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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

Pierangelo Ciurlia 1,

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.
Keywords: Installment options, integral representations., free boundary problem, incomplete Fourier transform.
Mathematics Subject Classification: Primary: 91B28, 35K20; Secondary: 42B1.
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.

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