"As we know, the value of a certain derivative can be expressed as a stochastic differential equation (SDE). Since stochastic differential equation can be transformed into a corresponding partial differential equation (PDE), it is worth to learn some numerical methods to solve PDEs. And Finite Difference Method (FDM) is the one widely used in this area. In this article, I will give a brief introduction to FDM and how it could be applied in option pricing. We focus on the famous Black-Scholes partial differential equation in this article. More details about FDM, Matrix Solver and stability analysis will be given in the later articles.
In the BS Model, the price of a European call option satisfies the following PDE:
where f is the price of European stock options, S is the price of the underlying stock, sigma is the volatility of the stock price per year and r is the riskless interest rate.
Initial and Boundary Conditions: In order to apply FDM, we also need to provide initial and boundary conditions. In this problem, the terminal condition is given, which is the payoff of the option at expiration time:
The terminal condition can be converted to initial condition by simply changing the sign of the first derivative with respect to time t in the original equation.
Then the PDE becomes:
Normally, we use Dirichlet boundary conditions to approximate this Cauchy problem, which can be expressed as:
(1) Explicit-Euler scheme
(2) Implicit-Euler scheme
(3) Crank-Nicolson scheme
Explicit-Euler scheme is an explicit method, which means the discretized system of equations can be solved explicitly. Therefore, this method runs fast on the computer. However, this method is only first-order accurate in time and has some stability issue. This means there is a restriction on the size of the time step for this method to be stable.
Implicit-Euler scheme is an implicit method, which means we need to solve a linear algebra system of equations. Fortunately, the matrix formed in this problem is tridiagonal, which, to some extent, reduces the storage and calculation cost. This method is also first-order accurate in time. But compared with Explicit-Euler, Implicit-Euler guarantees the stability.
The most popular scheme may be Crank-Nicolson, which is always stable and has second-order accuracy in time. Of course, this method is implicit and thus needs to deal with Matrix solving problem. This scheme seems perfect, but in fact it is not. We should notice that when the advection term is dominated, the so-called spurious oscillation may occur, which can cause great error. This issue will be addressed in detail in another following article." - Xiaohong Chen (May 2015 Graduate)