Volatility risk for regime-switching models. (English) Zbl 1085.62510

Summary: Regime-switching models have proven to be well-suited for capturing the time series behavior of many financial variables. In particular, they have become a popular framework for pricing equity linked insurance products. The success of these models demonstrates that realistic modeling of financial time series must allow for random changes in volatility. In the context of valuation of contingent claims, however, random volatility poses additional challenges when compared with the standard Black-Scholes framework. The main reason is the incompleteness of such models, which implies that contingent claims cannot be hedged perfectly and that a unique identification of the correct risk-neutral measure is not possible.
The objective of this paper is to provide tools for managing the volatility risk. First we present a formula for the expected value of a shortfall caused by misspecification of the realized cumulative variance. This, in particular, leads to a closed-form expression for the expected shortfall for any strategy a hedger may use to deal with the stochastic volatility. Next we identify a method of selection of the initial volatility that minimizes the expected shortfall. This strategy is the same as delta hedging based on the cumulative volatility that matches the Black-Scholes model with the stochastic volatility model. We also discuss methods of managing the volatility risk under model uncertainty. In these cases, super-hedging is a possible strategy but it is expensive. The results presented enable a more accurate analysis of the trade-off between the initial cost and the risk of a shortfall.


62P05 Applications of statistics to actuarial sciences and financial mathematics
91B84 Economic time series analysis
91B28 Finance etc. (MSC2000)
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[1] Biagini Fracesca, Mathematical Finance 10 pp 109– (2000) · Zbl 1014.91039
[2] Chow Yuan S., Probability Theory. Independence, Interchangeability, Martingales, 3. ed. (1997)
[3] Christensen Bent J., The European Journal of Finance 8 (2) pp 187– (2002)
[4] Cox John C., Econometrica 53 pp 385– (1985) · Zbl 1274.91447
[5] Di Masi Giovanni B., Theory of Probability and Applications 39 pp 172– (1994) · Zbl 04525926
[6] El Karoui Nicole, Mathematical Finance 8 pp 93– (1998) · Zbl 0910.90008
[7] Hardy Mary R., North American Actuarial Journal 5 (2) pp 41– (2001) · Zbl 1083.62530
[8] Hardy Mary R., Investment Guarantees: Modeling And Risk Management for Equity-Linked Life Insurance (2003) · Zbl 1092.91042
[9] Heath David, Mathematical Finance 11 (4) pp 385– (2001) · Zbl 1032.91058
[10] Heston Steven, Review of Financial Studies 6 pp 327– (1993) · Zbl 1384.35131
[11] Hobson David G, The Annals of Applied Probability 8 pp 193– (1998) · Zbl 0933.91012
[12] Hull John, Journal of Finance 42 pp 281– (1987)
[13] Kim Chang-Jin, State-Space Models with Regime Switching: Classical and Gibbs-Sampling Approaches with Applications (1999)
[14] Lamoureux Christopher G., Review of Financial Studies 6 pp 293– (1993)
[15] Møller Thomas, North American Actuarial Journal 5 (2) pp 79– (2001)
[16] Monroy , Patricia M. 2000 . ” Regime Switching Models witht-Distribution ” . Waterloo : Department of Statistics and Actuarial Science, University of Waterloo . Master essay
[17] Mykland Per A., The Annals of Applied Probability 10 pp 664– (2000) · Zbl 1065.91030
[18] Rebonato Riccardo, Volatility and Correlation in the Pricing of Equity, FX and Interest-Rate Options (1999)
[19] Scott Louis, Journal of Financial and Quantitative Analysis 22 pp 419– (1987)
[20] Shephard Neil, Time Series Models in Econometrics, Finance and Other Fields pp 1– (1996)
[21] Stein Elias, Review of Financial Studies 4 pp 727– (1991) · Zbl 1458.62253
[22] Taylor Stephen J., Mathematical Finance 4 pp 183– (1994) · Zbl 0884.90054
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