Figure 1.
Binomial tree
-
Previous Article
Optimizing 3-objective portfolio selection with equality constraints and analyzing the effect of varying constraints on the efficient sets
- JIMO Home
- This Issue
-
Next Article
A diagonal PRP-type projection method for convex constrained nonlinear monotone equations
doi: 10.3934/jimo.2020042
A lattice method for option evaluation with regime-switching asset correlation structure
Adam Smith Business School, University of Glasow, Glasgow G12 8QQ, UK |
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.
Keywords:
Option evaluation,
lattice method,
regime-switching,
correlation switching,
transition probabilities.
Mathematics Subject Classification:
Primary: 58F15, 58F17; Secondary: 53C35.
Citation:
Christoforidou Amalia, Christian-Oliver Ewald.
A lattice method for option evaluation with regime-switching asset correlation structure. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020042
![]()
References:
| [1] |
N. P. B. Bollen, Valuing options in regime-switching models, Journal of Derivatives, 6 (1998), 38-49. Google Scholar |
| [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. Google Scholar |
| [3] |
P. P. Boyle, A Lattice Framework for Option Pricing with Two State variables, Journal of Financial and Quantitative Analysis, 1988. Google Scholar |
| [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. Google Scholar |
| [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. Google Scholar |
| [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. Google Scholar |
| [12] |
V. Naik, Option valuation and hedging strategies with jumps in the volatility of asset returns, The Journal of Finance, 47 (1993), 1969-1984. Google Scholar |
| [13] |
G. W. Swhwert, Business cycles, financial crises, and risky asset volatility, Carnegie-Rochester Conference Series on Public Policy, 31 (1989), 83-126. Google Scholar |
| [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. Google Scholar |
| [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. |
show all references
References:
| [1] |
N. P. B. Bollen, Valuing options in regime-switching models, Journal of Derivatives, 6 (1998), 38-49. Google Scholar |
| [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. Google Scholar |
| [3] |
P. P. Boyle, A Lattice Framework for Option Pricing with Two State variables, Journal of Financial and Quantitative Analysis, 1988. Google Scholar |
| [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. Google Scholar |
| [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. Google Scholar |
| [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. Google Scholar |
| [12] |
V. Naik, Option valuation and hedging strategies with jumps in the volatility of asset returns, The Journal of Finance, 47 (1993), 1969-1984. Google Scholar |
| [13] |
G. W. Swhwert, Business cycles, financial crises, and risky asset volatility, Carnegie-Rochester Conference Series on Public Policy, 31 (1989), 83-126. Google Scholar |
| [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. Google Scholar |
| [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. |
Figure 2.
Pentanomial tree
Figure 3.
Sensitivity of Call Option value to Transition Probabilities
Figure 4.
Sensitivity of Call Option Value to the Regime Parameters
Figure 5.
Sensitivity of option price to the regime resistance of the two regimes
Figure 6.
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 |
| [1] |
Zhuo Jin, Linyi Qian. Lookback option pricing for regime-switching jump diffusion models. Mathematical Control & Related Fields, 2015, 5 (2) : 237-258. doi: 10.3934/mcrf.2015.5.237 |
| [2] |
Kun Fan, Yang Shen, Tak Kuen Siu, Rongming Wang. On a Markov chain approximation method for option pricing with regime switching. Journal of Industrial & Management Optimization, 2016, 12 (2) : 529-541. doi: 10.3934/jimo.2016.12.529 |
| [3] |
Fuke Wu, George Yin, Zhuo Jin. Kolmogorov-type systems with regime-switching jump diffusion perturbations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (7) : 2293-2319. doi: 10.3934/dcdsb.2016048 |
| [4] |
Jiaqin Wei. Time-inconsistent optimal control problems with regime-switching. Mathematical Control & Related Fields, 2017, 7 (4) : 585-622. doi: 10.3934/mcrf.2017022 |
| [5] |
Mourad Bellassoued, Raymond Brummelhuis, Michel Cristofol, Éric Soccorsi. Stable reconstruction of the volatility in a regime-switching local-volatility model. Mathematical Control & Related Fields, 2020, 10 (1) : 189-215. doi: 10.3934/mcrf.2019036 |
| [6] |
Zhuo Jin, George Yin, Hailiang Yang. Numerical methods for dividend optimization using regime-switching jump-diffusion models. Mathematical Control & Related Fields, 2011, 1 (1) : 21-40. doi: 10.3934/mcrf.2011.1.21 |
| [7] |
Yinghui Dong, Kam Chuen Yuen, Guojing Wang. Pricing credit derivatives under a correlated regime-switching hazard processes model. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1395-1415. doi: 10.3934/jimo.2016079 |
| [8] |
Jiapeng Liu, Ruihua Liu, Dan Ren. Investment and consumption in regime-switching models with proportional transaction costs and log utility. Mathematical Control & Related Fields, 2017, 7 (3) : 465-491. doi: 10.3934/mcrf.2017017 |
| [9] |
Jiaqin Wei, Zhuo Jin, Hailiang Yang. Optimal dividend policy with liability constraint under a hidden Markov regime-switching model. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1965-1993. doi: 10.3934/jimo.2018132 |
| [10] |
Chao Xu, Yinghui Dong, Zhaolu Tian, Guojing Wang. Pricing dynamic fund protection under a Regime-switching Jump-diffusion model with stochastic protection level. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-21. doi: 10.3934/jimo.2019072 |
| [11] |
Ping Chen, Haixiang Yao. Continuous-time mean-variance portfolio selection with no-shorting constraints and regime-switching. Journal of Industrial & Management Optimization, 2020, 16 (2) : 531-551. doi: 10.3934/jimo.2018166 |
| [12] |
Tak Kuen Siu, Yang Shen. Risk-minimizing pricing and Esscher transform in a general non-Markovian regime-switching jump-diffusion model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2595-2626. doi: 10.3934/dcdsb.2017100 |
| [13] |
Lin Xu, Rongming Wang, Dingjun Yao. Optimal stochastic investment games under Markov regime switching market. Journal of Industrial & Management Optimization, 2014, 10 (3) : 795-815. doi: 10.3934/jimo.2014.10.795 |
| [14] |
Nguyen Huu Du, Nguyen Thanh Dieu, Tran Dinh Tuong. Dynamic behavior of a stochastic predator-prey system under regime switching. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3483-3498. doi: 10.3934/dcdsb.2017176 |
| [15] |
Hongfu Yang, Xiaoyue Li, George Yin. Permanence and ergodicity of stochastic Gilpin-Ayala population model with regime switching. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3743-3766. doi: 10.3934/dcdsb.2016119 |
| [16] |
Rui Wang, Xiaoyue Li, Denis S. Mukama. On stochastic multi-group Lotka-Volterra ecosystems with regime switching. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3499-3528. doi: 10.3934/dcdsb.2017177 |
| [17] |
Jie Yu, Qing Zhang. Optimal trend-following trading rules under a three-state regime switching model. Mathematical Control & Related Fields, 2012, 2 (1) : 81-100. doi: 10.3934/mcrf.2012.2.81 |
| [18] |
Ka Chun Cheung, Hailiang Yang. Optimal investment-consumption strategy in a discrete-time model with regime switching. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 315-332. doi: 10.3934/dcdsb.2007.8.315 |
| [19] |
Li Zu, Daqing Jiang, Donal O'Regan. Persistence and stationary distribution of a stochastic predator-prey model under regime switching. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2881-2897. doi: 10.3934/dcds.2017124 |
| [20] |
Robert J. Elliott, Tak Kuen Siu. Stochastic volatility with regime switching and uncertain noise: Filtering with sub-linear expectations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (1) : 59-81. doi: 10.3934/dcdsb.2017003 |
2018 Impact Factor: 1.025
Tools
Metrics
Other articles
by authors
[Back to Top]