Journal of Applied Science and Engineering

Published by Tamkang University Press

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Hsien-Jen Lin This email address is being protected from spambots. You need JavaScript enabled to view it.1

1Department of Applied Mathematics, Aletheia University, Tamsui, Taiwan 251, R.O.C.


 

Received: March 30, 2012
Accepted: August 17, 2012
Publication Date: June 1, 2013

Download Citation: ||https://doi.org/10.6180/jase.2013.16.2.11  


ABSTRACT


We consider the problem of valuation of certain Asian options in the geometric jump-diffusion models with continuously dividend-paying assets. With the sources of diffusion risks and two primitive tradeable assets, the market in this model is, in general, incomplete, and so, there are more than one equivalent martingale measures and no-arbitrage prices. For this jump-diffusion model, we adopt the minimal martingale measure as the risk-neutral pricing measure for option valuation in a dynamicallyiincomplete market. A partial integro-differential equation satisfied by the no-arbitrage price of an Asian option is obtained by change of numeraire technique under the minimal martingale measure.


Keywords: Asian Options, Jump-diffusion Model, Dividend, Itô’s Formula, Partial Integro-differential Equations


REFERENCES


  1. [1] Black, F. and Scholes, M., “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy, Vol. 81, pp. 637654 (1973). doi: 10.1086/260062
  2. [2] Merton, R. C., “Theory of Rational Option Pricing,” Bell Journal of Economics and Management Science, Vol. 4, pp. 141183 (1973). doi: 10.2307/3003143
  3. [3] Merton, R. C., “Option Pricing When Underlying Stock Return are Discontinuous,” Journal of Financial Economics, Vol. 3, pp. 125144 (1976). doi: 10.1016/ 0304-405X(76)90022-2
  4. [4] Cont, R. and Tankov, P., Financial Modelling with Jump Processes, Chapman and Hall/CRC, Boca Raton (2004).
  5. [5] Geman, H. and Yor, M., “Bessel Processes, Asian Options, and Perpetuities,” Mathematical Finance, Vol. 3, No. 4, pp. 349375 (1993). doi: 10.1111/j.1467- 9965.1993.tb00092.x
  6. [6] Vcr, J., “Unified Asian Pricing,” Risk, Vol. 15, No. 6, pp. 113116 (2002).
  7. [7] Vcr, J. and Xu, M., “Pricing Asian Options in a Semimartingale Model,” Quantitative Finance, Vol. 4, pp. 170175 (2004). doi: 10.1080/14697680400000 021
  8. [8] Carr, P., Geman, H., Madan, D. and Yor, M., “The Fine Structure of Asset Returns: An Empirical Investigation,” Journal of Business, Vol. 75, pp. 305332 (2002). doi: 10.1086/338705
  9. [9] Eberlein, E. and Prause, K., The Generalized Hyperbolic Model: Financial Derivatives and Risk Measures, in: Geman, H., Madan, D., Pliska, S. and Vorst, T. (Ed.), In Mathematical Finance - Bachelier Congress 2000, Springer-Verlag, pp. 245267 (2000).
  10. [10] Andreasen, J., “The Pricing of Discretely Sampled Asian and Lookback Options: A Change of Numeraire Approach,” Journal of Computational Finance, Vol. 2, pp. 530 (1998).
  11. [11] Lipton, A., “Similarities via Self-Similarities,” Risk, Vol. 12, No. 9, pp. 101105 (1999).
  12. [12] Rogers, L. C. G. and Shi, Z., “The Value of an Asian Option,” Journal of Applied Probability, Vol. 32, pp. 10771088 (1995). doi: 10.2307/3215221
  13. [13] Föllmer, H. and Schweizer, M., Hedging of Contingent Claims under Incomplete Information, in: Davis, M. H. A. and Elliott, R. J. (Ed.), Applied Stochastic Analysis, Gordon and Breach, New York, pp. 389414 (1991).
  14. [14] Musiela, M. and Rutkowski, M., Martingale Methods in Financial Modeling, Springer-Verlag, Berlin (2005).
  15. [15] Lin, H. J., “Pricing Asian Options on Asset Driven by a Combined Geometric Brownian Motion and a Geometric Compound Poisson Process,” International Journal of Information and Management Sciences, Vol. 21, No. 2, pp. 113123 (2010).
  16. [16] Shreve, S. E., Stochastic Calculus for Finance II, Springer-Verlag, New York (2004).
  17. [17] Achdou, Y. and Pironneau, O., Computational Methods for Option Pricing, Frontiers Appl. Math. 30, SIAM, Philadelphia, PA (2005).
  18. [18] Tavella, D. and Randall, C., Pricing Financial Instruments: The Finite Difference Method, John Wiley & Sons, Chichester, UK (2000).
  19. [19] Cont, R. and Tankov, P., Financial Modelling with Jump Processes, Chapman and Hall/CRC, Boca Raton, FL (2004).
  20. [20] Duffy, D. J., Finite Difference Methods in Financial Engineering: A Partial Differential Equation Approach, John Wiley & Sons, Chichester, UK (2006).
  21. [21] Kemna, A. G. Z. and Vorst, A. C. F., “A Pricing Method Based on Average Asset Values,” Journal of Banking and Finance, Vol. 14, pp. 113129 (1990). doi: 10.1016/0378-4266(90)90039-5
  22. [22] Bénédicte Alziary, Jean-Paul Décamps, and PierreFrançois Koehl, “A P. D. E. Approach to Asia Options: Analytical and Numerical Evidence,” Journal of Banking and Finance, Vol. 21, No. 5, pp. 613640 (1997). doi: 10.1016/S0378-4266(96)00057-X