Analytical Upper Bounds for American Option Prices with Time-Changed Lévy Processes

AbstractThis paper generalizes and tightens Chen and Yeh's (2002) analytical upper bounds for American options under stochastic interest rates, stochastic volatility, and jumps, where American option prices are difficult to compute with accuracy. We first generalize Theorem 1 of Chen and Yeh (2002) and apply it to derive a tighter upper bound for American calls when the interest rate is greater than the dividend yield. Our upper bounds are not only tight, but also converge to accurate American call option prices when the dividend yield or strike price is small or when volatility is large. We then propose a general theorem that can be applied to derive upper bounds for American options whose payoffs depend on several risky assets. As a demonstration, we utilize our general theorem to derive upper bounds for American exchange options and American maximum options on two risky assets.

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APPROXIMATING LÉVY PROCESSES WITH A VIEW TO OPTION PRICING

International Journal of Theoretical and Applied Finance ◽

10.1142/s0219024910005681 ◽

2010 ◽

Vol 13 (01) ◽

pp. 63-91 ◽

Cited By ~ 25

Author(s):

JOHN CROSBY ◽

NOLWENN LE SAUX ◽

ALEKSANDAR MIJATOVIĆ

Keyword(s):

Option Pricing ◽

Lévy Processes ◽

Jump Diffusion ◽

Levy Processes ◽

Poisson Processes ◽

Exotic Options ◽

Option Prices ◽

Barrier Option ◽

Systematic Methodology ◽

The Mean

We examine how to approximate a Lévy process by a hyperexponential jump-diffusion (HEJD) process, composed of Brownian motion and of an arbitrary number of sums of compound Poisson processes with double exponentially distributed jumps. This approximation will facilitate the pricing of exotic options since HEJD processes have a degree of tractability that other Lévy processes do not have. The idea behind this approximation has been applied to option pricing by Asmussen et al. (2007) and Jeannin and Pistorius (2008). In this paper we introduce a more systematic methodology for constructing this approximation which allow us to compute the intensity rates, the mean jump sizes and the volatility of the approximating HEJD process (almost) analytically. Our methodology is very easy to implement. We compute vanilla option prices and barrier option prices using the approximating HEJD process and we compare our results to those obtained from other methodologies in the literature. We demonstrate that our methodology gives very accurate option prices and that these prices are more accurate than those obtained from existing methodologies for approximating Lévy processes by HEJD processes.

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