The Black-Scholes equation

The Black-Scholes equation is a parabolic partial differential equation (PDE) for option prices discovered by Black and Scholes in 1973. This discovery revolutionized financial markets in 1990-s. In 1997 Merton and Scholes were awarded the Nobel prize in economics. For any stock price s,

\(0<s<\infty\)

and time t,

\(0<t<T\)

the price u for an option expiring at time T satisfies the following Black-Scholes PDE:

\(\frac{\partial u}{\partial t} + \frac{1}{2} s^2 \sigma^2(s,t)\frac{\partial^2 u}{\partial s^2} + s\mu\frac{\partial u}{\partial s} -ru=0\)

where σ(s,t) is the volatility coefficient,

\(0<m<\sigma(s,t) < M < \infty\)

and μ and ρ are respectively the risk neutral drift and risk free interest which are assumed to be constants.

What happens in the case where more than options have to be studied at the same time? Well, simple extend the model to many variablesas following:

\(\frac{\partial u_1}{\partial t} + \frac{1}{2} s^2 \sigma_1^2(s,t)\frac{\partial^2 u_1}{\partial s^2} + s\mu_1\frac{\partial u_1}{\partial s} -r_1u_1=0\\ \frac{\partial u_2}{\partial t} + \frac{1}{2} s^2 \sigma_2^2(s,t)\frac{\partial^2 u_2}{\partial s^2} + s\mu_2\frac{\partial u_2}{\partial s} -r_2u_2=0\)

The coefficients σ, μ and r are of course different between the two equations. However, one can simply raise the question what happens when a coupling of the two dependent variables takes place. At the beginning we can assume a simple linear coupling of the two prices which will then lead to the Black-Scholes system:

\(\frac{\partial u_1}{\partial t} + \frac{1}{2} s^2 \sigma_1^2(s,t)\frac{\partial^2 u_1}{\partial s^2} + s\mu_1\frac{\partial u_1}{\partial s} =r_1u_1 +c_{12}u_2\\ \frac{\partial u_2}{\partial t} + \frac{1}{2} s^2 \sigma_2^2(s,t)\frac{\partial^2 u_2}{\partial s^2} + s\mu_2\frac{\partial u_2}{\partial s} =c_{21}u1 + r_2u_2\)

As we can see now both prices u1 and u2 appear in both equations. This means that we need some robust and at the same time flexible numerical method for the solution of the above system, which leads us of course to the finite element method.