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.

Inferring information from the S&P 500, CBOE VIX, and CBOE SKEW indices

“…(a) We first consider the stochastic volatility model with contemporaneous jumps in returns and volatility (SVCJ) in Broadie et al (2007) and Eraker et al (2003), that is, λ t ≡ λ 1 which is a constant. This is the most popular affine model in the literature, for example, Bakshi et al (1997), Duan and Yeh (2010), Lin and Chang (2010), Neuberger (2012), Neumann et al (2016), Zhu and Lian (2011, 2012) and others. (b) Aït‐Sahalia et al (2015) and Bates (2006) find that more jumps occur during more volatile periods, suggesting that λ t ≡ λ 1 + λ 2 v t , where λ 1 and λ 2 are two positive constants.…”

“…In this paper, we consider three typical models. The first model is the stochastic volatility model with contemporaneous jumps in returns and volatility (SVCJ), which is the most popular affine model in the literature, for example, Bakshi, Cao, and Chen (1997); Broadie, Chernov, and Johannes (2007); Da Fonseca and Ignatieva (2019); Duan and Yeh (2010); Eraker (2004); Eraker, Johannes, and Polson (2003); Kaeck, Rodrigues, and Seeger (2017); Lin and Chang (2010); Neuberger (2012); Neumann, Prokopczuk, and Simen (2016); Ruan and Zhang (2018); Zhu and Lian (2011, 2012); and others. Bakshi et al (1997), Broadie et al (2007), Eraker (2004), and Neumann et al (2016) document that the SVCJ model is good enough to fit options and returns data simultaneously.…”

Variance and Volatility Swaps Under a Two-Factor Stochastic Volatility Model With Regime Switching

“…which shows that the definition of the realized variance and volatility really matters when we try to price variance and volatility swaps. In the existing literature, one of the most widely adopted definitions for the realized variance and volatility (Elliott & Lian 2013, Howison et al 2004, Zhu & Lian 2011 are respectively…”

“…In this section, numerical experiments are carried out to study the influence of introducing the regime switching factor into the Heston model, which would be conducted through the comparison of the variance and volatility swap prices calculated through our formulae with those obtained under the Heston model with the formula for variance swap prices in Zhu & Lian (2011) and the formula for volatility swap prices in Zhu & Lian (2015).…”

“…Recently, a few studies on discretely sampled variance and volatility swaps have been conducted so that the pricing results can be directly applied in reality. By adopting the dimension reduction techniques, Zhu & Lian (2011) derived a closed-form pricing formula for variance swaps under the Heston stochastic volatility model for the first time. In addition, the analytical solution for volatility swap prices under the Heston model (Heston 1993) has also been found in Zhu & Lian (2015).…”