Abstract
In this article, the researchers obtained a recursive formula for the price of discrete single barrier option based on the Black–Scholes framework in which drift, dividend yield and volatility assumed as deterministic functions of time. With some general transformations, the partial differential equations (PDEs) corresponding to option value problem, in each monitoring time interval, were converted into well-known Black–Scholes PDE with constant coefficients. Finally, an innovative numerical approach was proposed to utilize the obtained recursive formula efficiently. Despite some claims, it has considerably low computational cost and could be competitive with the other introduced method. In addition, one advantage of this method, is that the Greeks of the contracts were also calculated.
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References
AitSahlia, F., & Lai, T. L. (1997). Valuation of discrete barrier and hindsight options. Journal of Financial Engineering, 6, 169–177.
Arnold, L. (1974). Stochastic differential equations., Theory and applications New York: Wiley.
Bertoldi, M., Bianchetti, M. (2003). Monte Carlo simulation of discrete barrier options. Internal Report, Financial Engineering Derivatives Modelling, Caboto SIM S.p.a., Banca Intesa Group, Milan, Italy.
Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637–659.
Boyle, P. P., & Tian, Y. S. (1998). An explicit finite difference approach to the pricing of barrier options. Applied Mathematical Finance, 5, 17–43.
Broadie, M., Glasserman, P., & Kou, S. (1999). Connecting discrete and continuous path-dependent options. Finance and Stochastics, 3(1), 55–82.
Broadie, M., Glasserman, P., & Kou, S. (1997). A continuity correction for discrete barrier options. Mathematical Finance, 7, 325–349.
Duan, J. C., Dudley, E., Gauthier, G., & Simonato, J. G. (2003). Pricing discretely monitored barrier options by a Markov chain. Journal of Derivatives Summer, 10, 9–32.
Evans, L. C. (2004). An introduction to stochastic differential equations Version 1.2.
Fusai, G., David Abrahams, I., & Sgarra, C. (2006). An exact analytical solution for discrete barrier options. Finance Stochastic, 10, 1–26.
Fusai, G., & Recchioni, M. C. (2008). Analysis of quadrature methods for pricing discrete barrier options. Journal of Economic Dynamics and Control, 31, 826–860.
Geman, H., & Yor, M. (1996). Pricing and hedging double barrier options: A probabilistic approach. Mathematical Finance, 6, 365–378.
Haug, E.G. (1999). Barrier put-call transformations, Tempus Financial Engineering Number 397, Norway, download at http://ssrn.com/abstract=150156, 150–156.
Heynen, R. C., & Kat, H. M. (1995). Lookback options with discrete and partial monitoring of the underlying price. Applied Mathematical Finance, 2, 273–284.
Hui, C. H., Lo, C. F., & Yuen, P. H. (2000). Comment on: ’pricing double barrier options using Laplace transforms’ by Antoon Pelsser. Finance and Stochastics, 4(1), 105–107.
Kunitomo, N., & Ikeda, M. (1992). Pricing options with curved boundaries. Mathematical Finance, 2(4), 275–298.
Lo, C. F., Lee, H. C., & Hui, C. H. E. N. (2003). A simple approach for pricing barrier options with time-dependent parameters. Quantitative Finance, 3, 98–107.
Marianito, R. R., & Rogemar, S. M. (2006). An alternative approach to solving the Black–Scholes equation with time-varying parameters. Applied Mathematics Letters, 19, 398–402.
Merton, R. C. (1973). Theory of rational option pricing. Bell Journal of Economics and Management Science, 4(1), 141–183.
Mutenga, S., & Staikouras, S. K. (2004). Insurance companies and firm-wide risk: A barrier option approach. Journal of Insurance Research and Practice, 19, 62–70.
Oksendal, B. (2003). Stochastic differential equations: An introduction with applications (6th ed.). Berlin: Springer.
Pelsser, A. (2000). Pricing double barrier options using Laplace transforms. Finance and Stochastics, 4, 95–104.
Sühan, A., Gerhold, S., Haidinger, R., & Hirhager, K. (2013). Digital double barrier options: Several barrier periods and structure floors. International Journal of Theoretical and Applied Finance, 16(8), 1350044.
Strauss, W. A. (1992). Partial differential equations, an introduction. New York: Wiley.
Wilmott, P., & Brandimarte, P. (1998). Derivatives: The theory and practice of financial engineering. New York: Wiley.
Wilmott, P., Dewynne, J., & Howison, S. (1993). Option pricing: Mathematical models and computation. Oxford: Oxford Financial Press.
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Farnoosh, R., Rezazadeh, H., Sobhani, A. et al. A Numerical Method for Discrete Single Barrier Option Pricing with Time-Dependent Parameters. Comput Econ 48, 131–145 (2016). https://doi.org/10.1007/s10614-015-9506-7
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DOI: https://doi.org/10.1007/s10614-015-9506-7