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Asset pricing using finite state Markov chain stochastic discount functions. (English) Zbl 1258.91079

From the abstract: “This article fuses two pieces of theory to make a tractable model for asset pricing. The first is the theory of asset pricing using a stochastic discounting function. The second is to model uncertainty in an economy using a Markov chain. Using the semi-martingale dynamics for the chain these models can be calibrated and asset valuations derived. Interest rate models, stock price models, futures pricing, exchange rates can all be introduced endogenously in this framework.”
Their use of stochastic discount functions is directly inspired from the works of L. C. G. Rogers and coauthors [L. C. G. Rogers, The potential approach to the term structure of interest rates and foreign exchange rates”, Math. Finance 7, No. 2, 157–176 (1997; Zbl 0884.90046)], using the so-called ‘potential approach’. Using this and defining this discount function via a finite-state Markov chain, they revisit the work of [J. H. Cochrane, Asset Pricing. Princeton University Press (2005)], and derive dynamics for stock prices, bond prices, exchange rates, future prices, forward rates and forward measures.

MSC:

91B25 Asset pricing models (MSC2010)
60J27 Continuous-time Markov processes on discrete state spaces
91G20 Derivative securities (option pricing, hedging, etc.)

Citations:

Zbl 0884.90046
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References:

[1] Cochrane J.H., Asset Pricing (rev. ed.) (2005)
[2] DOI: 10.1142/S0219024909005142 · Zbl 1182.91182 · doi:10.1142/S0219024909005142
[3] DOI: 10.1080/14697680802036168 · Zbl 1171.91337 · doi:10.1080/14697680802036168
[4] Elliott R.J., Stochastic Calculus and Applications (1982) · Zbl 0503.60062
[5] Elliott R.J., Hidden Markov Models: Estimation and Control, Applications of Mathematics (1994)
[6] Hull J., Options, Futures, and Other Derivatives (), 7. ed. (2008)
[7] Kluge T., The Potential Approach in Practice (2008)
[8] Kushner H.J., Probability Methods for Approximations in Stochastic Control and for Elliptic Equations (1977) · Zbl 0547.93076
[9] DOI: 10.1007/978-3-540-47856-0 · doi:10.1007/978-3-540-47856-0
[10] DOI: 10.1111/1467-9965.00029 · Zbl 0884.90046 · doi:10.1111/1467-9965.00029
[11] Rogers , L.C.G. , and Zane , O. 1997 .Fitting Potential Models to Interest Rates and Foreign Exchange Rates, Vasicek and Beyond(ed. L.P. Hughston ), 327 – 342 . RISK Publications , London .
[12] Rogers , L.C.G. , and Yousaf , F.A. 2002 .Markov Chains and the Potential Approach to Modelling Interest Rates and Exchange Rates. Mathematical finance–Bachelier Congress, 2000 (Paris) , 375 – 406 . Springer Finance, Springer , Berlin . · Zbl 1051.91038
[13] Rogers L.C.G., One for All: the Potential Approach to Pricing and Hedging (2006) · Zbl 1308.91171
[14] DOI: 10.1002/asmb.893 · Zbl 1286.91143 · doi:10.1002/asmb.893
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