March  2022, 1(1): 137-160. doi: 10.3934/fmf.2021005

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 Published  March 2022 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, 2022, 1 (1) : 137-160. doi: 10.3934/fmf.2021005
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. 

[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.

[3]

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

[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.

[5]

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

[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.

[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.

[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.

[9]

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

[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.

[11]

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

[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.

[13]

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

[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.

[15]

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

[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.

[17]

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

[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.

[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. 

[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. 

[21]

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

[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.

[23]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[33]

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

[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.

[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.

[36]

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

[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.

[38]

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

[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.

[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.

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

[43]

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

[44]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[52]

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

[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.

[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.

[55]

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

[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.

[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. 

[58]

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

[59]

A. Timmermann, Moments of Markov switching models, J. Econometrics, 96 (2000), 75-111.  doi: 10.1016/S0304-4076(99)00051-2.

[60]

V. Todorov, Econometric analysis of jump-driven stochastic volatility models, J. Econometrics, 160 (2011), 12-21.  doi: 10.1016/j.jeconom.2010.03.009.

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. 

[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.

[3]

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

[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.

[5]

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

[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.

[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.

[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.

[9]

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

[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.

[11]

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

[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.

[13]

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

[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.

[15]

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

[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.

[17]

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

[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.

[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. 

[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. 

[21]

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

[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.

[23]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[33]

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

[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.

[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.

[36]

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

[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.

[38]

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

[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.

[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.

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

[43]

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

[44]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[52]

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

[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.

[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.

[55]

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

[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.

[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. 

[58]

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

[59]

A. Timmermann, Moments of Markov switching models, J. Econometrics, 96 (2000), 75-111.  doi: 10.1016/S0304-4076(99)00051-2.

[60]

V. Todorov, Econometric analysis of jump-driven stochastic volatility models, J. Econometrics, 160 (2011), 12-21.  doi: 10.1016/j.jeconom.2010.03.009.

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\% $
[1]

Lorella Fatone, Francesca Mariani, Maria Cristina Recchioni, Francesco Zirilli. Pricing realized variance options using integrated stochastic variance options in the Heston stochastic volatility model. Conference Publications, 2007, 2007 (Special) : 354-363. doi: 10.3934/proc.2007.2007.354

[2]

Zhuo Jin, Linyi Qian. Lookback option pricing for regime-switching jump diffusion models. Mathematical Control and Related Fields, 2015, 5 (2) : 237-258. doi: 10.3934/mcrf.2015.5.237

[3]

Kun Fan, Yang Shen, Tak Kuen Siu, Rongming Wang. On a Markov chain approximation method for option pricing with regime switching. Journal of Industrial and Management Optimization, 2016, 12 (2) : 529-541. doi: 10.3934/jimo.2016.12.529

[4]

Wan-Hua He, Chufang Wu, Jia-Wen Gu, Wai-Ki Ching, Chi-Wing Wong. Pricing vulnerable options under a jump-diffusion model with fast mean-reverting stochastic volatility. Journal of Industrial and Management Optimization, 2022, 18 (3) : 2077-2094. doi: 10.3934/jimo.2021057

[5]

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 and Management Optimization, 2020, 16 (6) : 2603-2623. doi: 10.3934/jimo.2019072

[6]

Kai Zhang, Xiaoqi Yang, Kok Lay Teo. A power penalty approach to american option pricing with jump diffusion processes. Journal of Industrial and Management Optimization, 2008, 4 (4) : 783-799. doi: 10.3934/jimo.2008.4.783

[7]

Jia Yue, Nan-Jing Huang. Neutral and indifference pricing with stochastic correlation and volatility. Journal of Industrial and Management Optimization, 2018, 14 (1) : 199-229. doi: 10.3934/jimo.2017043

[8]

Yu Xing, Wei Wang, Xiaonan Su, Huawei Niu. Equilibrium valuation of currency options with stochastic volatility and systemic co-jumps. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022022

[9]

Hao Chang, Jiaao Li, Hui Zhao. Robust optimal strategies of DC pension plans with stochastic volatility and stochastic income under mean-variance criteria. Journal of Industrial and Management Optimization, 2022, 18 (2) : 1393-1423. doi: 10.3934/jimo.2021025

[10]

Shuang Li, Chuong Luong, Francisca Angkola, Yonghong Wu. Optimal asset portfolio with stochastic volatility under the mean-variance utility with state-dependent risk aversion. Journal of Industrial and Management Optimization, 2016, 12 (4) : 1521-1533. doi: 10.3934/jimo.2016.12.1521

[11]

Qing-Qing Yang, Wai-Ki Ching, Wanhua He, Tak-Kuen Siu. Pricing vulnerable options under a Markov-modulated jump-diffusion model with fire sales. Journal of Industrial and Management Optimization, 2019, 15 (1) : 293-318. doi: 10.3934/jimo.2018044

[12]

Wei Wang, Linyi Qian, Xiaonan Su. Pricing and hedging catastrophe equity put options under a Markov-modulated jump diffusion model. Journal of Industrial and Management Optimization, 2015, 11 (2) : 493-514. doi: 10.3934/jimo.2015.11.493

[13]

Lin Xu, Rongming Wang, Dingjun Yao. Optimal stochastic investment games under Markov regime switching market. Journal of Industrial and Management Optimization, 2014, 10 (3) : 795-815. doi: 10.3934/jimo.2014.10.795

[14]

María Teresa V. Martínez-Palacios, Adrián Hernández-Del-Valle, Ambrosio Ortiz-Ramírez. On the pricing of Asian options with geometric average of American type with stochastic interest rate: A stochastic optimal control approach. Journal of Dynamics and Games, 2019, 6 (1) : 53-64. doi: 10.3934/jdg.2019004

[15]

Robert J. Elliott, Tak Kuen Siu. Stochastic volatility with regime switching and uncertain noise: Filtering with sub-linear expectations. Discrete and Continuous Dynamical Systems - B, 2017, 22 (1) : 59-81. doi: 10.3934/dcdsb.2017003

[16]

Xingchun Wang, Yongjin Wang. Variance-optimal hedging for target volatility options. Journal of Industrial and Management Optimization, 2014, 10 (1) : 207-218. doi: 10.3934/jimo.2014.10.207

[17]

Tak Kuen Siu, Yang Shen. Risk-minimizing pricing and Esscher transform in a general non-Markovian regime-switching jump-diffusion model. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 2595-2626. doi: 10.3934/dcdsb.2017100

[18]

Ishak Alia, Mohamed Sofiane Alia. Open-loop equilibrium strategy for mean-variance Portfolio selection with investment constraints in a non-Markovian regime-switching jump-diffusion model. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022048

[19]

Mourad Bellassoued, Raymond Brummelhuis, Michel Cristofol, Éric Soccorsi. Stable reconstruction of the volatility in a regime-switching local-volatility model. Mathematical Control and Related Fields, 2020, 10 (1) : 189-215. doi: 10.3934/mcrf.2019036

[20]

Editorial Office. Retraction: Xiao-Qian Jiang and Lun-Chuan Zhang, A pricing option approach based on backward stochastic differential equation theory. Discrete and Continuous Dynamical Systems - S, 2019, 12 (4&5) : 969-969. doi: 10.3934/dcdss.2019065

 Impact Factor: 

Article outline

Figures and Tables

[Back to Top]