March  2022, 1(1): 53-79. doi: 10.3934/fmf.2021002

Semi-analytic pricing of double barrier options with time-dependent barriers and rebates at hit

1. 

Tandon School of Engineering, New York University, 1 Metro Tech Center, 10th floor, Brooklyn NY USA

2. 

Moscow State University, Moscow, Russia

* Corresponding author: Andrey Itkin

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

Fund Project: Dmitry Muravey acknowledges support by the Russian Science Foundation under the Grant number 20-68-47030

We continue a series of papers devoted to construction of semi-analytic solutions for barrier options. These options are written on underlying following some simple one-factor diffusion model, but all the parameters of the model as well as the barriers are time-dependent. We managed to show that these solutions are systematically more efficient for pricing and calibration than, e.g., the corresponding finite-difference solvers. In this paper we extend this technique to pricing double barrier options and present two approaches to solving it: the General Integral transform method and the Heat Potential method. Our results confirm that for double barrier options these semi-analytic techniques are also more efficient than the traditional numerical methods used to solve this type of problems.

Citation: Andrey Itkin, Dmitry Muravey. Semi-analytic pricing of double barrier options with time-dependent barriers and rebates at hit. Frontiers of Mathematical Finance, 2022, 1 (1) : 53-79. doi: 10.3934/fmf.2021002
References:
[1] L. Andersen and V. Piterbarg, Interest Rate Modeling, 2, Atlantic Financial Press, 2010. 
[2]

I. Bouchouev, Negative oil prices put spotlight on investors., Available from: Risk.net.

[3]

R. Brogan, Options traders adapt to electronic markets in pandemic, 2020., Available from: https://flextrade.com/options-traders-adapt-to-electronic-markets-in-pandemic/.

[4]

P. Carr and A. Itkin, Semi-closed form solutions for barrier and American options written on a time-dependent Ornstein Uhlenbeck process, Journal of Derivatives, Fall (2021). doi: 10.3905/jod.2020.1.113.

[5]

P. CarrA. Itkin and D. Muravey, Semi-closed form prices of barrier options in the time-dependent CEV and CIR models, Journal of Derivatives, 28 (2020), 26-50.  doi: 10.3905/jod.2020.1.113.

[6]

M. Costabel, Boundary integral operators for the heat equation, Integral Equations and Operator Theory, 13 (1990), 498-552.  doi: 10.1007/BF01210400.

[7]

J. C. CoxJ. E. IngersollJr . and S. A. Ross, A theory of the term structure of interest rates, Econometrica, 53 (1985), 385-407.  doi: 10.2307/1911242.

[8]

M. Craddock, Fundamental solutions, transition densities and the integration of Lie symmetries, Journal of Differential Equations, 246 (2009), 2538-2560.  doi: 10.1016/j.jde.2008.10.017.

[9]

M. Craddock and K. Lennox, Lie group symmetries as integral transforms of fundamental solutions, J. Differential Equations, 232 (2007), 652-674.  doi: 10.1016/j.jde.2006.07.011.

[10]

C. Dias, A method of recursive images to solve transient heat diffusionin multilayer materials, International Journal of Heat and Mass Transfer, 85 (2015), 1075-1083.  doi: 10.1016/j.ijheatmasstransfer.2015.01.138.

[11]

R. Doff, Valuing scenarios with real option pricing., Available from: Risk.net.

[12]

S. Farrington and M. Cesa, Podcast: Kaminski and Ronn on negative oil and options pricing., Available from: Risk.net.

[13]

A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, New Jersey, 1964.

[14]

N. Guinter, Potential Theory and Its Applications to Basic Problems of Mathematical Physics, Frederick Ungar, New York, 1967.

[15]

A. Itkin, Pricing Derivatives Under Lévy Models, 1$^st$ edition, Pseudo-Differential Operators, 12, Birkhauser, Basel, 2017. doi: 10.1007/978-1-4939-6792-6.

[16]

A. Itkin, A. Lipton and D. Muravey, From the Black-Karasinski to the Verhulst model to accommodate the unconventional Fed's policy, 2020., https://arXiv.org/abs/2006.11976.

[17]

A. Itkin, A. Lipton and D. Muravey, Multilayer heat equations: Application to finance, 2020, https://arXiv.org/abs/2102.08338.

[18]

A. Itkin and D. Muravey, Semi-analytic pricing of double barrier options with time-dependent barriers and rebates at hit, 2020., Available from: https://arXiv.org/abs/2009.09342.

[19]

A. Itkin and D. Muravey, Semi-closed form prices of barrier options in the Hull-White model, Risk, Dec. (2020).

[20]

E. M. Kartashov, Analytical methods for solution of non-stationary heat conductance boundary problems in domains with moving boundaries, Izvestiya RAS, Energetika, 5 (1999), 133-185. 

[21]

E. Kartashov, Analytical Methods in the Theory of Heat Conduction in Solids, Vysshaya Shkola, Moscow, 2001.

[22]

E. Kartashov and B. Y. Lyubov, Analytical methods in the theory of heat conduction in solids, Izv. Akad. Nauk SSSR, Energ. Trans., 83–111.

[23]

G. Kristensson, Jump conditions for single and doublelayer potentials, 2009., Available from: file:///C:/AndreyItkin/My Finance/FinPapers/BK/liter/JumpConditions.pdf.

[24]

A. Lipton, Mathematical Methods For Foreign Exchange: A Financial Engineer's Approach, World Scientific, 2001. doi: 10.1142/4694.

[25]

A. Lipton, The vol smile problem, Risk Magazine, 15 (2002), 61-65. 

[26]

A. Lipton and M. de Prado, A closed-form solution for optimal mean-reverting trading strategies, SSRN, (2020), 32 pp. doi: 10.2139/ssrn.3534445.

[27]

A. Lipton and V. Kaushansky, On the first hitting time density of an Ornstein-Uhlenbeck process, 2018., Available from: https://arXiv.org/pdf/1810.02390.pdf. doi: 10.1080/14697688.2020.1713394.

[28]

A. Lipton, V. Kaushansky and C. Reisinger, Semi-analytical solution of a McKean-Vlasov equation with feedback through hitting boundary, European Journal of Applied Mathematics, (2019), 1–34. doi: 10.1017/S0956792519000342.

[29]

A. Lyapunov, Works on the Theory of Potential, (Russian) Technical and Theoretical State Publishing House, Moscow - Leningrad, 1949.

[30]

A. Mijatovic, Local time and the pricing of time-dependent barrier options, Finance and Stochastics, 14 (2010), 13-48.  doi: 10.1007/s00780-008-0077-5.

[31]

D. Mumford, C. M. M. Nori, E. Previato and M. Stillman, Tata Lectures on Theta, Progress in Mathematics, Birkhäuser Boston, 1983. doi: 10.1007/978-0-8176-4578-6.

[32]

F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl and M. A. McClain, NIST Digital Library of Mathematical Functions., Available from: http://dlmf.nist.gov/.

[33]

A. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC, 2002.

[34]

B. Quaife, Fast Integral Equation Methods for the Modified Helmholtz Equation, Ph.D thesis, University of Calgary, 2011.

[35]

M. SpivakS. Veerapaneni and L. Greengard, The fast generalized gauss transform, SIAM Journal on Scientific Computing, 32 (2010), 3092-3107.  doi: 10.1137/100790744.

[36] A. Tikhonov and A. Samarskii, Equations of Mathematical Physics, Pergamon Press, Oxford, 1963. 
[37] B. van der Pol and H. Bremmer, Operational Calculus Based on the Two-Sided Laplace Integral, Cambridge University Press, Cambridge, UK, 1950. 
[38] A. M. Wazwaz, Linear and Nonlinear Integral Equations, Higher Education Press, Beijing and Springer-Verlag GmbH Berlin Heidelberg, 2011.  doi: 10.1007/978-3-642-21449-3.

show all references

References:
[1] L. Andersen and V. Piterbarg, Interest Rate Modeling, 2, Atlantic Financial Press, 2010. 
[2]

I. Bouchouev, Negative oil prices put spotlight on investors., Available from: Risk.net.

[3]

R. Brogan, Options traders adapt to electronic markets in pandemic, 2020., Available from: https://flextrade.com/options-traders-adapt-to-electronic-markets-in-pandemic/.

[4]

P. Carr and A. Itkin, Semi-closed form solutions for barrier and American options written on a time-dependent Ornstein Uhlenbeck process, Journal of Derivatives, Fall (2021). doi: 10.3905/jod.2020.1.113.

[5]

P. CarrA. Itkin and D. Muravey, Semi-closed form prices of barrier options in the time-dependent CEV and CIR models, Journal of Derivatives, 28 (2020), 26-50.  doi: 10.3905/jod.2020.1.113.

[6]

M. Costabel, Boundary integral operators for the heat equation, Integral Equations and Operator Theory, 13 (1990), 498-552.  doi: 10.1007/BF01210400.

[7]

J. C. CoxJ. E. IngersollJr . and S. A. Ross, A theory of the term structure of interest rates, Econometrica, 53 (1985), 385-407.  doi: 10.2307/1911242.

[8]

M. Craddock, Fundamental solutions, transition densities and the integration of Lie symmetries, Journal of Differential Equations, 246 (2009), 2538-2560.  doi: 10.1016/j.jde.2008.10.017.

[9]

M. Craddock and K. Lennox, Lie group symmetries as integral transforms of fundamental solutions, J. Differential Equations, 232 (2007), 652-674.  doi: 10.1016/j.jde.2006.07.011.

[10]

C. Dias, A method of recursive images to solve transient heat diffusionin multilayer materials, International Journal of Heat and Mass Transfer, 85 (2015), 1075-1083.  doi: 10.1016/j.ijheatmasstransfer.2015.01.138.

[11]

R. Doff, Valuing scenarios with real option pricing., Available from: Risk.net.

[12]

S. Farrington and M. Cesa, Podcast: Kaminski and Ronn on negative oil and options pricing., Available from: Risk.net.

[13]

A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, New Jersey, 1964.

[14]

N. Guinter, Potential Theory and Its Applications to Basic Problems of Mathematical Physics, Frederick Ungar, New York, 1967.

[15]

A. Itkin, Pricing Derivatives Under Lévy Models, 1$^st$ edition, Pseudo-Differential Operators, 12, Birkhauser, Basel, 2017. doi: 10.1007/978-1-4939-6792-6.

[16]

A. Itkin, A. Lipton and D. Muravey, From the Black-Karasinski to the Verhulst model to accommodate the unconventional Fed's policy, 2020., https://arXiv.org/abs/2006.11976.

[17]

A. Itkin, A. Lipton and D. Muravey, Multilayer heat equations: Application to finance, 2020, https://arXiv.org/abs/2102.08338.

[18]

A. Itkin and D. Muravey, Semi-analytic pricing of double barrier options with time-dependent barriers and rebates at hit, 2020., Available from: https://arXiv.org/abs/2009.09342.

[19]

A. Itkin and D. Muravey, Semi-closed form prices of barrier options in the Hull-White model, Risk, Dec. (2020).

[20]

E. M. Kartashov, Analytical methods for solution of non-stationary heat conductance boundary problems in domains with moving boundaries, Izvestiya RAS, Energetika, 5 (1999), 133-185. 

[21]

E. Kartashov, Analytical Methods in the Theory of Heat Conduction in Solids, Vysshaya Shkola, Moscow, 2001.

[22]

E. Kartashov and B. Y. Lyubov, Analytical methods in the theory of heat conduction in solids, Izv. Akad. Nauk SSSR, Energ. Trans., 83–111.

[23]

G. Kristensson, Jump conditions for single and doublelayer potentials, 2009., Available from: file:///C:/AndreyItkin/My Finance/FinPapers/BK/liter/JumpConditions.pdf.

[24]

A. Lipton, Mathematical Methods For Foreign Exchange: A Financial Engineer's Approach, World Scientific, 2001. doi: 10.1142/4694.

[25]

A. Lipton, The vol smile problem, Risk Magazine, 15 (2002), 61-65. 

[26]

A. Lipton and M. de Prado, A closed-form solution for optimal mean-reverting trading strategies, SSRN, (2020), 32 pp. doi: 10.2139/ssrn.3534445.

[27]

A. Lipton and V. Kaushansky, On the first hitting time density of an Ornstein-Uhlenbeck process, 2018., Available from: https://arXiv.org/pdf/1810.02390.pdf. doi: 10.1080/14697688.2020.1713394.

[28]

A. Lipton, V. Kaushansky and C. Reisinger, Semi-analytical solution of a McKean-Vlasov equation with feedback through hitting boundary, European Journal of Applied Mathematics, (2019), 1–34. doi: 10.1017/S0956792519000342.

[29]

A. Lyapunov, Works on the Theory of Potential, (Russian) Technical and Theoretical State Publishing House, Moscow - Leningrad, 1949.

[30]

A. Mijatovic, Local time and the pricing of time-dependent barrier options, Finance and Stochastics, 14 (2010), 13-48.  doi: 10.1007/s00780-008-0077-5.

[31]

D. Mumford, C. M. M. Nori, E. Previato and M. Stillman, Tata Lectures on Theta, Progress in Mathematics, Birkhäuser Boston, 1983. doi: 10.1007/978-0-8176-4578-6.

[32]

F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl and M. A. McClain, NIST Digital Library of Mathematical Functions., Available from: http://dlmf.nist.gov/.

[33]

A. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC, 2002.

[34]

B. Quaife, Fast Integral Equation Methods for the Modified Helmholtz Equation, Ph.D thesis, University of Calgary, 2011.

[35]

M. SpivakS. Veerapaneni and L. Greengard, The fast generalized gauss transform, SIAM Journal on Scientific Computing, 32 (2010), 3092-3107.  doi: 10.1137/100790744.

[36] A. Tikhonov and A. Samarskii, Equations of Mathematical Physics, Pergamon Press, Oxford, 1963. 
[37] B. van der Pol and H. Bremmer, Operational Calculus Based on the Two-Sided Laplace Integral, Cambridge University Press, Cambridge, UK, 1950. 
[38] A. M. Wazwaz, Linear and Nonlinear Integral Equations, Higher Education Press, Beijing and Springer-Verlag GmbH Berlin Heidelberg, 2011.  doi: 10.1007/978-3-642-21449-3.
Figure 1.  Contours of integration of Eq.(18) in the complex plane $ p \in \mathbb{C} $ with poles at $ p_1^\pm, p_2^\pm, \ldots $.
Figure 2.  Prices of Call double barrier options obtained in the test by using three methods.}
Figure 3.  Absolute difference in prices of Call double barrier options obtained in the test by using three methods.
Table 1.  Parameters of the test
$ S_0 $ $ \sigma_0 $ $ \sigma_K $ $ r $ $ a_L $ $ b_L $ $ a_H $ $ b_H $,
50 0.3 0.7 0.01 20 5 71 -2
$ S_0 $ $ \sigma_0 $ $ \sigma_K $ $ r $ $ a_L $ $ b_L $ $ a_H $ $ b_H $,
50 0.3 0.7 0.01 20 5 71 -2
Table 2.  Comparison of double barriers Call option prices obtained by various methods
T 0.038 0.083 0.333 0.500 1.000 2.000 5.000 0.038 0.083 0.333 0.500 1.000 2.000 5.000
K MC GIT
45 5.0174 5.0379 5.1500 5.2212 5.4473 5.8825 7.1820 5.0173 5.0375 5.1498 5.2244 5.4478 5.8912 7.1946
50 0.1732 0.2583 0.5216 0.6370 0.8941 1.2815 2.4866 0.1735 0.2587 0.5243 0.6407 0.8984 1.2932 2.5056
55 0.0000 0.0000 0.0000 0.0000 0.0012 0.0075 0.0766 0.0000 0.0000 0.0000 0.0000 0.0012 0.0083 0.0852
K FD HP
45 5.0173 5.0374 5.1483 5.2218 5.4416 5.8804 7.1465 5.0173 5.0375 5.1498 5.2244 5.4478 5.8912 7.1946
50 0.1757 0.2567 0.5224 0.6391 0.8972 1.2918 2.4999 0.1735 0.2587 0.5243 0.6407 0.8984 1.2932 2.5056
55 0.0000 0.0000 0.0000 0.0001 0.0017 0.0103 0.1221 0.0000 0.0000 0.0000 0.0000 0.0012 0.0083 0.0852
T 0.038 0.083 0.333 0.500 1.000 2.000 5.000 0.038 0.083 0.333 0.500 1.000 2.000 5.000
K MC GIT
45 5.0174 5.0379 5.1500 5.2212 5.4473 5.8825 7.1820 5.0173 5.0375 5.1498 5.2244 5.4478 5.8912 7.1946
50 0.1732 0.2583 0.5216 0.6370 0.8941 1.2815 2.4866 0.1735 0.2587 0.5243 0.6407 0.8984 1.2932 2.5056
55 0.0000 0.0000 0.0000 0.0000 0.0012 0.0075 0.0766 0.0000 0.0000 0.0000 0.0000 0.0012 0.0083 0.0852
K FD HP
45 5.0173 5.0374 5.1483 5.2218 5.4416 5.8804 7.1465 5.0173 5.0375 5.1498 5.2244 5.4478 5.8912 7.1946
50 0.1757 0.2567 0.5224 0.6391 0.8972 1.2918 2.4999 0.1735 0.2587 0.5243 0.6407 0.8984 1.2932 2.5056
55 0.0000 0.0000 0.0000 0.0001 0.0017 0.0103 0.1221 0.0000 0.0000 0.0000 0.0000 0.0012 0.0083 0.0852
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