American Call Options Under Jump‐Diffusion Processes – A Fourier Transform Approach

Abstract

We consider the American option pricing problem in the case where the underlying asset follows a jump‐diffusion process. We apply the method of Jamshidian to transform the problem of solving a homogeneous integro‐partial differential equation (IPDE) on a region restricted by the early exercise (free) boundary to that of solving an inhomogeneous IPDE on an unrestricted region. We apply the Fourier transform technique to this inhomogeneous IPDE in the case of a call option on a dividend paying underlying to obtain the solution in the form of a pair of linked integral equations for the free boundary and the option price. We also derive new results concerning the limit for the free boundary at expiry. Finally, we present a numerical algorithm for the solution of the linked integral equation system for the American call price, its delta and the early exercise boundary. We use the numerical results to quantify the impact of jumps on American call prices and the early exercise boundary.

Keywords:

• American options
• jump‐diffusion
• Volterra integral equation
• free boundary problem
• Fourier transform

Notes

1. The extension of McKean's approach to the jump‐diffusion case is provided by Chiarella and Ziogas (2006).

2. Note that one can also directly arrive at (5) from (3) by assuming that jump‐risk is fully diversifiable, and hence l(Y) = 0, as is done by Merton (1976). The extension of the traditional hedging and change of measure approaches to properly incorporate the l(Y) term is given by Cheang et al. (2006). We should point out that Pham (1997) seems to have been the first to report the IPDE (3) with the inclusion of the l(Y) term.

3. Since S⩾a(τ), we know that a(τ)/S⩽1.

4. It should be noted that C(SY, τ) = KV(ln(SY/K), τ) = KV(x+ln Y, τ).

5. We assume that the density function G is of the form that facilitates the reduction of the n‐dimensional integral to a single integral. This is certainly true of the log‐normal density function to be used later in the paper.

6. There, the dependence is sequential, that is first one solves for the free boundary, which then feeds into an integral expression for the option price.

7. Note that, since C(SY, τ) = SY−K for Y⩾a(τ)/S, we have

8. This is true because it is never optimal to exercise a call option if S<K.

9. While we note that these integral terms are expectations over the jump‐size density G(Y), this does not aid us when trying to provide a general analysis of f(a(0⁺)).

10. Briani et al. (2004) note that it is unclear how to select the stopping criteria when using iterative finite difference solutions for (5). Since we observe greater accuracy by using the average of the first and second iteration results than using the second iteration alone, the averaging scheme we use here seems more efficient than using a stopping criterion that involves three or more iterations.

11. By the global variance we mean [(dS/S−(μ−λk)dt)²] calculated from equation (1) to be in the case of a log‐normal jump density.