AN ALTERNATIVE VIEW ON OPTION FAIR VALUE

In Chapter 3 we talked of the fair value of an option and explained it in terms of a long run average. The analogy of playing a dice game was used to illustrate the concept of a probabilistic fair or expected value. In order to calculate the fair value one needed to have some idea of the different possible stock prices on the option expiry date. On expiry, it is a very easy matter to find the value of the option since, it it is in-the-money, it is worth the intrinsic value and, if it is out-of-thc-money.

then it is worth nothing. By "overlaying" these different expiring outcomes with a given stock price distribution it was a simple matter to come up with the expected value. We interpreted this

expected value as being equal to the price one would pay for the option if, in the long run, we wanted to break even. The notion of long run here can be thought of in two ways:

1. In one, the option player repeatedly buys the option at the fair price and waits for expiry. Some of the time the option will end up being worth something and some of the time the option will end up worthless. If the player always pays the fair value his "winnings" will just cover his bets—he will break even.

2. In another interpretation one can think of the option player being offered a large number of different options all expiring on the same day in the future. If the player buys all of the options in one go and pays the correct fair value for each, then the net proceeds, when they all expire, will just match his total outlay—he will break even.

In both of these interpretations of the option fair value, we assume that the player buys and holds until expiry. Between the purchase date and the expiry date, the player simply looks on as the price of his position changes every day, but he never acts. In this interpretation of fair value the option participant plays a passive role.

He either gets lucky with the stock price rising significantly or loses his stake.

The long volatility strategy now allows us to think of yet another concept of option fair value. Consider an individual setting up a delta neutral portfolio as outlined above and waiting for some volatility. Assume that the dealing costs are very small or zero. With no costs the long volatility player can afford to rehedge as frequently as he likes. Say he makes an arbitrary rule that he will rehedge each time the underlying price moves some small amount, say 10 cents. Will the player win or lose?

The answer depends on the price originally paid and the subsequent volatility experienced. If the price paid is low and the volatility is high, the player will win overall. If the price paid is high and the actual volatility is low then the player will lose. If the individual in question executes this trade on many occasions there will be a long-run average rehedging revenue. If this long-run revenue exactly matches the price paid, then we can say that the option price was fair in volatility terms.

This, then, gives us another way to see if an option is expensive or not.

Which is the correct fair value to use? The two approaches to fair value are completely different. The former is passive—a simple buy

and hold to expiry. The latter is active and involves continual dynamic rehedging.

What is interesting is that it turns out that both approaches give the same answer.

There is only one fair value but two different interpretations and the reason is not as complicated as it first seems. The key to understanding why these two different approaches give the same fair value lies in looking at the underlying stock price distribution in more detail. In Chapters 2 and 3 we used different models of stock price behaviour to explain different concepts.

In Chapter 2 we talked about measuring volatility. To illustrate how volatility was measured we considered how a certain stock price evolved from day to day. It was shown that volatility could be thought of as being a function of the sequence of daily price changes. We were not interested in where the stock price eventually went, just how it got there. In measuring volatility we were only interested in the magnitude of the positive and negative price changes.

In Chapter 3 we talked about the fair value of a call option and showed why the price profile should be curved. To do this we looked at a different aspect of the underlying stock price distribution. The fair value was shown to be a simple probabilistic average of the option prices on one day—the expiry day. The expiring option price is a direct function of the stock price on the expiry day and so we only required information about the stock price distribution on this one day. This approach completely ignores the path taken to expiry; it just uses the final expiry day price.

That is the difference in the two approaches. The static fair value model uses information about the stock price distribution on the one day in the future, whereas the long volatility fair value uses information about the day-to-day price fluctuations and completely ignores the future expiring price. Both approaches will give the same fair value if one assumes the same underlying stock price distribution. If we assume that the relevant distribution is lognormal and that the dispersion measure used is set equal to the expected volatility, then both fair value methods give identical results. If the volatility input into both models is high (low) then both fair values will be high (low). The volatility of the day-to-day price changes directly affects the distribution of prices on expiry. If the volatility is high then the distribution of stock prices on the one future expiry day will be very dispersed. If the volatility is high, then there will be a much higher chance of the expiring option ending up deep-in-the-money and being worth a considerable sum.

So an option on a highly volatile stock

should have a higher fair value even when one considers the static buy and hold strategy.

There is one final conceptual point to make on the two approaches to fair value.

Say we know for sure that the stock in question will exhibit a volatility of 15% for the foreseeable future. We consider two individuals buying a one-year call option and paying the correct fair value that has been calculated using the Black and Scholes model with a volatility value of 15%. The first individual follows the buy and hold strategy and the second executes the delta neutral long volatility strategy.

Say the stock price at the start is $99 and that the exercise price is also $100. The option in question happens to be the one used throughout this chapter so we know the price. It is $5.46 or $546 per contract. Both individuals should in the long run break even. Say the stock price on expiry is $90. The buy and hold player will have lost his entire stake of $546 but the volatility player will have recouped his $546 outlay through all the rehedging profits. The first individual will have lost 100% of his investment but the second will have broken even. And both paid the correct fair value. How can this be? Consider the individuals being given a second attempt at the same investment strategies. This time the stock price might end up at say, $114 in which case the option will end up being worth $14 or $1,400 per contract. The first individual will end up with $1,400 whereas the second will end up with $546.

In this second attempt the buy and hold strategy makes 1,400 -546 = $854 and the volatility strategy broke even. So that is the real difference in the two approaches to fair value. In the first approach, breaking even refers to a long run average and in the second breaking even means breaking even every time.