# American Institute of Mathematical Sciences

July  2013, 9(3): 549-560. doi: 10.3934/jimo.2013.9.549

## American type geometric step options

 1 School of Science, Hebei University of Technology, Tianjin, China 2 Department of Statistics and Actuarial Science, University of Hong Kong, Pokfulam Road, Hong Kong

Received  October 2011 Revised  September 2012 Published  April 2013

The step option is a special contact whose value decreases gradually in proportional to the spending time outside a barrier of the asset price. European step options were introduced and studied by Linetsky [11] and Davydov et al. [2]. This paper considers American step options, including perpetual case and finite expiration time case. In perpetual case, we find that the optimal exercise time is the first crossing time of the optimal level. The closed price formula for perpetual step option could be derived through Feynman-Kac formula. As for the latter, we present a system of variational inequalities satisfied by the option price. Using the explicit finite difference method we could get the numerical option price.
Citation: Xiaoyu Xing, Hailiang Yang. American type geometric step options. Journal of Industrial and Management Optimization, 2013, 9 (3) : 549-560. doi: 10.3934/jimo.2013.9.549
##### References:
 [1] R. Cont and E. Voltchkova, A finite difference scheme for option pricing in jump diffusion and expoential Lévy models, SIAM Journal on Numerical Analysis, 43 (2005), 1596-1626. doi: 10.1137/S0036142903436186. [2] D. Davydov and V. Linetsky, Structuring, pricing and hedging double-barrier step options, Journal of Computational Finance, 5 (2001), 55-88. [3] R. Douady, Closed-form formulas for extoic options and their lift time distribution, International Journal of Theoretical and Applied Finance, 2 (1999), 17-42. doi: 10.1142/S0219024999000030. [4] H. German and M. Yor, Pricing and hedging double barrier options: A probabilistic approach, Mathematical Finance, 6 (1996), 365-378. [5] C. H. Hui, C. F. Lo and P. H. Yuen, Comment on "Pricing double-barrier options using Laplace Transforms", Finance and Stochastics, 4 (2000), 105-107. doi: 10.1007/s007800050006. [6] M. Jeannin and M. Pistorius, A transform approach to calculate prices and greeks of barrier options driven by a class of Lévy processes, Quantitative Finance, 10 (2010), 629-644. doi: 10.1080/14697680902896057. [7] I. Karatzas and S. E. Shreve, "Brownian Motion and Stochastic Calculus," $2^{nd}$ edition, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0949-2. [8] I. Karatzas and S. E. Shreve, "Methods of Mathematical Finance," Springer-Verlag, New York, 1998. [9] S. G. Kou and H. Wang, Option Pricing under a double exponential jump diffusion model, Management Science, 50 (2004), 1178-1192. [10] N. Kunitomo and M. Ikeda, Pricing optons with curved boundaries, Mathematical Finance, 2 (1992), 275-198. [11] V. Linetsky, Step options, Mathematical Finance, 9 (1999), 55-96. doi: 10.1111/1467-9965.00063. [12] F. Longstaff and E. Schwartz, Valuing American options by simulation: A simple beast-squares approach, The Review of Financal Studies, 14 (2001), 113-147. [13] R. C. Merton, Theory of rational option pricing, Bell Journal of Economics and Management Science, 4 (1973), 141-183. doi: 10.2307/3003143. [14] D. Rich, The mathematical foundations of barrier option pricing theory, Advances in Futures and Options Research, 7 (1994), 267-312. [15] M. Rubinstein and E. Reiner, Breaking down the barriers, RISK, (1991), 28-35. [16] M. Schroder, On the valuation of double-barrier options: Computational aspects, Journal of Computaional Finance, 3 (2000), 1-34. [17] S. E. Shreve, "Stochastic Calculus for Fiance II: Continuous-Time Models," Springer-Verlag, New York, 2004. [18] J. Sidenius, Double barrier options: Valuation by path counting, Computational Finance, 1 (1998), 63-79. [19] J. Tsitsiklis and B. Van Roy, Regression methods for pricing complex American style options, IEEE Transactions on Neural Networks, 12 (2001), 694-703. doi: 10.1109/72.935083.

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##### References:
 [1] R. Cont and E. Voltchkova, A finite difference scheme for option pricing in jump diffusion and expoential Lévy models, SIAM Journal on Numerical Analysis, 43 (2005), 1596-1626. doi: 10.1137/S0036142903436186. [2] D. Davydov and V. Linetsky, Structuring, pricing and hedging double-barrier step options, Journal of Computational Finance, 5 (2001), 55-88. [3] R. Douady, Closed-form formulas for extoic options and their lift time distribution, International Journal of Theoretical and Applied Finance, 2 (1999), 17-42. doi: 10.1142/S0219024999000030. [4] H. German and M. Yor, Pricing and hedging double barrier options: A probabilistic approach, Mathematical Finance, 6 (1996), 365-378. [5] C. H. Hui, C. F. Lo and P. H. Yuen, Comment on "Pricing double-barrier options using Laplace Transforms", Finance and Stochastics, 4 (2000), 105-107. doi: 10.1007/s007800050006. [6] M. Jeannin and M. Pistorius, A transform approach to calculate prices and greeks of barrier options driven by a class of Lévy processes, Quantitative Finance, 10 (2010), 629-644. doi: 10.1080/14697680902896057. [7] I. Karatzas and S. E. Shreve, "Brownian Motion and Stochastic Calculus," $2^{nd}$ edition, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0949-2. [8] I. Karatzas and S. E. Shreve, "Methods of Mathematical Finance," Springer-Verlag, New York, 1998. [9] S. G. Kou and H. Wang, Option Pricing under a double exponential jump diffusion model, Management Science, 50 (2004), 1178-1192. [10] N. Kunitomo and M. Ikeda, Pricing optons with curved boundaries, Mathematical Finance, 2 (1992), 275-198. [11] V. Linetsky, Step options, Mathematical Finance, 9 (1999), 55-96. doi: 10.1111/1467-9965.00063. [12] F. Longstaff and E. Schwartz, Valuing American options by simulation: A simple beast-squares approach, The Review of Financal Studies, 14 (2001), 113-147. [13] R. C. Merton, Theory of rational option pricing, Bell Journal of Economics and Management Science, 4 (1973), 141-183. doi: 10.2307/3003143. [14] D. Rich, The mathematical foundations of barrier option pricing theory, Advances in Futures and Options Research, 7 (1994), 267-312. [15] M. Rubinstein and E. Reiner, Breaking down the barriers, RISK, (1991), 28-35. [16] M. Schroder, On the valuation of double-barrier options: Computational aspects, Journal of Computaional Finance, 3 (2000), 1-34. [17] S. E. Shreve, "Stochastic Calculus for Fiance II: Continuous-Time Models," Springer-Verlag, New York, 2004. [18] J. Sidenius, Double barrier options: Valuation by path counting, Computational Finance, 1 (1998), 63-79. [19] J. Tsitsiklis and B. Van Roy, Regression methods for pricing complex American style options, IEEE Transactions on Neural Networks, 12 (2001), 694-703. doi: 10.1109/72.935083.
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