Chapter 43 JUMP DIFFUSION MODEL. Abstract Jump diffusion processes have been used in modern finance to capture discontinuous behavior in asset pricing. Various jump diffusion models are considered in this chapter. | Chapter 43 JUMP DIFFUSION MODEL SHIU-HUEI WANG University of Southern California USA Abstract Jump diffusion processes have been used in modern finance to capture discontinuous behavior in asset pricing. Various jump diffusion models are considered in this chapter. Also the applications of jump diffusion processes on stocks bonds and interest rate are discussed. Keywords Black-Scholes model jump diffusion process mixed-jump process Bernoulli jump process Gauss-Hermite jump process conditional jump dynamics ARCH GARCH jump diffusion model affine jump diffusion model autoregressive jump process model jump diffusion with conditional heteroskedasticity. . Introduction In contrast to basic insights into continuous-time asset-pricing models that have been driven by stochastic diffusion processes with continuous sample paths jump diffusion processes have been used in finance to capture discontinuous behavior in asset pricing. As described in Merton 1976 the validity of Black-Scholes formula depends on whether the stock price dynamics can be described by a continuous-time diffusion process whose sample path is continuous with probability 1. Thus if the stock price dynamics cannot be represented by stochastic process with a continuous sample path the Black-Scholes solution is not valid. In other words as the price processes feature big jumps . not continuous continuous-time models cannot explain why the jumps occur and hence not adequate. In addition Ahn and Thompson 1986 also examined the effect of regulatory risks on the valuation of public utilities and found that those jump risks were priced even though they were uncorrelated with market factors. It shows that jump risks cannot be ignored in the pricing of assets. Thus a jump stochastic process defined in continuous time and also called as jump diffusion model was rapidly developed. The jump diffusion process is based on Poisson process which can be used for modeling systematic jumps caused by surprise effect. .
