Option Pricing Under The Jump To Default Extended Constant Elasticity Variance Model

The pricing theory of financial derivatives composes one of the three pillars of mod-ern financial theories, and is also one of the most substantial and fundamental areas in mathematical finance research. As a typical class of derivatives, option has drawn much attention that makes it the central research topic in financial derivatives pricing field. Since the invention of Black-Scholes option pricing formula, which initiated the second revolu-tion on Wall Street, the pricing theories on all kinds of exotic options such as American options, barrier options, Asian options,as well as a variety of financial innovations have embraced a thorough development. However, basing on the complete market hypothesis, as well as the presumption of constant volatility, these theories not only neglect the credit risk in stock market, but also can hardly be calibrated to the data observed in the real mar-ket. Therefore, in order to better describe the fluctuating characters of the underlying asset prices, reflect the volatility smile and skew phenomenon and the leverage effect between underlying assets' prices and observed price volatilities, as well as incorporate credit risk modeling in stock market, Carr and Linetsky proposed, basing on the reduced model, an extended CEV model involving the jump to default risk, and studied the pricing theory of European option in their settings, and also obtained an explicit pricing formula. Inspired by the Carr and Linetsky model, this paper further studies the American option and bar-rier option pricing theory within the extended CEV model incorporating jump to default risk. First of all, we discuss the barrier option pricing problem in extended CEV model involving jump to default risk. So as to obtain the pricing formula, we first solve and ob-tain a formula for a special perpetual barrier option which pays unit amount when knocked out, and discover that this formula can be expressed as a combination of first order and second order Whittaker functions. Using this result and with the help of Laplace transfor-mation, we obtain the pricing formula for down-and-out call options, and then the pricing formula of knock-in options, in light of the equating relationship among knock-out option, knock-in option and European option. Secondly, we study the American option pricing problem in the extended CEV model incorporating jump to default risk, and obtain a PDE for the American put option. Since the American option pricing problem in fact involves a free-boundary PDE problem, an explicit solution can hardly be achieved. As a result, how to improve the efficiency and accuracy of numerical calculation becomes the key part of our problem. Therefore in this paper, by employing the artificial boundary technique, we divide the half-boundless domain of the previous PDE into two adjacent regions with one bounded, the other half-boundless. Then we solve the price on the boundary of the half-boundless region by the Laplace transformation, using which as the start points for the implicit difference method solving the American option price within the bounded re-gion. With this method, an accurate solution for the American option price with arbitrary underlying price can be efficiently obtained. Also, a comparison between the calculation efficiency and accuracy of our improved method and classic implicit difference method is examined by a numerical experiment. Thirdly and lastly, as an application of the option pricing theory under JDCEV model, we discuss the pricing of a fixed-rate European con-vertible bond. Our result indicates that a convertible bond equals a plain vanilla bond plus a potential call option. We also obtain the pricing formula for the European convertible bond using the option pricing result in previous sections.