Bayesian statistical inference for European options with stock liquidity

Abstract

In the paper, the pricing of European options with stock liquidity is discussed. Since the liquidity discount factor leads to an analytically intractable likelihood function, we provide a new perspective to estimate the parameters entering the option pricing models with liquidity. A Bayesian statistical method is used to perform inference on model parameters and the option price. Although imperfect liquidity resulting in an incomplete market, the risk-neutral Esscher transforms can be used to obtain a European option price formula with stock liquidity. With the European option price formula being a prior, the posterior density of the option price is derived by using nonlinear transformation. A Metropolis-within-Gibbs algorithm is implemented to obtain samples from the posterior kernels. An application to S&P 500 index option is illustrated. Numerical experiments indicate that the Bayesian statistical method has its advantage comparing with traditional method in both parameter estimation and option pricing. By comparing with Black–Scholes model, we find that the Bayesian model with stock liquidity is more efficient in pricing options.