- 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.
- 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).
- The owner has the right to sell the stock to another investor in the secondary markets (more commonly known as stock exchanges).

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.

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.

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

(1) |

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

(2) |

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

(3) |

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.

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

- Liquidity - this means that all players are small relative to the size of the market and that there are no transaction costs.
- 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.
- Fairness - this means that there is no discrimination between different players in the market.
- Equal access to information - this means that all market players have equal access to all available 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 , then the price of A and B can have a difference of up to (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 value of the cash in portfolio A at expiration is . At some earlier time , it is . If the call option is exercised at time , the value of portfolio A is which is obviously less than when since . 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 is while that of portfolio B is . 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.

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
*Portfolio B :* one European put option plus
one share.

Both are worth at expiration of the options. Hence, the
portfolios must have the same value at the present time. Therefore,

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.