Pricing Variance Swaps for Stochastic Volatilities with Delay and Jumps

We study the valuation of the variance swaps under stochastic volatility with delay and jumps. In our model, the volatility of the underlying stock price process not only incorporates jumps, which are found to be active empirically, but also exhibits past dependence: the behavior of a stock price right after a given time depends not only on the situation at but also on the whole past (history) of the process up to time as well. The jump part in our model is finally represented by a general version of compound Poisson processes. We provide some analytical closed forms for the expectation of the realized variance for the stochastic volatility with delay and jumps. We also present a lower bound for delay as a measure of risk. As applications of our analytical solutions, a numerical example using S&P60 Canada Index (1998–2002) is then provided to price variance swaps.

[1]  Alan G. White,et al.  The Pricing of Options on Assets with Stochastic Volatilities , 1987 .

[2]  Yacine Ait-Sahalia.  Testing Continuous-Time Models of the Spot Interest Rate , 1995 .

[3]  J. Duan.  THE GARCH OPTION PRICING MODEL , 1995 .

[4]  B. LeBaron,et al.  Nonlinear Dynamics and Stock Returns , 2021, Cycles and Chaos in Economic Equilibrium.

[6]  A. Swishchuk.  Modeling of Variance and Volatility Swaps for Financial Markets with Stochastic Volatilities , 2004 .

[7]  N. Shephard,et al.  Variation, Jumps, Market Frictions and High Frequency Data in Financial Econometrics , 2005 .

[8]  Richard Bellman,et al.  Differential-Difference Equations , 1967 .

[9]  R. C. Merton,et al.  Theory of Rational Option Pricing , 2015, World Scientific Reference on Contingent Claims Analysis in Corporate Finance.

[10]  M. Tepper,et al.  The Equilibrium Allocation of Di ff usive and Jump Risks with Heterogeneous Agents 1 , 2004 .

[11]  Anatoliy Swishchuk.  Modeling and Pricing of Variance Swaps for Stochastic Volatilities with Delay , 2013 .

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

[13]  L. Rogers,et al.  Complete Models with Stochastic Volatility , 1998 .

[14]  Mou-Hsiung Chang,et al.  The European option with hereditary price structures: Basic theory , 1999, Appl. Math. Comput..

[15]  Saikat Nandi,et al.  Derivatives on Volatility: Some Simple Solutions Based on Observables , 2000 .

[16]  Michael S. Johannes,et al.  Model Specification and Risk Premia: Evidence from Futures Options , 2005 .

[17]  J. Hull.  Options, Futures, and Other Derivatives , 1989 .

[18]  Jean Jacod,et al.  Estimating the degree of activity of jumps in high frequency data , 2009, 0908.3095.

[19]  F. Black,et al.  The Pricing of Options and Corporate Liabilities , 1973, Journal of Political Economy.

[20]  Song-Ping Zhu,et al.  A CLOSED‐FORM EXACT SOLUTION FOR PRICING VARIANCE SWAPS WITH STOCHASTIC VOLATILITY , 2010 .

[21]  P. Wilmott,et al.  Option pricing: Mathematical models and computation , 1994 .

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

[23]  Jun Liu,et al.  Dynamic Asset Allocation with Event Risk , 2003 .

[24]  M. Avellaneda,et al.  Pricing and hedging derivative securities in markets with uncertain volatilities , 1995 .

[25]  R. Bellman,et al.  Differential-Difference Equations , 1967 .

[26]  D. Duffie,et al.  Transform Analysis and Asset Pricing for Affine Jump-Diffusions , 1999 .

[27]  David S. Bates.  Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Thephlx Deutschemark Options , 1993 .

[28]  M. Taqqu,et al.  Option Pricing in ARCH-type Models: with Detailed Proofs , 1995 .

[29]  K. Demeterfi,et al.  A Guide to Volatility and Variance Swaps , 1999 .

[30]  Jun Liu,et al.  Dynamic Derivative Strategies , 2002 .

[31]  Jan Kallsen,et al.  PRICING OPTIONS ON VARIANCE IN AFFINE STOCHASTIC VOLATILITY MODELS , 2010 .

[32]  Song-Ping Zhu,et al.  Pricing variance swaps with stochastic volatility , 2009 .

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

[34]  Jack K. Hale,et al.  Introduction to Functional Differential Equations , 1993, Applied Mathematical Sciences.

[35]  A. Swishchuk,et al.  A CONTINUOUS-TIME GARCH MODEL FOR STOCHASTIC VOLATILITY WITH DELAY , 2005 .

[36]  Jianhong Wu,et al.  The pricing of options for securities markets with delayed response , 2007, Math. Comput. Simul..

[37]  Jun-Ping Liua,et al.  Dynamic Derivative Strategies , 2001 .

[38]  Salah-Eldin A. Mohammed,et al.  Stochastic Differential Systems with Memory: Theory, Examples and Applications (Sixth Workshop on Stochastic Analysis) , 1996 .

[39]  W. Runggaldier,et al.  Diffusion Approximation in Past Dependent Models and Applications to Option Pricing , 1991 .

[40]  Nicholas G. Polson,et al.  Evidence for and the Impact of Jumps in Volatility and Returns , 2001 .

[41]  A. Swishchuk.  Modeling and Pricing of Variance Swaps for Multi-Factor Stochastic Volatilities with Delay , 2007 .

[42]  T. Bollerslev,et al.  Generalized autoregressive conditional heteroskedasticity , 1986 .

[43]  Gurdip Bakshi,et al.  Empirical Performance of Alternative Option Pricing Models , 1997 .

[44]  P. Wilmott.  Derivatives: The Theory and Practice of Financial Engineering , 1998 .

[45]  Yaozhong Hu,et al.  A Delayed Black and Scholes Formula , 2006, math/0604640.

[46]  Paul Wilmott,et al.  GARCH and volatility swaps , 2004 .

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

[48]  L. Clewlow,et al.  Energy Derivatives: Pricing and Risk Management , 2000 .

[49]  V. Akgiray.  Conditional Heteroscedasticity in Time Series of Stock Returns: Evidence and Forecasts , 1989 .

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

[51]  R. Cont,et al.  Financial Modelling with Jump Processes , 2003 .

[52]  Bernard Lapeyre,et al.  Introduction to Stochastic Calculus Applied to Finance , 2007 .

[53]  Tim Bollerslev,et al.  Risk, Jumps, and Diversification , 2007 .