Numerical pricing of geometric asian options with barriers

Abstract

In this article, a semianalytical method for pricing of barrier options is described and applied in the context of Asian options with geometric mean. The efficiency of the method is tested and compared with two finite difference methods.

Efficient BEM-Based Algorithm for Pricing Floating Strike Asian Barrier Options (with MATLAB® Code)

“…The existence of the solution of Problem (4)-(5) and its Feynman-Kac representation (10) are proven involving stochastic arguments without the need for exact boundary conditions. Boundary conditions at S = 0 and for A → ±∞ have to be empirically deduced analogously to what was done in [5,15]. At S = B, Condition (6) holds.…”

“…SABO is compared here (as in [5]) with two Finite Difference (FD) methods chosen among the wide class of FD methods available and deeply analyzed in [14]. This is because Equation (4) is proven to be hypoelliptic [10][11][12], a property that guarantees a smooth solution and that should benefit from approximations based on Taylor expansions.…”

“…In this context, SABO is a Semi-Analytical method conceived of for the pricing of Barrier Options, and its milestones are resumed in Section 4.1. It is quite a general method, applicable also to fixed strike payoffs [4,5], put options [6], time-dependent parameters [7] and double barriers [8].…”

“…..,N t , and the matrix entries are equal to the case of fixed strike Asian options deeply investigated also from a computational point of view in [4,5]:…”

Existence of a fundamental solution of partial differential equations associated to Asian options

“…Thus, the pricing problem for the simplest case of Arithmetic Average Asian Options can be solved by the argument outlined in the previous subsection when considering (1.6), but in this case the differential operator K needs to be replaced by L λ . As we can see from the explicit expression (1.21) of the fundamental solution Γ L λ of L λ , and as several authors point out (for instance, see [1,17,19,20,43]), the explicit representation of the Asian option prices given by Geman and Yor in [24] is hardly numerically treatable, in particular when pricing Asian Options with short maturities or small volatilities. We quote [49,24] for an exhaustive presentation of the topic, other related works are due to Matsumoto, Geman and Yor [35,24,34], Carr and Schröder [10], Bally and Kohatsu-Higa [6].…”