# American Institute of Mathematical Sciences

July  2011, 10(4): 1205-1224. doi: 10.3934/cpaa.2011.10.1205

## On a general class of free boundary problems for European-style installment options with continuous payment plan

 1 Department of Economics, Faculty of Economics "Federico Caffè", University of Rome III, Via Silvio D'Amico 77, 00145 Rome, Italy

Received  June 2010 Revised  December 2010 Published  April 2011

In this paper we present an integral equation approach for the valuation of European-style installment derivatives when the payment plan is assumed to be a continuous function of the asset price and time. The contribution of this study is threefold. First, we show that in the Black-Scholes model the option pricing problem can be formulated as a free boundary problem under very general conditions on payoff structure and payment schedule. Second, by applying a Fourier transform-based solution technique, we derive a recursive integral equation for the free boundary along with an analytic representation of the option price. Third, based on these results, we propose a unified framework which generalizes the existing methods and is capable of dealing with a wide range of monotonic payoff functions and continuous payment plans. Finally, by using the illustrative example of European vanilla installment call options, an explicit pricing formula is obtained for time-varying payment schedules.
Citation: Pierangelo Ciurlia. On a general class of free boundary problems for European-style installment options with continuous payment plan. Communications on Pure and Applied Analysis, 2011, 10 (4) : 1205-1224. doi: 10.3934/cpaa.2011.10.1205
##### References:
 [1] G. Alobaidi, R. Mallier and A. S. Deakin, Laplace transforms and installment options, Mathematical Models and Methods in Applied Sciences, 14 (2004), 1167-1189. doi: 10.1142/S0218202504003581. [2] G. Alobaidi and R. Mallier, Installment options close to expiry, Journal of Applied Mathematics and Stochastic Analysis, (2006), Art. ID 60824, 1-9. doi: 10.1155/JAMSA/2006/60824. [3] H. Ben-Ameur, M. Breton and P. François, A dynamic programming approach to price installment options, European Journal of Operational Research, 169 (2006), 667-676. doi: 10.1016/j.ejor.2004.05.009. [4] P. Carr, R. Jarrow and R. Myneni, Alternative characterizations of American put options, Mathematical Finance, 2 (1992), 87-106. doi: 10.1111/j.1467-9965.1992.tb00040.x. [5] P. Ciurlia, On the evaluation of European continuous-installment options, Department of Economics Working paper, No 113 (2010), University of Rome III. [6] P. Ciurlia and C. Caperdoni, A note on the pricing of perpetual continuous-installment options, Mathematical Methods in Economics and Finance, 4 (2009), 11-26. [7] P. Ciurlia and I. Roko, Valuation of American continuous-installment options, Computational Economics, 25 (2005), 143-165. doi: 10.1007/s10614-005-6279-4. [8] M. Davis, W. Schachermayer and R. Tompkins, Pricing, no-arbitrage bounds and robust hedging of instalment options, Quantitative Finance, 1 (2001), 597-610. doi: 10.1088/1469-7688/1/6/302. [9] M. Davis, W. Schachermayer and R. Tompkins, Installment options and static hedging, Journal of Risk Finance, 3 (2002), 46-52. doi: 10.1108/eb043487. [10] S. Griebsch, C. Kühn and U. Wystup, Instalment options: A closed-form solution and the limiting case, in "Mathematical Control Theory and Finance" (eds. A. Sarychev, A. Shiryaev, M. Guerra and M.d.R. Grossinho), Springer-Verlag, Berlin, (2008), 211-229. doi: 10.1007/978-3-540-69532-5_12. [11] S. D. Jacka, Optimal stopping and the American put, Mathematical Finance, 1 (1991), 1-14. doi: 10.1111/j.1467-9965.1991.tb00007.x. [12] F. Karsenty and J. Sikorav, Installment plan, Risk, 6 (1993), 36-40. [13] I. J. Kim, The analytical valuation of American options, Review of Financial Studies, 3 (1990), 547-572. doi: 10.1093/rfs/3.4.547. [14] T. Kimura, American continuous-installment options: valuation and premium decomposition, SIAM Journal on Applied Mathematics, 70 (2009), 803-824. doi: 10.1137/080740969. [15] T. Kimura, Valuing continuous-installment options, European Journal of Operational Research, 201 (2010), 222-230. doi: 10.1016/j.ejor.2009.02.010. [16] H. P. McKean, Appendix: A free boundary problem for the heat equation arising from a problem in mathematical economics, Industrial Management Review, 6 (1965), 32-39. [17] Z. Yang and F. Yi, A variational inequality arising from American installment call options pricing, Journal of Mathematical Analysis and Applications, 357 (2009), 54-68. doi: 10.1016/j.jmaa.2009.03.045. [18] F. Yi, Z. Yang and X. Wang, A variational inequality arising from European installment call options pricing, SIAM Journal on Mathematical Analysis, 40 (2008), 306-326. doi: 10.1137/060670353.

show all references

##### References:
 [1] G. Alobaidi, R. Mallier and A. S. Deakin, Laplace transforms and installment options, Mathematical Models and Methods in Applied Sciences, 14 (2004), 1167-1189. doi: 10.1142/S0218202504003581. [2] G. Alobaidi and R. Mallier, Installment options close to expiry, Journal of Applied Mathematics and Stochastic Analysis, (2006), Art. ID 60824, 1-9. doi: 10.1155/JAMSA/2006/60824. [3] H. Ben-Ameur, M. Breton and P. François, A dynamic programming approach to price installment options, European Journal of Operational Research, 169 (2006), 667-676. doi: 10.1016/j.ejor.2004.05.009. [4] P. Carr, R. Jarrow and R. Myneni, Alternative characterizations of American put options, Mathematical Finance, 2 (1992), 87-106. doi: 10.1111/j.1467-9965.1992.tb00040.x. [5] P. Ciurlia, On the evaluation of European continuous-installment options, Department of Economics Working paper, No 113 (2010), University of Rome III. [6] P. Ciurlia and C. Caperdoni, A note on the pricing of perpetual continuous-installment options, Mathematical Methods in Economics and Finance, 4 (2009), 11-26. [7] P. Ciurlia and I. Roko, Valuation of American continuous-installment options, Computational Economics, 25 (2005), 143-165. doi: 10.1007/s10614-005-6279-4. [8] M. Davis, W. Schachermayer and R. Tompkins, Pricing, no-arbitrage bounds and robust hedging of instalment options, Quantitative Finance, 1 (2001), 597-610. doi: 10.1088/1469-7688/1/6/302. [9] M. Davis, W. Schachermayer and R. Tompkins, Installment options and static hedging, Journal of Risk Finance, 3 (2002), 46-52. doi: 10.1108/eb043487. [10] S. Griebsch, C. Kühn and U. Wystup, Instalment options: A closed-form solution and the limiting case, in "Mathematical Control Theory and Finance" (eds. A. Sarychev, A. Shiryaev, M. Guerra and M.d.R. Grossinho), Springer-Verlag, Berlin, (2008), 211-229. doi: 10.1007/978-3-540-69532-5_12. [11] S. D. Jacka, Optimal stopping and the American put, Mathematical Finance, 1 (1991), 1-14. doi: 10.1111/j.1467-9965.1991.tb00007.x. [12] F. Karsenty and J. Sikorav, Installment plan, Risk, 6 (1993), 36-40. [13] I. J. Kim, The analytical valuation of American options, Review of Financial Studies, 3 (1990), 547-572. doi: 10.1093/rfs/3.4.547. [14] T. Kimura, American continuous-installment options: valuation and premium decomposition, SIAM Journal on Applied Mathematics, 70 (2009), 803-824. doi: 10.1137/080740969. [15] T. Kimura, Valuing continuous-installment options, European Journal of Operational Research, 201 (2010), 222-230. doi: 10.1016/j.ejor.2009.02.010. [16] H. P. McKean, Appendix: A free boundary problem for the heat equation arising from a problem in mathematical economics, Industrial Management Review, 6 (1965), 32-39. [17] Z. Yang and F. Yi, A variational inequality arising from American installment call options pricing, Journal of Mathematical Analysis and Applications, 357 (2009), 54-68. doi: 10.1016/j.jmaa.2009.03.045. [18] F. Yi, Z. Yang and X. Wang, A variational inequality arising from European installment call options pricing, SIAM Journal on Mathematical Analysis, 40 (2008), 306-326. doi: 10.1137/060670353.
 [1] Barbara Brandolini, Francesco Chiacchio, Jeffrey J. Langford. Estimates for sums of eigenvalues of the free plate via the fourier transform. Communications on Pure and Applied Analysis, 2020, 19 (1) : 113-122. doi: 10.3934/cpaa.2020007 [2] Toyohiko Aiki. A free boundary problem for an elastic material. Conference Publications, 2007, 2007 (Special) : 10-17. doi: 10.3934/proc.2007.2007.10 [3] Yang Zhang. A free boundary problem of the cancer invasion. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1323-1343. doi: 10.3934/dcdsb.2021092 [4] Hayk Mikayelyan, Henrik Shahgholian. Convexity of the free boundary for an exterior free boundary problem involving the perimeter. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1431-1443. doi: 10.3934/cpaa.2013.12.1431 [5] Y. T. Li, R. Wong. Integral and series representations of the dirac delta function. Communications on Pure and Applied Analysis, 2008, 7 (2) : 229-247. doi: 10.3934/cpaa.2008.7.229 [6] Alexander Alekseenko, Jeffrey Limbacher. Evaluating high order discontinuous Galerkin discretization of the Boltzmann collision integral in $\mathcal{O}(N^2)$ operations using the discrete fourier transform. Kinetic and Related Models, 2019, 12 (4) : 703-726. doi: 10.3934/krm.2019027 [7] Juan H. Arredondo, Francisco J. Mendoza, Alfredo Reyes. On the norm continuity of the hk-fourier transform. Electronic Research Announcements, 2018, 25: 36-47. doi: 10.3934/era.2018.25.005 [8] Georgi Grahovski, Rossen Ivanov. Generalised Fourier transform and perturbations to soliton equations. Discrete and Continuous Dynamical Systems - B, 2009, 12 (3) : 579-595. doi: 10.3934/dcdsb.2009.12.579 [9] Xiaoshan Chen, Fahuai Yi. Free boundary problem of Barenblatt equation in stochastic control. Discrete and Continuous Dynamical Systems - B, 2016, 21 (5) : 1421-1434. doi: 10.3934/dcdsb.2016003 [10] Naoki Sato, Toyohiko Aiki, Yusuke Murase, Ken Shirakawa. A one dimensional free boundary problem for adsorption phenomena. Networks and Heterogeneous Media, 2014, 9 (4) : 655-668. doi: 10.3934/nhm.2014.9.655 [11] Yongzhi Xu. A free boundary problem model of ductal carcinoma in situ. Discrete and Continuous Dynamical Systems - B, 2004, 4 (1) : 337-348. doi: 10.3934/dcdsb.2004.4.337 [12] Anna Lisa Amadori. Contour enhancement via a singular free boundary problem. Conference Publications, 2007, 2007 (Special) : 44-53. doi: 10.3934/proc.2007.2007.44 [13] Shihe Xu. Analysis of a delayed free boundary problem for tumor growth. Discrete and Continuous Dynamical Systems - B, 2011, 15 (1) : 293-308. doi: 10.3934/dcdsb.2011.15.293 [14] Hiroshi Matsuzawa. A free boundary problem for the Fisher-KPP equation with a given moving boundary. Communications on Pure and Applied Analysis, 2018, 17 (5) : 1821-1852. doi: 10.3934/cpaa.2018087 [15] Micah Webster, Patrick Guidotti. Boundary dynamics of a two-dimensional diffusive free boundary problem. Discrete and Continuous Dynamical Systems, 2010, 26 (2) : 713-736. doi: 10.3934/dcds.2010.26.713 [16] Chonghu Guan, Xun Li, Rui Zhou, Wenxin Zhou. Free boundary problem for an optimal investment problem with a borrowing constraint. Journal of Industrial and Management Optimization, 2022, 18 (3) : 1915-1934. doi: 10.3934/jimo.2021049 [17] Alexander Arguchintsev, Vasilisa Poplevko. An optimal control problem by parabolic equation with boundary smooth control and an integral constraint. Numerical Algebra, Control and Optimization, 2018, 8 (2) : 193-202. doi: 10.3934/naco.2018011 [18] Kanghui Guo and Demetrio Labate. Sparse shearlet representation of Fourier integral operators. Electronic Research Announcements, 2007, 14: 7-19. doi: 10.3934/era.2007.14.7 [19] Elena Cordero, Fabio Nicola, Luigi Rodino. Time-frequency analysis of fourier integral operators. Communications on Pure and Applied Analysis, 2010, 9 (1) : 1-21. doi: 10.3934/cpaa.2010.9.1 [20] Constantin N. Beli. Representations of integral quadratic forms over dyadic local fields. Electronic Research Announcements, 2006, 12: 100-112.

2020 Impact Factor: 1.916