Pricing Options on Realized Variance in the Heston Model with Jumps in Returns and Volatility - Part II: An Approximate Distribution of Discrete Variance

We analyse the effect of the discrete sampling on the valuation of options on the realized variance in the Heston (1993) stochastic volatility model. It has been known for a while (Buehler (2006)) that, even though the quadratic variance can serve as an approximation to the discrete variance for valuing longer-term options on the realized variance, this approximation underestimates option values for short-term maturities (with maturities up to three months). We propose a method of mixing of the discrete variance in a log-normal model and the quadratic variance in a stochastic volatility model, which allows to accurately approximate the distribution of the discrete variance in the Heston model. As a result, we can apply semi-analytical Fourier transform methods developed by Sepp (2008) for pricing shorter-term options on the realized variance.

[1]  The Value of a Variance Swap - A Question of Interest , 2009 .

[2]  Leif Andersen.  Simple and efficient simulation of the Heston stochastic volatility model , 2008 .

[3]  Johannes Muhle-Karbe,et al.  Asymptotic and exact pricing of options on variance , 2013, Finance Stochastics.

[4]  R. C. Merton,et al.  Option pricing when underlying stock returns are discontinuous , 1976 .

[5]  N. Shephard,et al.  Econometric analysis of realised volatility and its use in estimating stochastic volatility models , 2000 .

[6]  Peter Carr,et al.  Volatility Derivatives , 2009 .

[7]  K. Demeterfi,et al.  More than You ever Wanted to Know about Volatility Swaps , 1999 .

[8]  Artur Sepp.  An approximate distribution of delta-hedging errors in a jump-diffusion model with discrete trading and transaction costs , 2010 .

[9]  H. Buhler.  Volatility Markets Consistent modeling, hedging and practical implementation , 2006 .

[10]  A. Lipton,et al.  Stochastic volatility models and Kelvin waves , 2008 .

[11]  Alexander Lipton,et al.  Mathematical Methods for Foreign Exchange: A Financial Engineer's Approach , 2001 .

[12]  Donald W. K. Andrews,et al.  Identification and Inference for Econometric Models , 2005 .

[13]  Alan L. Lewis.  Option Valuation under Stochastic Volatility , 2000 .

[14]  P. Carr,et al.  Robust Replication of Volatility Derivatives , 2008 .

[15]  S. Howison,et al.  On the pricing and hedging of volatility derivatives , 2004 .

[16]  P. Carr,et al.  Pricing swaps and options on quadratic variation under stochastic time change models—discrete observations case , 2010 .

[17]  P. Carr,et al.  Option Pricing, Interest Rates and Risk Management: Towards a Theory of Volatility Trading , 2001 .

[18]  E. Stein,et al.  Stock Price Distributions with Stochastic Volatility: An Analytic Approach , 1991 .

[19]  Mark Broadie,et al.  The Effect of Jumps and Discrete Sampling on Volatility and Variance Swaps , 2008 .

[20]  N. Shephard,et al.  Econometric analysis of realized volatility and its use in estimating stochastic volatility models , 2002 .

[21]  S. Heston.  A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options , 1993 .

[22]  N. Shephard,et al.  How accurate is the asymptotic approximation to the distribution of realised variance , 2001 .

[23]  Artur Sepp.  Pricing Options on Realized Variance in the Heston Model with Jumps in Returns and Volatility , 2008 .

[24]  Peter Forsyth,et al.  Pricing methods and hedging strategies for volatility derivatives , 2006 .

[25]  Volatility and Dividends - Volatility Modelling with Cash Dividends and Simple Credit Risk , 2010 .