doi: 10.3934/fmf.2021005
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Option pricing under a discrete-time Markov switching stochastic volatility with co-jump model

1. 

Smith School of Business & Institute for Systems Research, University of Maryland, College Park, MD 20742, USA

2. 

School of Insurance, Southwestern University of Finance and Economics, 611130 Chengdu, China

3. 

Capital One Financial Corporation, McLean, Virginia 22102

4. 

School of Finance, Nankai University, 300350 Tianjin, China

* Corresponding author: Michael C. Fu

Received  February 2021 Revised  July 2021 Early access July 2021

We consider option pricing using a discrete-time Markov switching stochastic volatility with co-jump model, which can capture asset price features such as leptokurtosis, skewness, volatility clustering, and varying mean-reversion speed of volatility. For pricing European options, we develop a computationally efficient method for obtaining the probability distribution of average integrated variance (AIV), which is key to option pricing under stochastic-volatility-type models. Building upon the efficiency of the European option pricing approach, we are able to price an American-style option, by converting its pricing into the pricing of a portfolio of European options. Our work also provides constructive guidance for analyzing derivatives based on variance, e.g., the variance swap. Numerical results indicate our methods can be implemented very efficiently and accurately.

Citation: Michael C. Fu, Bingqing Li, Rongwen Wu, Tianqi Zhang. Option pricing under a discrete-time Markov switching stochastic volatility with co-jump model. Frontiers of Mathematical Finance, doi: 10.3934/fmf.2021005
References:
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D. D. AingworthS. R. Das and R. Motwani, A simple approach for pricing equity options with Markov switching state variables, Quant. Finance, 6 (2006), 95-105.  doi: 10.1080/14697680500511215.  Google Scholar

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S. AlizadehM. W. Brandt and F. X. Diebold, Range-based estimation of stochastic volatility models, J. Finance, 57 (2002), 1047-1091.   Google Scholar

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G. BakshiN. Ju and H. Ou-Yang, Estimation of continuous-time models with an application to equity volatility dynamics, J. Financial Econom., 82 (2006), 227-249.  doi: 10.1016/j.jfineco.2005.09.005.  Google Scholar

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F. M. Bandi and R. Renò, Price and volatility co-jumps, J. Financial Econom., 119 (2016), 107-146.   Google Scholar

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O. E. Barndorff-Nielsen and N. Shephard, Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics, J. R. Stat. Soc. Ser. B Stat. Methodol., 63 (2001), 167-241.  doi: 10.1111/1467-9868.00282.  Google Scholar

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M. Britten-Jones and A. Neuberger, Option prices, implied price processes, and stochastic volatility, J. Finance, 55 (2000), 839-866.  doi: 10.1111/0022-1082.00228.  Google Scholar

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M. BroadieM. Chernov and M. Johannes, Model specification and risk premia: Evidence from futures options, J. Finance, 62 (2007), 1453-1490.   Google Scholar

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M. Broadie and J. Detemple, American option valuation: New bounds, approximations, and a comparison of existing methods, Rev. Financial Stud., 9 (1996), 1211-1250.  doi: 10.1093/rfs/9.4.1211.  Google Scholar

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M. Broadie and P. Glasserman, Monte Carlo methods for pricing high-dimensional American options: An overview, Net Exposure, 3 (1997), 15-37.   Google Scholar

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M. Broadie and P. Glasserman, Pricing American-style securities using simulation, J. Econom. Dynam. Control, 21 (1997), 1323-1352.  doi: 10.1016/S0165-1889(97)00029-8.  Google Scholar

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J. Buffington and R. J. Elliott, American options with regime switching, Int. J. Theor. Appl. Finance, 5 (2002), 497-514.  doi: 10.1142/S0219024902001523.  Google Scholar

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N. Cai and S. G. Kou, Option pricing under a mixed-exponential jump diffusion model, Management Sci., 57 (2011), 2067-2081.   Google Scholar

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N. CaiY. Song and S. Kou, A general framework for pricing Asian options under Markov processes, Oper. Res., 63 (2015), 540-554.  doi: 10.1287/opre.2015.1385.  Google Scholar

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P. Collin-DufresneR. S. Goldstein and F. Yang, On the relative pricing of long-maturity index options and collateralized debt obligations, J. Finance, 67 (2012), 1983-2014.  doi: 10.1111/j.1540-6261.2012.01779.x.  Google Scholar

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R. Cont and P. Tankov, Nonparametric calibration of jump-diffusion option pricing models, J. Computational Finance, 7 (2004), 1-49.   Google Scholar

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Z. CuiJ. L. Kirkby and D. Nguyen, A general framework for discretely sampled realized variance derivatives in stochastic volatility models with jumps, European J. Oper. Res., 262 (2017), 381-400.  doi: 10.1016/j.ejor.2017.04.007.  Google Scholar

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D. Du and D. Luo, The pricing of jump propagation: Evidence from spot and options markets, Management Sci., 65 (2019), 2360-2387.  doi: 10.1287/mnsc.2017.2885.  Google Scholar

[26]

J.-C. DuanI. Popova and P. Ritchken, Option pricing under regime switching, Quant. Finance, 2 (2002), 116-132.  doi: 10.1088/1469-7688/2/2/303.  Google Scholar

[27]

D. DuffieJ. Pan and K. Singleton, Transform analysis and asset pricing for affine jump-diffusions, Econometrica, 68 (2000), 1343-1376.  doi: 10.1111/1468-0262.00164.  Google Scholar

[28]

R. J. ElliottT. K. Siu and L. Chan, Option pricing for GARCH models with Markov switching, Int. J. Theo. Appl. Finance, 9 (2006), 825-841.  doi: 10.1142/S0219024906003846.  Google Scholar

[29]

B. Eraker, Do stock prices and volatility jump? Reconciling evidence from spot and option prices, J. Finance, 59 (2004), 1367-1403.  doi: 10.1111/j.1540-6261.2004.00666.x.  Google Scholar

[30]

B. ErakerM. Johannes and N. Polson, The impact of jumps in volatility and returns, J. Finance, 58 (2003), 1269-1300.  doi: 10.1111/1540-6261.00566.  Google Scholar

[31]

M. C. FuS. B. LapriseD. B. MadanY. Su and R. Wu, Pricing American options: A comparison of Monte Carlo simulation approaches, J. Comput. Finance, 4 (2001), 39-88.  doi: 10.21314/JCF.2001.066.  Google Scholar

[32]

M. C. FuB. LiG. Li and R. Wu, Option pricing for a jump-diffusion model with general discrete jump-size distributions, Management Sci., 63 (2017), 3961-3977.  doi: 10.1287/mnsc.2016.2522.  Google Scholar

[33]

C.-D. FuhK. W. R. HoI. Hu and R. H. Wang, Option pricing with Markov switching, J. Data Sci., 10 (2012), 483-509.   Google Scholar

[34]

S. Galluccio and Y. Lecam, Implied calibration and moments asymptotics in stochastic volatility jump diffusion models, Available at SSRN: https://ssrn.com/abstract=831784, 2008. doi: 10.2139/ssrn.831784.  Google Scholar

[35]

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[36]

X. Guo, Information and option pricings, Quant. Finance, 1 (2001), 38-44.  doi: 10.1080/713665550.  Google Scholar

[37]

S. L. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Rev. Financial Stud., 6 (1993), 327-343.  doi: 10.1093/rfs/6.2.327.  Google Scholar

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C. Homescu, Local stochastic volatility models: Calibration and pricing, Available at SSRN: https://ssrn.com/abstract=2448098, 2014. doi: 10.2139/ssrn.2448098.  Google Scholar

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J. Hull and A. White, The pricing of options on assets with stochastic volatilities, J. Finance, 42 (1987), 281-300.  doi: 10.1111/j.1540-6261.1987.tb02568.x.  Google Scholar

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M. S. JohannesN. G. Polson and J. R. Stroud, Optimal filtering of jump diffusions: Extracting latent states from asset prices, Rev. Financial Stud., 22 (2009), 2759-2799.  doi: 10.1093/rfs/hhn110.  Google Scholar

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S. G. Kou, A jump-diffusion model for option pricing, Management Sci., 48 (2002), 1086-1101.   Google Scholar

[44]

S. G. Kou and H. Wang, Option pricing under a double exponential jump diffusion model, Management Sci., 50 (2004), 1178-1192.   Google Scholar

[45]

S. KouC. Yu and H. Zhong, Jumps in equity index returns before and during the recent financial crisis: A Bayesian analysis, Management Sci., 63 (2017), 988-1010.  doi: 10.1287/mnsc.2015.2359.  Google Scholar

[46]

S. B. LapriseM. C. FuS. I. MarcusA. E. B. Lim and H. Zhang, Pricing American-style derivatives with European call options, Management Sci., 52 (2006), 95-110.  doi: 10.1287/mnsc.1050.0447.  Google Scholar

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H. LiM. T. Wells and C. L. Yu, A Bayesian analysis of return dynamics with Lévy jumps, Rev. Financial Stud., 21 (2008), 2345-2378.  doi: 10.1093/rfs/hhl036.  Google Scholar

[48]

R. H. Liu, Q. Zhang and G. Yin, Option pricing in a regime-switching model using the fast Fourier transform, J. Appl. Math. Sto. Anal., 2006 (2006), Art. ID 18109, 22 pp. doi: 10.1155/JAMSA/2006/18109.  Google Scholar

[49]

C. C. Lo and K. Skindilias, An improved Markov chain approximation methodology: Derivatives pricing and model calibration, Int. J. Theo. Appl. Finance, 17 (2014), 1450047, 22 pp. doi: 10.1142/S0219024914500472.  Google Scholar

[50]

F. A. Longstaff and E. S. Schwartz, Valuing American options by simulation: A simple least-squares approach, Rev. Financial Stud., 14 (2001), 113-147.  doi: 10.1093/rfs/14.1.113.  Google Scholar

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R. C. Merton, Option pricing when underlying stock returns are discontinuous, J. Financial Econom., 3 (1976), 125-144.  doi: 10.1016/0304-405X(76)90022-2.  Google Scholar

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J. P. Morgan/Reuters, RiskMetrics$^{TM}$-Technical Document, Fourth Edition, Morgan Guaranty Trust Company of New York, New York, 1996. Google Scholar

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V. Naik, Option valuation and hedging strategies with jumps in the volatility of asset returns, J. Finance, 48 (1993), 1969-1984.  doi: 10.1111/j.1540-6261.1993.tb05137.x.  Google Scholar

[54]

J. Pan, The jump-risk premia implicit in options: evidence from an integrated time-series study, J. Financial Econom., 63 (2002), 3-50.  doi: 10.1016/S0304-405X(01)00088-5.  Google Scholar

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M. Perlin, MS_Regress-the MATLAB package for Markov regime switching models, Available at SSRN: http://ssrn.com/abstract=1714016, 2015. Google Scholar

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A. Rossi and G. M. Gallo, Volatility estimation via hidden Markov models, J. Empirical Finance, 13 (2006), 203-230.  doi: 10.1016/j.jempfin.2005.09.003.  Google Scholar

[57]

T. RydénT. Teräsvirta and S. Åsbrink, Stylized facts of daily return series and the hidden Markov model, J. Appl. Econ., 13 (1998), 217-244.   Google Scholar

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show all references

References:
[1]

T. Adrian and J. Rosenberg, Stock returns and volatility: Pricing the short-run and long-run components of market risk, J. Finance, 63 (2008), 2997-3030.   Google Scholar

[2]

D. D. AingworthS. R. Das and R. Motwani, A simple approach for pricing equity options with Markov switching state variables, Quant. Finance, 6 (2006), 95-105.  doi: 10.1080/14697680500511215.  Google Scholar

[3]

S. AlizadehM. W. Brandt and F. X. Diebold, Range-based estimation of stochastic volatility models, J. Finance, 57 (2002), 1047-1091.   Google Scholar

[4]

G. BakshiN. Ju and H. Ou-Yang, Estimation of continuous-time models with an application to equity volatility dynamics, J. Financial Econom., 82 (2006), 227-249.  doi: 10.1016/j.jfineco.2005.09.005.  Google Scholar

[5]

F. M. Bandi and R. Renò, Price and volatility co-jumps, J. Financial Econom., 119 (2016), 107-146.   Google Scholar

[6]

O. E. Barndorff-Nielsen and N. Shephard, Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics, J. R. Stat. Soc. Ser. B Stat. Methodol., 63 (2001), 167-241.  doi: 10.1111/1467-9868.00282.  Google Scholar

[7]

D. S. Bates, Jumps and stochastic volatility: Exchange rate processes implicit in Deutsche Mark options, Rev. Financial Stud., 9 (1996), 69-107.  doi: 10.1093/rfs/9.1.69.  Google Scholar

[8]

M. Britten-Jones and A. Neuberger, Option prices, implied price processes, and stochastic volatility, J. Finance, 55 (2000), 839-866.  doi: 10.1111/0022-1082.00228.  Google Scholar

[9]

M. BroadieM. Chernov and M. Johannes, Model specification and risk premia: Evidence from futures options, J. Finance, 62 (2007), 1453-1490.   Google Scholar

[10]

M. Broadie and J. Detemple, American option valuation: New bounds, approximations, and a comparison of existing methods, Rev. Financial Stud., 9 (1996), 1211-1250.  doi: 10.1093/rfs/9.4.1211.  Google Scholar

[11]

M. Broadie and P. Glasserman, Monte Carlo methods for pricing high-dimensional American options: An overview, Net Exposure, 3 (1997), 15-37.   Google Scholar

[12]

M. Broadie and P. Glasserman, Pricing American-style securities using simulation, J. Econom. Dynam. Control, 21 (1997), 1323-1352.  doi: 10.1016/S0165-1889(97)00029-8.  Google Scholar

[13]

P. J. Brockwell, Lévy-Driven CARMA processes, Ann. Inst. Statist. Math., 53 (2001), 113-124.  doi: 10.1023/A:1017972605872.  Google Scholar

[14]

J. Buffington and R. J. Elliott, American options with regime switching, Int. J. Theor. Appl. Finance, 5 (2002), 497-514.  doi: 10.1142/S0219024902001523.  Google Scholar

[15]

N. Cai and S. G. Kou, Option pricing under a mixed-exponential jump diffusion model, Management Sci., 57 (2011), 2067-2081.   Google Scholar

[16]

N. CaiY. Song and S. Kou, A general framework for pricing Asian options under Markov processes, Oper. Res., 63 (2015), 540-554.  doi: 10.1287/opre.2015.1385.  Google Scholar

[17]

G. Casella and R. L. Berger, Statistical Inference, Thomson Learning, CT, 2002. doi: 10.2307/2532634.  Google Scholar

[18]

K. Chourdakis and G. Dotsis, Maximum likelihood estimation of non-affine volatility processes, J. Empirical Finance, 18 (2011), 533-545.  doi: 10.1016/j.jempfin.2010.10.006.  Google Scholar

[19]

P. ChristoffersenS. Heston and K. Jacobs, The shape and term structure of the index option smirk: Why multifactor stochastic volatility models work so well, Management Sci., 55 (2009), 1914-1932.   Google Scholar

[20]

P. ChristoffersenK. Jacobs and K. Mimouni, Volatility dynamics for the S & P500: Evidence from realized volatility, daily returns and option prices, Rev. Financial Stud., 23 (2010), 3141-3189.   Google Scholar

[21]

I. J. Clark, Foreign Exchange Option Pricing: A Practitioner's Guide, Wiley, West Sussex, United Kingdom, 2011. doi: 10.1002/9781119208679.  Google Scholar

[22]

P. Collin-DufresneR. S. Goldstein and F. Yang, On the relative pricing of long-maturity index options and collateralized debt obligations, J. Finance, 67 (2012), 1983-2014.  doi: 10.1111/j.1540-6261.2012.01779.x.  Google Scholar

[23]

R. Cont and P. Tankov, Nonparametric calibration of jump-diffusion option pricing models, J. Computational Finance, 7 (2004), 1-49.   Google Scholar

[24]

Z. CuiJ. L. Kirkby and D. Nguyen, A general framework for discretely sampled realized variance derivatives in stochastic volatility models with jumps, European J. Oper. Res., 262 (2017), 381-400.  doi: 10.1016/j.ejor.2017.04.007.  Google Scholar

[25]

D. Du and D. Luo, The pricing of jump propagation: Evidence from spot and options markets, Management Sci., 65 (2019), 2360-2387.  doi: 10.1287/mnsc.2017.2885.  Google Scholar

[26]

J.-C. DuanI. Popova and P. Ritchken, Option pricing under regime switching, Quant. Finance, 2 (2002), 116-132.  doi: 10.1088/1469-7688/2/2/303.  Google Scholar

[27]

D. DuffieJ. Pan and K. Singleton, Transform analysis and asset pricing for affine jump-diffusions, Econometrica, 68 (2000), 1343-1376.  doi: 10.1111/1468-0262.00164.  Google Scholar

[28]

R. J. ElliottT. K. Siu and L. Chan, Option pricing for GARCH models with Markov switching, Int. J. Theo. Appl. Finance, 9 (2006), 825-841.  doi: 10.1142/S0219024906003846.  Google Scholar

[29]

B. Eraker, Do stock prices and volatility jump? Reconciling evidence from spot and option prices, J. Finance, 59 (2004), 1367-1403.  doi: 10.1111/j.1540-6261.2004.00666.x.  Google Scholar

[30]

B. ErakerM. Johannes and N. Polson, The impact of jumps in volatility and returns, J. Finance, 58 (2003), 1269-1300.  doi: 10.1111/1540-6261.00566.  Google Scholar

[31]

M. C. FuS. B. LapriseD. B. MadanY. Su and R. Wu, Pricing American options: A comparison of Monte Carlo simulation approaches, J. Comput. Finance, 4 (2001), 39-88.  doi: 10.21314/JCF.2001.066.  Google Scholar

[32]

M. C. FuB. LiG. Li and R. Wu, Option pricing for a jump-diffusion model with general discrete jump-size distributions, Management Sci., 63 (2017), 3961-3977.  doi: 10.1287/mnsc.2016.2522.  Google Scholar

[33]

C.-D. FuhK. W. R. HoI. Hu and R. H. Wang, Option pricing with Markov switching, J. Data Sci., 10 (2012), 483-509.   Google Scholar

[34]

S. Galluccio and Y. Lecam, Implied calibration and moments asymptotics in stochastic volatility jump diffusion models, Available at SSRN: https://ssrn.com/abstract=831784, 2008. doi: 10.2139/ssrn.831784.  Google Scholar

[35]

M. Grasselli, The 4/2 stochastic volatility model: A unified approach for the Heston and the 3/2 model, Math. Finance, 27 (2017), 1013-1034.  doi: 10.1111/mafi.12124.  Google Scholar

[36]

X. Guo, Information and option pricings, Quant. Finance, 1 (2001), 38-44.  doi: 10.1080/713665550.  Google Scholar

[37]

S. L. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Rev. Financial Stud., 6 (1993), 327-343.  doi: 10.1093/rfs/6.2.327.  Google Scholar

[38]

C. Homescu, Local stochastic volatility models: Calibration and pricing, Available at SSRN: https://ssrn.com/abstract=2448098, 2014. doi: 10.2139/ssrn.2448098.  Google Scholar

[39]

J. Hull and A. White, The pricing of options on assets with stochastic volatilities, J. Finance, 42 (1987), 281-300.  doi: 10.1111/j.1540-6261.1987.tb02568.x.  Google Scholar

[40]

M. S. JohannesN. G. Polson and J. R. Stroud, Optimal filtering of jump diffusions: Extracting latent states from asset prices, Rev. Financial Stud., 22 (2009), 2759-2799.  doi: 10.1093/rfs/hhn110.  Google Scholar

[41] F. C. Klebaner, Introduction to Stochastic Calculus with Applications, second edition, Imperial College Press, 2005.  doi: 10.1142/p386.  Google Scholar
[42]

C. KlüppelbergA. Lindner and R. Maller, A continuous-time GARCH process driven by a Lévy process: Stationarity and second-order behaviour, J. Appl. Probab., 41 (2004), 601-622.  doi: 10.1239/jap/1091543413.  Google Scholar

[43]

S. G. Kou, A jump-diffusion model for option pricing, Management Sci., 48 (2002), 1086-1101.   Google Scholar

[44]

S. G. Kou and H. Wang, Option pricing under a double exponential jump diffusion model, Management Sci., 50 (2004), 1178-1192.   Google Scholar

[45]

S. KouC. Yu and H. Zhong, Jumps in equity index returns before and during the recent financial crisis: A Bayesian analysis, Management Sci., 63 (2017), 988-1010.  doi: 10.1287/mnsc.2015.2359.  Google Scholar

[46]

S. B. LapriseM. C. FuS. I. MarcusA. E. B. Lim and H. Zhang, Pricing American-style derivatives with European call options, Management Sci., 52 (2006), 95-110.  doi: 10.1287/mnsc.1050.0447.  Google Scholar

[47]

H. LiM. T. Wells and C. L. Yu, A Bayesian analysis of return dynamics with Lévy jumps, Rev. Financial Stud., 21 (2008), 2345-2378.  doi: 10.1093/rfs/hhl036.  Google Scholar

[48]

R. H. Liu, Q. Zhang and G. Yin, Option pricing in a regime-switching model using the fast Fourier transform, J. Appl. Math. Sto. Anal., 2006 (2006), Art. ID 18109, 22 pp. doi: 10.1155/JAMSA/2006/18109.  Google Scholar

[49]

C. C. Lo and K. Skindilias, An improved Markov chain approximation methodology: Derivatives pricing and model calibration, Int. J. Theo. Appl. Finance, 17 (2014), 1450047, 22 pp. doi: 10.1142/S0219024914500472.  Google Scholar

[50]

F. A. Longstaff and E. S. Schwartz, Valuing American options by simulation: A simple least-squares approach, Rev. Financial Stud., 14 (2001), 113-147.  doi: 10.1093/rfs/14.1.113.  Google Scholar

[51]

R. C. Merton, Option pricing when underlying stock returns are discontinuous, J. Financial Econom., 3 (1976), 125-144.  doi: 10.1016/0304-405X(76)90022-2.  Google Scholar

[52]

J. P. Morgan/Reuters, RiskMetrics$^{TM}$-Technical Document, Fourth Edition, Morgan Guaranty Trust Company of New York, New York, 1996. Google Scholar

[53]

V. Naik, Option valuation and hedging strategies with jumps in the volatility of asset returns, J. Finance, 48 (1993), 1969-1984.  doi: 10.1111/j.1540-6261.1993.tb05137.x.  Google Scholar

[54]

J. Pan, The jump-risk premia implicit in options: evidence from an integrated time-series study, J. Financial Econom., 63 (2002), 3-50.  doi: 10.1016/S0304-405X(01)00088-5.  Google Scholar

[55]

M. Perlin, MS_Regress-the MATLAB package for Markov regime switching models, Available at SSRN: http://ssrn.com/abstract=1714016, 2015. Google Scholar

[56]

A. Rossi and G. M. Gallo, Volatility estimation via hidden Markov models, J. Empirical Finance, 13 (2006), 203-230.  doi: 10.1016/j.jempfin.2005.09.003.  Google Scholar

[57]

T. RydénT. Teräsvirta and S. Åsbrink, Stylized facts of daily return series and the hidden Markov model, J. Appl. Econ., 13 (1998), 217-244.   Google Scholar

[58]

C. Tan, Market Practice in Financial Modelling, World Scientific, Singapore, 2012. doi: 10.1142/8257.  Google Scholar

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Figure 1.  Sample Path $ \omega $: $ \sigma_{0} = u_{2} $, $ \sigma_{1} = u_{2} $, $ \sigma_{2} = u_{m-1} , \ldots, \sigma_{L-2} = u_{2} $, $ \sigma_{L-1} = u_{2} $, $ \sigma_{L} = u_{3} $
Figure 2.  $ \Omega_{l} $, Subsample Path Space of $ \omega_{l} $ Up to Step $ L = 3 $
Figure 3.  $ \Psi_{l} $, Value Space of $ [|\omega_{l}|, l, \sigma_{l}] $ Up to Step $ L = 3 $
Figure 4.  Recursion Recombination Relationship From $ \Psi_{2} $ to $ \Psi_{3} $
Figure 5.  Computation Time as a Function of the Number of Time Steps $ L $ (Log Scales)
Figure 6.  Comparison for Option Price under Market and Model
Table 1.  RR Algorithm: Obtaining the Value Space $ \Psi $ and the Probability Distribution $ \{p_{V}(\cdot)\} $
Input: State set $ \{u_{1},\cdots,u_{m}\} $, transition probabilities $ p_{ij} $, $ i,j\in\{1,\cdots,m\}, $ initial state $ \sigma_{0} $, number of time steps $ L $.
Initialization: Set $\Psi_{1}=\{[\sigma_{0}^{2}, 1, \sigma_{1}]: \sigma_{1} \in \{u_{1}, \cdots, u_{m}\}\}$,
$p([\sigma_{0}^{2}, 1, \sigma_{1}]) =p_{\sigma_{0}\sigma_{1}}$.
Recursion: For $l=1$ to $L-1$
Let $\Psi_{l+1}=\{[z, l+1, \sigma_{l+1}]: z=\frac{x\cdot l+\sigma_{l}^{2}}{l+1}, [x, l, \sigma_{l}]\in \Psi_{l}, \sigma_{l+1} \in \{u_{1}, \cdots, u_{m}\}\}$,
$p([z, l+1, \sigma_{l+1}])=\sum\limits_{[x, l, \sigma_{l}]\in \Psi_{l}: x=(z\cdot (l+1)-\sigma_{l}^{2})/{l}}p([x, l, \sigma_{l}])p_{\sigma_{l}\sigma_{l+1}}$.
Output: $\Psi=\{x: [x, L, \sigma_{L}] \in \Psi_{L}\}$,
$p_{V}(v)=\sum\limits_{[x, L, \sigma_{L}]\in \Psi_{L}:x=v}p(x, L, \sigma_{L})$.
Input: State set $ \{u_{1},\cdots,u_{m}\} $, transition probabilities $ p_{ij} $, $ i,j\in\{1,\cdots,m\}, $ initial state $ \sigma_{0} $, number of time steps $ L $.
Initialization: Set $\Psi_{1}=\{[\sigma_{0}^{2}, 1, \sigma_{1}]: \sigma_{1} \in \{u_{1}, \cdots, u_{m}\}\}$,
$p([\sigma_{0}^{2}, 1, \sigma_{1}]) =p_{\sigma_{0}\sigma_{1}}$.
Recursion: For $l=1$ to $L-1$
Let $\Psi_{l+1}=\{[z, l+1, \sigma_{l+1}]: z=\frac{x\cdot l+\sigma_{l}^{2}}{l+1}, [x, l, \sigma_{l}]\in \Psi_{l}, \sigma_{l+1} \in \{u_{1}, \cdots, u_{m}\}\}$,
$p([z, l+1, \sigma_{l+1}])=\sum\limits_{[x, l, \sigma_{l}]\in \Psi_{l}: x=(z\cdot (l+1)-\sigma_{l}^{2})/{l}}p([x, l, \sigma_{l}])p_{\sigma_{l}\sigma_{l+1}}$.
Output: $\Psi=\{x: [x, L, \sigma_{L}] \in \Psi_{L}\}$,
$p_{V}(v)=\sum\limits_{[x, L, \sigma_{L}]\in \Psi_{L}:x=v}p(x, L, \sigma_{L})$.
Table 2.  Parameter Values for MS-SVCJ Model
ParameterValueParameterValueParameterValue
Maturity$T=0.25$Jump variance$\varepsilon^{2}=0.005$Initial state$\sigma_{0}^{2}=0.04$
Strike price$K=55$Max # jumps$N_{max}=10$State space$\sigma_{k}^{2} \in \{0.02, 0.04, 0.06, 0.08\} $
Risk-free rate$r=0.05$Time step$\tau=0.25/30$Transition probability matrix$ P=\left[ \begin{array}{cccc} 0.70 & 0.15 & 0.10 & 0.05 \\ 0.03 & 0.90 & 0.06 & 0.01 \\ 0.05 & 0.05 & 0.85 & 0.05 \\ 0.03 & 0.07 & 0.10 & 0.80\\ \end{array} \right] $
Asset price$S_{0}=50$Duration$\Delta=0.02$
Jump intensity$\lambda=3$Attenuating factor$\beta=250$
Jump mean$\mu=-0.025$Proportional coefficient$b=2$
max # jumps truncated at $N_{max}$ such that $P(N>N_{max}) < \epsilon$; for $\epsilon=5.5\times10^{-5}$ with $\lambda$ and $T$ values, $N_{max}=10$.
ParameterValueParameterValueParameterValue
Maturity$T=0.25$Jump variance$\varepsilon^{2}=0.005$Initial state$\sigma_{0}^{2}=0.04$
Strike price$K=55$Max # jumps$N_{max}=10$State space$\sigma_{k}^{2} \in \{0.02, 0.04, 0.06, 0.08\} $
Risk-free rate$r=0.05$Time step$\tau=0.25/30$Transition probability matrix$ P=\left[ \begin{array}{cccc} 0.70 & 0.15 & 0.10 & 0.05 \\ 0.03 & 0.90 & 0.06 & 0.01 \\ 0.05 & 0.05 & 0.85 & 0.05 \\ 0.03 & 0.07 & 0.10 & 0.80\\ \end{array} \right] $
Asset price$S_{0}=50$Duration$\Delta=0.02$
Jump intensity$\lambda=3$Attenuating factor$\beta=250$
Jump mean$\mu=-0.025$Proportional coefficient$b=2$
max # jumps truncated at $N_{max}$ such that $P(N>N_{max}) < \epsilon$; for $\epsilon=5.5\times10^{-5}$ with $\lambda$ and $T$ values, $N_{max}=10$.
Table 3.  Option Valuation Comparison
MS-SVCJ MC Simulation
$ N $=600 750 900 1200 1500
Option Price 0.9696 0.9680 0.9683 0.9684 0.9687 0.9689
(Std Err) (.0063) (.0044) (.0050) (.0066) (.0054)
Computation Time (seconds) 30 16575 20213 25014 28411 41034
MS-SVCJ MC Simulation
$ N $=600 750 900 1200 1500
Option Price 0.9696 0.9680 0.9683 0.9684 0.9687 0.9689
(Std Err) (.0063) (.0044) (.0050) (.0066) (.0054)
Computation Time (seconds) 30 16575 20213 25014 28411 41034
Table 4.  Bermudan Call Option Pricing Under MS-SV Model
Option Price Computation Time
Algorithm $ n $ $ S_{0} $=60 90 100 110 140 (seconds)
Tangent 50 1.294 9.846 14.867 20.848 43.204 0.41
100 1.302 9.861 14.883 20.861 43.212 0.98
200 1.305 9.864 14.886 20.864 43.213 2.97
Secant 200 1.307 9.868 14.890 20.867 43.215 1.64
100 1.311 9.875 14.897 20.873 43.219 0.55
50 1.328 9.904 14.925 20.899 43.234 0.23
LSM 1.306 9.866 14.888 20.860 43.210 415
(Std Err) (.016) (.019) (.043) (.020) (.074)
Option Price Computation Time
Algorithm $ n $ $ S_{0} $=60 90 100 110 140 (seconds)
Tangent 50 1.294 9.846 14.867 20.848 43.204 0.41
100 1.302 9.861 14.883 20.861 43.212 0.98
200 1.305 9.864 14.886 20.864 43.213 2.97
Secant 200 1.307 9.868 14.890 20.867 43.215 1.64
100 1.311 9.875 14.897 20.873 43.219 0.55
50 1.328 9.904 14.925 20.899 43.234 0.23
LSM 1.306 9.866 14.888 20.860 43.210 415
(Std Err) (.016) (.019) (.043) (.020) (.074)
Table 5.  Bermudan Call Option Pricing Under MS-SVCJ Model
Option Price Computation Time
Algorithm $ n $ $ S_{0} $=60 90 100 110 140 (seconds)
Tangent 20 1.970 11.624 16.815 22.845 44.864 68
50 2.040 11.723 16.911 22.932 44.924 390
100 2.050 11.737 16.924 22.945 44.933 1504
Secant 100 2.060 11.752 16.938 22.957 44.941 895
50 2.080 11.780 16.965 22.982 44.958 230
20 2.223 11.977 17.154 23.157 45.078 39
LSM 2.066 11.773 16.957 22.972 44.903 23934
(Std Err) (.028) (.082) (.082) (.079) (.095)
Option Price Computation Time
Algorithm $ n $ $ S_{0} $=60 90 100 110 140 (seconds)
Tangent 20 1.970 11.624 16.815 22.845 44.864 68
50 2.040 11.723 16.911 22.932 44.924 390
100 2.050 11.737 16.924 22.945 44.933 1504
Secant 100 2.060 11.752 16.938 22.957 44.941 895
50 2.080 11.780 16.965 22.982 44.958 230
20 2.223 11.977 17.154 23.157 45.078 39
LSM 2.066 11.773 16.957 22.972 44.903 23934
(Std Err) (.028) (.082) (.082) (.079) (.095)
Table 6.  Computation Time for CE (seconds, '*' indicates out of memory)
15 16 17 18 19 20 25 30
2 0.007 0.01 0.04 0.04 0.07 0.13 4.5 151
3 1.3 4.1 12 38 119 365 * *
4 92 * * * * * * *
5 * * * * * * * *
6 * * * * * * * *
15 16 17 18 19 20 25 30
2 0.007 0.01 0.04 0.04 0.07 0.13 4.5 151
3 1.3 4.1 12 38 119 365 * *
4 92 * * * * * * *
5 * * * * * * * *
6 * * * * * * * *
Table 7.  Computation Time for RR Algorithm (seconds)
20 25 30 35 40 45 50
2 0.005 0.006 0.007 0.008 0.011 0.013 0.014
3 0.009 0.013 0.019 0.024 0.033 0.040 0.050
4 0.05 0.11 0.21 0.39 0.66 1.05 1.58
5 0.27 0.8 1.9 4.0 7.3 12 19
6 1.2 4.2 10 22 38 60 87
20 25 30 35 40 45 50
2 0.005 0.006 0.007 0.008 0.011 0.013 0.014
3 0.009 0.013 0.019 0.024 0.033 0.040 0.050
4 0.05 0.11 0.21 0.39 0.66 1.05 1.58
5 0.27 0.8 1.9 4.0 7.3 12 19
6 1.2 4.2 10 22 38 60 87
Table 8.  Call Option Prices
Market Model
Strike Bid Ask Mid-Price Price Bias
125 15.05 16.85 15.95 15.83 $ -0.75\% $
130 11.60 11.80 11.70 11.44 $ -2.22\% $
135 7.60 7.70 7.65 7.52 $ -1.70\% $
140 4.30 4.45 4.375 4.32 $ -1.26\% $
145 2.07 2.16 2.115 2.10 $ -0.71\% $
150 0.80 0.84 0.82 0.84 $ 2.44\% $
155 0.28 0.29 0.285 0.29 $ 1.75\% $
160 0.08 0.10 0.09 0.09 $ 0.00\% $
Market Model
Strike Bid Ask Mid-Price Price Bias
125 15.05 16.85 15.95 15.83 $ -0.75\% $
130 11.60 11.80 11.70 11.44 $ -2.22\% $
135 7.60 7.70 7.65 7.52 $ -1.70\% $
140 4.30 4.45 4.375 4.32 $ -1.26\% $
145 2.07 2.16 2.115 2.10 $ -0.71\% $
150 0.80 0.84 0.82 0.84 $ 2.44\% $
155 0.28 0.29 0.285 0.29 $ 1.75\% $
160 0.08 0.10 0.09 0.09 $ 0.00\% $
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