Modeling of Financial Asset Prices with Hyperbolic-Sine Stochastic Model

We propose an analytically tractable local volatility model for asset price dynamics leading to volatility smile/skew and fatter-tailed probability distribution. The proposed local volatility model is based on stochastic process of hyperbolic-sine type. We derive the transition probability density function for hyperbolic-sine model and justify that this function has delta function terminal condition at initial time. We compare the probability density functions in Black-Scholes and hyperbolic-sine models to demonstrate that the probability distribution in hyperbolic sine model has some features of fat-tailed distributions. Risk neutral valuation technique is applied to find explicit valuation formula for European call option price in hyperbolic-sine model. In hyperbolic-sine model European call option is more valuable than an identical option in Black-Scholes model for ATM options. We verify that in hyperbolic-sine model Breeden-Litzenberger formula (relation between European call option price and probability density function) holds true. We also examine that Dupire formula correctly recovers volatility function from European call option price in hyperbolic-sine model.