Hostname: page-component-8448b6f56d-wq2xx Total loading time: 0 Render date: 2024-04-23T07:56:32.968Z Has data issue: false hasContentIssue false
  • English
  • Français

PRICING TIMER OPTIONS: SECOND-ORDER MULTISCALE STOCHASTIC VOLATILITY ASYMPTOTICS

Part of: Mathematical finance Equations of mathematical physics and other areas of application Mathematical economics

Published online by Cambridge University Press:  23 August 2021

XUHUI WANG Open the ORCID record for XUHUI WANG [Opens in a new window] ,
SHENG-JHIH WU Open the ORCID record for SHENG-JHIH WU [Opens in a new window]  and
XINGYE YUE Open the ORCID record for XINGYE YUE [Opens in a new window]
XUHUI WANG*
Affiliation:
Center for Advanced Statistics and Econometrics Research, School of Mathematical Sciences, Soochow University, 1 Shi-Zi Street, Suzhou, 215000, China
SHENG-JHIH WU
Affiliation:
Independent Researcher, Taiwan; e-mail: shemsjw@gmail.com.
XINGYE YUE
Affiliation:
Center for Financial Engineering, School of Mathematical Sciences, Soochow University, 1 Shi-Zi Street, Suzhou, 215000, China; e-mail: xyyue@suda.edu.cn.
*
e-mail: xhwang@alu.suda.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the pricing of timer options in a class of stochastic volatility models, where the volatility is driven by two diffusions—one fast mean-reverting and the other slowly varying. Employing singular and regular perturbation techniques, full second-order asymptotics of the option price are established. In addition, we investigate an implied volatility in terms of effective maturity for the timer options, and derive its second-order expansion based on our pricing asymptotics. A numerical experiment shows that the price approximation formula has a high level of accuracy, and the implied volatility in terms of its effective maturity is illustrated.

Keywords

timer optionstochastic volatilityimplied volatilitymultiscale asymptoticssingular perturbation

MSC classification

Primary: 91B70: Stochastic models
Secondary: 91G20: Derivative securities 35Q91: PDEs in connection with game theory, economics, social and behavioral sciences
Type
Research Article
Information
The ANZIAM Journal , Volume 63 , Issue 2 , April 2021 , pp. 249 - 267
Check for updates
Copyright
© Australian Mathematical Society 2021

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Carr, P. and Madan, D., “Towards a theory of volatility trading”, in: Handbooks in mathematical finance: Option pricing, interest rates and risk management (Eds Jouini, E., Cvitanic, J. and Musiela, M.), (Cambridge University Press, Cambridge, 2001) 458–476; doi:10.1017/CBO9780511569708.013.CrossRefGoogle Scholar
Chen, W.-T. and Zhu, S.-P., “Pricing perpetual American puts under multi-scale stochastic volatility”, Asymptot. Anal. 80 (2012) 133–148; doi:10.3233/ASY-2012-1110.CrossRefGoogle Scholar
Chernov, M., Gallant, A. R., Ghysels, E. and Tauchen, G., “Alternative models for stock price dynamics”, J. Econometrics 116 (2003) 225–257; doi:10.1016/S0304-4076(03)00108-8.CrossRefGoogle Scholar
Cui, Z., Kirkby, J. L., Lian, G. and Nguyen, D., “Integral representation of probability density of stochastic volatility models and timer options”, Int. J. Theor. Appl. Finance 20 (2017) 1750055; doi:10.1142/S0219024917500558.CrossRefGoogle Scholar
Durrett, R., Brownian motion and Martingales in analysis, Wadsworth Math. Ser. (Wadsworth International Group, Belmont, CA, 1984); ISBN-10 0534030653.Google Scholar
Fouque, J.-P. and Han, C.-H., “Asian options under multiscale stochastic volatility”, in: Mathematics of finance, Volume 351 of Contemp. Math. (Eds Yin, G. and Zhang, Q.), (American Mathematical Society, Providence, RI, 2004) 125–138; doi:10.1090/conm/351/06398.CrossRefGoogle Scholar
Fouque, J.-P. and Han, C.-H., “Evaluation of compound options using perturbation approximation”, J. Comput. Finance 9 (2005) 41–61; doi:10.21314/JCF.2005.125.CrossRefGoogle Scholar
Fouque, J.-P., Lorig, M. and Sircar, R., “Second order multiscale stochastic volatility asymptotics: stochastic terminal layer analysis and calibration”, Finance Stoch. 20 (2016) 543–588; doi:10.1007/s00780-016-0298-y.CrossRefGoogle Scholar
Fouque, J.-P., Papanicolaou, G., Sircar, R. and Solna, K., “Singular perturbations in option pricing”, SIAM J. Appl. Math. 63 (2003) 1648–1665; doi:10. 1137/S0036139902401550.CrossRefGoogle Scholar
Fouque, J.-P., Papanicolaou, G., Sircar, R. and Solna, K., “Multiscale stochastic volatility asymptotics”, Multiscale Model. Simul. 2 (2003) 22–42; doi:10.1137/030600291.CrossRefGoogle Scholar
Fouque, J.-P., Papanicolaou, G., Sircar, R. and Solna, K., “Short time-scale in S&P500 volatility”, J. Comput. Finance 6 (2003) 1–24; doi:10.21314/JCF.2003.103.CrossRefGoogle Scholar
Fouque, J.-P., Papanicolaou, G., Sircar, R. and Solna, K., Multiscale stochastic volatility for equity, interest rate, and credit derivatives (Cambridge University Press, Cambridge, 2011); doi:10.1017/CBO9781139020534.CrossRefGoogle Scholar
Heston, S. L., “A closed-form solution for options with stochastic volatility with applications to bond and currency options”, Rev. Finance Stud. 6 (1993) 327–343; doi:10.1093/rfs/6.2.327.CrossRefGoogle Scholar
Kuo, I.-D. and Wang, K.-L., “Implied deterministic volatility functions: an empirical test for Euribor options”, J. Futures Mark. 29 (2009) 319–347; doi:10.1002/fut.20363.CrossRefGoogle Scholar
Li, C., “Managing volatility risk: innovation of financial derivatives, stochastic models and their analytical implementation”, Ph. D. Thesis, Columbia University, 2010, available at http://www.math.columbia.edu/~thaddeus/theses/2010/li.pdf.Google Scholar
Liang, L., Lemmens, D. and Tempere, J., “Path integral approach to the pricing of timer options with the Duru–Kleinert time transformation”, Phys. Rev. E 83 (2011) 056112; doi:10.1103/PhysRevE.83.056112.CrossRefGoogle ScholarPubMed
Saunders, D., “Pricing timer options under fast mean-reverting stochastic volatility”, Can. Appl. Math. Q. 17 (2009) 737–753, available at http://www.math.ualberta.ca/ami/CAMQ/pdf_files/ vol_17/17_4/17_4f.pdf.Google Scholar
Sawyer, N., “SG CIB launches timer options” (CME Group, 2007), available at https://www.risk.net/derivatives/structured-products/1506473/sg-cib-launches-timer-options.Google Scholar
Yin, G., “Asymptotic expansions of option price under regime-switching diffusions with a fast-varying switching process”, Asymptot. Anal. 65 (2009) 203–222; doi:10.3233/ASY-2009-0953.CrossRefGoogle Scholar
Zeto, S. and Man, Y., “Pricing and hedging American fixed-income derivatives with implied volatility structures in the two-factor Heath–Jarrow–Morton model”, J. Futures Mark. 22 (2002) 839–875; doi:10.1002/fut.10031.CrossRefGoogle Scholar
Zhang, J., Lu, X. and Han, Y., “Pricing perpetual timer option under the stochastic volatility model of Hull–White”, ANZIAM J. 58 (2017) 406–416; doi:10.1017/S1446181117000177.Google Scholar
Zheng, W. and Zeng, P., “Pricing timer options and variance derivatives with closed-form partial transform under the 3/2 mode”, Appl. Math. Finance 23 (2016) 344–373; doi:10.1080/1350486X.2017.1285242.CrossRefGoogle Scholar