Certainly one of the biggest breakthroughs in quantitative finance, the Black-Scholes (pronounced like “Black-Shoales” not “Black-Skoales”) model was introduced in 1973 and provides a mathematically principled approach to options pricing. While the original models relies on partial differential equations it subsequently found a different interpretation through stochastic processes (martingales) by describing stock prices through a geometric Brownian motion. In this post we want to show that the Black-Scholes formula can be derived using very basic arguments for the special case of European call options.
Question:Can you derive the Black-Scholes formula for the price of a European call option.
The payoff of a European call option at time with strike price and asset value is given by . This formula is quite intuitive; if the asset price is larger than we will execute the option. Otherwise the option has no value and no rational trader would execute a call.
From the above, the value of the option at the time of expiry is straightforward. However, how much should I pay at an earlier time where the final value of is uncertain? Underlying the Black-Scholes model is the assumption that follows a geometric Brownian motion. This can be seen as a process analogue of a log-normal distribution. For us, the only relevant information we need is that the final value follows a log-normal distribution, meaning that . We will come back to the question what and are.
We can now calculate the expectation of the payoff at time :
Note that the integral is whenever . We can use that to rewrite
Now we can compute
and using a change of variable we can compute
This gives us the main part of the famous Black-Scholes formula for a European call:Now, is often assumed to have mean , which is taken to be the risk neutral rate of return and variance , i.e. the variance is proportional to the time horizon.
Plugging these values in we obtain
While this is the expected value at time $T$, we need to discount this expectation to produce the present value of this expectation. With continuous discounting at rate we then haveFinal note: Often, this formula is written for the expectation at time time between and . The calculation is completely equivalent, we just need to substitute with and accordingly the time to maturity from to .