American Institute of Mathematical Sciences

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March  2022, 1(1): 1-51. doi: 10.3934/fmf.2021001

Geometric step options and Lévy models: Duality, PIDEs, and semi-analytical pricing

Walter Farkas 1,2,3,, and  Ludovic Mathys 1,2,3,

1. 

Department of Banking and Finance, University of Zurich, Switzerland
2. 

Department of Mathematics, ETH Zurich, Switzerland
3. 

Swiss Finance Institute, Switzerland

* Corresponding author: Walter Farkas (walter.farkas@bf.uzh.ch)

Received  January 2021 Revised  April 2021 Published  March 2022 Early access  May 2021

The present article studies geometric step options in exponential Lévy markets. Our contribution is manifold and extends several aspects of the geometric step option pricing literature. First, we provide symmetry and duality relations and derive various characterizations for both European-type and American-type geometric double barrier step options. In particular, we are able to obtain a jump-diffusion disentanglement for the early exercise premium of American-type geometric double barrier step contracts and its maturity-randomized equivalent as well as to characterize the diffusion and jump contributions to these early exercise premiums separately by means of partial integro-differential equations and ordinary integro-differential equations. As an application of our characterizations, we derive semi-analytical pricing results for (regular) European-type and American-type geometric down-and-out step call options under hyper-exponential jump-diffusion models. Lastly, we use the latter results to discuss the early exercise structure of geometric step options once jumps are added and to subsequently provide an analysis of the impact of jumps on the price and hedging parameters of (European-type and American-type) geometric step contracts.

Keywords: Geometric step options, American-type options, Lévy markets, jump-diffusion disentanglement, maturity-randomization.
Mathematics Subject Classification: 91-08, 91B25, 91B70, 91G20, 91G60, 91G80.
Citation: Walter Farkas, Ludovic Mathys. Geometric step options and Lévy models: Duality, PIDEs, and semi-analytical pricing. Frontiers of Mathematical Finance, 2022, 1 (1) : 1-51. doi: 10.3934/fmf.2021001
References:
[1]

J. Abate and W. Whitt, A unified framework for numerically inverting Laplace transforms, INFORMS Journal on Computing, 18 (2006), 408-421.  doi: 10.1287/ijoc.1050.0137.  Google Scholar

[2]

D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511809781.  Google Scholar

[3]

O. E. Barndorff-Nielsen, Normal inverse Gaussian distributions and stochastic volatility modelling, Scandinavian Journal of Statistics, 24 (1997), 1-13.  doi: 10.1111/1467-9469.00045.  Google Scholar

[4]

F. Black and M. Scholes, The pricing of options and corporate liabilities, Journal of Political Economy, 81 (1973), 637-654.  doi: 10.1086/260062.  Google Scholar

[5]

M. Boyarchenko and S. Boyarchenko, Double barrier options in regime-switching hyper-exponential jump-diffusion models, International Journal of Theoretical and Applied Finance, 14 (2011), 1005-1043.  doi: 10.1142/S0219024911006620.  Google Scholar

[6]

S. Boyarchenko and S. Levendorskii, Efficient Laplace inversion, Wiener-Hopf factorization and pricing lookbacks, International Journal of Theoretical and Applied Finance, 16 (2013), 1350011. doi: 10.1142/S0219024913500118.  Google Scholar

[7]

M. Boyarchenko and S. Levendorskii, Static and semi-static hedging as contrarian or conformist bets, preprint, arXiv: 1902.02854. doi: 10.1111/mafi.12240.  Google Scholar

[8]

N. Cai, On first passage times of a hyper-exponential jump diffusion process, Operations Reseearch Letters, 37 (2009), 127-134.  doi: 10.1016/j.orl.2009.01.002.  Google Scholar

[9]

N. Cai, N. Chen and X. Wan, Pricing double-barrier options under a flexible jump diffusion model, Operations Research Letters, 37 (2009), 163-167.  doi: 10.1016/j.orl.2009.02.006.  Google Scholar

[10]

N. Cai, N. Chen and X. Wan, Occupation times of jump-diffusion processes with double exponential jumps and the pricing of options, Mathematics of Operations Research, 35 (2010), 412-437.  doi: 10.1287/moor.1100.0447.  Google Scholar

[11]

N. Cai and S. G. Kou, Option pricing under a mixed-exponential jump diffusion model, Management Science, 57 (2011), 2067-2081.  doi: 10.1287/opre.1110.1006.  Google Scholar

[12]

N. Cai and S. G. Kou, Pricing Asian options under a hyper-exponential jump-diffusion model, Operations Research, 60 (2012), 64-77.  doi: 10.1287/opre.1110.1006.  Google Scholar

[13]

G. Campolieti, R. N. Makarov and K. Wouterloot, Pricing step options under the CEV and other solvable diffusion models, International Journal of Theoretical and Applied Finance, 16 (2013), 1350027. doi: 10.1142/S0219024913500271.  Google Scholar

[14]

P. P. Carr, Randomization and the American put, The Review of Financial Studies, 11 (1998), 597-626.   Google Scholar

[15]

P. P. Carr, H. Geman, D. B. Madan and M. Yor, The fine structure of asset returns: An empirical investigation, Journal of Business, 75 (2002), 305-332.   Google Scholar

[16]

M. Chesney and N. Vasiljevic, Parisian options with jumps: A maturity-excursion randomization approach, Quant. Finance, 18 (2018), 1887-1908.  doi: 10.1080/14697688.2018.1444785.  Google Scholar

[17]

Y. Chuancun, S. Ying and W. Yuzhen, Exit problems for jump processes with applications to dividend problems, Journal of Computational and Applied Mathematics, 245 (2013), 30-52.  doi: 10.1016/j.cam.2012.12.004.  Google Scholar

[18]

D. Davydov and V. Linetsky, Structuring, pricing and hedging double barrier step options, Journal of Computational Finance, 5 (2002), 55-87.   Google Scholar

[19]

J. Detemple, A. S. Laminou and F. Moraux, American step options, European Journal of Operational Research, 282 (2020), 363-385.  doi: 10.1016/j.ejor.2019.09.009.  Google Scholar

[20]

J. Fajardo and E. Mordecki, Symmetry and duality in Lévy markets, Quant. Finance, 6 (2006), 219-227.  doi: 10.1080/14697680600680068.  Google Scholar

[21]

J. Fajardo and E. Mordecki, Skewness premium with Lévy processes, Quant. Finance, 14 (2014), 1619-1616.  doi: 10.1080/14697688.2011.618809.  Google Scholar

[22]

W. Farkas, L. Mathys and N. Vasiljević, Intra-Horizon expected shortfall and risk structure in models with jumps, Mathematical Finance, 31 (2021), 772-823.  doi: 10.1111/mafi.12302.  Google Scholar

[23]

M. B. Garman and S. W. Kohlhagen, Foreign currency option values, Journal of International Money and Finance, 2 (1983), 231-237.   Google Scholar

[24]

H. U. Gerber and E. S. W. Shiu, Option pricing by Esscher transforms, Transactions of the Society of Actuaries, 46 (1994), 99-191.   Google Scholar

[25]

M. J. Harrison and S. R. Pliska, Martingales and stochastic integrals in the theory of continous trading, Stochastic Process. Appl., 11 (1981), 215-260.  doi: 10.1016/0304-4149(81)90026-0.  Google Scholar

[26]

M. Hofer and P. Mayer, Pricing and hedging of lookback options in hyperexponential jump diffusion models, Applied Mathematical Finance, 20 (2013), 489-511.  doi: 10.1080/1350486X.2013.774985.  Google Scholar

[27]

K. R. Jackson, S. Jaimungal and V. Surkov, Fourier space time-stepping for option pricing with Lévy models, Journal of Computational Finance, 12 (2008), 1-29.  doi: 10.21314/JCF.2008.178.  Google Scholar

[28]

M. Jeanblanc and M. Chesney, Pricing American currency options in an exponential Lévy model, Applied Mathematical Finance, 11 (2004), 207-225.   Google Scholar

[29]

M. Jeanblanc, M. Yor and M. Chesney, Mathematical Methods for Financial Markets, Springer Finance, Springer, Berlin, 2006. doi: 10.1007/978-1-84628-737-4.  Google Scholar

[30]

T. Kimura, Alternative randomization for valuing American options, Asia-Pacific Journal of Operational Research, 27 (2010), 167-187.  doi: 10.1142/S0217595910002624.  Google Scholar

[31]

S. G. Kou, A jump-diffusion model for option pricing, Management Science, 48 (2002), 1086-1101.   Google Scholar

[32]

S. G. Kou and H. Wang, First passage times of a jump diffusion process, Advances in Applied Probability, 35 (2003), 504-531.  doi: 10.1239/aap/1051201658.  Google Scholar

[33]

S. G. Kou and H. Wang, Option pricing under a double exponential jump diffusion model, Management Science, 50 (2004), 1178-1192.  doi: 10.1287/opre.1110.1006.  Google Scholar

[34]

A. Kuznetsov, On the convergence of the Gaver-Stehfest algorithm, Siam Journal on Numerical Analysis, 51 (2013), 2984-2998.  doi: 10.1137/13091974X.  Google Scholar

[35]

D. Lamberton and M. Mikou, The smooth-fit property in an exponential Lévy model, Journal of Applied Probability, 49 (2011), 137-149.  doi: 10.1017/S0021900200008901.  Google Scholar

[36]

M. Leippold and N. Vasiljević, Pricing and disentanglement of American puts in the hyper-exponential jump-diffusion model, Journal of Banking and Finance, 77 (2017), 78-94.   Google Scholar

[37]

M. Leippold and N. Vasiljević, Option-Implied intra-horizon value-at-risk, Management Science, 66 (2019), 397-414.  doi: 10.1287/mnsc.2018.3157.  Google Scholar

[38]

S. Levendorskii, Pricing of the American put under Lévy processes, International Journal of Theoretical and Applied Finance, 7 (2004), 303-335.  doi: 10.1142/S0219024904002463.  Google Scholar

[39]

J. Lin and K. Palmer, Convergence of barrier option prices in the binomial model, Mathematical Finance, 23 (2013), 318-338.  doi: 10.1111/j.1467-9965.2011.00501.x.  Google Scholar

[40]

V. Linetsky, Step options, Mathematical Finance, 9 (1999), 55-96.  doi: 10.1111/1467-9965.00063.  Google Scholar

[41]

D. B. Madan, P. P. Carr and E. C. Chang, The variance gamma process and option pricing, European Finance Review, 2 (1998), 79-105.   Google Scholar

[42]

D. B. Madan and E. Seneta, The variance gamma model for share market returns, The Journal of Business, 63 (1990), 511-524.   Google Scholar

[43]

L. Mathys, Valuing tradeability in exponential Lévy models, Quantitative Finance and Economics, 4 (2020), 459-488.  doi: 10.3934/QFE.2020021.  Google Scholar

[44]

L. Mathys, On extensions of the Barone-Adesi & Whaley method to price American-type options, Journal of Computational Finance, 24 (2020), 33-76.  doi: 10.21314/JCF.2020.397.  Google Scholar

[45]

J. P. V. Nunes, J. P. Ruas and J. C. Dias, Early exercise boundaries for American-style knock-out options, European Journal of Operational Research, 285 (2020), 753-766.  doi: 10.1016/j.ejor.2020.02.006.  Google Scholar

[46]

G. Peskir and A. N. Shiryaev, Optimal Stopping and Free-Boundary Problems, Lectures in Mathematics, ETH Zürich, Bikhäuser, 2006.  Google Scholar

[47]

N. Rodosthenous and H. Zhang, Beating the omega clock: An optimal stopping problem with random time-horizon under spectrally negative Lévy models, Annals of Applied Probability, 28 (2018), 2105-2140.  doi: 10.1214/17-AAP1322.  Google Scholar

[48]

K.-I. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1999.  Google Scholar

[49]

L. Trigeorgis and A. E. Tsekrekos, Real options in operations research: A review, European Journal of Operational Research, 270 (2018), 1-24.  doi: 10.1016/j.ejor.2017.11.055.  Google Scholar

[50]

P. P. Valko and J. Abate, Comparison of sequence accelerators for the Gaver method of numerical Laplace transform inversion, Computers & Mathematics with Applications, 48 (2004), 629-636.  doi: 10.1016/j.camwa.2002.10.017.  Google Scholar

[51]

H. Y. Wong and J. Zhao, Valuing American options under the CEV model by Laplace-Carson transforms, Operations Research Letters, 38 (2010), 474-481.  doi: 10.1016/j.orl.2010.07.006.  Google Scholar

[52]

L. Wu and J. Zhou, Occupation times of hyper-exponential jump diffusion processes with application to price step options, Journal of Computational and Applied Mathematics, 294 (2016), 251-274.  doi: 10.1016/j.cam.2015.09.001.  Google Scholar

[53]

L. Wu, J. Zhou and Y. Bai, Occupation times of Lévy-driven Ornstein-Uhlenbeck processes with two-sided exponential jumps and applications, Statistics and Probability Letters, 125 (2017), 80-90.  doi: 10.1016/j.spl.2017.01.021.  Google Scholar

[54]

X. Xing and H. Yang, American type geometric step options, Journal of Industrial and Management Optimization, 9 (2013), 549-560.  doi: 10.3934/jimo.2013.9.549.  Google Scholar

show all references

References:
[1]

J. Abate and W. Whitt, A unified framework for numerically inverting Laplace transforms, INFORMS Journal on Computing, 18 (2006), 408-421.  doi: 10.1287/ijoc.1050.0137.  Google Scholar

[2]

D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511809781.  Google Scholar

[3]

O. E. Barndorff-Nielsen, Normal inverse Gaussian distributions and stochastic volatility modelling, Scandinavian Journal of Statistics, 24 (1997), 1-13.  doi: 10.1111/1467-9469.00045.  Google Scholar

[4]

F. Black and M. Scholes, The pricing of options and corporate liabilities, Journal of Political Economy, 81 (1973), 637-654.  doi: 10.1086/260062.  Google Scholar

[5]

M. Boyarchenko and S. Boyarchenko, Double barrier options in regime-switching hyper-exponential jump-diffusion models, International Journal of Theoretical and Applied Finance, 14 (2011), 1005-1043.  doi: 10.1142/S0219024911006620.  Google Scholar

[6]

S. Boyarchenko and S. Levendorskii, Efficient Laplace inversion, Wiener-Hopf factorization and pricing lookbacks, International Journal of Theoretical and Applied Finance, 16 (2013), 1350011. doi: 10.1142/S0219024913500118.  Google Scholar

[7]

M. Boyarchenko and S. Levendorskii, Static and semi-static hedging as contrarian or conformist bets, preprint, arXiv: 1902.02854. doi: 10.1111/mafi.12240.  Google Scholar

[8]

N. Cai, On first passage times of a hyper-exponential jump diffusion process, Operations Reseearch Letters, 37 (2009), 127-134.  doi: 10.1016/j.orl.2009.01.002.  Google Scholar

[9]

N. Cai, N. Chen and X. Wan, Pricing double-barrier options under a flexible jump diffusion model, Operations Research Letters, 37 (2009), 163-167.  doi: 10.1016/j.orl.2009.02.006.  Google Scholar

[10]

N. Cai, N. Chen and X. Wan, Occupation times of jump-diffusion processes with double exponential jumps and the pricing of options, Mathematics of Operations Research, 35 (2010), 412-437.  doi: 10.1287/moor.1100.0447.  Google Scholar

[11]

N. Cai and S. G. Kou, Option pricing under a mixed-exponential jump diffusion model, Management Science, 57 (2011), 2067-2081.  doi: 10.1287/opre.1110.1006.  Google Scholar

[12]

N. Cai and S. G. Kou, Pricing Asian options under a hyper-exponential jump-diffusion model, Operations Research, 60 (2012), 64-77.  doi: 10.1287/opre.1110.1006.  Google Scholar

[13]

G. Campolieti, R. N. Makarov and K. Wouterloot, Pricing step options under the CEV and other solvable diffusion models, International Journal of Theoretical and Applied Finance, 16 (2013), 1350027. doi: 10.1142/S0219024913500271.  Google Scholar

[14]

P. P. Carr, Randomization and the American put, The Review of Financial Studies, 11 (1998), 597-626.   Google Scholar

[15]

P. P. Carr, H. Geman, D. B. Madan and M. Yor, The fine structure of asset returns: An empirical investigation, Journal of Business, 75 (2002), 305-332.   Google Scholar

[16]

M. Chesney and N. Vasiljevic, Parisian options with jumps: A maturity-excursion randomization approach, Quant. Finance, 18 (2018), 1887-1908.  doi: 10.1080/14697688.2018.1444785.  Google Scholar

[17]

Y. Chuancun, S. Ying and W. Yuzhen, Exit problems for jump processes with applications to dividend problems, Journal of Computational and Applied Mathematics, 245 (2013), 30-52.  doi: 10.1016/j.cam.2012.12.004.  Google Scholar

[18]

D. Davydov and V. Linetsky, Structuring, pricing and hedging double barrier step options, Journal of Computational Finance, 5 (2002), 55-87.   Google Scholar

[19]

J. Detemple, A. S. Laminou and F. Moraux, American step options, European Journal of Operational Research, 282 (2020), 363-385.  doi: 10.1016/j.ejor.2019.09.009.  Google Scholar

[20]

J. Fajardo and E. Mordecki, Symmetry and duality in Lévy markets, Quant. Finance, 6 (2006), 219-227.  doi: 10.1080/14697680600680068.  Google Scholar

[21]

J. Fajardo and E. Mordecki, Skewness premium with Lévy processes, Quant. Finance, 14 (2014), 1619-1616.  doi: 10.1080/14697688.2011.618809.  Google Scholar

[22]

W. Farkas, L. Mathys and N. Vasiljević, Intra-Horizon expected shortfall and risk structure in models with jumps, Mathematical Finance, 31 (2021), 772-823.  doi: 10.1111/mafi.12302.  Google Scholar

[23]

M. B. Garman and S. W. Kohlhagen, Foreign currency option values, Journal of International Money and Finance, 2 (1983), 231-237.   Google Scholar

[24]

H. U. Gerber and E. S. W. Shiu, Option pricing by Esscher transforms, Transactions of the Society of Actuaries, 46 (1994), 99-191.   Google Scholar

[25]

M. J. Harrison and S. R. Pliska, Martingales and stochastic integrals in the theory of continous trading, Stochastic Process. Appl., 11 (1981), 215-260.  doi: 10.1016/0304-4149(81)90026-0.  Google Scholar

[26]

M. Hofer and P. Mayer, Pricing and hedging of lookback options in hyperexponential jump diffusion models, Applied Mathematical Finance, 20 (2013), 489-511.  doi: 10.1080/1350486X.2013.774985.  Google Scholar

[27]

K. R. Jackson, S. Jaimungal and V. Surkov, Fourier space time-stepping for option pricing with Lévy models, Journal of Computational Finance, 12 (2008), 1-29.  doi: 10.21314/JCF.2008.178.  Google Scholar

[28]

M. Jeanblanc and M. Chesney, Pricing American currency options in an exponential Lévy model, Applied Mathematical Finance, 11 (2004), 207-225.   Google Scholar

[29]

M. Jeanblanc, M. Yor and M. Chesney, Mathematical Methods for Financial Markets, Springer Finance, Springer, Berlin, 2006. doi: 10.1007/978-1-84628-737-4.  Google Scholar

[30]

T. Kimura, Alternative randomization for valuing American options, Asia-Pacific Journal of Operational Research, 27 (2010), 167-187.  doi: 10.1142/S0217595910002624.  Google Scholar

[31]

S. G. Kou, A jump-diffusion model for option pricing, Management Science, 48 (2002), 1086-1101.   Google Scholar

[32]

S. G. Kou and H. Wang, First passage times of a jump diffusion process, Advances in Applied Probability, 35 (2003), 504-531.  doi: 10.1239/aap/1051201658.  Google Scholar

[33]

S. G. Kou and H. Wang, Option pricing under a double exponential jump diffusion model, Management Science, 50 (2004), 1178-1192.  doi: 10.1287/opre.1110.1006.  Google Scholar

[34]

A. Kuznetsov, On the convergence of the Gaver-Stehfest algorithm, Siam Journal on Numerical Analysis, 51 (2013), 2984-2998.  doi: 10.1137/13091974X.  Google Scholar

[35]

D. Lamberton and M. Mikou, The smooth-fit property in an exponential Lévy model, Journal of Applied Probability, 49 (2011), 137-149.  doi: 10.1017/S0021900200008901.  Google Scholar

[36]

M. Leippold and N. Vasiljević, Pricing and disentanglement of American puts in the hyper-exponential jump-diffusion model, Journal of Banking and Finance, 77 (2017), 78-94.   Google Scholar

[37]

M. Leippold and N. Vasiljević, Option-Implied intra-horizon value-at-risk, Management Science, 66 (2019), 397-414.  doi: 10.1287/mnsc.2018.3157.  Google Scholar

[38]

S. Levendorskii, Pricing of the American put under Lévy processes, International Journal of Theoretical and Applied Finance, 7 (2004), 303-335.  doi: 10.1142/S0219024904002463.  Google Scholar

[39]

J. Lin and K. Palmer, Convergence of barrier option prices in the binomial model, Mathematical Finance, 23 (2013), 318-338.  doi: 10.1111/j.1467-9965.2011.00501.x.  Google Scholar

[40]

V. Linetsky, Step options, Mathematical Finance, 9 (1999), 55-96.  doi: 10.1111/1467-9965.00063.  Google Scholar

[41]

D. B. Madan, P. P. Carr and E. C. Chang, The variance gamma process and option pricing, European Finance Review, 2 (1998), 79-105.   Google Scholar

[42]

D. B. Madan and E. Seneta, The variance gamma model for share market returns, The Journal of Business, 63 (1990), 511-524.   Google Scholar

[43]

L. Mathys, Valuing tradeability in exponential Lévy models, Quantitative Finance and Economics, 4 (2020), 459-488.  doi: 10.3934/QFE.2020021.  Google Scholar

[44]

L. Mathys, On extensions of the Barone-Adesi & Whaley method to price American-type options, Journal of Computational Finance, 24 (2020), 33-76.  doi: 10.21314/JCF.2020.397.  Google Scholar

[45]

J. P. V. Nunes, J. P. Ruas and J. C. Dias, Early exercise boundaries for American-style knock-out options, European Journal of Operational Research, 285 (2020), 753-766.  doi: 10.1016/j.ejor.2020.02.006.  Google Scholar

[46]

G. Peskir and A. N. Shiryaev, Optimal Stopping and Free-Boundary Problems, Lectures in Mathematics, ETH Zürich, Bikhäuser, 2006.  Google Scholar

[47]

N. Rodosthenous and H. Zhang, Beating the omega clock: An optimal stopping problem with random time-horizon under spectrally negative Lévy models, Annals of Applied Probability, 28 (2018), 2105-2140.  doi: 10.1214/17-AAP1322.  Google Scholar

[48]

K.-I. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1999.  Google Scholar

[49]

L. Trigeorgis and A. E. Tsekrekos, Real options in operations research: A review, European Journal of Operational Research, 270 (2018), 1-24.  doi: 10.1016/j.ejor.2017.11.055.  Google Scholar

[50]

P. P. Valko and J. Abate, Comparison of sequence accelerators for the Gaver method of numerical Laplace transform inversion, Computers & Mathematics with Applications, 48 (2004), 629-636.  doi: 10.1016/j.camwa.2002.10.017.  Google Scholar

[51]

H. Y. Wong and J. Zhao, Valuing American options under the CEV model by Laplace-Carson transforms, Operations Research Letters, 38 (2010), 474-481.  doi: 10.1016/j.orl.2010.07.006.  Google Scholar

[52]

L. Wu and J. Zhou, Occupation times of hyper-exponential jump diffusion processes with application to price step options, Journal of Computational and Applied Mathematics, 294 (2016), 251-274.  doi: 10.1016/j.cam.2015.09.001.  Google Scholar

[53]

L. Wu, J. Zhou and Y. Bai, Occupation times of Lévy-driven Ornstein-Uhlenbeck processes with two-sided exponential jumps and applications, Statistics and Probability Letters, 125 (2017), 80-90.  doi: 10.1016/j.spl.2017.01.021.  Google Scholar

[54]

X. Xing and H. Yang, American type geometric step options, Journal of Industrial and Management Optimization, 9 (2013), 549-560.  doi: 10.3934/jimo.2013.9.549.  Google Scholar

Figure 1.  Relative early exercise contribution, diffusion contribution to the early exercise premium, and absolute early exercise premium of the geometric down-and-out step call as functions of the underlying price $ S_{0} \in [85,115] $ and the knock-out rate $ \rho_{L} \in [-1000,0] $, when the remaining parameters are chosen as: $ \mathcal{T} = 1.0 $, $ \sigma_{X} = 0.2 $, $ r = 0.05 $, $ \delta = 0.07 $, $ K = 100 $, $ L = 95 $, $ \lambda = 5 $, $ p = 0.5 $, $ \xi = 25 $, $ \eta = 50 $
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Figure 2.  Difference in the prices, deltas, and gammas for the geometric down-and-out step calls with and without jumps as functions of the underlying price $ S_{0} \in [85.115] $ and the intensity parameter $ \lambda \in [0,20] $, when the remaining parameters are chosen as: $ \mathcal{T} = 1.0 $, $ \sigma_{X} = 0.2 $, $ r = 0.05 $, $ \delta = 0.07 $, $ K = 100 $, $ L = 95 $, $ \rho_{L} = -26.34 $, $ p = 0.5 $, $ \xi = 25 $, $ \eta = 50 $
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Figure 3.  Difference in the prices, deltas, and gammas for the geometric down-and-out step calls with and without jumps as functions of the underlying price $ S_{0} \in [85.115] $ and the positive jump parameter $ \xi \in [5,100] $, when the remaining parameters are chosen as: $ \mathcal{T} = 1.0 $, $ \sigma_{X} = 0.2 $, $ r = 0.05 $, $ \delta = 0.07 $, $ K = 100 $, $ L = 95 $, $ \rho_{L} = -26.34 $, $ \lambda = 5 $, $ p = 0.5 $, $ \eta = 50 $
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Table 1.  Theoretical (down-and-out) call values and diffusion contributions to the early exercise premium for $r = 0.05$, $\delta = 0.07$, $S_{0} = 100$, $K = 100$, $L = 95$, $\rho_{L} = -26.34$, $p = 0.7$, $\xi = 25$ and $\eta = 50$
(Down-and-Out) Call Option Prices
Parameters Standard Call Price Step Call Price Barrier Call Price
$\lambda$ Euro Amer DC (%) Euro Amer DC (%) Euro Amer DC (%)
$1$ $6.833$ $7.040$ $91.52 \%$ $4.596$ $4.789$ $91.71 \% $ $3.374$ $3.551$ $91.88 \%$
$S_{0} = 100$ $0.1$ $6.622$ $6.822$ $99.07 \%$ $4.519$ $4.706$ $99.09 \%$ $3.338$ $3.514$ $99.12 \%$
$\sigma_{X}=0.2$ $0.01$ $6.600$ $6.800$ $99.91 \% $ $4.511$ $4.698$ $99.91 \%$ $3.334$ $3.510$ $99.91 \%$
$\mathcal{T} = 1.0$ $0.001$ $6.598$ $6.797$ $99.99 \%$ $4.510$ $4.697$ $99.99 \%$ $3.333$ $3.509$ $99.99 \%$
$0.0001$ $6.598$ $6.797$ $100.00 \%$ $4.510$ $4.697$ $100.00\% $ $3.333$ $3.509$ $100.00 \% $
B & S Values - 6.598 6.885 - 4.511 4.745 - 3.332 3.529 -
Rel. Error (%) - 0.001% -1.277% - 0.015% -1.025% - 0.025% -0.568% -
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(Down-and-Out) Call Option Prices
Parameters Standard Call Price Step Call Price Barrier Call Price
$\lambda$ Euro Amer DC (%) Euro Amer DC (%) Euro Amer DC (%)
$1$ $6.833$ $7.040$ $91.52 \%$ $4.596$ $4.789$ $91.71 \% $ $3.374$ $3.551$ $91.88 \%$
$S_{0} = 100$ $0.1$ $6.622$ $6.822$ $99.07 \%$ $4.519$ $4.706$ $99.09 \%$ $3.338$ $3.514$ $99.12 \%$
$\sigma_{X}=0.2$ $0.01$ $6.600$ $6.800$ $99.91 \% $ $4.511$ $4.698$ $99.91 \%$ $3.334$ $3.510$ $99.91 \%$
$\mathcal{T} = 1.0$ $0.001$ $6.598$ $6.797$ $99.99 \%$ $4.510$ $4.697$ $99.99 \%$ $3.333$ $3.509$ $99.99 \%$
$0.0001$ $6.598$ $6.797$ $100.00 \%$ $4.510$ $4.697$ $100.00\% $ $3.333$ $3.509$ $100.00 \% $
B & S Values - 6.598 6.885 - 4.511 4.745 - 3.332 3.529 -
Rel. Error (%) - 0.001% -1.277% - 0.015% -1.025% - 0.025% -0.568% -
Table 2.  Theoretical American-type (down-and-out) call values and structure of the early exercise premium for $\delta = 0.05$, $K = 100$, $\sigma_{X} = 0.2$, and $\mathcal{T} = 0.25$
(Down-and-Out) Barrier Call Option Prices
Parameters Barrier Option Values
$\lambda$ $p$ $\xi$ $\eta$ FST Amer DC (%)
$3.029$ $0.381$ $26$ $24$ $1.016$ $1.017$ $80.08\%$
$3.051$ $0.386$ $51$ $24$ $0.920$ $0.921$ $96.04\%$
(1) $2.991$ $0.386$ $26$ $49$ $0.934$ $0.934$ $78.03\%$
$S_{0}=90$ $3.013$ $0.390$ $51$ $49$ $0.837$ $0.838$ $95.22\%$
$r = 0.07$ $7.067$ $0.381$ $26$ $24 $ $1.337$ $1.338$ $69.81\%$
$L = 78.261$ $7.120$ $0.386$ $51$ $24 $ $1.126$ $1.127$ $93.44\%$
$6.978$ $0.386$ $26$ $49 $ $1.141$ $1.141$ $67.16\%$
$7.031$ $0.390$ $51$ $49 $ $0.924$ $0.924$ $92.04\%$
$3.029$ $0.381$ $26$ $24$ $3.850$ $3.850$ $85.02\%$
$3.051$ $0.386$ $51$ $24$ $3.741$ $3.728$ $96.73\%$
(1) $2.991$ $0.386$ $26$ $49$ $3.693$ $3.677$ $84.91\%$
$S_{0}=100$ $3.013$ $0.390$ $51$ $49$ $3.580$ $3.555$ $96.66\%$
$r = 0.00$ $7.067$ $0.381$ $26$ $24 $ $4.316$ $4.366$ $72.44\%$
$L = 86.957$ $7.120$ $0.386$ $51$ $24 $ $4.089$ $4.103$ $93.13\%$
$6.978$ $0.386$ $26$ $49 $ $3.970$ $3.977$ $71.33\%$
$7.031$ $0.390$ $51$ $49 $ $3.719$ $3.699$ $92.77\%$
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(Down-and-Out) Barrier Call Option Prices
Parameters Barrier Option Values
$\lambda$ $p$ $\xi$ $\eta$ FST Amer DC (%)
$3.029$ $0.381$ $26$ $24$ $1.016$ $1.017$ $80.08\%$
$3.051$ $0.386$ $51$ $24$ $0.920$ $0.921$ $96.04\%$
(1) $2.991$ $0.386$ $26$ $49$ $0.934$ $0.934$ $78.03\%$
$S_{0}=90$ $3.013$ $0.390$ $51$ $49$ $0.837$ $0.838$ $95.22\%$
$r = 0.07$ $7.067$ $0.381$ $26$ $24 $ $1.337$ $1.338$ $69.81\%$
$L = 78.261$ $7.120$ $0.386$ $51$ $24 $ $1.126$ $1.127$ $93.44\%$
$6.978$ $0.386$ $26$ $49 $ $1.141$ $1.141$ $67.16\%$
$7.031$ $0.390$ $51$ $49 $ $0.924$ $0.924$ $92.04\%$
$3.029$ $0.381$ $26$ $24$ $3.850$ $3.850$ $85.02\%$
$3.051$ $0.386$ $51$ $24$ $3.741$ $3.728$ $96.73\%$
(1) $2.991$ $0.386$ $26$ $49$ $3.693$ $3.677$ $84.91\%$
$S_{0}=100$ $3.013$ $0.390$ $51$ $49$ $3.580$ $3.555$ $96.66\%$
$r = 0.00$ $7.067$ $0.381$ $26$ $24 $ $4.316$ $4.366$ $72.44\%$
$L = 86.957$ $7.120$ $0.386$ $51$ $24 $ $4.089$ $4.103$ $93.13\%$
$6.978$ $0.386$ $26$ $49 $ $3.970$ $3.977$ $71.33\%$
$7.031$ $0.390$ $51$ $49 $ $3.719$ $3.699$ $92.77\%$
Table 3.  Theoretical (down-and-out) call values and structure of the early exercise premium for $r = 0.05$, $\delta = 0.07$, $K = 100$, $L = 95$, $\rho_{L} = -26.34$, $p = 0.5$, $\xi = 50$ and $\eta = 25$
(Down-and-Out) Call Option Prices
Parameters Standard Call Price Step Call Price Barrier Call Price
$S_{0}$ Euro EEP EEP (%) DC (%) Euro EEP EEP (%) DC (%) Euro EEP EEP (%) DC (%)
$90$ $3.500$ $0.062$ $1.74 \%$ $94.20 \%$ $0.268$ $0.009$ $3.07 \% $ $94.32 \%$ $0$ $0$ - -
(1) $95$ $5.241$ $0.112$ $2.09 \%$ $94.27 \%$ $1.757$ $0.059$ $3.23 \%$ $94.33 \% $ $0$ $0$ - -
$\sigma_{X}=0.2$ $100$ $7.416$ $0.190$ $2.50 \%$ $94.34 \%$ $4.992$ $0.178$ $3.45 \%$ $94.36 \%$ $3.686$ $0.165$ $4.28 \%$ $94.37 \%$
$\lambda = 5.0$ $105$ $10.011$ $0.305$ $2.96 \% $ $94.40 \% $ $8.309$ $0.330$ $3.82 \%$ $94.39 \%$ $7.305$ $0.353$ $4.61 \%$ $94.40 \%$
$\mathcal{T} = 1.0$ $110$ $12.992$ $0.469$ $3.48 \% $ $94.46 \%$ $11.804$ $0.535$ $4.34 \%$ $94.44 \%$ $11.037$ $0.597$ $5.13 \%$ $94.44 \%$
$115$ $16.314$ $0.691$ $4.07 \% $ $94.52 \%$ $15.492$ $0.811$ $4.98 \%$ $94.50 \% $ $14.914$ $0.920$ $5.81 \%$ $94.54 \% $
$90$ $4.098$ $0.065$ $1.57 \%$ $89.68 \%$ $0.344$ $0.010$ $2.79 \% $ $89.87 \%$ $0$ $0$ - -
(2) $95$ $5.933$ $0.113$ $1.87 \%$ $89.80 \%$ $2.012$ $0.061$ $2.93 \%$ $89.89 \% $ $0$ $0$ - -
$\sigma_{X}=0.2$ $100$ $8.169$ $0.186$ $2.22 \%$ $89.90 \%$ $5.413$ $0.175$ $3.12 \%$ $89.93 \%$ $3.990$ $0.161$ $3.88 \%$ $89.95 \%$
$\lambda = 10.0$ $105$ $10.791$ $0.290$ $2.62 \% $ $90.00 \% $ $8.791$ $0.314$ $3.44 \%$ $89.99 \%$ $7.683$ $0.334$ $4.17 \%$ $89.99 \%$
$\mathcal{T} = 1.0$ $110$ $13.767$ $0.435$ $3.06 \% $ $90.08 \%$ $12.313$ $0.497$ $3.88 \%$ $90.05 \%$ $11.442$ $0.552$ $4.60 \%$ $90.05 \%$
$115$ $17.056$ $0.628$ $3.55 \% $ $90.17 \%$ $16.004$ $0.738$ $4.41 \%$ $90.12 \% $ $15.325$ $0.835$ $5.16 \%$ $90.13 \% $
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(Down-and-Out) Call Option Prices
Parameters Standard Call Price Step Call Price Barrier Call Price
$S_{0}$ Euro EEP EEP (%) DC (%) Euro EEP EEP (%) DC (%) Euro EEP EEP (%) DC (%)
$90$ $3.500$ $0.062$ $1.74 \%$ $94.20 \%$ $0.268$ $0.009$ $3.07 \% $ $94.32 \%$ $0$ $0$ - -
(1) $95$ $5.241$ $0.112$ $2.09 \%$ $94.27 \%$ $1.757$ $0.059$ $3.23 \%$ $94.33 \% $ $0$ $0$ - -
$\sigma_{X}=0.2$ $100$ $7.416$ $0.190$ $2.50 \%$ $94.34 \%$ $4.992$ $0.178$ $3.45 \%$ $94.36 \%$ $3.686$ $0.165$ $4.28 \%$ $94.37 \%$
$\lambda = 5.0$ $105$ $10.011$ $0.305$ $2.96 \% $ $94.40 \% $ $8.309$ $0.330$ $3.82 \%$ $94.39 \%$ $7.305$ $0.353$ $4.61 \%$ $94.40 \%$
$\mathcal{T} = 1.0$ $110$ $12.992$ $0.469$ $3.48 \% $ $94.46 \%$ $11.804$ $0.535$ $4.34 \%$ $94.44 \%$ $11.037$ $0.597$ $5.13 \%$ $94.44 \%$
$115$ $16.314$ $0.691$ $4.07 \% $ $94.52 \%$ $15.492$ $0.811$ $4.98 \%$ $94.50 \% $ $14.914$ $0.920$ $5.81 \%$ $94.54 \% $
$90$ $4.098$ $0.065$ $1.57 \%$ $89.68 \%$ $0.344$ $0.010$ $2.79 \% $ $89.87 \%$ $0$ $0$ - -
(2) $95$ $5.933$ $0.113$ $1.87 \%$ $89.80 \%$ $2.012$ $0.061$ $2.93 \%$ $89.89 \% $ $0$ $0$ - -
$\sigma_{X}=0.2$ $100$ $8.169$ $0.186$ $2.22 \%$ $89.90 \%$ $5.413$ $0.175$ $3.12 \%$ $89.93 \%$ $3.990$ $0.161$ $3.88 \%$ $89.95 \%$
$\lambda = 10.0$ $105$ $10.791$ $0.290$ $2.62 \% $ $90.00 \% $ $8.791$ $0.314$ $3.44 \%$ $89.99 \%$ $7.683$ $0.334$ $4.17 \%$ $89.99 \%$
$\mathcal{T} = 1.0$ $110$ $13.767$ $0.435$ $3.06 \% $ $90.08 \%$ $12.313$ $0.497$ $3.88 \%$ $90.05 \%$ $11.442$ $0.552$ $4.60 \%$ $90.05 \%$
$115$ $17.056$ $0.628$ $3.55 \% $ $90.17 \%$ $16.004$ $0.738$ $4.41 \%$ $90.12 \% $ $15.325$ $0.835$ $5.16 \%$ $90.13 \% $
Table 4.  Theoretical (down-and-out) call values and structure of the early exercise premium for $r = 0.05$, $\delta = 0.07$, $K = 100$, $L = 95$, $\rho_{L} = -26.34$, $p = 0.5$, $\xi = 50$ and $\eta = 50$
(Down-and-Out) Call Option Prices
Parameters Standard Call Price Step Call Price Barrier Call Price
$S_{0}$ Euro EEP EEP (%) DC (%) Euro EEP EEP (%) DC (%) Euro EEP EEP (%) DC (%)
$90$ $3.163$ $0.064$ $1.98 \%$ $93.97 \%$ $0.232$ $0.008$ $3.46 \% $ $94.10 \%$ $0$ $0$ - -
(1) $95$ $4.835$ $0.117$ $2.37 \%$ $94.05 \%$ $1.588$ $0.060$ $3.63 \%$ $94.12 \% $ $0$ $0$ - -
$\sigma_{X}=0.2$ $100$ $6.958$ $0.202$ $2.82 \%$ $94.12 \%$ $4.679$ $0.188$ $3.87 \%$ $94.14 \%$ $3.432$ $0.174$ $4.82\%$ $94.15 \%$
$\lambda = 5.0$ $105$ $9.523$ $0.328$ $3.33 \% $ $94.19 \% $ $7.949$ $0.355$ $4.28 \%$ $94.18 \%$ $6.983$ $0.382$ $5.18 \%$ $94.18 \%$
$\mathcal{T} = 1.0$ $110$ $12.498$ $0.509$ $3.91 \% $ $94.25 \%$ $11.430$ $0.583$ $4.85 \%$ $94.23 \%$ $10.702$ $0.654$ $5.76 \%$ $94.24 \%$
$115$ $15.835$ $0.758$ $4.57 \% $ $94.33 \%$ $15.122$ $0.891$ $5.57 \%$ $94.31 \% $ $14.586$ $1.017$ $6.52 \%$ $94.43 \% $
$90$ $3.441$ $0.068$ $1.94 \%$ $88.95 \%$ $0.268$ $0.009$ $3.40 \% $ $89.16 \%$ $0$ $0$ - -
(2) $95$ $5.155$ $0.121$ $2.30 \%$ $89.08 \%$ $1.685$ $0.062$ $3.55 \%$ $89.19 \% $ $0$ $0$ - -
$\sigma_{X}=0.2$ $100$ $7.303$ $0.204$ $2.72 \%$ $89.20 \%$ $4.836$ $0.190$ $3.77 \%$ $89.23 \%$ $3.522$ $0.174$ $4.71 \%$ $89.25 \%$
$\lambda = 10.0$ $105$ $9.875$ $0.325$ $3.19 \% $ $89.30 \% $ $8.138$ $0.352$ $4.14 \%$ $89.29 \%$ $7.107$ $0.377$ $5.04 \%$ $89.29 \%$
$\mathcal{T} = 1.0$ $110$ $12.839$ $0.497$ $3.72 \% $ $89.40 \%$ $11.636$ $0.569$ $4.66 \%$ $89.36 \%$ $10.845$ $0.638$ $5.56 \%$ $89.37 \%$
$115$ $16.152$ $0.729$ $4.32 \% $ $89.53 \%$ $15.330$ $0.859$ $5.31 \%$ $89.46 \% $ $14.737$ $0.981$ $6.24 \%$ $89.57 \% $
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(Down-and-Out) Call Option Prices
Parameters Standard Call Price Step Call Price Barrier Call Price
$S_{0}$ Euro EEP EEP (%) DC (%) Euro EEP EEP (%) DC (%) Euro EEP EEP (%) DC (%)
$90$ $3.163$ $0.064$ $1.98 \%$ $93.97 \%$ $0.232$ $0.008$ $3.46 \% $ $94.10 \%$ $0$ $0$ - -
(1) $95$ $4.835$ $0.117$ $2.37 \%$ $94.05 \%$ $1.588$ $0.060$ $3.63 \%$ $94.12 \% $ $0$ $0$ - -
$\sigma_{X}=0.2$ $100$ $6.958$ $0.202$ $2.82 \%$ $94.12 \%$ $4.679$ $0.188$ $3.87 \%$ $94.14 \%$ $3.432$ $0.174$ $4.82\%$ $94.15 \%$
$\lambda = 5.0$ $105$ $9.523$ $0.328$ $3.33 \% $ $94.19 \% $ $7.949$ $0.355$ $4.28 \%$ $94.18 \%$ $6.983$ $0.382$ $5.18 \%$ $94.18 \%$
$\mathcal{T} = 1.0$ $110$ $12.498$ $0.509$ $3.91 \% $ $94.25 \%$ $11.430$ $0.583$ $4.85 \%$ $94.23 \%$ $10.702$ $0.654$ $5.76 \%$ $94.24 \%$
$115$ $15.835$ $0.758$ $4.57 \% $ $94.33 \%$ $15.122$ $0.891$ $5.57 \%$ $94.31 \% $ $14.586$ $1.017$ $6.52 \%$ $94.43 \% $
$90$ $3.441$ $0.068$ $1.94 \%$ $88.95 \%$ $0.268$ $0.009$ $3.40 \% $ $89.16 \%$ $0$ $0$ - -
(2) $95$ $5.155$ $0.121$ $2.30 \%$ $89.08 \%$ $1.685$ $0.062$ $3.55 \%$ $89.19 \% $ $0$ $0$ - -
$\sigma_{X}=0.2$ $100$ $7.303$ $0.204$ $2.72 \%$ $89.20 \%$ $4.836$ $0.190$ $3.77 \%$ $89.23 \%$ $3.522$ $0.174$ $4.71 \%$ $89.25 \%$
$\lambda = 10.0$ $105$ $9.875$ $0.325$ $3.19 \% $ $89.30 \% $ $8.138$ $0.352$ $4.14 \%$ $89.29 \%$ $7.107$ $0.377$ $5.04 \%$ $89.29 \%$
$\mathcal{T} = 1.0$ $110$ $12.839$ $0.497$ $3.72 \% $ $89.40 \%$ $11.636$ $0.569$ $4.66 \%$ $89.36 \%$ $10.845$ $0.638$ $5.56 \%$ $89.37 \%$
$115$ $16.152$ $0.729$ $4.32 \% $ $89.53 \%$ $15.330$ $0.859$ $5.31 \%$ $89.46 \% $ $14.737$ $0.981$ $6.24 \%$ $89.57 \% $
Table 5.  Theoretical (down-and-out) call values and structure of the early exercise premium for $r = 0.05$, $\delta = 0.07$, $K = 100$, $L = 95$, $\rho_{L} = -26.34$, $p = 0.5$, $\xi = 25$ and $\eta = 50$
(Down-and-Out) Call Option Prices
Parameters Standard Call Price Step Call Price Barrier Call Price
$S_{0}$ Euro EEP EEP (%) DC (%) Euro EEP EEP (%) DC (%) Euro EEP EEP (%) DC (%)
$90$ $3.645$ $0.080$ $2.15 \%$ $75.53 \%$ $0.294$ $0.012$ $3.75 \% $ $76.36 \%$ $0$ $0$ - -
(1) $95$ $5.362$ $0.137$ $2.49 \%$ $75.97 \%$ $1.685$ $0.067$ $3.82 \%$ $76.45 \% $ $0$ $0$ - -
$\sigma_{X}=0.2$ $100$ $7.501$ $0.222$ $2.88 \%$ $76.37 \%$ $4.854$ $0.202$ $3.99 \%$ $76.61 \%$ $3.506$ $0.182$ $4.94\%$ $76.81 \%$
$\lambda = 5.0$ $105$ $10.054$ $0.345$ $3.31 \% $ $76.71 \% $ $8.177$ $0.368$ $4.31 \%$ $76.94 \%$ $7.110$ $0.391$ $5.21 \%$ $77.25 \%$
$\mathcal{T} = 1.0$ $110$ $12.994$ $0.514$ $3.80 \% $ $76.98 \%$ $11.685$ $0.585$ $4.77 \%$ $77.55 \%$ $10.861$ $0.652$ $5.67 \%$ $78.24 \%$
$115$ $16.279$ $0.740$ $4.35 \% $ $77.14 \%$ $15.381$ $0.870$ $5.35 \%$ $78.79 \% $ $14.759$ $0.989$ $6.28 \%$ $80.55 \% $
$90$ $4.347$ $0.096$ $2.16 \%$ $62.58 \%$ $0.391$ $0.015$ $3.78 \% $ $63.44 \%$ $0$ $0$ - -
(2) $95$ $6.141$ $0.155$ $2.45 \%$ $63.00 \%$ $1.865$ $0.074$ $3.82 \%$ $63.47 \% $ $0$ $0$ - -
$\sigma_{X}=0.2$ $100$ $8.321$ $0.238$ $2.78 \%$ $63.38 \%$ $5.152$ $0.212$ $3.94 \%$ $63.56 \%$ $3.649$ $0.188$ $4.89\%$ $63.68 \%$
$\lambda = 5.0$ $105$ $10.878$ $0.354$ $3.15 \% $ $63.72 \% $ $8.549$ $0.374$ $4.19 \%$ $63.76 \%$ $7.328$ $0.393$ $5.09 \%$ $63.90 \%$
$\mathcal{T} = 1.0$ $110$ $13.788$ $0.508$ $3.55 \% $ $64.01 \%$ $12.099$ $0.577$ $4.55 \%$ $64.10 \%$ $11.126$ $0.640$ $5.44 \%$ $64.46 \%$
$115$ $17.019$ $0.709$ $4.00 \% $ $64.18 \%$ $15.809$ $0.834$ $5.01 \%$ $64.74 \% $ $15.047$ $0.946$ $5.91 \%$ $65.93 \% $
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(Down-and-Out) Call Option Prices
Parameters Standard Call Price Step Call Price Barrier Call Price
$S_{0}$ Euro EEP EEP (%) DC (%) Euro EEP EEP (%) DC (%) Euro EEP EEP (%) DC (%)
$90$ $3.645$ $0.080$ $2.15 \%$ $75.53 \%$ $0.294$ $0.012$ $3.75 \% $ $76.36 \%$ $0$ $0$ - -
(1) $95$ $5.362$ $0.137$ $2.49 \%$ $75.97 \%$ $1.685$ $0.067$ $3.82 \%$ $76.45 \% $ $0$ $0$ - -
$\sigma_{X}=0.2$ $100$ $7.501$ $0.222$ $2.88 \%$ $76.37 \%$ $4.854$ $0.202$ $3.99 \%$ $76.61 \%$ $3.506$ $0.182$ $4.94\%$ $76.81 \%$
$\lambda = 5.0$ $105$ $10.054$ $0.345$ $3.31 \% $ $76.71 \% $ $8.177$ $0.368$ $4.31 \%$ $76.94 \%$ $7.110$ $0.391$ $5.21 \%$ $77.25 \%$
$\mathcal{T} = 1.0$ $110$ $12.994$ $0.514$ $3.80 \% $ $76.98 \%$ $11.685$ $0.585$ $4.77 \%$ $77.55 \%$ $10.861$ $0.652$ $5.67 \%$ $78.24 \%$
$115$ $16.279$ $0.740$ $4.35 \% $ $77.14 \%$ $15.381$ $0.870$ $5.35 \%$ $78.79 \% $ $14.759$ $0.989$ $6.28 \%$ $80.55 \% $
$90$ $4.347$ $0.096$ $2.16 \%$ $62.58 \%$ $0.391$ $0.015$ $3.78 \% $ $63.44 \%$ $0$ $0$ - -
(2) $95$ $6.141$ $0.155$ $2.45 \%$ $63.00 \%$ $1.865$ $0.074$ $3.82 \%$ $63.47 \% $ $0$ $0$ - -
$\sigma_{X}=0.2$ $100$ $8.321$ $0.238$ $2.78 \%$ $63.38 \%$ $5.152$ $0.212$ $3.94 \%$ $63.56 \%$ $3.649$ $0.188$ $4.89\%$ $63.68 \%$
$\lambda = 5.0$ $105$ $10.878$ $0.354$ $3.15 \% $ $63.72 \% $ $8.549$ $0.374$ $4.19 \%$ $63.76 \%$ $7.328$ $0.393$ $5.09 \%$ $63.90 \%$
$\mathcal{T} = 1.0$ $110$ $13.788$ $0.508$ $3.55 \% $ $64.01 \%$ $12.099$ $0.577$ $4.55 \%$ $64.10 \%$ $11.126$ $0.640$ $5.44 \%$ $64.46 \%$
$115$ $17.019$ $0.709$ $4.00 \% $ $64.18 \%$ $15.809$ $0.834$ $5.01 \%$ $64.74 \% $ $15.047$ $0.946$ $5.91 \%$ $65.93 \% $
Table 6.  Theoretical (down-and-out) call values and structure of the early exercise premium for $r = 0.05$, $\delta = 0.07$, $K = 100$, $L = 95$, $\rho_{L} = -26.34$, $p = 0.5$, $\xi = 25$ and $\eta = 25$
(Down-and-Out) Call Option Prices
Parameters Standard Call Price Step Call Price Barrier Call Price
$S_{0}$ Euro EEP EEP (%) DC (%) Euro EEP EEP (%) DC (%) Euro EEP EEP (%) DC (%)
$90$ $3.966$ $0.077$ $1.91 \%$ $76.61 \%$ $0.330$ $0.012$ $3.37 \% $ $77.40 \%$ $0$ $0$ - -
(1) $95$ $5.745$ $0.131$ $2.23 \%$ $77.04 \%$ $1.845$ $0.066$ $3.44 \%$ $77.48 \% $ $0$ $0$ - -
$\sigma_{X}=0.2$ $100$ $7.931$ $0.210$ $2.58 \%$ $77.42 \%$ $5.151$ $0.192$ $3.60 \%$ $77.62 \%$ $3.748$ $0.174$ $4.44\%$ $77.79 \%$
$\lambda = 5.0$ $105$ $10.514$ $0.323$ $2.98 \% $ $77.75 \% $ $8.516$ $0.346$ $3.90 \%$ $77.89 \%$ $7.415$ $0.366$ $4.70 \%$ $78.11 \%$
$\mathcal{T} = 1.0$ $110$ $13.463$ $0.479$ $3.43 \% $ $78.01 \%$ $12.037$ $0.544$ $4.32 \%$ $78.35 \%$ $11.178$ $0.603$ $5.12 \%$ $78.81 \%$
$115$ $16.740$ $0.685$ $3.93 \% $ $78.17 \%$ $15.730$ $0.803$ $4.85 \%$ $79.21 \% $ $15.069$ $0.907$ $5.68 \%$ $80.34 \% $
$90$ $4.950$ $0.091$ $1.81 \%$ $64.97 \%$ $0.468$ $0.016$ $3.20 \% $ $65.77 \%$ $0$ $0$ - -
(2) $95$ $6.842$ $0.144$ $2.07 \%$ $65.37 \%$ $2.166$ $0.073$ $3.25 \%$ $65.81 \% $ $0$ $0$ - -
$\sigma_{X}=0.2$ $100$ $9.098$ $0.220$ $2.36 \%$ $65.74 \%$ $5.678$ $0.198$ $3.37 \%$ $65.91 \%$ $4.077$ $0.177$ $4.16\%$ $66.01 \%$
$\lambda = 5.0$ $105$ $11.704$ $0.322$ $2.68 \% $ $66.08 \% $ $9.138$ $0.341$ $3.60 \%$ $66.09 \%$ $7.852$ $0.358$ $4.35 \%$ $66.17 \%$
$\mathcal{T} = 1.0$ $110$ $14.634$ $0.458$ $3.03 \% $ $66.37 \%$ $12.709$ $0.518$ $3.92 \%$ $66.34 \%$ $11.668$ $0.571$ $4.67 \%$ $66.50 \%$
$115$ $17.857$ $0.632$ $3.42 \% $ $66.60 \%$ $16.419$ $0.741$ $4.32 \%$ $66.73 \% $ $15.583$ $0.834$ $5.08 \%$ $67.20 \% $
Table Options

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(Down-and-Out) Call Option Prices
Parameters Standard Call Price Step Call Price Barrier Call Price
$S_{0}$ Euro EEP EEP (%) DC (%) Euro EEP EEP (%) DC (%) Euro EEP EEP (%) DC (%)
$90$ $3.966$ $0.077$ $1.91 \%$ $76.61 \%$ $0.330$ $0.012$ $3.37 \% $ $77.40 \%$ $0$ $0$ - -
(1) $95$ $5.745$ $0.131$ $2.23 \%$ $77.04 \%$ $1.845$ $0.066$ $3.44 \%$ $77.48 \% $ $0$ $0$ - -
$\sigma_{X}=0.2$ $100$ $7.931$ $0.210$ $2.58 \%$ $77.42 \%$ $5.151$ $0.192$ $3.60 \%$ $77.62 \%$ $3.748$ $0.174$ $4.44\%$ $77.79 \%$
$\lambda = 5.0$ $105$ $10.514$ $0.323$ $2.98 \% $ $77.75 \% $ $8.516$ $0.346$ $3.90 \%$ $77.89 \%$ $7.415$ $0.366$ $4.70 \%$ $78.11 \%$
$\mathcal{T} = 1.0$ $110$ $13.463$ $0.479$ $3.43 \% $ $78.01 \%$ $12.037$ $0.544$ $4.32 \%$ $78.35 \%$ $11.178$ $0.603$ $5.12 \%$ $78.81 \%$
$115$ $16.740$ $0.685$ $3.93 \% $ $78.17 \%$ $15.730$ $0.803$ $4.85 \%$ $79.21 \% $ $15.069$ $0.907$ $5.68 \%$ $80.34 \% $
$90$ $4.950$ $0.091$ $1.81 \%$ $64.97 \%$ $0.468$ $0.016$ $3.20 \% $ $65.77 \%$ $0$ $0$ - -
(2) $95$ $6.842$ $0.144$ $2.07 \%$ $65.37 \%$ $2.166$ $0.073$ $3.25 \%$ $65.81 \% $ $0$ $0$ - -
$\sigma_{X}=0.2$ $100$ $9.098$ $0.220$ $2.36 \%$ $65.74 \%$ $5.678$ $0.198$ $3.37 \%$ $65.91 \%$ $4.077$ $0.177$ $4.16\%$ $66.01 \%$
$\lambda = 5.0$ $105$ $11.704$ $0.322$ $2.68 \% $ $66.08 \% $ $9.138$ $0.341$ $3.60 \%$ $66.09 \%$ $7.852$ $0.358$ $4.35 \%$ $66.17 \%$
$\mathcal{T} = 1.0$ $110$ $14.634$ $0.458$ $3.03 \% $ $66.37 \%$ $12.709$ $0.518$ $3.92 \%$ $66.34 \%$ $11.668$ $0.571$ $4.67 \%$ $66.50 \%$
$115$ $17.857$ $0.632$ $3.42 \% $ $66.60 \%$ $16.419$ $0.741$ $4.32 \%$ $66.73 \% $ $15.583$ $0.834$ $5.08 \%$ $67.20 \% $
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