Pricing American Options Under Jump-dif Fusion Models

The amount of financial option trading has grown to enormous scale since the pioneering work by Black, Scholes and Merton on the pricing of options in1973. But their models are widely recognized to be flawed, a lot of people want to overcome these problems. Already1976Merton proposed to add jumps which have normally distributed size to the behavior of asset prices. In2002the model proposed by Kou assumed the distribution of jump size to be a log-double-exponential function. The aim of this paper is to propose an analytical expansion formula for pricing American options under Kou’s and Merton’s jump-diffusion models. In this paper we use barrier options to estimate the price of American options under Kou’s and Merton’s jump-diffusion models.First we introduce a convenient parameterization, denoted by: the ratio is called normalization moneyness. The barrier option that will be exercised as soon as the normalized moneyness reaches a certain barrier level. If the barrier level y is chosen when maturity is short (τ=T－t is small), the barrier option price can be denoted by P(θ,τ;y). With different barrier level y we can get different barrier option prices. The short-maturity American option price is then approximated by the maximum over these option prices.Let us show the second-order expansion of the short-maturity barrier op- tion price: whereThough our expansion is only of second-order, in financial market there are a lot of short-term options whose maturity usually is less than one year, our calibration formula obviously is simple and useful for practical purposes. By virtue of our method, we have run several numerical experiments to show the effect of our expansion formula. We have found that it performs extremely well when the time to maturity is small.