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July
2021, 17(4): 1729-1752.
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 and Management Optimization,
2021, 17
(4)
: 1729-1752.
doi: 10.3934/jimo.2020042
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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. |
show all references
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. |
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 |
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