We propose optimal mean-variance dynamic hedging strategies in discrete time under a multivariate Gaussian regime-switching model. The methodology, which also performs pricing, is robust to time-varying and clustering risk observed in financial time series. As such, it overcomes the main theoretical drawbacks of the Black-Scholes model. To support our approach, we provide goodness-of-fit tests to validate the model and for choosing the appropriate number of regimes, and we illustrate the methodology using monthly S & P 500 vanilla options prices. Then, we present the associated out-of-sample hedging results in the context of harvesting the implied versus realized volatility premium. Using the proposed methodology, the Sharpe ratio derived from the strategy doubles over the Black-Scholes delta-hedging methodology.
Black, F. and Scholes, M. (1973) The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 81, 637-654. https://doi.org/10.1086/260062
Fama, E.F. (1965) The Behavior of Stock-Market Prices. Journal of Business, 38, 34-105. https://doi.org/10.1086/294743
Mandelbrot, B. (1963) The Variation of Certain Speculative Prices. Journal of Business, 36, 307-332. https://doi.org/10.1086/294632
Schwert, G.W. (1989) Why Does Stock Market Volatility Change over Time? Journal of Finance, 44, 1115-1153. https://doi.org/10.1111/j.1540-6261.1989.tb02647.x
Boyle, P.P. and Emanuel, D. (1980) Discretely Adjusted Option Hedges. Journal of Financial Economics, 8, 259-282. https://doi.org/10.1016/0304-405X(80)90003-3
Gilster, J.E.J. (1990) The Systematic Risk of Discretely Rebalanced Option Hedges. Journal of Financial and Quantitative Analysis, 25, 507-516. https://doi.org/10.2307/2331013
Mello, A.S. and Neuhaus, H.J. (1998) A Portfolio Approach to Risk Reduction in Discretely Rebalanced Option Hedges. Management Science, 44, 921-934. https://doi.org/10.1287/mnsc.44.7.921
Buraschi, A. and Jackwerth, J. (2001) The Price of a Smile: Hedging and Spanning in Option Markets. The Review of Financial Studies, 14, 495-527. https://doi.org/10.1093/rfs/14.2.495
Owen, M.P. (2002) Utility Based Optimal Hedging in Incomplete Markets. Annals of Applied Probability, 12, 691-709. https://doi.org/10.1214/aoap/1026915621
Potters, M., Bouchaud, J.P. and Sestovic, D. (2001) Hedged Monte-Carlo: Low Variance Derivative Pricing with Objective Probabilities. Journal of Physics A, 289, 517-525. https://doi.org/10.1016/S0378-4371(00)00554-9
Pochart, B. and Bouchaud, J.P. (2004) Option Pricing and Hedging with Minimum Local Expected Shortfall. Quantitative Finance, 4, 607-618. https://doi.org/10.1080/14697680400000042
Cox, J. and Ross, S. (1976) The Valuation of Options for Alternative Stochastic Processes. Journal of Financial Economics, 3, 145-166. https://doi.org/10.1016/0304-405X(76)90023-4
Harrison, J.M. and Kreps, D.M. (1979) Martingales and Arbitrage in Multiperiod Securities Markets. Journal of Economic Theory, 20, 381-408. https://doi.org/10.1016/0022-0531(79)90043-7
Follmer, H. and Schweizer, M. (1991) Hedging of Contingent Claims under Incomplete Information. In: Davis, M. and Elliott, R., Eds., Applied Stochastic Analysis, Volume 5 of Stochastics Monographs, Gordon and Breach Science Publishers, 389-414.
Schweizer, M. (1992) Mean-Variance Hedging for General Claims. Annals of Applied Probability, 2, 171-179. https://doi.org/10.1214/aoap/1177005776
Schweizer, M. (1995) Variance-Optimal Hedging in Discrete Time. Mathematics of Operations Research, 20, 1-32. https://doi.org/10.1287/moor.20.1.1
Papageorgiou, N., Rémillard, B. and Hocquard, A. (2008) Replicating the Properties of Hedge Fund Returns. Journal of Alternative Investments, 11, 8-38. https://doi.org/10.3905/jai.2008.712595
Rémillard, B. and Rubenthaler, S. (2013) Optimal Hedging in Discrete Time. Quantitative Finance, 13, 819-825. https://doi.org/10.1080/14697688.2012.745012
Christoffersen, P. and Jacobs, K. (2004) Which GARCH Model for Option Valuation? Management Science, 50, 1204-1221. https://doi.org/10.1287/mnsc.1040.0276
Hull, J.C. and White, A. (1987) The Pricing of Options on Assets with Stochastic Volatilities. The Journal of Finance, 42, 281-300. https://doi.org/10.1111/j.1540-6261.1987.tb02568.x
Wiggins, J.B. (1987) Option Values under Stochastic Volatility: Theory and Empirical Estimates. Journal of Financial Economics, 19, 351-372. https://doi.org/10.1016/0304-405X(87)90009-2
Heston, S. (1993) A Closed-Form Solution for Options with Stochastic Volatility with Application to Bond and Currency Options. Review of Financial Studies, 6, 327-343. https://doi.org/10.1093/rfs/6.2.327
Kou, S.G. (2002) A Jump-Diffusion Model for Option Pricing. Management Science, 48, 1086-1101. https://doi.org/10.1287/mnsc.48.8.1086.166
Kou, S.G. and Wang, H. (2004) Option Pricing under a Double Exponential Jump Diffusion Model. Management Science, 50, 1178-1192. https://doi.org/10.1287/mnsc.1030.0163
Hamilton, J.D. (1990) Analysis of Time Series Subject to Changes in Regime. Journal of Econometrics, 45, 39-70. https://doi.org/10.1016/0304-4076(90)90093-9
Kim, C.J., Piger, J. and Startz, R. (2008) Estimation of Markov Regime-Switching Regression Models with Endogenous Switching. Journal of Econometrics, 143, 263-273. https://doi.org/10.1016/j.jeconom.2007.10.002
Bansal, R. and Zhou, H. (2002) Term Structure of Interest Rates with Regime Shifts. The Journal of Finance, 57, 1997-2043. https://doi.org/10.1111/0022-1082.00487
So, M.K.P., Lam, K. and Li, W.K. (1998) A Stochastic Volatility Model with Markov Switching. Journal of Business & Economic Statistics, 16, 244-253. https://doi.org/10.1080/07350015.1998.10524758
Fong, W.M. and See, K.H. (2001) Modelling the Conditional Volatility of Commodity Index Futures as a Regime Switching Process. Journal of Applied Econometrics, 16, 133-163. https://doi.org/10.1002/jae.590
Dempster, A.P., Laird, N.M. and Rubin, D.B. (1977) Maximum Likelihood from Incomplete Data via the EM Algorithm. Journal of the Royal Statistical Society, Series B, 39, 1-38.
Genest, C. and Rémillard, B. (2008) Validity of the Parametric Bootstrap for Goodness-of-Fit Testing in Semiparametric Models. Annales de l’Institut Henri Poincaré, 44, 1096-1127. https://doi.org/10.1214/07-AIHP148
Carr, P. and Wu, L. (2008) Variance Risk Premiums. Review of Financial Studies, 22, 1311-1341. https://doi.org/10.1093/rfs/hhn038
French, K.R., Schwert, G. and Stambaugh, R.F. (1987) Expected Stock Returns and Volatility. Journal of Financial Economics, 19, 3-29. https://doi.org/10.1016/0304-405X(87)90026-2
Glosten, L.R., Jagannathan, R. and Runkle, D.E. (1993) On the Relation between the Expected Value and the Volatility of the Nominal Excess Return on Stocks. The Journal of Finance, 48, 1779-1801. https://doi.org/10.1111/j.1540-6261.1993.tb05128.x
Bakshi, G. and Kapadia, N. (2003) Delta-Hedged Gains and the Negative Market Volatility Risk Premium. Review of Financial Studies, 16, 527-566. https://doi.org/10.1093/rfs/hhg002
Cappé, O., Moulines, E. and Rydén, T. (2005) Inference in Hidden Markov Models. Springer Series in Statistics. Springer, New York.
Rémillard, B. (2011) Validity of the Parametric Bootstrap for Goodness-of-Fit Testing in Dynamic Models. Technical Report. SSRN Working Paper Series No. 1966476. Social Science Research Network, Amsterdam.
Durbin, J. (1973) Weak Convergence of the Sample Distribution Function When Parameters Are Estimated. Annals of Statistics, 1, 279-290. https://doi.org/10.1214/aos/1176342365
Diebold, F.X., Gunther, T.A. and Tay, A.S. (1998) Evaluating Density Eorecasts with Applications to Financial Risk Management. International Economic Review, 39, 863-883. https://doi.org/10.2307/2527342
Bai, J. (2003) Testing Parametric Conditional Distributions of Dynamic Models. The Review of Economics and Statistics, 85, 531-549. https://doi.org/10.1162/003465303322369704
Bai, J. and Chen, Z. (2008) Testing Multivariate Distributions in GARCH Models. Journal of Econometrics, 143, 19-36. https://doi.org/10.1016/j.jeconom.2007.08.012
Bollerslev, T., Litvinova, J. and Tauchen, G. (2006) Leverage and Volatility Feedback Effects in High-Frequency Data. Journal of Financial Econometrics, 4, 353-384. https://doi.org/10.1093/jjfinec/nbj014
Rémillard, B. and Rubenthaler, S. (2016) Option Pricing and Hedging for Regime-Switching Geometric Brownian Motion Models. Working Paper Series, SSRN Working Paper Series No. 2599064. Social Science Research Network, Amsterdam.
Fleming, J., Kirby, C. and Ostdiek, B. (2001) The Economic Value of Volatility Timing. The Journal of Finance, 56, 329-352. https://doi.org/10.1111/0022-1082.00327
Merton, R.C. (1973) An Intertemporal Capital Asset Pricing Model. Econometrica, 41, 867-887. https://doi.org/10.2307/1913811
Rémillard, B. (2013) Statistical Methods for Financial Engineering. Chapman and Hall/CRC Financial Mathematics Series. Taylor & Francis Group, Abingdon-on-Thames.
Del Moral, P., Rémillard, B. and Rubenthaler, S. (2012) Monte Carlo Approximations of American Options that Preserve Monotonicity and Convexity. In: Camona, R., Moral, P.D., Hu, P. and Oudjane, N., Eds., Numerical Methods in Finance, Springer, New York, 117-145. https://doi.org/10.1007/978-3-642-25746-9_4