibrium level to another so that the change in price in response to new information will be independent of the prior change in price. In other words, the price of a security follows a Markov process which is only dependent on the latest information released to the market. This is an extremely important conclusion which we shall use extensively later. Most option pricing models assume that the stock prices follow certain specific Markov processes.

Empirical tests have shown that real markets satisfy most, though not all, of the conditions necessary for an efficient market. Please refer to Jacob and Pettit[3] for a summary of these tests.

The no arbitrage principle can now be easily seen to hold in an efficient market. For, if we assume that two equivalent securities A and B have different prices (say A has a lower price than B) then a market participant can make a risk-less profit by buying A, changing it to B (since A and B are equivalent, this is possible) at no cost and then selling B. (If one is uncomfortable with changing security A to B, one can just buy A and sell B. Since A and B are equivalent, the final payoffs are the same and cancel each other leaving one with the initial profit of the difference in the prices.) Hence, an upward pressure on the price of A is created (as A is being bought by the participant) as is a downward pressure on the price of B. This will continue until the prices of A and B become equal. In an efficient market, this will be a very short period of time as all the participants are aware of arbitrage opportunities when they arise. Note that the absence of transaction costs is of fundamental importance in this discussion. If we assume that a single transaction has a fixed cost $\delta$, then the price of A and B can have a difference of up to $2\delta$ (since two transactions are required to benefit from the arbitrage opportunity) as the transaction costs will wipe out the arbitrage profit.

The principle of no arbitrage is actually more general than the efficient market hypothesis. This can easily be seen by considering a market where only some of the people have all the available information. These people will still try to exploit arbitrage opportunities as they arise and will exert pressure on the market to satisfy the principle of no arbitrage.

Hence, in an ideal market, arbitrage opportunities will not exist. We can use this fact for option valuation. This is done by replicating an option using a portfolio consisting of zero-coupon bonds and the underlying security. If we can make this portfolio replicate the payoff of the option, then the principle of no arbitrage assures us that the option will have the same value as the portfolio.

The Equivalence of European and American Calls on a Stock that does not Pay Dividends

We can use the principle of no arbitrage to show that the value of an European call option is the same as the value of an American call option as long as no dividends are paid. To do so, we consider the following portfolios :
Portfolio A : one American call option plus an amount of cash equal to $Xe^{-r(T-t)}$ ($X$ is the strike price, $r$ is the risk-free interest rate, $T$ is the maturity time and $t$ is the present time).
Portfolio B : one share.

The value of the cash in portfolio A at expiration is $X$. At some earlier time $\tau$, it is $Xe^{-r(T-\tau)}$. If the call option is exercised at time $\tau$, the value of portfolio A is $$S - X + Xe^{-r(T-\tau)}$$ which is obviously less than $S$ when $\tau < T$ since $r>0$. Hence, portfolio A is always worth less than portfolio B if the call option is exercised prior to maturity. If the call option is held to expiration, the value of portfolio A at time $T$ is $$\max(S_T, X)$$ while that of portfolio B is $S_T$. Hence, at maturity, portfolio A is greater than or equal in value to B. Therefore, the value of portfolio A is less than that of portfolio B if the call is exercised early while it is at least as much as the value of portfolio B if the call is only exercised at maturity. Hence, early exercise of the call option is never optimal and the value of the American and European call options are the same.

Put-Call Parity

We can use the principle of no arbitrage to derive an important relationship between European put and call prices with the same strike price. To do so, we consider the following portfolios :
Portfolio A : one European call option plus an amount of cash equal to $Xe^{-r(T-t)}$
Portfolio B : one European put option plus one share.

Both are worth $\max(S_T, X)$ at expiration of the options. Hence, the portfolios must have the same value at the present time. Therefore,

$$C + Xe^{-r(T-t)} = P + S$$ (4)

where $C$ is the value of the European call option and $P$ is the value of the European put option.

Hence, calculating the value of the call option also gives us the value of the put option with the same strike price. Therefore, in the rest of this thesis, we will only consider how to calculate the value of the call options.