Maximum Likelihood Estimation of Asymmetric Jump-Diffusion Processes: Application to Security Prices

Chapter 2 Jump-Diffusion Models for Asset Pricing in Financial Engineering

“…Jump-diffusion models are able to reproduce the leptokurtic feature of the return distribution, and the "volatility smile" observed in option prices (see Kou, 2002). The empirical tests performed in Ramezani and Zeng (2002) suggest that the double exponential jumpdiffusion model fits stock data better than the normal jump-diffusion model, and both of them fit the data better than the classical geometric Brownian motion model. (3) A model must be simple enough to be amenable to computation.…”

“…For notational simplicity and in order to get analytical solutions for various option pricing problems, the drift μ and the volatility σ are assumed to be constants, and the Brownian motion and jumps are assumed to be one-dimensional. Ramezani and Zeng (2002) independently propose the double exponential jump-diffusion model from an econometric viewpoint as a way of improving the empirical fit of Merton's normal jumpdiffusion model to stock price data. There are two interesting properties of the double exponential distribution that are crucial for the model.…”

Maximum likelihood estimation of the double exponential jump-diffusion process

“…In this section we first present the PBJD model of Ramezani and Zeng (1998). The PBJD assumes that good and bad news are generated by two independent Poisson processes and jump magnitudes are drawn from the Pareto and the Beta distributions.…”

“…Under Kou's (2002) DEJD specification, a single Poisson process with fixed intensity generates the jumps in prices, but the jump magnitudes are drawn from two independent exponential distributions. Ramezani and Zeng (1998) independently propose the Pareto-Beta jump-diffusion (PBJD), assuming that good and bad news are generated by two independent Poisson processes and jump magnitudes are drawn from the Pareto and Beta distributions. Below we show that the two models are closely related in that the parameters of one model can be exactly recovered from the other.…”

“…Ramezani and Zeng (1998) propose their model from an econometric viewpoint. These papers are clearly complementary: WhileRamezani and Zeng (1998) focus on the problem of parameter estimation,Kou (2002) andKou and Wang (2004) develop the DEJD option pricing formula, which would require the estimated parameters as inputs.…”