Figure 1.
Binomial tree
-
Previous Article
Channel leadership and recycling channel in closed-loop supply chain: The case of recycling price by the recycling party
- 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] |
Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020317 |
| [2] |
Shuyang Dai, Fengru Wang, Jerry Zhijian Yang, Cheng Yuan. A comparative study of atomistic-based stress evaluation. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020322 |
| [3] |
Agnaldo José Ferrari, Tatiana Miguel Rodrigues de Souza. Rotated $ A_n $-lattice codes of full diversity. Advances in Mathematics of Communications, 2020 doi: 10.3934/amc.2020118 |
| [4] |
Tian Ma, Shouhong Wang. Topological phase transition III: Solar surface eruptions and sunspots. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020350 |
| [5] |
Chao Xing, Jiaojiao Pan, Hong Luo. Stability and dynamic transition of a toxin-producing phytoplankton-zooplankton model with additional food. Communications on Pure & Applied Analysis, 2021, 20 (1) : 427-448. doi: 10.3934/cpaa.2020275 |
| [6] |
Mehdi Bastani, Davod Khojasteh Salkuyeh. On the GSOR iteration method for image restoration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 27-43. doi: 10.3934/naco.2020013 |
| [7] |
Hong Niu, Zhijiang Feng, Qijin Xiao, Yajun Zhang. A PID control method based on optimal control strategy. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 117-126. doi: 10.3934/naco.2020019 |
| [8] |
Li-Bin Liu, Ying Liang, Jian Zhang, Xiaobing Bao. A robust adaptive grid method for singularly perturbed Burger-Huxley equations. Electronic Research Archive, 2020, 28 (4) : 1439-1457. doi: 10.3934/era.2020076 |
| [9] |
Zexuan Liu, Zhiyuan Sun, Jerry Zhijian Yang. A numerical study of superconvergence of the discontinuous Galerkin method by patch reconstruction. Electronic Research Archive, 2020, 28 (4) : 1487-1501. doi: 10.3934/era.2020078 |
| [10] |
Yuxia Guo, Shaolong Peng. A direct method of moving planes for fully nonlinear nonlocal operators and applications. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020462 |
| [11] |
Noah Stevenson, Ian Tice. A truncated real interpolation method and characterizations of screened Sobolev spaces. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5509-5566. doi: 10.3934/cpaa.2020250 |
| [12] |
Yue Feng, Yujie Liu, Ruishu Wang, Shangyou Zhang. A conforming discontinuous Galerkin finite element method on rectangular partitions. Electronic Research Archive, , () : -. doi: 10.3934/era.2020120 |
| [13] |
Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020319 |
| [14] |
Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056 |
| [15] |
Gang Bao, Mingming Zhang, Bin Hu, Peijun Li. An adaptive finite element DtN method for the three-dimensional acoustic scattering problem. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020351 |
| [16] |
Abdollah Borhanifar, Maria Alessandra Ragusa, Sohrab Valizadeh. High-order numerical method for two-dimensional Riesz space fractional advection-dispersion equation. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020355 |
| [17] |
Gervy Marie Angeles, Gilbert Peralta. Energy method for exponential stability of coupled one-dimensional hyperbolic PDE-ODE systems. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020108 |
| [18] |
Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077 |
2019 Impact Factor: 1.366
Tools
Article outline
Figures and Tables
[Back to Top]