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Asymptotics for the discrete-time average of the geometric Brownian motion and Asian options

Part of: Limit theorems Mathematical finance

Published online by Cambridge University Press:  26 June 2017

Dan Pirjol  and
Lingjiong Zhu
Dan Pirjol*
Affiliation:
J. P. Morgan
Lingjiong Zhu*
Affiliation:
Florida State University
*
* Postal address: J. P. Morgan, New York, NY 10172, USA. Email address: dpirjol@gmail.com
** Postal address: Department of Mathematics, Florida State University, 1017 Academic Way, Tallahassee, FL 32306, USA. Email address: zhu@math.fsu.edu
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Abstract

The time average of geometric Brownian motion plays a crucial role in the pricing of Asian options in mathematical finance. In this paper we consider the asymptotics of the discrete-time average of a geometric Brownian motion sampled on uniformly spaced times in the limit of a very large number of averaging time steps. We derive almost sure limit, fluctuations, large deviations, and also the asymptotics of the moment generating function of the average. Based on these results, we derive the asymptotics for the price of Asian options with discrete-time averaging in the Black–Scholes model, with both fixed and floating strike.

Keywords

Asian optioncentral limit theoremBerry–Esseen boundlarge deviations

MSC classification

Primary: 91G20: Derivative securities
Secondary: 91G80: Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems) 60F05: Central limit and other weak theorems 60F10: Large deviations
Type
Research Article
Information
Advances in Applied Probability , Volume 49 , Issue 2 , June 2017 , pp. 446 - 480
Copyright
Copyright © Applied Probability Trust 2017 

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References

[1] Andreasen, J. (1998). The pricing of discretely sampled Asian and lookback options: a change of numeraire approach. J. Comput. Finance 2, 5–30. CrossRefGoogle Scholar
[2] Alziary, B., Decamps, J.-P. and Koehl, P.-F. (1997). A P.D.E. approach to Asian options: analytical and numerical evidence. J. Banking Finance 21, 613–640. CrossRefGoogle Scholar
[3] Asmussen, S., Jensen, J. L. and Rojas-Nandayapa, L. (2011). A literature review on log-normal sums. Preprint, University of Queensland. Google Scholar
[4] Berestycki, H., Busca, J. and Florent, I. (2004). Computing the implied volatility in stochastic volatility models. Commun. Pure Appl. Math. 57, 1352–1373. CrossRefGoogle Scholar
[5] Bikelis, A. (1966). Estimates of the remainder term in the central limit theorem. Litovsk. Mat. Sb. 6, 323–346. Google Scholar
[6] Carr, P. and Madan, D. B. (1999). Option valuation using the fast Fourier transform. J. Comput. Finance 2, 61–73. CrossRefGoogle Scholar
[7] Carr, P. and Schröder, M. (2004). Bessel processes, the integral of geometric Brownian motion, and Asian options. Theory Prob. Appl. 48, 400–425. CrossRefGoogle Scholar
[8] Chung, S. L., Shackleton, M. and Wojakowski, R. (2000). Efficient quadratic approximation of floating strike Asian option values. Working paper, Lancaster University Management School. Google Scholar
[9] Curran, M. (1994). Valuing Asian options and portfolio options by conditioning on the geometric mean price. Manag. Sci. 40, 1705–1711. CrossRefGoogle Scholar
[10] Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications (Appl. Math. 38), 2nd edn. Springer, New York. CrossRefGoogle Scholar
[11] Dewynne, J. N. and Shaw, W. T. (2008). Differential equations and asymptotic solutions for arithmetic Asian options: 'Black-Scholes formulae' for Asian rate calls. Europ. J. Appl. Math. 19, 353–391. CrossRefGoogle Scholar
[12] Dewynne, J. N. and Wilmott, P. (1995). A note on average rate options with discrete sampling. SIAM J. Appl. Math. 55, 267–276. CrossRefGoogle Scholar
[13] Donati-Martin, C., Ghomrasni, R. and Yor, M. (2001). On certain Markov processes attached to exponential functionals of Brownian motion; application to Asian options. Rev. Math. Iberoam. 17, 179–193. CrossRefGoogle Scholar
[14] Dufresne, D. (1990). The distribution of a perpetuity, with applications to risk theory and pension funding. Scand. Actuarial J. 9, 39–79. CrossRefGoogle Scholar
[15] Dufresne, D. (2000). Laguerre series for Asian and other options. Math. Finance 10, 407–428. CrossRefGoogle Scholar
[16] Dufresne, D. (2004). The log-normal approximation in financial and other computations. Adv. Appl. Prob. 36, 747–773. CrossRefGoogle Scholar
[17] Dufresne, D. (2005). Bessel processes and a functional of Brownian motion. In Numerical Methods in Finance, eds M. Michele and H. Ben-Ameur, Springer, New York, pp. 35–57. CrossRefGoogle Scholar
[18] Feng, J., Forde, M. and Fouque, J.-P. (2010). Short maturity asymptotics for a fast mean-reverting Heston stochastic volatility model. SIAM J. Financial Math. 1, 126–141. CrossRefGoogle Scholar
[19] Forde, M. and Jacquier, A. (2009). Small-time asymptotics for implied volatility under the Heston model. Internat. J. Theoret. Appl. Finance 12, 861–876. CrossRefGoogle Scholar
[20] Foschi, P., Pagliarani, S. and Pascucci, A. (2013). Approximations for Asian options in local volatility models. J. Comput. Appl. Math. 237, 442–459. CrossRefGoogle Scholar
[21] Fu, M. C., Madan, D. B. and Wang, T. (1998). Pricing continuous Asian options: a comparison of Monte Carlo and Laplace transform inversion methods. J. Comput. Finance 2, 49–74. CrossRefGoogle Scholar
[22] Fusai, G. and Meucci, A. (2008). Pricing of discretely monitored Asian options under Lévy processes. J. Banking Finance 32, 2076–2088. CrossRefGoogle Scholar
[23] Fusai, G., Marazzina, D. and Marena, M. (2011). Pricing discretely monitored Asian options by maturity randomization. SIAM J. Finance Math. 2, 383–403. CrossRefGoogle Scholar
[24] Fusai, G., Marena, M. and Roncoroni, A. (2008). Analytical pricing of discretely monitored Asian-Style options: theory and application to commodity markets. J. Banking Finance 32, 2033–2045. CrossRefGoogle Scholar
[25] Gatheral, J. et al. (2012). Asymptotics of implied volatility in local volatility models. Math. Finance 22, 591–620. CrossRefGoogle Scholar
[26] Geman, H. and Yor, M. (1993). Bessel processes, Asian options, and perpetuities. Math. Finance 3, 349–375. CrossRefGoogle Scholar
[27] Guasoni, P. and Robertson, S. (2008). Optimal importance sampling with explicit formulas in continuous time. Finance Stoch. 12, 1–19. CrossRefGoogle Scholar
[28] Henderson, V. and Wojakowski, R. (2002). On the equivalence of floating- and fixed-strike Asian options. J. Appl. Prob. 39, 391–394. CrossRefGoogle Scholar
[29] Ingersoll, J. E. (1988). Theory of Financial Decision Making. Rowman and Littlefeild, Totowa, NJ. Google Scholar
[30] Kemna, A. G. Z. and Vorst, A. C. F. (1990). A pricing method for options based on average asset values. J. Banking Finance 14, 113–129. CrossRefGoogle Scholar
[31] Lapeyre, B. and Teman, E. (1999). Competitive Monte Carlo methods for pricing Asian options. Working paper, CERMICS, Ecole Nationale des Ponts et Chaussees. Google Scholar
[32] Laub, P. J., Asmussen, S., Jensen, J. L. and Rojas-Nandayapa, L. (2015). Approximating the Laplace transform of the sum of dependent lognormals. In Probability, Analysis and Number Theory (Adv. Appl. Spec. Vol. 48A), eds C. M. Goldie and A. Mijatović, Applied Probability Trust, Sheffield, pp. 203–215. Google Scholar
[33] Levy, E. (1992). Pricing European average rate currency options. J. Internat. Money Finance 11, 474–491. CrossRefGoogle Scholar
[34] Lewis, A. (2002). Asian connections. Wilmott Mag. 7, 57–63. Google Scholar
[35] Linetsky, V. (2002). Exotic spectra. Risk 15, 85–89. Google Scholar
[36] Linetsky, V. (2004). Spectral expansions for Asian (average price) options. Operat. Res. 52, 856–867. CrossRefGoogle Scholar
[37] Lord, R. (2006). Partially exact and bounded approximations for arithmetic Asian options. J. Comput. Finance 10, 1–52. CrossRefGoogle Scholar
[38] Milevsky, M. A. and Posner, S. E. (1998). Asian options, the sum of lognormals, and the reciprocal gamma distribution. J. Financial Quant. Anal. 33, 409–442. CrossRefGoogle Scholar
[39] Nagaev, S. V. (1965). Some limit theorems for large deviations. Teor. Verojat. Primen. 10, 231–254. Google Scholar
[40] Pinelis, I. (2013). On the non-uniform Berry–Esseen bound. Preprint. Available at https://arxiv.org/abs/1301.2828v5. Google Scholar
[41] Pirjol, D. and Zhu, L. (2016). Discrete sums of geometric Brownian motions, annuities and Asian options. Insurance Math. Econom. 70, 19–37. CrossRefGoogle Scholar
[42] Pirjol, D. and Zhu, L. (2017). Short maturity Asian options in local volatility models.SIAM J. Financial Math. 7, 947–992. CrossRefGoogle Scholar
[43] Ritchken, P., Sankarasubramanian, L and Vijh, A. M. (1993). The valuation of path dependent contracts on the average. Manag. Sci. 39, 1202–1213. CrossRefGoogle Scholar
[44] Rogers, L. C. G. and Shi, Z. (1995). The value of an Asian option. J. Appl. Prob. 32, 1077–1088. CrossRefGoogle Scholar
[45] Roper, M. and Rutkowski, M. (2009). On the relationship between the call price surface and the implied volatility surface close to expiry. Internat. J. Theoret. Appl. Finance 12, 427–441. CrossRefGoogle Scholar
[46] Shaw, W. T. (2003). Pricing Asian options by contour integration, including asymptotic methods for low volatility. Working paper. Google Scholar
[47] Tavella, D. and Randall, C. (2000). Pricing Financial Instruments: The Finite Difference Method. John Wiley. Google Scholar
[48] Tehranchi, M. R. (2009). Asymptotics of implied volatility far from maturity. J. Appl. Prob. 46, 629–650. CrossRefGoogle Scholar
[49] Vanmaele, M. et al. (2006). Bounds for the price of discrete arithmetic Asian options. J. Comput. Appl. Math. 185, 51–90. CrossRefGoogle Scholar
[50] Varadhan, S. R. S. (1984). Large Deviations and Applications. Society for Industrial and Applied Mathematics, Philadelphia, PA. CrossRefGoogle Scholar
[51] Vecer, J. (2001). A new PDE approach for pricing arithmetic average Asian options. J. Comput. Finance 4, 105–113. CrossRefGoogle Scholar
[52] Vecer, J. (2002). Unified Asian pricing. Risk 15, 113–116. Google Scholar
[53] Vecer, J. (2014). Black-Scholes representation for Asian options. Math. Finance 24, 598–626. CrossRefGoogle Scholar
[54] Vecer, J. and Xu, M. (2004). Pricing Asian options in a semimartingale model. Quant. Finance 4, 170–175. CrossRefGoogle Scholar
[55] Wong, E. (1964). The construction of a class of stationary Markoff processes. In Proc. Symp. Appl. Math., Vol. XVI, American Mathematical Society, Providence, RI, pp. 264–276. Google Scholar