Valuation of volatility derivatives with time-varying volatility: An analytical probabilistic approach using a mixture distribution for pricing nonlinear payoff volatility derivatives in discrete observation case

Abstract

In this paper, we present an analytical probabilistic approach for pricing nonlinear payoff volatility derivatives with discrete sampling by assuming that the underlying asset price evolves according to the Black–Scholes model with time-varying volatility. A major difficulty to solve the pricing problem analytically is the volatility of the underlying asset price is time-varying, resulting the realized variance is distributed according to a mixture distribution, the probability density function of which is unknown. By utilizing the properties of a linear combination of noncentral chi-square random variables, we can calculate the expectation of square root of the realized variance analytically and provide formulas for pricing volatility swaps, volatility options, and variance options, including put–call parity relationships. Furthermore, we demonstrate an interesting application of our formulas by constructing simple closed-form approximate formulas for pricing the volatility derivatives for the constant elasticity of variance model. Finally, Monte Carlo simulations are conducted to illustrate the performance our approach, and effects of price volatility on the fair strike prices of the volatility derivatives are investigated and analyzed through several numerical experiments.