Abstract
This paper considers a partial differential equation (PDE) approach to evaluate coherent risk measures for derivative instruments when the dynamics of the risky underlying asset are governed by a Markov-modulated geometric Brownian motion (GBM); that is, the appreciation rate and the volatility of the underlying risky asset switch over time according to the state of a continuous-time hidden Markov chain model which describes the state of an economy. The PDE approach provides market practitioners with a flexible and effective way to evaluate risk measures in the Markov-modulated Black–Scholes model. We shall derive the PDEs satisfied by the risk measures for European-style options, barrier options and American-style options.
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References
Aggoun L., Elliott R.J., Moore J.B. (1994) Hidden Markov Models: Estimation and Control. Springer, Berlin Heidelberg New York
Artzner P., Delbaen F., Eber J., Heath D. (1999) Coherent measures of risk. Math Financ 9(3): 203–228
Buffington J., Elliott R.J. (2002a) Regime switching and European options. In: Lawrence K.S. (ed) Stochastic Theory and Control, Proceedings of a Workshop. Springer, Berlin Heidelberg New York, pp 73–81
Buffington J., Elliott R.J. (2002b) American options with regime switching. Int J Theor App Financ 5, 497–514
Chan L.L., Elliott R.J., Siu T.K. (2005) Option pricing and Esscher transform under regime switching. Ann Financ 1(4): 423–432
Duffie, D., Pan, J.: An overview of value at risk. J Derivatives Spring, 7–49 (1997)
Elliott R.J., van der Hoek J. (1997) An application of hidden Markov models to asset allocation problems. Financ Stochastics 3, 229–238
Elliott R.J., Hunter W.C., Jamieson B.M. (2001) Financial signal processing. Int J Theor Appl Financ 4, 567–584
Elliott R.J., Hinz J. (2002) Portfolio analysis, hidden Markov models and chart analysis by PF-Diagrams. Int J Theor App Financ 5, 385–399
Elliott R.J., Malcolm W.P., Tsoi A.H. (2003) Robust parameter estimation for asset price models with Markov modulated volatilities. J Econ Dyn Control 27(8): 1391–1409
Elliott R.J., Kopp P.E. (2004) Mathematics of Financial Markets. Springer, Berlin Heidelberg New York
Elliott, R.J., Royal, A.J.: Asset prices with regime switching variance Gamma dynamics. In: Bensoussan, A., Zhang, Q. (eds.) Hand Book on Mathematical Finance. Elsevier, Amsterdam (2007) (To appear)
Embrechts, P.: Value at risk, Lecture Notes, Centre of Financial Time Series. The University of Hong Kong (2000)
Föllmer H., Schweizer M. (1991) Hedging of contingent claims under incomplete information. In: Davis M.H.A., Elliott R.J. (eds) Applied Stochastic Analysis. Gordon and Breach, London, pp. 389–414
Guo X. (2001) Information and option pricings. Quant Finance 1, 38–44
Jahel, E., Perraudin, W., Sellin, P.: Value at risk for derivatives. J Derivatives, Spring 7–26 (1999)
Kallsen J. (2000) Utility-based derivative pricing in incomplete markets. In: German H., Madan D., Pliska S.R., Vorst T. (eds) Mathematical Finance – Bachelier congress. Springer, Berlin Heidelberg New York
Morgan J.P. (1996) Riskmetrics – technical document. J.P Morgan, New York
Pliska S.R. (1997) Introduction to mathematical finance. Blackwell, Oxford
Schweizer M. (1992) Mean-variance hedging for general claims. Ann App Probab 2, 171–179
Siu T.K., Yang H. (2000) A p.d.e. approach for measuring risk of derivatives. App Math Financ 7(3): 211–228
Siu T.K., Yang H. (2001) Risk measures for derivatives under Black-Scholes economy. Int J Theor App Financ 4(5): 819–835
Siu T.K., Tong H., Yang H. (2001) Bayesian risk measures for derivatives via random Esscher transform. North Am Actuar J 5(3): 78–91
Stojanovic, S.D.: Higher dimensional fair option pricing and hedging under HARA and CARA utilities, (Preprint August 2005; Available at SSRN: http://ssrn.com/abstract=912763)
Wilmott P. (1998) Derivatives: The Theory and Practice of Financial Engineering. Wiley, London
Yao D.D., Zhang Q., Zhou X.Y. (2006) A regime-switching model for European options. In: Yan H., Yin G., Zhang Q.(eds) Stochastic processes, optimization, and control theory applications in financial engineearing, queueing networks, and manufacturing systems. Springer, Berlin Heidelberg New York, pp. 281–300
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Elliott, R.J., Siu, T.K. & Chan, L. A PDE approach for risk measures for derivatives with regime switching. Annals of Finance 4, 55–74 (2008). https://doi.org/10.1007/s10436-006-0068-5
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DOI: https://doi.org/10.1007/s10436-006-0068-5
Keywords
- Risk measures
- Regime-switching PDE
- Regime-switching HJB equation
- Stochastic optimal control
- Esscher transform
- Delta-neutral hedging
- Jump risk
- American options
- Exotic options