# American Institute of Mathematical Sciences

July  2021, 17(4): 1729-1752. doi: 10.3934/jimo.2020042

## A lattice method for option evaluation with regime-switching asset correlation structure

Received  January 2016 Revised  January 2017 Published  July 2021 Early access  March 2020

Fund Project: The first authors are supported by the Adam Smith Business School of the University of Glasow

This paper develops a lattice method for option evaluation in the presence of regime shifts in the correlation structure of assets, aiming at investigating whether the option prices reflect such shifts. We try to investigate whether option prices reflect switches in the correlation between the underlying asset of an option and risk-free rates.We develop and test two models.In the first model we allow all the parameters to follow a regime-switching process while in the second model, in order to isolate the regime-switching correlation effect on the option prices, we allow only the correlation to follow a regime-switching process. We use pentanomial lattices to represent the evolution of the regime-switching underlying assets. This is then applied in our empirical analysis, which focuses on crude oil. We use grid- and patternsearch based techniques to fit our models. Our findings suggest that prices of market traded options reflect the regime-switches and that a model which considers these switches produces significantly more accurate results than a single-regime model. We demonstrate that there is an asymmetry between parameter values obtained from historical data (backward looking) and those that are implied by traded options (for- ward looking) by employing the Kim filter to estimate our model.

Citation: Christoforidou Amalia, Christian-Oliver Ewald. A lattice method for option evaluation with regime-switching asset correlation structure. Journal of Industrial and Management Optimization, 2021, 17 (4) : 1729-1752. doi: 10.3934/jimo.2020042
##### References:
 [1] N. P. B. Bollen, Valuing options in regime-switching models, Journal of Derivatives, 6 (1998), 38-49. [2] N. P. B. Bollen, S. F. Gray and R. E. Whaley, Regime switching in foreign exchange rates: Evidence from currency option prices, Journal of Econometrics, 94 (2000), 239-276. [3] P. P. Boyle, A Lattice Framework for Option Pricing with Two State variables, Journal of Financial and Quantitative Analysis, 1988. [4] J. C. Cox, S. A. Ross and M. Rubinstein, Option pricing: A Simplified approach, Journal of Financial Economics, 7 (1979), 229-263. [5] J. C. Duan, A GARCH option pricing model, Mathematical Finance, 5 (1995), 13-32.  doi: 10.1111/j.1467-9965.1995.tb00099.x. [6] J. C. Duan, I. Popova and P. Ritchken, Option pricing under regime switching, Quantitative Finance, 2 (2002), 116-132.  doi: 10.1088/1469-7688/2/2/303. [7] S. F. Gray, Modeling the conditional distribution of interest rates as a regime-switching process, Journal of Financial Economics, 42 (1996), 27-62. [8] J. D. Hamilton, Rational expectations econometric analysis of changes in regime: An investigation of the term structure of interest rates,, Journal og Econometric Dynamics and Control, 12 (1988), 385-423.  doi: 10.1016/0165-1889(88)90047-4. [9] J. Hull and A. White, The pricing of options assets with stochastic volatilities, The Journal of Finance, 42 (1987), 281-300. [10] C. J. Kim, Dynamic linear models with Markov-switching,, Journal of Econometrics, 60 (1994), 1-22.  doi: 10.1016/0304-4076(94)90036-1. [11] C. J. Kim, Charles R.Nelson State-Space Models with Regime Switching. Classical and Gibbs-Sampling Approaches with Applications, Massachusetts Institution of Technology, United States of America, 1999. [12] V. Naik, Option valuation and hedging strategies with jumps in the volatility of asset returns, The Journal of Finance, 47 (1993), 1969-1984. [13] G. W. Swhwert, Business cycles, financial crises, and risky asset volatility, Carnegie-Rochester Conference Series on Public Policy, 31 (1989), 83-126. [14] C. M. Turner, R. Startz and C. R. Nelson, A markov model of heteroscedasticity, riskm and learning in the risky asset market, Journal of Financial Economics, 25 (1989), 3-22. [15] D. D. Yao, Q. Zhang and X. Y. Zhou, A regime-switching model for european option pricing,, Stochastic Processes, Optimization, and Control Theory: Applications in Financial Engineering, Queueing Networks, and Manufacturing Systems, Springer US, 94 (2006), 281–300. doi: 10.1007/0-387-33815-2_14.

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##### References:
 [1] N. P. B. Bollen, Valuing options in regime-switching models, Journal of Derivatives, 6 (1998), 38-49. [2] N. P. B. Bollen, S. F. Gray and R. E. Whaley, Regime switching in foreign exchange rates: Evidence from currency option prices, Journal of Econometrics, 94 (2000), 239-276. [3] P. P. Boyle, A Lattice Framework for Option Pricing with Two State variables, Journal of Financial and Quantitative Analysis, 1988. [4] J. C. Cox, S. A. Ross and M. Rubinstein, Option pricing: A Simplified approach, Journal of Financial Economics, 7 (1979), 229-263. [5] J. C. Duan, A GARCH option pricing model, Mathematical Finance, 5 (1995), 13-32.  doi: 10.1111/j.1467-9965.1995.tb00099.x. [6] J. C. Duan, I. Popova and P. Ritchken, Option pricing under regime switching, Quantitative Finance, 2 (2002), 116-132.  doi: 10.1088/1469-7688/2/2/303. [7] S. F. Gray, Modeling the conditional distribution of interest rates as a regime-switching process, Journal of Financial Economics, 42 (1996), 27-62. [8] J. D. Hamilton, Rational expectations econometric analysis of changes in regime: An investigation of the term structure of interest rates,, Journal og Econometric Dynamics and Control, 12 (1988), 385-423.  doi: 10.1016/0165-1889(88)90047-4. [9] J. Hull and A. White, The pricing of options assets with stochastic volatilities, The Journal of Finance, 42 (1987), 281-300. [10] C. J. Kim, Dynamic linear models with Markov-switching,, Journal of Econometrics, 60 (1994), 1-22.  doi: 10.1016/0304-4076(94)90036-1. [11] C. J. Kim, Charles R.Nelson State-Space Models with Regime Switching. Classical and Gibbs-Sampling Approaches with Applications, Massachusetts Institution of Technology, United States of America, 1999. [12] V. Naik, Option valuation and hedging strategies with jumps in the volatility of asset returns, The Journal of Finance, 47 (1993), 1969-1984. [13] G. W. Swhwert, Business cycles, financial crises, and risky asset volatility, Carnegie-Rochester Conference Series on Public Policy, 31 (1989), 83-126. [14] C. M. Turner, R. Startz and C. R. Nelson, A markov model of heteroscedasticity, riskm and learning in the risky asset market, Journal of Financial Economics, 25 (1989), 3-22. [15] D. D. Yao, Q. Zhang and X. Y. Zhou, A regime-switching model for european option pricing,, Stochastic Processes, Optimization, and Control Theory: Applications in Financial Engineering, Queueing Networks, and Manufacturing Systems, Springer US, 94 (2006), 281–300. doi: 10.1007/0-387-33815-2_14.
Binomial tree
Pentanomial tree
Sensitivity of Call Option value to Transition Probabilities
Sensitivity of Call Option Value to the Regime Parameters
Sensitivity of option price to the regime resistance of the two regimes
Sensitivity of Option Price to the regime switching correlation
 Jump Probability Underlying Asset Price Up $\pi_{u}$ $S_{u}$ Horizontal $\pi_{0}$ $S$ Down $\pi_{d}$ $S_{d}$
 Jump Probability Underlying Asset Price Up $\pi_{u}$ $S_{u}$ Horizontal $\pi_{0}$ $S$ Down $\pi_{d}$ $S_{d}$
 Event Probability Underlying Asset Price Asset 1 Asset 2 $E_{1}$ $\pi_{1}$ $S_{p}u_{p}$ $S_{b}u_{b}$ $E_{2}$ $\pi_{2}$ $S_{p}u_{p}$ $S_{b}d_{b}$ $E_{3}$ $\pi_{3}$ $S_{p}d_{p}$ $S_{b}d_{b}$ $E_{4}$ $\pi_{4}$ $S_{p}d_{p}$ $S_{b}u_{b}$ $E_{5}$ $\pi_{5}$ $S_{p}$ $S_{b}$
 Event Probability Underlying Asset Price Asset 1 Asset 2 $E_{1}$ $\pi_{1}$ $S_{p}u_{p}$ $S_{b}u_{b}$ $E_{2}$ $\pi_{2}$ $S_{p}u_{p}$ $S_{b}d_{b}$ $E_{3}$ $\pi_{3}$ $S_{p}d_{p}$ $S_{b}d_{b}$ $E_{4}$ $\pi_{4}$ $S_{p}d_{p}$ $S_{b}u_{b}$ $E_{5}$ $\pi_{5}$ $S_{p}$ $S_{b}$
 Strike Price Market Price Predicted by Lattice Predicted by BS 16 4.9 3.881616 3.9936 17 2.85 3.127282 3.0613 18 2.35 2.373772 2.2162 19 1.65 1.629733 1.5029 20 0.95 0.949999 0.9500 21 0.4 0.399999 0.5587 22 0.2 0.200003 0.3059 23 0.2 0.136097 0.1565 24 0.1 0.105053 0.0750 25 0.15 0.088113 0.0339 Sum of absolute differences 1.110163 2.573241
 Strike Price Market Price Predicted by Lattice Predicted by BS 16 4.9 3.881616 3.9936 17 2.85 3.127282 3.0613 18 2.35 2.373772 2.2162 19 1.65 1.629733 1.5029 20 0.95 0.949999 0.9500 21 0.4 0.399999 0.5587 22 0.2 0.200003 0.3059 23 0.2 0.136097 0.1565 24 0.1 0.105053 0.0750 25 0.15 0.088113 0.0339 Sum of absolute differences 1.110163 2.573241
 Strike Price Market Price Predicted by Lattice Predicted by BS 16 5 4.4153631 4.3208036 17 3.8 3.7998973 3.5464351 18 2.45 2.3996588 2.8585724 19 2.25 1.9270998 2.2636265 20 1.6 1.5997621 1.7622281 21 1.35 1.3087541 1.3499542 22 0.95 1.0205032 1.0186664 23 1.05 1.1713058 0.7580323 24 0.6 0.6000609 0.5569055 25 0.5 0.4960518 0.4043912 Sum of absolute differences 0.4890502 1.1145081
 Strike Price Market Price Predicted by Lattice Predicted by BS 16 5 4.4153631 4.3208036 17 3.8 3.7998973 3.5464351 18 2.45 2.3996588 2.8585724 19 2.25 1.9270998 2.2636265 20 1.6 1.5997621 1.7622281 21 1.35 1.3087541 1.3499542 22 0.95 1.0205032 1.0186664 23 1.05 1.1713058 0.7580323 24 0.6 0.6000609 0.5569055 25 0.5 0.4960518 0.4043912 Sum of absolute differences 0.4890502 1.1145081
 Strike Price Market Price Predicted by Lattice Predicted by BS 16 5 4.4153631 4.3208036 17 3.8 3.7998973 3.5464351 18 2.45 2.3996588 2.8585724 19 2.25 1.9270998 2.2636265 20 1.6 1.5997621 1.7622281 21 1.35 1.3087541 1.3499542 22 0.95 1.0205032 1.0186664 23 1.05 1.1713058 0.7580323 24 0.6 0.6000609 0.5569055 25 0.5 0.4960518 0.4043912 Sum of absolute differences 0.4890502 1.1145081
 Strike Price Market Price Predicted by Lattice Predicted by BS 16 5 4.4153631 4.3208036 17 3.8 3.7998973 3.5464351 18 2.45 2.3996588 2.8585724 19 2.25 1.9270998 2.2636265 20 1.6 1.5997621 1.7622281 21 1.35 1.3087541 1.3499542 22 0.95 1.0205032 1.0186664 23 1.05 1.1713058 0.7580323 24 0.6 0.6000609 0.5569055 25 0.5 0.4960518 0.4043912 Sum of absolute differences 0.4890502 1.1145081
 Strike Price Predicted 16 4.9 1.625144372 17 2.85 1.269327353 18 2.35 0.940833993 19 1.65 0.626509398 20 0.95 0.420399871 21 0.4 0.245524212 22 0.2 0.158808617 23 0.2 0.079890983 24 0.1 0.045583587 25 0.15 0.023566284
 Strike Price Predicted 16 4.9 1.625144372 17 2.85 1.269327353 18 2.35 0.940833993 19 1.65 0.626509398 20 0.95 0.420399871 21 0.4 0.245524212 22 0.2 0.158808617 23 0.2 0.079890983 24 0.1 0.045583587 25 0.15 0.023566284
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