The Pricing of Dual-Expiry Exotics with Mean Reversion and Jumps
- 1 Department of Mathematics and Statistics, University of Ottawa, Ottawa, Canada
- 2 Enterprise Risk and Portfolio Management, Bank of Montreal, Toronto, Canada
- 3 Equity Derivative Structuring, CIBC World Market Inc., Toronto, Canada
Abstract
This paper develops a new class of models for pricing dual-expiry options that are characterized by two expiry dates. The underlying asset price is modeled by a time changed exponential Ornstein Uhlenbeck (OU) process, where the time change process is a Lévy subordinator. The new models can capture both mean reversion and jumps often observed in various types of underlying assets of exotics. The pricing method exploits the observation that dual expiry options have payoffs that can be perfectly replicated by a particular set of first and second order binary options. The novelty of the paper is that we are able to derive the analytical solutions to the prices of these binaries through eigenfunction expansion method. Based on that, we can obtain the formulas for dual-expiry exotics through static replication. We also numerically investigate the sensitivities of prices of chooser, compound and extendable options with respect to the parameters of the models.
- Buchen, P.W. (2004) The Pricing of Dual-Expiry Exotics. Quantitative Finance, 4, 101-108. https://doi.org/10.1088/1469-7688/4/1/009
- O, H.C. and Kim, M.C. (2013) Higher Order Binary Options and Multiple-Expiry Exotics. Electronic Journal of Mathematical Analysis and Applications, 1, 247-259.
- Agliardi, R. (2009) The Quintessential Option Pricing Formula under Lévy Processes. Applied Mathematics Letters, 22, 1626-1631. https://doi.org/10.1016/j.aml.2009.05.008
- Agliardi, R. (2011) A Comprehensive Mathematical Approach to Exotic Option Pricing. Mathematical Methods in the Applied Sciences, 35, 1256-1268. https://doi.org/10.1002/mma.2519
- Boyarchenko, S.I. and Levendorskii, S.Z. (2002) Non-Gaussian Merton-Black-Scholes Theory. World Scientific, Singapore. https://doi.org/10.1142/4955
- Buchen, P.W. (2012) An Introduction to Exotic Option Pricing. CRC Press, Boca Raton. https://doi.org/10.1201/b11589
- Lim, D., Li, L. and Linetsky, V. (2012) Evaluating Callable and Putable Bonds: An Eigenfunction Expansion Approach. Journal of Economic Dynamics & Control, 36, 1888-1908. https://doi.org/10.1016/j.jedc.2012.06.002
- Li, L. and Linetsky, V. (2014) Time-Changed Ornstein-Uhlenbeck Processes and Their Applications in Commodity Derivative Models. Mathematical Finance, 24, 289-330. https://doi.org/10.1111/mafi.12003
- Li, L., Mendoza-Arriaga, R., Mo, Z. and Mitchell, D. (2016) Modelling Electricity Prices: A Time Change Approach. Quantitative Finance, 16, 1089-1109. https://doi.org/10.1080/14697688.2015.1125521
- Tong, Z. and Liu, A. (2017) Analytical Pricing Formulas for Discretely Sampled Generalized Variance Swaps under Stochastic Time Change. International Journal of Financial Engineering, 4, 1-24. https://doi.org/10.1142/S2424786317500281
- Linetsky, V. and Mitchell, D. (2008) Spectral Methods in Derivatives Pricing, in Birge, J.R. and Linetsky, V. (editors). Handbook of Financial Engineering, Elsevier, Amsterdam, 223-299.
- Mendoza-Arriaga, R., Carr, P. and Linetsky, V. (2010) Time Changed Markov Processes in Unified Credit-Equity Modeling. Mathematical Finance, 20, 527-569. https://doi.org/10.1111/j.1467-9965.2010.00411.x
- Mendoza-Arriaga, R. and Linetsky, V. (2013) Time-Changed CIR Default Intensities with Two-Sided Mean-Reverting Jumps. The Annals of Applied Probability, 24, 811-856. https://doi.org/10.1214/13-AAP936