A Closed‐form Exact Solution for Pricing Variance Swaps With Stochastic Volatility

Abstract

In this paper, we present a highly efficient approach to price variance swaps with discrete sampling times. We have found a closed-form exact solution for the partial differential equation (PDE) system based on the Heston (1993) two-factor stochastic volatility model embedded in the framework proposed by Little and Pant (2001). In comparison with all the previous approximation models based on the assumption of continuous sampling time, the current research of working out a closed-form exact solution for variance swaps with discrete sampling times at least serves for two major purposes: (i) to verify the degree of validity of using a continuous-sampling-time approximation for variance swaps of relatively short sampling period; (ii) to demonstrate that significant errors can result from still adopting such an assumption for a variance swap with small sampling frequencies or long tenor. Other key features of our new solution approach include: (a) with the newly found analytic solution, all the hedging ratios of a variance swap can also be analytically derived; (b) numerical values can be very efficiently computed from the newly found analytic formula.

Pricing Variance Swaps Under Stochastic Volatility and Stochastic Interest Rate

“…Our solutions extend the corresponding results in [119], where only stochastic volatility in the pricing model was considered.…”

“…Overall, all of these researchers assumed continuous sampling time, whereas the discrete sampling is the actual practice in financial markets. In fact, options of discretely-sampled variance swaps were mis-valued when the continuous sampling were used as approximations, and produce huge inaccuracies in certain sampling periods, as discussed in [9,34,80,119].…”

“…Thus, they suggested that institutions hedging the swaps propose clients as natural counter-parties to reduce the transaction costs. An extension of the approach in [80] was made by Zhu and Lian in [119] through incorporating Heston two-factor stochastic volatility for pricing discretely-sampled variance swaps. Levels of validity for short periods when using the continuous-time sampling were provided through significant errors, along with analytical hedging derivations and numerical simulations.…”

Pricing variance swaps under stochastic volatility and stochastic interest rate

“…Among stochastic volatility models, the one proposed by [66] has received a lot of attentions, since it gives a satisfactory description of the underlying asset dynamics [34,37]. Recently, Zhu and Lian [119,120] used Heston model to derive a closed form exact solution to the price of variance swaps.…”

“…The formula for the measure of realized variance used in this thesis and several other authors [80,119] is 24) whereas in the market, a typical measure of the realized variance is defined as…”