Esscher transforms and consumption-based models. (English) Zbl 1231.91423
Summary: The Esscher transform is an important tool in actuarial science. Since the pioneering work of H. U. Gerber and E. S. W. Shiu [Mitt., Schweiz. Ver. Versicherungsmath. 1994, No. 2, 143–166 (1994; Zbl 0816.90012)], the use of the Esscher transform for option valuation has also been investigated extensively. However, the relationships between the asset pricing model based on the Esscher transform and some fundamental equilibrium-based asset pricing models, such as consumption-based models, have so far not been well-explored. In this paper, we attempt to bridge the gap between consumption-based models and asset pricing models based on Esscher-type transformations in a discrete-time setting. Based on certain assumptions for the distributions of asset returns, changes in aggregate consumptions and returns on the market portfolio, we construct pricing measures that are consistent with those arising from Esscher-type transformations. Explicit relationships between the market price of risk, and the risk preference parameters are derived for some particular cases.
MSC:
| 91G20 | Derivative securities (option pricing, hedging, etc.) |
| 91B25 | Asset pricing models (MSC2010) |
| 91B30 | Risk theory, insurance (MSC2010) |
Keywords:
Esscher transform; Esscher-Girsanov transform; consumption-based model; stochastic discount factor; exponential affine form; Euler equation; Radon-Nikodym derivative; utility functionCitations:
Zbl 0816.90012References:
| [1] | Badescu, A. M.; Kulperger, R. J., GARCH option pricing: A semiparametric approach, Insurance: Mathematics and Economics, 43, 1, 69-84 (2008) · Zbl 1140.91374 |
| [2] | Badescu, A. M.; Kulperger, R. J.; Lazar, E., Option valuation with normal mixture GARCH models, Studies in Nonlinear Dynamics & Econometrics, 12, 2 (2008), Article 5 · Zbl 1193.91148 |
| [3] | Barndorff-Nielsen, O. E., Normal inverse Gaussian distributions and stochastic volatility modelling, Scandinavian Journal of Statistics, 24, 1-13 (1997) · Zbl 0934.62109 |
| [4] | Bertholon, H.; Monfort, A.; Pegoraro, F., Econometric asset pricing modelling, Journal of Financial Econometrics, 6, 4, 407-458 (2008) |
| [5] | Black, F.; Scholes, M., The pricing of options and corporate liabilities, Journal of Political Economy, 81, 637-659 (1973) · Zbl 1092.91524 |
| [6] | Breeden, D. T., An intertemporal asset pricing model with stochastic consumption and investment opportunities, Journal of Financial Economics, 7, 3, 265-296 (1979) · Zbl 1131.91330 |
| [7] | Brennan, M. J., The pricing of contingent claims in discrete time models, Journal of Finance, 24, 1, 53-68 (1979) |
| [8] | Brown, D. P.; Gibbons, M. R., A simple econometric approach for utility-based asset pricing models, Journal of Finance, 40, 2, 359-381 (1985) |
| [9] | Bühlmann, H., An economic premium principle, ASTIN Bulletin, 11, 1, 52-60 (1980) |
| [10] | Bühlmann, H., The general economic premium principle, ASTIN Bulletin, 14, 1, 13-21 (1984) |
| [11] | Bühlmann, H.; Delbaen, F.; Embrechts, P.; Shiryaev, A. N., No-arbitrage, change of measures and conditional Esscher transforms, CWI Quarterly, 9, 4, 291-317 (1996) · Zbl 0943.91037 |
| [12] | Bühlmann, H.; Delbaen, F.; Embrechts, P.; Shiryaev, A. N., On Esscher transforms in discrete finance models, ASTIN Bulletin, 28, 2, 171-186 (1998) · Zbl 1162.91367 |
| [13] | Camara, A., An extended set of risk neutral valuation relationships for the pricing of contingent claims, Review of Derivative Research, 3, 67-83 (1999) · Zbl 1274.91404 |
| [14] | Camara, A., A generalization of the Brennan-Rubinstein approach for the pricing derivatives, Journal of Finance, 2, 805-819 (2003) |
| [15] | Campbell, J. Y.; Lo, A. W.; MacKinlay, A. C., The Econometrics of Financial Markets (1997), Princeton University Press: Princeton University Press Princeton · Zbl 0927.62113 |
| [16] | Chan, T., Pricing contingent claims on stocks driven by Lévy processes, Annals of Applied Probability, 9, 2, 504-528 (1999) · Zbl 1054.91033 |
| [17] | Christoffersen, P., Redouane, E., Feunou, B., Jacobs, K., 2009. Option valuation with conditional heteroskedasticity and non-normality. Review of Financial Studies (forthcoming); Christoffersen, P., Redouane, E., Feunou, B., Jacobs, K., 2009. Option valuation with conditional heteroskedasticity and non-normality. Review of Financial Studies (forthcoming) |
| [18] | Cox, J.; Ingersoll, J.; Ross, S., A theory of the term structure of interest rates, Econometrica, 53, 2, 385-407 (1985) · Zbl 1274.91447 |
| [19] | Duan, J.-C., The GARCH option pricing model, Mathematical Finance, 5, 13-32 (1995) · Zbl 0866.90031 |
| [20] | Duan, J.-C., 1999. Conditionally fat-tailed distributions and the volatility smile in options, unpublished manuscript. University of Toronto; Duan, J.-C., 1999. Conditionally fat-tailed distributions and the volatility smile in options, unpublished manuscript. University of Toronto |
| [21] | Duffie, D.; Zame, B., The consumption-based capital asset pricing model, Econometrica, 57, 1279-1298 (1989) · Zbl 0684.90007 |
| [22] | Elliott, R. J.; Chan, L.; Siu, T. K., Option pricing and Esscher transform under regime switching, Annals of Finance, 1, 4, 423-432 (2005) · Zbl 1233.91270 |
| [23] | Elliott, R. J.; Hunter, W. C.; Kopp, P. E.; Madan, D. B., Pricing via multiplicative price decomposition, Journal of Financial Engineering, 4, 247-262 (1995) |
| [24] | Elliott, R. J.; Madan, D. B., A discrete time equivalent martingale measure, Mathematical Finance, 9, 6, 825-841 (1998) |
| [25] | Embrechts, P., Actuarial versus financial pricing of insurance, Risk Finance, 1, 17-26 (2000) |
| [26] | Elliott, R. J.; Siu, T. K.; Chan, L., Option pricing for GARCH models with Markov switching, International Journal of Theoretical and Applied Finance, 9, 6, 825-841 (2006) · Zbl 1138.91437 |
| [27] | Epstein, L.; Zin, S., Substitution, risk aversion, and temporal behavior of consumption and asset returns: A theoretical framework, Econometrica, 57, 937-969 (1989) · Zbl 0683.90012 |
| [28] | Epstein, L.; Zin, S., Substitution, risk aversion, and temporal behavior of consumption and asset returns: An empirical analysis, Journal of Political Economy, 99, 263-286 (1991) |
| [29] | Esscher, F., On the probability function in the collective theory of risk, Skandinavisk Aktuarietidskrift, 15, 175-195 (1932) · JFM 58.1177.05 |
| [30] | Garcia, R.; Renault, E.; Semenov, A., Disentangling risk aversion and intertemporal substitution through a reference level, Finance Research Letters, 3, 181-193 (2006) |
| [31] | Gerber, H. U.; Shiu, E. S.W., Option pricing by Esscher transforms (with discussions), Transactions of the Society of Actuaries, 46, 99-191 (1994) |
| [32] | Gerber, H. U.; Goovaerts, M. J., On the representation of additive principles of premium calculation, Scandinavian Actuarial Journal, 4, 221-227 (1981) · Zbl 0484.62109 |
| [33] | Goovaerts, M. J.; Laeven, R., Actuarial risk measures for financial derivative pricing, Insurance: Mathematics and Economics, 42, 2, 540-547 (2008) · Zbl 1152.91444 |
| [34] | Goovaerts, M. J.; Kaas, R.; Laeven, R. J.A.; Tang, Q., A comonotonic image of independence for additive risk measures, Insurance: Mathematics and Economics, 35, 3, 581-594 (2004) · Zbl 1122.91341 |
| [35] | Gourieroux, C.; Monfort, A., Econometric specifications of stochastic discount factor models, Journal of Econometrics, 136, 509-530 (2007) · Zbl 1420.91461 |
| [36] | Hansen, L. P.; Richard, S., The role of Conditioning information in deducing testable restrictions implied by dynamic asset pricing models, Econometrica, 55, 587-613 (1987) · Zbl 0613.90004 |
| [37] | Hansen, L. P.; Jagannathan, R., Assesing specification errors in stochastic discount factor models, Journal of Finance, 52, 2, 557-590 (1992) |
| [38] | Harrison, M. J.; Kreps, D. M., Martingales and arbitrage in multiperiod security markets, Journal of Economic Theory, 20, 381-408 (1979) · Zbl 0431.90019 |
| [39] | Harrison, J.; Pliska, S., Martingales and stochastic integrals in the theory of continuous trading, Stochastic Processes and Their Applications, 11, 3, 215-260 (1981) · Zbl 0482.60097 |
| [40] | Harrison, J.; Pliska, S., A stochastic calculus model of continuous trading: Complete markets, Stochastic Processes and their Applications, 15, 3, 313-316 (1983) · Zbl 0511.60094 |
| [41] | Iwaki, H.; Kijima, M.; Morimoto, Y., An economic premium principle in a multiperiod economy, Insurance: Mathematics and Economics, 28, 3, 325-339 (2001) · Zbl 1071.91025 |
| [42] | Iwaki, H., An economic premium principle in a continuous-time economy, Journal of the Operations Research Society of Japan, 45, 4, 346-361 (2002) · Zbl 1068.91031 |
| [43] | Lucas, R., Asset pricing in an exchange economy, Econometrica, 46, 1429-1445 (1978) · Zbl 0398.90016 |
| [44] | Merton, R. C., An intertemporal capital asset pricing model, Econometrica, 41, 867-887 (1973) · Zbl 0283.90003 |
| [45] | Rubinstein, M., The valuation of uncertain income streams and the pricing of options, Bell Journal of Economics, 7, 407-425 (1976) |
| [46] | Siu, T. K.; Tong, H.; Yang, H., On pricing derivatives under GARCH models: A dynamic Gerber-Shiu approach, North American Actuarial Journal, 8, 3, 17-32 (2004) · Zbl 1085.91531 |
| [47] | Siu, T. K., Fair valuation of participating policies with surrender options and regime switching, Insurance: Mathematics and Economics, 37, 3, 533-552 (2005) · Zbl 1129.60062 |
| [48] | Siu, T.K., 2009. Esscher transform: From actuarial science to quantitative Finance. Encyclopedia of Quantitative Finance (forthcoming); Siu, T.K., 2009. Esscher transform: From actuarial science to quantitative Finance. Encyclopedia of Quantitative Finance (forthcoming) |
| [49] | Schroder, M., Risk-neutral parameter shifts and derivatives pricing in discrete time, Journal of Finance, 59, 5, 2375-2402 (2004) |
| [50] | Taylor, S. J., Modeling Financial Time Series (1986), John Wiley & Sons: John Wiley & Sons New York · Zbl 1130.91345 |
| [51] | Wang, S. S., A class of distortion operators for pricing financial and insurance risks, Journal of Risk and Insurance, 67, 1, 15-36 (2000) |
| [52] | Wang, S. S., Equilibrium pricing transforms: New results on Bühlmann’s 1980 economic model, ASTIN Bulletin, 32, 213-234 (2003) · Zbl 1098.91551 |
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