Analytically pricing volatility swaps and volatility options with discrete sampling: Nonlinear payoff volatility derivatives

Abstract

This paper presents the first analytical pricing formulas for volatility swaps and volatility options with discrete sampling under the Black-Scholes model with time varying risk-free interest rate. Despite numerous analytical works on the pricing of variance swaps with discrete sampling under different models of asset prices, an analytical pricing formula for volatility swaps as well as volatility options had not been well addressed until now. The main challenge in pricing volatility swaps and volatility options is that payoff functions contain a square root operator, making their expectations nonlinear. By utilizing properties of noncentral chi random variables, we can compute expectations of payoff functions analytically and obtain formulas for pricing volatility swaps and volatility options, including variance swaps and variance options. Furthermore, we investigate the accuracy of the well-known convexity correction formula. Most interestingly, we extend our results to the Black-Scholes model with time varying parameters and to the Heston stochastic volatility model in which the variance process is assumed to follow the extended Cox-Ingersoll-Ross process by constructing simple closed-form approximate formulas for pricing volatility swaps and demonstrate the accuracy and efficiency of this approach by comparing the approximated prices against those obtained with Monte Carlo simulations.