Turn to any textbook treatment of the Black-Scholes model, and you will find a list of things that the model ‘assumes’. Wikipedia is no exception. These divide into assumptions about the market, such as no arbitrage, ability to short sell etc., which I shall set aside for now, and assumptions about the asset process. Foremost among these are that
- Future returns are independent of past values, i.e. the process is random
- Log returns are Gaussian, or normally distributed
- Volatility is constant
- Drift is constant
Now it is true that if these conditions are satisfied, then the model will work (I shall discuss an appropriate sense of ‘work’ below). That is, these are sufficient conditions (p implies q). But it is also commonly assumed1 that if they are not satisfied, then the model will not work (not p implies not q), i.e. it is assumed that the conditions are necessary, as well as sufficient.
Nothing could be further from the truth.
None of these conditions, apart (possibly) from returns being the logarithm of the underlying process, are necessary for Black Scholes to work. Black Scholes works when the realised profit or loss from replicating the option matches the assumptions used when pricing the option. For those not familiar with the technique of dynamic hedging, we start by computing the ‘delta’, or sensitivity of the option price to changes in the underlying price, then take a position equal and opposite to the delta, so that changes in the price of the option are offset by changes in the hedge position. Since the delta itself changes somewhat with the underlying price (an effect known as gamma), we re-hedge where necessary.
Thus you can show, by simulating all kinds of non-standard price distributions, that Black Scholes is incredibly robust.
Today I have taken a simulated normal or Gaussian distribution, but sorted it in order of magnitude, so that while it is still Gaussian, it is no longer random. That means the ‘Brownian’ bit of ‘Geometric Brownian Motion’ no longer applies. As you can see from the green line of the chart above, the price falls throughout the first half of the series, first rapidly, then more slowly, with negative values drawn from the distribution, then rises, first slowly, then more rapidly, with positive values drawn from the distribution.
The blue line shows the price of a put option price struck at 90, computed using the standard Black 76 option formula. As you expect, the price rises as the underlying falls below 95, then falls back as the underlying increases, ending at zero as the option expires out of the money (underlying about 97).
The red line shows the value of the synthetic or replicating option constructed using the delta of the put. The (short) delta starts at 18%, since the underlying price begins at 95, rises to nearly 100% as the put is increasingly in the money, then falls back to zero at the end.
Clearly the hedge is not perfect – the synthetic position loses towards the end as we are hedging at the same frequency even though approaching expiry – option traders call this ‘pin risk’. But the same thing could easily have happened even if the process was completely random. Thus while the assumption of randomness may be a sufficient condition for deriving the Black formula, it is by no means necessary, and it is a myth that it is.
Note also that the assumption of constant volatility is unnecessary – the changes at the beginning and end of the process are obviously sharper than those in the middle. I shall discuss this in a later post.
- For example, ‘It is recognised, for example, that the geometric Brownian Motion (GBM) underlying Black Scholes does not truly represent the ERM risks’.