Generalized trapezoidal formulas for the black–scholes equation of option pricing

Abstract

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(α) 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.

Keywords:

• Black–Scholes equation
• Option pricing
• European options
• Crank–Nicolson scheme
• Generalized trapezoidal formula schemes
• Unconditional stability