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 & 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.  doi: 10.1137/S0036142903436186.  Google Scholar

[2]

D. Davydov and V. Linetsky, Structuring, pricing and hedging double-barrier step options,, Journal of Computational Finance, 5 (2001), 55.   Google Scholar

[3]

R. Douady, Closed-form formulas for extoic options and their lift time distribution,, International Journal of Theoretical and Applied Finance, 2 (1999), 17.  doi: 10.1142/S0219024999000030.  Google Scholar

[4]

H. German and M. Yor, Pricing and hedging double barrier options: A probabilistic approach,, Mathematical Finance, 6 (1996), 365.   Google Scholar

[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.  doi: 10.1007/s007800050006.  Google Scholar

[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.  doi: 10.1080/14697680902896057.  Google Scholar

[7]

I. Karatzas and S. E. Shreve, "Brownian Motion and Stochastic Calculus,", $2^{nd}$ edition, (1991).  doi: 10.1007/978-1-4612-0949-2.  Google Scholar

[8]

I. Karatzas and S. E. Shreve, "Methods of Mathematical Finance,", Springer-Verlag, (1998).   Google Scholar

[9]

S. G. Kou and H. Wang, Option Pricing under a double exponential jump diffusion model,, Management Science, 50 (2004), 1178.   Google Scholar

[10]

N. Kunitomo and M. Ikeda, Pricing optons with curved boundaries,, Mathematical Finance, 2 (1992), 275.   Google Scholar

[11]

V. Linetsky, Step options,, Mathematical Finance, 9 (1999), 55.  doi: 10.1111/1467-9965.00063.  Google Scholar

[12]

F. Longstaff and E. Schwartz, Valuing American options by simulation: A simple beast-squares approach,, The Review of Financal Studies, 14 (2001), 113.   Google Scholar

[13]

R. C. Merton, Theory of rational option pricing,, Bell Journal of Economics and Management Science, 4 (1973), 141.  doi: 10.2307/3003143.  Google Scholar

[14]

D. Rich, The mathematical foundations of barrier option pricing theory,, Advances in Futures and Options Research, 7 (1994), 267.   Google Scholar

[15]

M. Rubinstein and E. Reiner, Breaking down the barriers,, RISK, (1991), 28.   Google Scholar

[16]

M. Schroder, On the valuation of double-barrier options: Computational aspects,, Journal of Computaional Finance, 3 (2000), 1.   Google Scholar

[17]

S. E. Shreve, "Stochastic Calculus for Fiance II: Continuous-Time Models,", Springer-Verlag, (2004).   Google Scholar

[18]

J. Sidenius, Double barrier options: Valuation by path counting,, Computational Finance, 1 (1998), 63.   Google Scholar

[19]

J. Tsitsiklis and B. Van Roy, Regression methods for pricing complex American style options,, IEEE Transactions on Neural Networks, 12 (2001), 694.  doi: 10.1109/72.935083.  Google Scholar

show all references

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.  doi: 10.1137/S0036142903436186.  Google Scholar

[2]

D. Davydov and V. Linetsky, Structuring, pricing and hedging double-barrier step options,, Journal of Computational Finance, 5 (2001), 55.   Google Scholar

[3]

R. Douady, Closed-form formulas for extoic options and their lift time distribution,, International Journal of Theoretical and Applied Finance, 2 (1999), 17.  doi: 10.1142/S0219024999000030.  Google Scholar

[4]

H. German and M. Yor, Pricing and hedging double barrier options: A probabilistic approach,, Mathematical Finance, 6 (1996), 365.   Google Scholar

[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.  doi: 10.1007/s007800050006.  Google Scholar

[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.  doi: 10.1080/14697680902896057.  Google Scholar

[7]

I. Karatzas and S. E. Shreve, "Brownian Motion and Stochastic Calculus,", $2^{nd}$ edition, (1991).  doi: 10.1007/978-1-4612-0949-2.  Google Scholar

[8]

I. Karatzas and S. E. Shreve, "Methods of Mathematical Finance,", Springer-Verlag, (1998).   Google Scholar

[9]

S. G. Kou and H. Wang, Option Pricing under a double exponential jump diffusion model,, Management Science, 50 (2004), 1178.   Google Scholar

[10]

N. Kunitomo and M. Ikeda, Pricing optons with curved boundaries,, Mathematical Finance, 2 (1992), 275.   Google Scholar

[11]

V. Linetsky, Step options,, Mathematical Finance, 9 (1999), 55.  doi: 10.1111/1467-9965.00063.  Google Scholar

[12]

F. Longstaff and E. Schwartz, Valuing American options by simulation: A simple beast-squares approach,, The Review of Financal Studies, 14 (2001), 113.   Google Scholar

[13]

R. C. Merton, Theory of rational option pricing,, Bell Journal of Economics and Management Science, 4 (1973), 141.  doi: 10.2307/3003143.  Google Scholar

[14]

D. Rich, The mathematical foundations of barrier option pricing theory,, Advances in Futures and Options Research, 7 (1994), 267.   Google Scholar

[15]

M. Rubinstein and E. Reiner, Breaking down the barriers,, RISK, (1991), 28.   Google Scholar

[16]

M. Schroder, On the valuation of double-barrier options: Computational aspects,, Journal of Computaional Finance, 3 (2000), 1.   Google Scholar

[17]

S. E. Shreve, "Stochastic Calculus for Fiance II: Continuous-Time Models,", Springer-Verlag, (2004).   Google Scholar

[18]

J. Sidenius, Double barrier options: Valuation by path counting,, Computational Finance, 1 (1998), 63.   Google Scholar

[19]

J. Tsitsiklis and B. Van Roy, Regression methods for pricing complex American style options,, IEEE Transactions on Neural Networks, 12 (2001), 694.  doi: 10.1109/72.935083.  Google Scholar

[1]

Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079

[2]

Anton A. Kutsenko. Isomorphism between one-Dimensional and multidimensional finite difference operators. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020270

[3]

Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077

[4]

Lingfeng Li, Shousheng Luo, Xue-Cheng Tai, Jiang Yang. A new variational approach based on level-set function for convex hull problem with outliers. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020070

[5]

Hong Niu, Zhijiang Feng, Qijin Xiao, Yajun Zhang. A PID control method based on optimal control strategy. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 117-126. doi: 10.3934/naco.2020019

[6]

Yue Feng, Yujie Liu, Ruishu Wang, Shangyou Zhang. A conforming discontinuous Galerkin finite element method on rectangular partitions. Electronic Research Archive, , () : -. doi: 10.3934/era.2020120

[7]

Bernard Bonnard, Jérémy Rouot. Geometric optimal techniques to control the muscular force response to functional electrical stimulation using a non-isometric force-fatigue model. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2020032

[8]

Gang Bao, Mingming Zhang, Bin Hu, Peijun Li. An adaptive finite element DtN method for the three-dimensional acoustic scattering problem. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020351

[9]

Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296

[10]

Andrew D. Lewis. Erratum for "nonholonomic and constrained variational mechanics". Journal of Geometric Mechanics, 2020, 12 (4) : 671-675. doi: 10.3934/jgm.2020033

[11]

Meng Chen, Yong Hu, Matteo Penegini. On projective threefolds of general type with small positive geometric genus. Electronic Research Archive, , () : -. doi: 10.3934/era.2020117

[12]

Shipra Singh, Aviv Gibali, Xiaolong Qin. Cooperation in traffic network problems via evolutionary split variational inequalities. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020170

[13]

José Madrid, João P. G. Ramos. On optimal autocorrelation inequalities on the real line. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020271

[14]

Wenbin Li, Jianliang Qian. Simultaneously recovering both domain and varying density in inverse gravimetry by efficient level-set methods. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020073

[15]

Mehdi Bastani, Davod Khojasteh Salkuyeh. On the GSOR iteration method for image restoration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 27-43. doi: 10.3934/naco.2020013

[16]

Tommi Brander, Joonas Ilmavirta, Petteri Piiroinen, Teemu Tyni. Optimal recovery of a radiating source with multiple frequencies along one line. Inverse Problems & Imaging, 2020, 14 (6) : 967-983. doi: 10.3934/ipi.2020044

[17]

Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020046

[18]

Li-Bin Liu, Ying Liang, Jian Zhang, Xiaobing Bao. A robust adaptive grid method for singularly perturbed Burger-Huxley equations. Electronic Research Archive, 2020, 28 (4) : 1439-1457. doi: 10.3934/era.2020076

[19]

Zexuan Liu, Zhiyuan Sun, Jerry Zhijian Yang. A numerical study of superconvergence of the discontinuous Galerkin method by patch reconstruction. Electronic Research Archive, 2020, 28 (4) : 1487-1501. doi: 10.3934/era.2020078

[20]

Yuxia Guo, Shaolong Peng. A direct method of moving planes for fully nonlinear nonlocal operators and applications. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020462

2019 Impact Factor: 1.366

Metrics

  • PDF downloads (25)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]