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Computation of Greeks Using Binomial Tree

DOI: 10.4236/jmf.2017.73031, PP. 597-623

Keywords: Options, Greeks, Binomial Tree

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Abstract:

This paper proposes a new efficient algorithm for the computation of Greeks for options using the binomial tree. We also show that Greeks for European options introduced in this article are asymptotically equivalent to the discrete version of Malliavin Greeks. This fact enables us to show that our Greeks converge to Malliavin Greeks in the continuous time model. The computation algorithm of Greeks for American options using the binomial tree is also given in this article. There are three advantageous points to use binomial tree approach for the computation of Greeks. First, mathematics is much simpler than using the continuous time Malliavin calculus approach. Second, we can construct a simple algorithm to obtain the Greeks for American options. Third, this algorithm is very efficient because one can compute the price and Greeks (delta, gamma, vega, and rho) at once. In spite of its importance, only a few previous studies on the computation of Greeks for American options exist, because performing sensitivity analysis for the optimal stopping problem is difficult. We believe that our method will become one of the popular ways to compute Greeks for options.

References

[1]  Hull, J. (2008) Options, Futures and Other Derivatives. 7th Edition, Prentice Hall, Upper Saddle River, New Jersey.
[2]  Fournié, E., Laszry, J., Lebuchoux, J., Lions, P. and Touzi, N. (1999) Applications of Malliavin Calculus to Monte Carlo Methods in Finance. Finance and Stochastics, 3, 391-412.
https://doi.org/10.1007/s007800050068
[3]  Kohatsu-Higa, A. and Montero, M. (2003) Malliavin Calculus Applied to Finance. Physica A, 320, 548-570.
https://doi.org/10.1016/S0378-4371(02)01531-5
[4]  Bernis, G., Gobet E. and Kohatsu-Higa, A. (2003) Monte Carlo Evaluation of Greeks for Multidimensional Barrier and Lookback Options. Mathematical Finance, 13, 99-113.
https://doi.org/10.1111/1467-9965.00008
[5]  Gobet, E. (2004) Revisiting the Greeks for European and American Options. Proceedings of the International Symposium on Stochastic Processes and Mathematical Finance at Ritsumeikan University, Kusatsu, 5-9 March 2003, 53-71.
https://doi.org/10.1142/9789812702852_0003
[6]  Bally, V., Caramellino, L. and Zanette, A. (2005) Pricing and Hedging American Options by Monte Carlo Methods Using a Malliavin Calculus Approach. Monte Carlo Methods and Applications, 11, 97-133.
https://doi.org/10.1515/156939605777585944
[7]  Longstaff, F. and Schwartz, E.S. (2001) Valuing American Options by Simulation: A Simple Least Squares Approach. Review of Financial Studies, 14, 113-147.
https://doi.org/10.1093/rfs/14.1.113
[8]  Muroi, Y. and Suda, S. (2013) Discrete Malliavin Calculus and Computations of Greeks in the Binomial Tree. European Journal of Operational Research, 231, 349-361.
https://doi.org/10.1016/j.ejor.2013.05.038
[9]  Muroi, Y. and Suda, S. (2017) Computation of Greeks in the Jump-Diffusion Model using Discrete Malliavin Calculus. Mathematics and Computers in Simulation, 140, 69-93.
https://doi.org/10.1016/j.matcom.2017.03.002
[10]  Cox, J.C., Ross, S.A. and Rubinstein, M. (1979) Option Pricing: A Simplified Approach. Journal of Financial Economics, 7, 229-263.
https://doi.org/10.1016/0304-405X(79)90015-1
[11]  Leitz-Martini, M. (2000) A Discrete Clark-Ocone Formula. Maphysto Research Report, 29.
[12]  Privault, N. (2008) Stochastic Analysis of Bernoulli Processes. Probability Surveys, 5, 435-483.
https://doi.org/10.1214/08-PS139
[13]  Privault, N. (2009) Stochastic Analysis in Discrete and Continuous Settings. Springer, Berlin.
https://doi.org/10.1007/978-3-642-02380-4
[14]  Privault, N. and Schoutens, W. (2002) Discrete Chaotic Calculus and Covariance Identities. Stochatics and Stochastic Reports, 72, 289-315.
https://doi.org/10.1080/10451120290019230
[15]  Suda, S. and Muroi, Y. (2015) Computation of Greeks Using Binomial Trees in a Jump-Diffusion Model. Journal of Economic Dynamics and Control, 51, 93-110.
https://doi.org/10.1016/j.jedc.2014.09.032
[16]  Chung, S.L., Hung, W., Lee, H.H. and Shih, P.T. (2011) On the Rate of Convergence of Binomial Greeks. Journal of Futures Markets, 31, 562-597.
https://doi.org/10.1002/fut.20484
[17]  Pelsser, A. and Vorst, T. (1994) The Binomial Model and the Greeks. Journal of Derivatives, 1, 45-49.
https://doi.org/10.3905/jod.1994.407888
[18]  Chung, S.L. and Shackleton, M. (2002) The Binomial Black-Scholes Model and the Greeks. Journal of Futures Markets, 22, 143-153.
https://doi.org/10.1002/fut.2211
[19]  Tian, Y. (1993) A Modified Lattice Approach to Option Pricing. Journal of Futures Markets, 13, 563-577.
https://doi.org/10.1002/fut.3990130509
[20]  Leisen, D. and Reimer, M. (1996) Binomial Models for Option Valuation-Examining and Improving Convergence. Applied Mathematical Finance, 3, 319-346.
https://doi.org/10.1080/13504869600000015
[21]  Heston, S. and Zhou, G. (2000) On the Rate of Convergence of Discrete-Time Contingent Claims. Mathematical Finance, 10, 53-75.
https://doi.org/10.1111/1467-9965.00080
[22]  Walsh, J. (2003) The Rate of Convergence of the Binomial Tree Scheme. Finance and Stochastics, 7, 337-361.
https://doi.org/10.1007/s007800200094

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