Article Contents
Article Contents

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

• 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.
Mathematics Subject Classification: Primary: 91B28, 35K20; Secondary: 42B10.

 Citation:

•  [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.