Pricing discretely monitored Asian options under Lévy processes

Abstract

We present methodologies to price discretely monitored Asian options when the underlying evolves according to a generic Lévy process. For geometric Asian options we provide closed-form solutions in terms of the Fourier transform and we study in particular these formulas in the Lévy-stable case. For arithmetic Asian options we solve the valuation problem by recursive integration and derive a recursive theoretical formula for the moments to check the accuracy of the results. We compare the implementation of our method to Monte Carlo simulation implemented with control variates and using different parametric Lévy processes. We also discuss model risk issues.

Pricing of Discretely Sampled Asian Options Under Levy Processes

“…For discretely sampled Asian options, Benhamou [9], Fusai and Meucci [36] enhance the method of approximating the joint probability density function of [26], and price Asians under Lévy processes. Since the transition density function of a Lévy process is generally unknown in the closed form, the accuracy of this approach crucially rely on the accuracy on an approximation of the transition density function.Černý and Kyriakou [27] reduce the pricing problem to a sequence of European options in the one-factor model, and use trapezoidal rule as the numerical realization of the (inverse) Fourier transform to approximate the prices of European options.…”

“…The set of numerical parameters guarantee a small error not larger than 10 −10 . [36]. In this subsection , we compare the performance of our algorithm with the performance of MC method and the method developed by Fusai and Meucci [36], for calculating the prices of discretely monitored Asian call options.…”

“…In this subsection , we compare the performance of our algorithm with the performance of MC method and the method developed by Fusai and Meucci [36], for calculating the prices of discretely monitored Asian call options. As in [36], we assume that under a chosen EMM, the log-spot price, X t = ln S t , of the underlying follows a KoBoL process (see (2.12)) with parameters ν = 1.2945, c = 0.0244, λ + = 0.0765, and λ − = −7.5515. Assume the interest rate is r = 0.0367, which allows us to find the remaining parameter µ ≈ 0.138736 from the EMM condition r + ψ(−i) = 0 (where ψ(ξ) is the characteristic exponent of {X t }).…”

“…(We use the acronym "MC", "LX(f)" and "FM" to label the results obtained using our method (implemented with flat (refined) iFFT) and the method in [36]. )…”

“…In this subsection, we compare the performance of our method with the performance of the method developed in [36] and [27], for pricing discretely monitored Asian call option under the Brownian motion. The results are summarized in Table 8 and 11.…”