Abstract
A new exactly solvable option pricing model has been introduced and elaborated. It is assumed that a stock price follows a Geometric shot noise process. An arbitrage-free integro-differential option pricing equation has been obtained and solved. The new Greeks have been analytically calculated. It has been shown that in diffusion approximation the developed option pricing model incorporates the well-known Black–Scholes equation and its solution. The stochastic dynamic origin of the Black–Scholes volatility has been uncovered.To model the observed market stock price patterns consisting of high frequency small magnitude and low frequency large magnitude jumps, the superposition of two Geometric shot noises has been implemented. A new generalized option pricing equation has been obtained and its exact solution was found. Merton’s jump–diffusion formula for option price was recovered in diffusion approximation.Despite the non-Gaussian nature of probability distributions involved, the new option pricing model has the same degree of analytical tractability as the Black–Scholes model and the Merton jump–diffusion model. This attractive feature allows one to derive exact formulas to value options and option related instruments in the market with jump-like price patterns.
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