Methods for Finite Differences
In this chapter, we present methods to obtain the European call price by solving the Heston PDE along a two-dimensional grid representing the stock price and the volatility. We first show how to construct uniform and non-uniform grids for the discretization of the stock price and the volatility, and present formulas for finite difference approximations to the derivatives in the Heston PDE. We then present the weighted method, a popular method which includes the implicit scheme, explicit scheme, and Crank-Nicolson scheme as special cases. We encountered the explicit scheme briefly in Chapter 8, when we applied this method to the pricing of American options. Next, we explain the boundary conditions of the PDE for a European call. Finally, we present the Alternating Direction Implicit (ADI) method, which produces accurate results with very few time points.
The methods can easily be modified to allow for the pricing of European puts, which requires a reformulation of the boundary conditions. In many cases, however, it is simpler to use put-call parity to obtain the put price.
THE PDE IN TERMS OF AN OPERATOR
Recall from Chapter 1 the Heston PDE for the value U(S, v, t) of an option on a dividend-paying stock, with λ = 0, when the spot price is S and the volatility is v, and when the maturity is t
Recall also that, since we are using t to represents ...