This work presents a comparison study of different numerical methods to solve Black-Scholes-type partial differential equations (PDE) in the convection-dominated case, i.e., for European options, if the ratio of the risk-free interest rate and the squared volatility-known in fluid dynamics as Péclet number is high. For Asian options, additional similar problems arise when the "spatial" variable, the stock price, is close to zero.
Here we focus on three methods:
the exponentially fitted scheme, a modification of Wang's finite volume method
specially designed for the Black-Scholes equation, and the Kurganov-Tadmor scheme
for a general convection-diffusion equation, that is applied for the first time
to option pricing problems. Special emphasis is put on the Kurganov-Tadmor because
its flexibility allows the simulation of a great variety of types of options and
it exhibits quadratic convergence. For the reduction technique proposed by Wilmott,
a put-call parity is presented based on the similarity reduction and the put-call
parity expression for Asian options.
Finally, we present experiments and comparisons with different (non)linear Black-Scholes PDEs.