Option Pricing Models
The purpose of an option pricing model is to determine the theoretical fair value for a call or put option given certain known variables. In other words - to determine an option's expected return.
Basically, the expected return of an option contract is a function of two variables:
- The payoff of the option at maturity date
- The probability of the option being in-the-money at maturity
These two values multiplied together give you the theoretical price.
Calculating the option payoff is quite easy: for call options it is the maximum of either 0 or the underlying price minus the strike price. For put options it is the maximum of either 0 or the strike price minus the underlying price. More simply:
Call Option Payoff = Max (0, (Underlying Price - Strike Price))
Put Option Payoff = Max (0, (Strike Price - Underlying Price))
But it is in determining the probability of the payoff that becomes a little more difficult.
Essentially, you want to know where the underlying price is likely to be trading at by the expiration date. To determine this probability is no easy task.
For example, say that a stock is currently trading at 100 and you are trying to value a call option on this stock with a strike price of 100 and maturity date of 1 month. Imagine that you know the exact probabilities of where this stock will be trading at the maturity date:
50% chance it will be trading at 95
50% chance it will be trading at 105
If these were the only two outcomes available and you knew the probabilities of these outcomes, then pricing this option is very easy.
First, you know that for a call option, if he underlying is trading below the strike price than the call option is worthless. Second, if the underlying is trading above the strike price then the payoff of the option is the underlying price minus the strike price - i.e. 5 (105 - 100).
So now we have two outcomes and two payoffs.
A 50% chance of making 0 and a 50% chance of making 5.
Then we can construct a simple formula to describe the expected return of our option contract:
(Probability of Stock Trading at 95) x (Option Payoff at 95) + (Probability of Stock Trading at 105) x (Option Payoff at 105)
Which becomes: (0.50 x 0) + (0.50 x 5) = 2.50
Of course in the real world, there is a much larger set of price outcomes and we will never know for sure what the true probability really is. That was the challenge Fisher and Black had when they ventured into writing their paper on pricing real options.