Abstract
A three factor variance model introduced by Gatheral in 2008, called the double mean reverting (DMR) model, is well-known to reflect the empirical dynamics of the variance and prices of options on both SPX and VIX consistently with the market. One drawback of the DMR model is that calibration may not be easy as no closed form solution for European options exists, not like the Heston model. In this paper, we still use the double mean reverting nature to extend the Heston model and study the pricing of variance swaps given by simple returns in discrete sampling times. The constant mean level of Heston's stochastic volatility is extended to a slowly varying process which is specified in two different ways in terms of the Ornstein-Uhlenbeck (OU) and Cox-Ingersoll-Ross (CIR) processes. So, two types of double mean reversion are considered and the corresponding models are called the double mean reverting Heston-OU model and the double mean reverting Heston-CIR models. We solve Riccati type nonlinear equations and derive closed form exact solutions or closed form approximations of the fair strike prices of the variance swaps depending on the correlation structure of the three factors. We verify the accuracy of our analytic solutions by comparing with values computed by Monte Carlo simulation. The impact of the double mean reverting formulation on the fair strike prices of the variance swaps are also scrutinized in the paper.
| Original language | English |
|---|---|
| Pages (from-to) | 130-144 |
| Number of pages | 15 |
| Journal | Chaos, Solitons and Fractals |
| Volume | 114 |
| DOIs | |
| Publication status | Published - 2018 Sep |
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All Science Journal Classification (ASJC) codes
- Statistical and Nonlinear Physics
- Mathematics(all)
- Physics and Astronomy(all)
- Applied Mathematics
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Analytic solutions for variance swaps with double-mean-reverting volatility. / Kim, See Woo; Kim, Jeong Hoon.
In: Chaos, Solitons and Fractals, Vol. 114, 09.2018, p. 130-144.Research output: Contribution to journal › Article
TY - JOUR
T1 - Analytic solutions for variance swaps with double-mean-reverting volatility
AU - Kim, See Woo
AU - Kim, Jeong Hoon
PY - 2018/9
Y1 - 2018/9
N2 - A three factor variance model introduced by Gatheral in 2008, called the double mean reverting (DMR) model, is well-known to reflect the empirical dynamics of the variance and prices of options on both SPX and VIX consistently with the market. One drawback of the DMR model is that calibration may not be easy as no closed form solution for European options exists, not like the Heston model. In this paper, we still use the double mean reverting nature to extend the Heston model and study the pricing of variance swaps given by simple returns in discrete sampling times. The constant mean level of Heston's stochastic volatility is extended to a slowly varying process which is specified in two different ways in terms of the Ornstein-Uhlenbeck (OU) and Cox-Ingersoll-Ross (CIR) processes. So, two types of double mean reversion are considered and the corresponding models are called the double mean reverting Heston-OU model and the double mean reverting Heston-CIR models. We solve Riccati type nonlinear equations and derive closed form exact solutions or closed form approximations of the fair strike prices of the variance swaps depending on the correlation structure of the three factors. We verify the accuracy of our analytic solutions by comparing with values computed by Monte Carlo simulation. The impact of the double mean reverting formulation on the fair strike prices of the variance swaps are also scrutinized in the paper.
AB - A three factor variance model introduced by Gatheral in 2008, called the double mean reverting (DMR) model, is well-known to reflect the empirical dynamics of the variance and prices of options on both SPX and VIX consistently with the market. One drawback of the DMR model is that calibration may not be easy as no closed form solution for European options exists, not like the Heston model. In this paper, we still use the double mean reverting nature to extend the Heston model and study the pricing of variance swaps given by simple returns in discrete sampling times. The constant mean level of Heston's stochastic volatility is extended to a slowly varying process which is specified in two different ways in terms of the Ornstein-Uhlenbeck (OU) and Cox-Ingersoll-Ross (CIR) processes. So, two types of double mean reversion are considered and the corresponding models are called the double mean reverting Heston-OU model and the double mean reverting Heston-CIR models. We solve Riccati type nonlinear equations and derive closed form exact solutions or closed form approximations of the fair strike prices of the variance swaps depending on the correlation structure of the three factors. We verify the accuracy of our analytic solutions by comparing with values computed by Monte Carlo simulation. The impact of the double mean reverting formulation on the fair strike prices of the variance swaps are also scrutinized in the paper.
UR - http://www.scopus.com/inward/record.url?scp=85049772484&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85049772484&partnerID=8YFLogxK
U2 - 10.1016/j.chaos.2018.06.024
DO - 10.1016/j.chaos.2018.06.024
M3 - Article
AN - SCOPUS:85049772484
VL - 114
SP - 130
EP - 144
JO - Chaos, Solitons and Fractals
JF - Chaos, Solitons and Fractals
SN - 0960-0779
ER -