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LAPLACE BOUNDS APPROXIMATION FOR AMERICAN OPTIONS

Published online by Cambridge University Press:  09 October 2020

Jingtang Ma ,
Zhenyu Cui Open the ORCID record for Zhenyu Cui [Opens in a new window]  and
Wenyuan Li
Jingtang Ma
Affiliation:
School of Economic Mathematics and Fintech Innovation Center, Southwestern University of Finance and Economics, Chengdu611130, China E-mail: [email protected]
Zhenyu Cui
Affiliation:
School of Business, Stevens Institute of Technology, Castle Point on Hudson, Hoboken, NJ07030, USA E-mail: [email protected]
Wenyuan Li
Affiliation:
School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu611130, China E-mail: [email protected]
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Abstract

In this paper, we develop the lower–upper-bound approximation in the space of Laplace transforms for pricing American options. We construct tight lower and upper bounds for the price of a finite-maturity American option when the underlying stock is modeled by a large class of stochastic processes, e.g. a time-homogeneous diffusion process and a jump diffusion process. The novelty of the method is to first take the Laplace transform of the price of the corresponding “capped (barrier) option” with respect to the time to maturity, and then carry out optimization procedures in the Laplace space. Finally, we numerically invert the Laplace transforms to obtain the lower bound of the price of the American option and further utilize the early exercise premium representation in the Laplace space to obtain the upper bound. Numerical examples are conducted to compare the method with a variety of existing methods in the literature as benchmark to demonstrate the accuracy and efficiency.

Keywords

American option pricingearly exercise boundaryjump diffusionsLaplace transformoption bounds
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 36 , Issue 2 , April 2022 , pp. 514 - 547
Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press

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References

Amin, K. (1993). Jump diffusion option valuation in dicsrete time. Journal of Finance 48: 1833–1863.CrossRefGoogle Scholar
Andersen, L. & Broadie, M. (2004). Primal-dual simulation algorithm for pricing multidimensional American options. Management Science 50: 1222–1234.CrossRefGoogle Scholar
Barone-Adesi, G. & Whaley, R.E. (1987). Efficient analytic approximations of American option values. Journal of Finance 42: 301–320.CrossRefGoogle Scholar
Barraclough, K. & Whaley, R.E. (2012). Early exercise of put options on stocks. Journal of Finance 67: 1423–1456.CrossRefGoogle Scholar
Berros, J. (2010). American option pricing in a jump-diffusion model. Saarbrücken, Germany: LAP Lambert Academic Publishing.Google Scholar
Boyle, P. (1998). A lattice framework for option pricing with two state variables. Journal of Financial and Quantitative Analysis 23: 1–12.CrossRefGoogle Scholar
Brennan, M. & Schwartz, E. (1977). The valuation of American put options. Journal of Finance 32: 449–462.CrossRefGoogle Scholar
Broadie, M. & Cao, M. (2008). Improved lower and upper bound algorithms for pricing American options by simulation. Quantitative Finance 8: 845–861.CrossRefGoogle Scholar
Broadie, M. & Detemple, J. (1996). American option valuation: New bounds, approximations, and a comparison of existing methods. Review of Financial Studies 9: 1211–1250.CrossRefGoogle Scholar
Broadie, M. & Detemple, J. (2004). Option pricing: Valuation models and applications. Management Science 50: 1145–1177.CrossRefGoogle Scholar
Cai, N. & Sun, L. (2014). Valuation of stock loans with jump risk. Journal of Economic Dynamics and Control 40: 213–241.CrossRefGoogle Scholar
Carr, P. (1998). Randomization and the American put. Review of Financial Studies 11: 597–626.CrossRefGoogle Scholar
Carr, P., Jarrow, R., & Myneni, R. (1992). Alternative characterizations of American put options. Mathematical Finance 2: 87–106.CrossRefGoogle Scholar
Chiarella, C. & Ziogas, A. (2005). Evaluation of American strangles. Journal of Economic Dynamics and Control 29: 31–62.CrossRefGoogle Scholar
Chiarella, C. & Ziogas, A. (2009). American call options under jump diffusion processes: A Fourier transform approach. Applied Mathematical Finance 16: 37–79.CrossRefGoogle Scholar
Chiarella, C., El-Hassan, N., & Kucera, A. (1999). Evaluation of American option prices in a path integral framework using Fourier-Hermite series expansions. Journal of Economic Dynamics and Control 23: 1387–1424.CrossRefGoogle Scholar
Chiarella, C., Kang, B., Meyer, G.H., & Ziogas, A. (2009). The evaluation of American option prices under stochastic volatility and jump-diffusion dynamics using the method of lines. International Journal of Theoretical and Applied Finance 12: 393–425.CrossRefGoogle Scholar
Chung, S.L. & Shih, P.T. (2009). Static hedging and pricing American options. Journal of Banking and Finance 33: 2140–2149.CrossRefGoogle Scholar
Chung, S.L., Hung, M.W., & Wang, J.W. (2010). Tight bounds on American option prices. Journal of Banking and Finance 34: 77–89.CrossRefGoogle Scholar
Chung, S.L., Shih, P.T., & Tsai, W.C. (2013). Static hedging and pricing American knock-in put options. Journal of Banking and Finance 37: 191–205.CrossRefGoogle Scholar
Cox, J., Ross, S., & Rubinstein, M. (1979). Option pricing: A simplified approach. Journal of Financial Economics 7: 229–263.CrossRefGoogle Scholar
Davydov, D. & Linetsky, V. (2001). Pricing and hedging path-dependent options under the CEV process. Management Science 47: 949–965.CrossRefGoogle Scholar
Detemple, J. (2005). American-style derivatives: Valuation and computation. London, UK: Chapman & Hall/CRC Financial Mathematics Series.CrossRefGoogle Scholar
Detemple, J. (2014). Optimal exercise for derivative securities. Annual Review of Financial Economics 6: 459–487.CrossRefGoogle Scholar
Detemple, J. & Tian, W. (2002). The valuation of American options for a class of diffusion processes. Management Science 48: 917–937.CrossRefGoogle Scholar
Fang, F. & Oosterlee, C.W. (2009). Pricing early-exercise and discrete barrier options by Fourier-cosine series expansions. Numerische Mathematik 114: 27–62.CrossRefGoogle Scholar
Haugh, M. & Kogan, L. (2004). Pricing American options: A duality approach. Operations Research 52: 258–270.CrossRefGoogle Scholar
Huang, J., Subrahmanyam, M., & Yu, G. (1996). Pricing and hedging American options: A recursive integration method. Review of Financial Studies 9: 277–330.CrossRefGoogle Scholar
Hull, J.C. & White, A. (1998). The use of the control variate technique in option pricing. Journal of Financial and Quantitative Analysis 23: 237–251.CrossRefGoogle Scholar
Jensen, M.V. & Pedersen, L.H. (2016). Early option exercise: Never say never. Journal of Financial Economics 121: 278–299.CrossRefGoogle Scholar
Ju, N. (1998). Pricing an American option by approximating its early exercise boundary as a multipiece exponential function. Review of Financial Studies 11: 627–646.CrossRefGoogle Scholar
Kim, I.J. (1990). The analytic valuation of American options. Review of Financial Studies 3: 547–572.CrossRefGoogle Scholar
Kou, S.G. (2002). A jump-diffusion model for option pricing. Management Science 48: 1086–1101.CrossRefGoogle Scholar
Kou, S.G. & Wang, H. (2004). Option pricing under a double-exponential jump diffusion model. Management Science 50: 1178–1192.CrossRefGoogle Scholar
Kuhlman, K.L. (2012). Review of inverse Laplace transform algorithms for Laplace-space numerical approaches. Numerical Algorithms 63: 339–355.CrossRefGoogle Scholar
Leippold, M. & Vasiljevic, N. (2017). Pricing and disentanglement of American puts in the hyper-exponential jump-diffusion model. Journal of Banking and Finance 77: 78–94.CrossRefGoogle Scholar
Liu, Y., Cui, Z., & Zhang, N. (2016). Integral representation of vega for American put options. Finance Research Letters 19: 204–208.CrossRefGoogle Scholar
Longstaff, F.A. & Schwartz, E.S. (2001). Valuing American options by simulation: A simple least-squares approach. Review of Financial Studies 14: 113–147.CrossRefGoogle Scholar
Muthuraman, K. (2008). A moving boundary approach to American option pricing. Journal of Economic Dynamics and Control 32: 3520–3537.CrossRefGoogle Scholar
Nunes, J.P.V., Ruas, J.P., & Dias, J.C. (2015). Pricing and static hedging of American-style knock-in options. Journal of Banking and Finance 58: 343–360.CrossRefGoogle Scholar
Ruas, J.P., Dias, J.C., & Nunes, J.P.V. (2013). Pricing and static hedging of American-style options under the jump to default extended CEV model. Journal of Banking and Finance 37: 4059–4072.CrossRefGoogle Scholar
Sepp, A. (2004). Analytical pricing of double barrier options under a double-exponential jump diffusion process: Applications of Laplace transform. International Journal of Theoretical and Applied Finance 7: 151–175.CrossRefGoogle Scholar
Wong, H.Y. & Zhao, J. (2010). Valuing American options under the CEV model by Laplace-Carson transforms. Operations Research Letters 38: 474–481.CrossRefGoogle Scholar
Zhang, X.L. (1997). Numerical analysis of American option pricing in a jump diffusion model. Mathematics of Operation Research 22: 668–690.CrossRefGoogle Scholar
Zhu, S.P. (2006). An exact and explicit solution for the valuation of American put options. Quantitative Finance 6: 229–242.CrossRefGoogle Scholar
Zhu, S.P. (2006). A new analytical approximation formula for the optimal exercise boundary of the American put options. International Journal of Theoretical and Applied Finance 7: 1141–1177.CrossRefGoogle Scholar