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 Peclet 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 in 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.