 ## Option Pricing

The Nobel Prize in Economics in 1997 was awarded to Robert Merton, Fischer Black and Myron Scholes for their pioneering work in establishing the foundation for the financial engineering that has revolutionized contemporary finance.

They developed an intellectually elegant model that enables traders to take a set of observable prices and to calculate mathematically a price for an option. While some may criticize it for the assumptions that it makes, the model has been instrumental in allowing traders to understand intuitively the behavioral characteristics of the product and it has facilitated the exponential growth of derivatives worldwide.

Indeed, one could argue that because of its assumptions, the model has forced traders to understand every aspect of options pricing.

Let’s take a look at an investor who wants to buy IBM stock. The buyer of an IBM call option has the right, but not the obligation, to buy IBM stock at the maturity date at a pre-set strike price. He will exercise this option if the option is “in-the-money.” That is to say, the buyer will exercise his right to purchase IBM stock if the strike price is less than the prevailing cash price for IBM stock at the time the option expires. The question the modeler has to answer is this: How do we evaluate what the stock price is going to be at maturity, an event that could take place months from now? The first assumption has to do with the way in which we model the process that characterizes the movement of the underlying price of IBM. We know that the price of IBM stock fluctuates. Perhaps it follows a trend. We have observed the volatility of the stock price. If we have a way of describing the statistical process underlying the IBM price, then we have taken the first critical step towards pricing the option.

## Option Pricing: Assumptions and Considerations

Let us assume that the stock price follows a Markov process. This is simply a process in which only the current value of a variable is useful in forecasting what the future value of the variable could be.

### Assumption 1: The underlying price is lognormally distributed

In fact, let us assume that the logarithm of the price is normally distributed such that it has specific a trend and a specific volatility. Doing this enables us to predict what the expected value of the stock price will be and it also makes the model mathematically manageable. Using the normal distribution makes it easier to find a closed-form solution to the problem. A closed-form solution is simply an equation that we can use to determine option pricing. This means that volatility for the underlying price is constant and the same for all maturities. Different maturities will trade with different volatility, in practice, because of cash flow events, expectations of political instability, political events, or management changes.

### Assumption 2: The short selling of securities with the full use of proceeds is permitted

When discussing hedging, it is assumed that we could buy and sell stock against our options position in order to capture the effects of a volatile underlying price, thereby paying for the option premium over the life of the instrument. In order to sell stock, there can be no restriction on short sales.

### Assumption 3: There are no transaction costs or taxes. All securities are perfectly desirable

Taxes and transaction costs, such as brokerage, would distort the simple problem of trying to understand option pricing. In practice, the investor or the options professional accounts for these factors in the course of doing business. Transaction costs and taxes will distort the delta hedging decision, providing a disincentive for delta hedging and altering the way in which we determine the option’s value.

### Assumption 4: There are no dividends during the life of the derivative security

Again, we ignore dividends in the derivation of the Black-Scholes model because of the distortionary effects these can have on our delta hedging decision. If we buy a call option and need to delta hedge it by short selling securities, we may be hesitant to short sell securities that pay a dividend.

### Assumption 5: There are no risk-less arbitrage opportunities

This is an assumption of efficient markets theory. An arbitrage opportunity exists when one can buy and sell the same instrument (or virtually the same instrument) simultaneously for different prices, thereby locking in a “riskless” profit. Because the transaction is instantaneous, there is no risk to the individual. A market may be said to be efficient if there are no such opportunities. To put it another way, as soon as such opportunities arise, some astute investor immediately realizes them. You would not expect them to last for long, as markets will correct themselves rationally; or, so says the theory.

### Assumption 6: Security Trading is continuous

Prices of stocks on North American exchanges move in discrete increments, such as 1/32. By assuming that prices can trade in a mathematically continuous fashion, the model is more mathematically manageable.

### Assumption 7: The risk-free rate of interest, r, is the constant and the same for all maturities

This is a big assumption. It states that the government yield curve is flat. Just from common sense, we know that is not true. But, in order to solve the model for a wasting asset, it is important to model rates as constants.

### Intuition

Having made all of these assumptions, how did Black, Scholes and Merton apply them to option pricing? They modelled a portfolio that consisted of one unit of the option and a fraction (the delta) of shares in the underlying instrument of the option, choosing the delta so that the portfolio did not change in value for small movements in the price of the underlying price. By doing so, there was no longer any risk in the portfolio. Therefore, they reasoned, the test portfolio must have the same return as a riskless portfolio, r.

Solving this differential equation using boundary conditions fixed by whether or not the derivative was a call or a put resulted in the closed-form solution for the options price. Now, when you want to price an option, you input the following parameters into your Black-Scholes calculator and you have the price:

6 years ago