How a friend of Napoleon’s helped the modern day market
Black-Scholes equation describes the price of a European call or put option over time. How do we solve it using Fourier transform to derive the Black-Scholes model to estimate the price of an option contract?
Let’s say I have a bunch of scented candles that I don’t need and want to sell. You are not sure if you need a scented candle, but the holidays are almost here and you will need to buy presents for your friends and family sooner or later. It is pretty obvious that the prices for holiday presents may go up closer to the holiday season, so will you do the shopping now or buy an opportunity (not an obligation) to buy some candles for a certain price?
If you choose to buy this opportunity, how much should you pay for it? And what are we going to do if the prices go down? Or up?
Enter Black-Scholes model.
Back in 1973 three economists, Fischer Black, Myron Scholes, and Robert Merton, introduced a partial differential equation and a mathematical model that allows to answer all these questions. For their work they won a Nobel Prize in 1997. Modern day traders and investors are using this model all the time.
To estimate the price of an option we need to take some information into consideration. For example, how long do you want to wait until you make a decision (time to expiration)? What is the current stock price? What is going to be the exercise price? How volatile is this stock? Let’s introduce some notations:
Let’s say we have two stocks, one with higher volatility, one with lower volatility (sigma):
By looking at the mathematical definition of the distributions we can see that the higher is the volatility, the higher is the price going to be.
This model was derived from a so-called Black-Scholes equation, a PDE that takes partial derivatives as an input and after certain mathematical manipulations makes a prediction that turns out to be very close to the actual price. The Black-Scholes PDE looks like this:
Now we can see that the long and complicated Black-Scholes PDE turns into one of the fundamental PDE’s — the heat equation! From math textbooks we remember that the heat (or diffusion equation) looks like this:
Joseph Fourier (1768–1830) was a French mathematician and physicist, who was a friend and scientific advisor of Napoleon Bonaparte. His work gave us Fourier Series and Fourier Transform that simplified the process of solving PDE’s. Generally speaking, Joseph Fourier suggested that a function can be transformed, then used to simplify and easily solve the equation, then transformed back to see the actual solution.
Here are the rules that we are going to use to solve the heat equation with Fourier transform:
Here is what we’ll do to solve our heat equation:
Or by using Sympy:
Solving this integral gives us the Black-Scholes model, which was mentioned above, but once again:
Summary:
- Given Black-Scholes equation we can transform it using the change of variables
- Now Black-Scholes equation turns into diffusion equation
- Diffusion equation can be solved with Fourier transform
- We have a Black-Scholes model for pricing an options contract!
