What does the Black-Scholes Option Pricing Formula really tell us?
June 6, 2009
Black-Scholes option pricing formula is perhaps the most celebrated and famous formula in the theory of finance. Even though it is now well accepted that the formula contains a lot of crucial flaws, such as the assumption of constant volatility, it is used by option traders and banks all over the world to price vanilla options. The formula, at a first glance is quite daunting and any one not well versed with the advanced mathematics of financial derivatives may feel intimidated.
The price of a plain vanilla Call option for a non-dividend paying stock, with a price of and volatility of , in an economy with short term interest rate, , the Black-Scholes formula is given by:
Where, and are given by:
Equation (1) is truly intimidating and does not appear at all intuitive.
In the above equation, the strike price (of the stock), , is pre-determined and fixed and and are cumulative probability distribution functions for a Normal (Gaussian) distribution.
Since and are two different numbers and is always greater than by a factor of , under ordinary circumstances, for any non-degenerate asset both the probability measures and should have different values. And indeed, both are different in practice as well. is the delta (hedge ratio) and is the probability of that the call option will finish in the money (i.e. the option buyer will make money).
Therefore, even though both and are cumulative probability measures (for a Normal distribution) their meaning and implication are different.
However, if the term is extremely small, then and would become equal. This could be the case for very small maturities, say, one day. If the option maturity is one day then even for moderately high volatility both and would be very close to each other in value. Else, if the volatility is very low then, even longer maturities, the values of and would be very close to each other.
Assuming, the theoretical case of we will get: . For such an eventuality, and ignoring the discounting factor for the strike price, the Black-Scholes equation will collapse to:
The above formula is quite intuitive and exactly in keeping with the definition of a Call option. What the above formula says is that the value of the call option today is the sum of all probability weighted payoffs at maturity. In other words, if we assume that the asset returns follow a Normal distribution then the sum of all the call option payoffs at maturity, i.e. the terminal value of the spot minus the strike, multiplied by the probability of the occurrence of those payoffs is the value of the option today (ignoring discounting).
The call option has a payoff of at maturity. However, there could be infinite number of payoffs, because there are infinite possible values of the asset between zero and infinity. And each of these payoffs will have a distinct probability, (drawn from a Normal distribution) associated with it. is the probability associated with the ith payoff.
Therefore, applying discounting, the value today will be:
The call option will have a value only if the spot (asset) finishes in the money, i.e. which is equivalent of saying that is greater than zero. But at time t = 0, today, we do not know whether the spot will finish in the money or not. All we can say is that the asset returns at maturity will have a Normal probability distribution. This means, that for all terminal asset values from 0 to infinity, each call option payoff associated with that particular terminal asset value will have definite probability of occurrence and that this probability is drawn from a Normal distribution.
This is exactly what the Black-Scholes equation shown above tells us.
Functionally, both equation (2) and (3) are same. In continuous time form, the summation sign in equation (3) will get replaced by an integral sign and one can write the equation (3) as:
If we apply the random walk model of asset price and the Gaussian probability distribution then the above integral can be easily solved to give a closed form equation (formula) like the equation (1) above (Black-Scholes formula).
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