Open Access
2013 Asymptotics of European Double-Barrier Option with Compound Poisson Component
R. Carrada-Herrera, S. M. Grudsky, C. Palomino-Jiménez, R. M. Porter
Commun. Math. Anal. 14(2): 40-66 (2013).
Abstract

We consider standard European as well as double-barrier European options for underlyings that are given by the superposition of a Guassian and a compound Poisson (jump) process with discrete values. We derive a formula for calculating such options and furthermore show that as the barriers tend to $\pm\infty$, the value of the double-barrier option tends asymptotically to that of the standard option. Numerical examples are provided.

References

1.

N. I. Akhiezer, Lectures on Approximation Theory, Frederick Ungar Publishing Co., New York 1956. N. I. Akhiezer, Lectures on Approximation Theory, Frederick Ungar Publishing Co., New York 1956.

2.

O. E. Barndorff-Neilsen, Processes of normal inverse Gaussian type. Finance Stoch. 2, No. 1 (1998), pp 41-68.  MR1804664 0894.90011 10.1007/s007800050032 O. E. Barndorff-Neilsen, Processes of normal inverse Gaussian type. Finance Stoch. 2, No. 1 (1998), pp 41-68.  MR1804664 0894.90011 10.1007/s007800050032

3.

A. Böttcher, S. Grudsky, and R. M. Porter, European double-barrier options with a compound Poisson component. Progress in Economics Research 18 (A. Tavidze, ed.), Nova Science Publishers, Huntington, New York (2011), pp 315-331. A. Böttcher, S. Grudsky, and R. M. Porter, European double-barrier options with a compound Poisson component. Progress in Economics Research 18 (A. Tavidze, ed.), Nova Science Publishers, Huntington, New York (2011), pp 315-331.

4.

M. Boyarchenko and S.I. Boyarchenko, Double barrier options in regime-switching hyper-exponential jump-diffusion models. Int J. Theor. Appl. Finance 14, No. 7 (2011), pp 1005-1043.  MR2864297 1233.91257 10.1142/S0219024911006620 M. Boyarchenko and S.I. Boyarchenko, Double barrier options in regime-switching hyper-exponential jump-diffusion models. Int J. Theor. Appl. Finance 14, No. 7 (2011), pp 1005-1043.  MR2864297 1233.91257 10.1142/S0219024911006620

5.

M. Boyarchenko and S. Z. Levendorskiĭ, Valuation of continuously monitored double barrier options and related securities. Math. Finance 22, No. 3 (2012), pp 419-444.  MR2943179 10.1111/j.1467-9965.2010.00469.x 06139575 M. Boyarchenko and S. Z. Levendorskiĭ, Valuation of continuously monitored double barrier options and related securities. Math. Finance 22, No. 3 (2012), pp 419-444.  MR2943179 10.1111/j.1467-9965.2010.00469.x 06139575

6.

M. Boyarchenko and S. Z. Levendorskiĭ, Non-Gaussian Merton-Black-Scholes Theory. (Advanced Series on Statistical Science and Applied Probability 9), World Scientific Publishing Co., Inc., River Edge, NJ and Singapore 2002.  MR1904936 0997.91031 M. Boyarchenko and S. Z. Levendorskiĭ, Non-Gaussian Merton-Black-Scholes Theory. (Advanced Series on Statistical Science and Applied Probability 9), World Scientific Publishing Co., Inc., River Edge, NJ and Singapore 2002.  MR1904936 0997.91031

7.

M. Boyarchenko and S. Z. Levendorskiĭ, Perpetual American options under Lévy processes. SIAM Journal Control Optimization 40, No. 6 (2002), pp 1663-1696.  MR1897191 10.1137/S0363012900373987 M. Boyarchenko and S. Z. Levendorskiĭ, Perpetual American options under Lévy processes. SIAM Journal Control Optimization 40, No. 6 (2002), pp 1663-1696.  MR1897191 10.1137/S0363012900373987

8.

M. Boyarchenko and S. Z. Levendorskiĭ, Barrier options and touch-and-out options under regular Lévy processes of exponential type. Annals of Applied Probability, 12, No. 4 (2002), pp 1261-1298.  MR1936593 10.1214/aoap/1037125863 euclid.aoap/1037125863 M. Boyarchenko and S. Z. Levendorskiĭ, Barrier options and touch-and-out options under regular Lévy processes of exponential type. Annals of Applied Probability, 12, No. 4 (2002), pp 1261-1298.  MR1936593 10.1214/aoap/1037125863 euclid.aoap/1037125863

9.

P. Carr and J. Crosby, A class of Lévy process models with almost exact calibration to both barrier and vanilla FX options. Quant. Finance 10, No. 10 (2010), pp 1115-1136.  MR2739090 10.1080/14697680903413605 P. Carr and J. Crosby, A class of Lévy process models with almost exact calibration to both barrier and vanilla FX options. Quant. Finance 10, No. 10 (2010), pp 1115-1136.  MR2739090 10.1080/14697680903413605

10.

E. Eberlein, Application of generalized hyperbolic Lévy motions to finance. Lévy Processes: Theory and Applications, O. E. Barndorff-Nielsen, T. Mikosh, and S. Resnil (eds.), Birkhäuser (2001), pp 319-337.  MR1833703 E. Eberlein, Application of generalized hyperbolic Lévy motions to finance. Lévy Processes: Theory and Applications, O. E. Barndorff-Nielsen, T. Mikosh, and S. Resnil (eds.), Birkhäuser (2001), pp 319-337.  MR1833703

11.

E. Eberlein and J. Jacod, On the range of options prices. Finance Stoch. 1 (1997), pp 131-140. E. Eberlein and J. Jacod, On the range of options prices. Finance Stoch. 1 (1997), pp 131-140.

12.

F. Esscher, On the probability function in the collective theory of risk. Skandinavisk Aktuarietidskrift 15 (1932), pp 175-95. F. Esscher, On the probability function in the collective theory of risk. Skandinavisk Aktuarietidskrift 15 (1932), pp 175-95.

13.

M. V. Fedoryuk, The Saddle Point Method, Nauka, Moscow (in Russian) 1988.  MR507923 0463.41020 M. V. Fedoryuk, The Saddle Point Method, Nauka, Moscow (in Russian) 1988.  MR507923 0463.41020

14.

L. Feng and V. Linetsky, Pricing discretely monitored barrier options and defaultable bonds. Lévy process models: a fast Hilbert transform approach, Math. Finance 18, No. 3 (2008), pp 337-384.  MR2427727 10.1111/j.1467-9965.2008.00338.x L. Feng and V. Linetsky, Pricing discretely monitored barrier options and defaultable bonds. Lévy process models: a fast Hilbert transform approach, Math. Finance 18, No. 3 (2008), pp 337-384.  MR2427727 10.1111/j.1467-9965.2008.00338.x

15.

H. Geman and M. Yor, Pricing and hedging double-barrier options: a probabilistic approach. Math. Finance 6, No. 4 (1996), pp 365-378. H. Geman and M. Yor, Pricing and hedging double-barrier options: a probabilistic approach. Math. Finance 6, No. 4 (1996), pp 365-378.

16.

J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford University Press, New York 1985.  MR790497 J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford University Press, New York 1985.  MR790497

17.

G. A. Grinberg, A Collection of Problems in the Mathematical Theory of Electrical and Magnetic Phenomena, Moscow-Leningrad (in Russian) 1948. G. A. Grinberg, A Collection of Problems in the Mathematical Theory of Electrical and Magnetic Phenomena, Moscow-Leningrad (in Russian) 1948.

18.

C. H. Hui, One-touch double barrier binary option values. Applied Financial Economics 6 (1996), pp 343-346. C. H. Hui, One-touch double barrier binary option values. Applied Financial Economics 6 (1996), pp 343-346.

19.

C. H. Hui, Time dependent barrier option values. J. Futures Markets 17 (1997), pp 667-688. C. H. Hui, Time dependent barrier option values. J. Futures Markets 17 (1997), pp 667-688.

20.

J. D. Jackson, Classical Electrodynamics, John Wiley and Sons, New York 1962.  MR436782 J. D. Jackson, Classical Electrodynamics, John Wiley and Sons, New York 1962.  MR436782

21.

J. Jeans, The Mathematical Theory of Electricity and Magnetism, Cambridge University Press, New York 1963.  MR115577 J. Jeans, The Mathematical Theory of Electricity and Magnetism, Cambridge University Press, New York 1963.  MR115577

22.

T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin and Heidelberg 1995.  MR1335452 T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin and Heidelberg 1995.  MR1335452

23.

N. E. Kochin, I. A. Kibel', and N. V. Roze, Theoretical Hydrodynamics, Interscience (Translated from Russian) 1964. N. E. Kochin, I. A. Kibel', and N. V. Roze, Theoretical Hydrodynamics, Interscience (Translated from Russian) 1964.

24.

S. G. Krein, Linear Differential Equation in Banach Space, Transl. Math. Monogr. 29 1971.  MR342804 S. G. Krein, Linear Differential Equation in Banach Space, Transl. Math. Monogr. 29 1971.  MR342804

25.

N. Kunimoto and M. Ikeda, Pricing options with curved boundaries. Math. Finance 2 (1997), pp 275-298. N. Kunimoto and M. Ikeda, Pricing options with curved boundaries. Math. Finance 2 (1997), pp 275-298.

26.

B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Cambridge University Press, New York 1982.  MR690064 B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Cambridge University Press, New York 1982.  MR690064

27.

A. Lipton, Mathematical Methods for Foreign Exchange, A Financial Engineer's Approach, World Scientific, New Jersy 2001.  MR1878627 0989.91002 A. Lipton, Mathematical Methods for Foreign Exchange, A Financial Engineer's Approach, World Scientific, New Jersy 2001.  MR1878627 0989.91002

28.

D. B. Madan, P. Carr, and E. C. Chang, The variance gamma process and option pricing. European Finance Review 2 (1997), pp 79-105. D. B. Madan, P. Carr, and E. C. Chang, The variance gamma process and option pricing. European Finance Review 2 (1997), pp 79-105.

29.

A. Matacz, Financial modelling and option theory with the truncated Lévy process. Int. J. Theor. Appl. Finance 3, No. 1 (2000), pp 143-160. A. Matacz, Financial modelling and option theory with the truncated Lévy process. Int. J. Theor. Appl. Finance 3, No. 1 (2000), pp 143-160.

30.

E. Mordecki, Optimal stopping and perpetual options for Lévy processes. Talk presented at the I World Congress of the Bachelier Finance Society (2000). E. Mordecki, Optimal stopping and perpetual options for Lévy processes. Talk presented at the I World Congress of the Bachelier Finance Society (2000).

31.

F. W. J. Olver, Asymptotics and Special Functions, Academic Press, New York 1974.  MR435697 F. W. J. Olver, Asymptotics and Special Functions, Academic Press, New York 1974.  MR435697

32.

A. Pelsser, Pricing double barrier options using Laplace transform. Finance Stoch. 4, No. 1 (1997), pp 95-104.  MR1790136 0940.91026 10.1007/s007800050005 A. Pelsser, Pricing double barrier options using Laplace transform. Finance Stoch. 4, No. 1 (1997), pp 95-104.  MR1790136 0940.91026 10.1007/s007800050005

33.

M. Potters, J-P. Bouchaud, and L. Laloux, Theory of Financial Risks: From Statistical Physics to Risk Management, Cambridge University Press, Cambridge 2000.  MR1787145 M. Potters, J-P. Bouchaud, and L. Laloux, Theory of Financial Risks: From Statistical Physics to Risk Management, Cambridge University Press, Cambridge 2000.  MR1787145

34.

R. Raible, Lévy Processes in Finance: Theory, Numerics, and Empirical Facts. Dissertation, Mathematische Fakultät, Universität Freiberg im Breisgau 2000. R. Raible, Lévy Processes in Finance: Theory, Numerics, and Empirical Facts. Dissertation, Mathematische Fakultät, Universität Freiberg im Breisgau 2000.

35.

A. Sepp, Analytical pricing of double-barrier options under a double-exponential jump diffusion process: applications of Laplace transform. Int. J. Theor. Appl. Finance 7, No. 2 (2004), pp 151-175.  MR2053597 1107.91345 10.1142/S0219024904002402 A. Sepp, Analytical pricing of double-barrier options under a double-exponential jump diffusion process: applications of Laplace transform. Int. J. Theor. Appl. Finance 7, No. 2 (2004), pp 151-175.  MR2053597 1107.91345 10.1142/S0219024904002402

36.

A. N. Shiryaev, Essentials of Stochastic Finance: Facts, Models, Theory, World Scientific, Singapore-Jersey-London-Hong Kong 1999.  MR1695318 0926.62100 A. N. Shiryaev, Essentials of Stochastic Finance: Facts, Models, Theory, World Scientific, Singapore-Jersey-London-Hong Kong 1999.  MR1695318 0926.62100

37.

J. Sidenius, Double barrier options: valuation by path counting. Journal of Computational Finance 1, No. 3 (1998), pp 63-79. J. Sidenius, Double barrier options: valuation by path counting. Journal of Computational Finance 1, No. 3 (1998), pp 63-79.

38.

P. Wilmott, J. Dewynne, and S. Howison, Option Pricing: Mathematical Models and Computations, Oxford Financial Press, Oxford 1993. P. Wilmott, J. Dewynne, and S. Howison, Option Pricing: Mathematical Models and Computations, Oxford Financial Press, Oxford 1993.
Copyright © 2013 Mathematical Research Publishers
R. Carrada-Herrera, S. M. Grudsky, C. Palomino-Jiménez, and R. M. Porter "Asymptotics of European Double-Barrier Option with Compound Poisson Component," Communications in Mathematical Analysis 14(2), 40-66, (2013). https://doi.org/
Published: 2013
Vol.14 • No. 2 • 2013
Back to Top