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COMPUTATIONAL METHOD FOR PROBABILITY DISTRIBUTION ON RECURSIVE RELATIONSHIPS IN FINANCIAL APPLICATIONS

Published online by Cambridge University Press:  26 January 2019

Jong Jun Park  and
Kyungsub Lee Open the ORCID record for Kyungsub Lee [Opens in a new window]
Jong Jun Park
Affiliation:
Department of Mathematical Sciences, KAIST, 291 Daehak-ro, Daejeon34141, Republic of Korea
Kyungsub Lee
Affiliation:
Department of Statistics, Yeungnam University, Gyeongsan, Gyeongbuk38541, Republic of Korea E-mail: [email protected]
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Abstract

In quantitative finance, it is often necessary to analyze the distribution of the sum of specific functions of observed values at discrete points of an underlying process. Examples include the probability density function, the hedging error, the Asian option, and statistical hypothesis testing. We propose a method to calculate such a distribution, utilizing a recursive method, and examine it using various examples. The results of the numerical experiment show that our proposed method has high accuracy.

Keywords

applied probabilitycomputational probabilitymanagement sciencemathematical financestochastic modeling
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 34 , Issue 2 , April 2020 , pp. 258 - 278
Copyright
Copyright © Cambridge University Press 2019

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References

1.Barndorff-Nielsen, O.E. (2002). Econometric analysis of realized volatility and its use in estimating stochastic volatility models. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 64: 253–280.CrossRefGoogle Scholar
2.Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31: 307–327.CrossRefGoogle Scholar
3.Breeden, D.T. & Litzenberger, R.H. (1978). Prices of state-contingent claims implicit in option prices. Journal of Business 51: 621–651.CrossRefGoogle Scholar
4.Broadie, M. & Kaya, Ö. (2006). Exact simulation of stochastic volatility and other affine jump diffusion processes. Operations Research 54: 217–231.CrossRefGoogle Scholar
5.Carr, P. & Madan, D. (1999). Option valuation using the fast Fourier transform. Journal of Computational Finance 2: 61–73.CrossRefGoogle Scholar
6.Choe, G.H. & Lee, K. (2014). High moment variations and their application. Journal of Futures Markets 34: 1040–1061.CrossRefGoogle Scholar
7.Christoffersen, P., Heston, S., & Jacobs, K. (2006). Option valuation with conditional skewness. Journal of Econometrics 131: 253–284.CrossRefGoogle Scholar
8.Christoffersen, P., Jacobs, K., & Mimouni, K. (2010). Volatility dynamics for the S&P500: evidence from realized volatility, daily returns, and option prices. Review of Financial Studies 23: 3141–3189.CrossRefGoogle Scholar
9.Cont, R. (2001). Empirical properties of asset returns: stylized facts and statistical issues. Quantitative Finance 1: 223–236.CrossRefGoogle Scholar
10Cont, R., Tankov, P., & Voltchkova, E. (2007). Hedging with options in models with jumps. In Benth, F.E., Di Nunno, G., Lindstrom, T., Øksendal, B., & Zhang, T. (eds.), Stochastic Analysis and Applications. Berlin, Heidelberg, New York: Springer, pp. 197–217.CrossRefGoogle Scholar
11.Cox, J.C. & Ross, S.A. (1976). The valuation of options for alternative stochastic processes. Journal of Financial Economics 3: 145–166.CrossRefGoogle Scholar
12.Cox, J.C., JrIngersoll, J.E., & Ross, S.A. (1985). A theory of the term structure of interest rates. Econometrica 53: 385–407.CrossRefGoogle Scholar
13.Fama, E.F. (1965). The behavior of stock-market prices. The Journal of Business 38: 34–105.CrossRefGoogle Scholar
14Föllmer, H. & Sondermann, D. (1986). Hedging of non-redundant contingent claims. In Mas-Colell, A. & Hildenbrand, W. (eds.), Contributions to Mathematical Economics: In Honor of Gérard Debreu. Amsterdam, New York, Oxford, Tokyo: North Holland, pp. 205–224.Google Scholar
15.French, K.R., Schwert, G.W., & Stambaugh, R.F. (1987). Expected stock returns and volatility. Journal of Financial Economics 19: 3–29.CrossRefGoogle Scholar
16.Harvey, C.R. & Siddique, A. (2000). Conditional skewness in asset pricing tests. Journal of Finance 55: 1263–1295.CrossRefGoogle Scholar
17.Heston, S.L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies 6: 327–343.CrossRefGoogle Scholar
18.Kemna, A.G. & Vorst, A. (1990). A pricing method for options based on average asset values. Journal of Banking & Finance 14: 113–129.CrossRefGoogle Scholar
19.Lee, K. (2014). Recursive formula for arithmetic Asian option prices. Journal of Futures Markets 34: 220–234.CrossRefGoogle Scholar
20.Lee, K. (2016). Probabilistic and statistical properties of moment variations and their use in inference and estimation based on high frequency return data. Studies in Nonlinear Dynamics & Econometrics 20: 19–36.Google Scholar
21.Lee, K. & Seo, B.K. (2017). Modeling microstructure price dynamics with symmetric Hawkes and diffusion model using ultra-high-frequency stock data. Journal of Economic Dynamics and Control 79: 154–183.CrossRefGoogle Scholar
22.Madan, D.B., Carr, P.P., & Chang, E.C. (1998). The variance gamma process and option pricing. European Finance Review 2: 79–105.CrossRefGoogle Scholar
23.Musiela, M. & Rutkowski, M. (2006). Martingale Methods in Financial Modelling. Vol. 36. Berlin, Heidelberg, New York: Springer Science & Business Media.Google Scholar
24.Park, M., Lee, K., & Choe, G.H. (2016). Distribution of discrete time delta-hedging error via a recursive relation. East Asian Journal on Applied Mathematics 6: 314–336.CrossRefGoogle Scholar
25.Sepp, A. (2012). An approximate distribution of delta-hedging errors in a jump-diffusion model with discrete trading and transaction costs. Quantitative Finance 12: 1119–1141.CrossRefGoogle Scholar
26.Věcěr, J. (2002). Unified Asian pricing. Risk 15: 113–116.Google Scholar