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SHORT MATURITY ASIAN OPTIONS FOR THE CEV MODEL

Published online by Cambridge University Press:  05 June 2018

Dan Pirjol  and
Lingjiong Zhu
Dan Pirjol
Affiliation:
J. P. Morgan 277 Park Avenue, New York, NY-10172, USA E-mail: [email protected]
Lingjiong Zhu
Affiliation:
Department of MathematicsFlorida State University1017 Academic Way, Tallahassee, FL-32306, USA E-mail: [email protected]
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Abstract

We present a rigorous study of the short maturity asymptotics for Asian options with continuous-time averaging, under the assumption that the underlying asset follows the constant elasticity of variance (CEV) model. The leading order short maturity limit of the Asian option prices under the CEV model is obtained in closed form. We propose an analytical approximation for the Asian options prices which reproduces the exact short maturity asymptotics, and demonstrate good numerical agreement of the asymptotic results with Monte Carlo simulations and benchmark test cases for option parameters relevant for practical applications.

Keywords

Asian optionsCEV modellarge deviationsshort maturityvariational problem
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 33 , Issue 2 , April 2019 , pp. 258 - 290
Copyright
Copyright © Cambridge University Press 2018 

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References

1.Abramowitz, M. & Stegun, I.A. (1972). Handbook of mathematical functions with formulas, graphs and mathematical tables. New York: Dover Publications.Google Scholar
2.Alòs, E., Léon, J., & Vives, J. (2007). On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility. Finance and Stochastics 11: 571–589.Google Scholar
3.Alziary, B., Decamps, J.P., & Koehl, P.F. (1997). A PDE approach to Asian options: Analytical and Numerical evidence. Journal of Banking and Finance 21: 613–640.Google Scholar
4.Andersen, L. & Lipton, A. (2013). Asymptotics for exponential Lévy processes and their volatility smile: Survey and new results. International Journal of Theoretical and Applied Finance 16: 1350001.Google Scholar
5.Arguin, L.P., Liu, N.-L., & Wang, T.-H. (2017). Most-likely-path in Asian option pricing under local volatility models, arXiv:1706.02408[q-fin.CP].Google Scholar
6.Armstrong, J., Forde, M., Lorig, M., & Zhang, H. (2016). Small-time asymptotics under Local-Stochastic volatility with a jump-to-default curvature and the heat kernel expansion. SIAM Journal on Financial Mathematics 8: 82–113.Google Scholar
7.Baldi, P. & Caramellino, L. (2011). General Freidlin-Wentzell large deviations and positive diffusions. Statistics and Probability Letters 81: 1218–1229.Google Scholar
8.Berestycki, H., Busca, J., & Florent, I. (2002). Asymptotics and calibration of local volatility models. Quantitative Finance 2: 61–69.Google Scholar
9.Berestycki, H., Busca, J., & Florent, I. (2004). Computing the implied volatility in stochastic volatility models. Communications On Pure And Applied Mathematics 57: 1352–1373.Google Scholar
10.Cai, Y. & Wang, S. (2015). Central limit theorem and moderate deviation principle for CKLS model with small random perturbation. ‘Statistics and Probability Letters’ 98: 6–11.Google Scholar
11.Cai, N., Li, C., & Shi, C. (2014). Closed-form expansions of discretely monitored Asian options in diffusion models. Mathematics of Operations Research 39: 789–822.Google Scholar
12.Cai, N., Song, Y., & Kou, S. (2015). A general framework for pricing Asian options under Markov processes. Operations Research 63: 540–554.Google Scholar
13.Carr, P. & Schröder, M. (2003). Bessel processes, the integral of geometric Brownian motion, and Asian options. Theory of Probability and its Applications 48: 400–425.Google Scholar
14.Carr, P. & Sun, J. (2007). A new approach for option pricing under stochastic volatility. Review of Derivatives Research 10: 87–150.Google Scholar
15.Carr, P. & Wu, L. (2003). What type of process underlies options? A simple robust test. Journal of Finance 58: 2581–2610.Google Scholar
16.Cheng, W., Costanzino, N., Liechty, J., Mazzucato, A., & Nistor, V. (2011). Closed-form asymptotics and numerical approximations of 1D parabolic equations with applications to option pricing. SIAM Journal on Financial Mathematics 2: 901–934.Google Scholar
17.Cox, J.C. (1996). Notes on Option Pricing I: Constant elasticity of variance diffusions. Reprinted in the Journal of Portfolio Management. 23: 15–17.Google Scholar
18.Cox, J.C., Ingersoll, J.E., & Ross, S.A. (1985). A theory of the term structure of interest rates. Econometrica 53: 385–407.Google Scholar
19.Cox, J.C. & Ross, S.A. (1976). The valuation of options for alternative stochastic processes. Journal of Financial Economics 3: 145–166.Google Scholar
20.Dassios, A. & Nagaradjasarma, J. (2006). The square-root process and Asian options. Quantitative Finance 6: 337–347.Google Scholar
21.Dembo, A. & Zeitouni, O. (1998). Large deviations techniques and applications, 2nd ed. New York: Springer.Google Scholar
22.Donati-Martin, C., Rouault, A., Yor, M., & Zani, M. (2004). Large deviations for squares of Bessel and Ornstein-Uhlenbeck processes. Probability Theory Related Fields 129: 261–289.Google Scholar
23.Drimus, G.G. (2012). Options on realized variance by transform methods: a non-affine stochastic volatility model. Quantitative Finance 12: 1679–1694.Google Scholar
24.Dufresne, D. (2000). Laguerre series for Asian and other options. Mathematical Finance 10: 407–428.Google Scholar
25.Dufresne, D. (2001). The integrated square-root process. Technical report, University of Montreal.Google Scholar
26.Dufresne, D. (2005). Bessel processes and a functional of Brownian motion. In Michele, M. & Ben-Ameur, H. (eds.), Numerical methods in finance. New York: Springer, pp. 35–57.Google Scholar
27.Feller, W. (1951). Two singular diffusion problems. Annals of Mathematics 54: 173–182.Google Scholar
28.Feng, J., Forde, M., & Fouque, J.-P. (2010). Short maturity asymptotics for a fast mean-reverting Heston stochastic volatility model. SIAM Journal on Financial Mathematics 1: 126–141.Google Scholar
29.Feng, J., Fouque, J.-P., & Kumar, R. (2012). Small-time asymptotics for fast mean-reverting stochastic volatility models. Annals of Applied Probability 22: 1541–1575.Google Scholar
30.Figueroa-López, J.E. & Forde, M. (2012). The small-maturity smile for exponential Lévy models. SIAM Journal on Financial Mathematics 3: 33–65.Google Scholar
31.Forde, M. & Jacquier, A. (2009). Small time asymptotics for implied volatility under the Heston model. International Journal of Theoretical and Applied Finance 12: 861.Google Scholar
32.Forde, M. & Jacquier, A. (2011). Small time asymptotics for an uncorrelated Local-Stochastic volatility model. Applied Mathematical Finance 18: 517–535.Google Scholar
33.Forde, M., Jacquier, A., & Lee, R. (2012). The small-time smile and term structure of implied volatility under the Heston model. SIAM Journal on Financial Mathematics 3: 690–708.Google Scholar
34.Foschi, P., Pagliarani, S., & Pascucci, A. (2013). Approximations for Asian options in local volatility models. Journal of Computational and Applied Mathematics 237: 442–459.Google Scholar
35.Fu, M., Madan, D., & Wang, T. (1998). Pricing continuous time Asian options: a comparison of Monte Carlo and Laplace transform inversion methods. Journal of Computational Finance 2: 49–74.Google Scholar
36.Fusai, M., Marena, M., & Roncoroni, A. (2008). Analytical pricing of discretely monitored Asian-style options: Theory and applications to commodity markets. Journal of Banking and Finance 32: 2033–2045.Google Scholar
37.Fusai, M. & Kyriakou, I. (2016). General optimized lower and upper bounds for discrete and continuous arithmetic Asian options. Mathematics of Operations Research 41: 531–559.Google Scholar
38.Gao, K. & Lee, R. (2014). Asymptotics of implied volatility to arbitrary order. Finance and Stochastics 18: 349–392.Google Scholar
39.Gatheral, J., Hsu, E.P., Laurent, P., Ouyang, C., & Wang, T.-H. (2012). Asymptotics of implied volatility in local volatility models. Mathematical Finance 22: 591–620.Google Scholar
40.Gatheral, J. & Wang, T.-H (2012). The heat-kernel most-likely-path approximation. IJTAF. 15: 1250001.Google Scholar
41.Geman, H. & Yor, M. (1993). Bessel processes, Asian options and perpetuities. Mathematical Finance 3: 349–375.Google Scholar
42.Gobet, E. & Miri, M. (2014). Weak approximation of averaged diffusion processes. Stochastic Processes and their Applications 124: 475–504.Google Scholar
43.Hagan, P. & Woodward, D. (1999). Equivalent Black volatilities. Applied Mathematical Finance 6: 147–157.Google Scholar
44.Henderson, V. & Wojakowski, R. (2002). On the equivalence of floating and fixed-strike Asian options. Journal of Applied Probability 39: 391–394.Google Scholar
45.Henry-Labordère, P. (2009). Analysis, geometry and modeling in finance: advanced methods in option pricing. New York: Chapman and Hall/CRC.Google Scholar
46.Linetsky, V. (2004). Spectral expansions for Asian (Average price) options. Operations Research 52: 856–867.Google Scholar
47.Linetsky, V. & Mendoza, R. (2009). Constant elasticity of variance (CEV) diffusion model. In Cont, R. (ed.) Encyclopedia of quantitative finance. New York: Wiley, pp. 328–334.Google Scholar
48.Mazzon, A. (2011) Processo square root. PhD Thesis, University of Bologna.Google Scholar
49.Muhle-Karbe, J. & Nutz, M. (2011). Small-time asymptotics of option prices and first absolute moments. Journal of Applied Probability 48: 1003–1020.Google Scholar
50.Pirjol, D. & Zhu, L. (2016). Short maturity Asian options in local volatility models. SIAM Journal on Financial Mathematics 7: 947–992.Google Scholar
51.Rogers, L. & Shi, Z. (1995). The value of an Asian option. Journal of Applied Probability 32: 1077–1088.Google Scholar
52.Shiraya, K., Takahashi, A., & Toda, M. (2011). Pricing barrier and average options under stochastic volatility environment. Journal of Computational Finance 15: 111–148.Google Scholar
53.Tankov, P. (2010). Pricing and hedging in exponential Lévy models: review of recent results. In Carmona, R., Çinlar, E., Ekeland, I., Jouini, E., Scheinkman, J.A. & Touzi, N. (eds) Paris-Princeton lectures on mathematical Finance 2010, volume 2003 of lecture notes in math. Berlin: Springer, pp. 319–359.Google Scholar
54.Varadhan, S.R.S. (1967). Diffusion processes in a small time interval. Communications on Pure and Applied Mathematics 20: 659–685.Google Scholar
55.Varadhan, S.R.S. (1984). Large deviations and applications. Philadelphia: SIAM.Google Scholar
56.Vecer, J. (2001). A new PDE approach for pricing arithmetic average Asian options. Journal of Computational Finance 4: 105–113.Google Scholar
57.Vecer, J. & Xu, M. (2002). Unified Asian pricing. Risk 15: 113–116.Google Scholar