Variance Swaps Under Multiscale Stochastic Volatility of Volatility
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Variance Swaps Under Multiscale Stochastic Volatility of Volatility Min-Ku Lee1 · See-Woo Kim2 · Jeong-Hoon Kim2 Received: 19 August 2019 / Revised: 10 September 2020 / Accepted: 26 October 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Many hedge funds and retail investors demand volatility and variance derivatives in order to manage their exposure to volatility and volatility-of-volatility risk associated with their trading positions. The Heston model is a standard popular stochastic volatility model for pricing volatility and variance derivatives. However, it may fail to capture some important empirical features of the relevant market data due to the fact that the elasticity of volatility of volatility of the underlying price takes a special value, i.e., 1/2, whereas it has a merit of analytical tractability. We exploit a multiscale stochastic extension of volatility of volatility to obtain a better agreement with the empirical data while taking analytical advantage of the original Heston dynamics as much as possible in the context of pricing discrete variance swaps. By using an asymptotic technique with two small parameters, we derive a quasi-closed form formula for the fair strike price of variance swap and find useful pricing properties with respect to the stochastic extension parameters. Keywords Variance swap · Stochastic volatility · Stochastic volatility of volatility · Asymptotic expansion Mathematics Subject Classification (2010) 91G20 · 60J60 · 35Q91
Jeong-Hoon Kim
[email protected] Min-Ku Lee [email protected] See-Woo Kim [email protected] 1
Department of Mathematics, Kunsan National University, Kunsan 54150, Republic of Korea
2
Department of Mathematics, Yonsei University, Seoul 03722, Republic of Korea
Methodology and Computing in Applied Probability
1 Introduction The management of exposure to volatility and volatility-of-volatility risk associated with trading positions becomes important for many investors and hedge funds. One popular forward contract on discrete realized variance is variance swap. Its payoff is decided by how much difference between the realized variance and a fixed level of variance called the fair strike price occurs. Here, the realized variance is associated with the arithmetic average of the square of the returns. Thus the important key of the variance swap contract is how to valuate the strike price fairly. The Heston stochastic volatility model (Heston 1993) is a dominant model for the volatility to obtain the fair strike prices of variance swaps. The model is given by √ x dXt = rXt dt + Vt Xt dW √ t, v dVt = κ(θ − Vt )dt + ξ Vt dWt under a risk-neutral probability measure, where Wtx and Wtv are Brownian motions. Some of relevant research works are as follows. Swishchuk (2004) studied volatility and variance swaps under Heston’s stochastic volatility by using a probabilistic approach. Broadie and Jain (2008) presented a closed-form solution for both volatility and variance swaps under the Heston model. Sepp (2008) pri
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