Generalized trapezoidal formulas for the Black-Scholes equation of option pricing

For the celebrated Black-Scholes parabolic equation of option pricing, we present new time integration schemes based on the generalized trapezoidal formulas introduced by Chawla et al. [3]. The resulting GTF(alpha) schemes are unconditionally stable and second order in both space and time. Interestingly, since the Black-Scholes equation is linear, GTF (1/3) attains order three in time. The computational performance of the obtained schemes is compared with the Crank-Nicolson scheme for the case of European option valuation. Since the payoff is nondifferentiable having a "corner" on expiry at the exercise price, the classical trapezoidal formula used in the Crank-Nicolson scheme can experience oscillations at this corner. It is demonstrated that our present GTF (1/3) scheme can cope with this situation and performs consistently superior than the Crank-Nicolson scheme.