Coherent acceptability measures in multiperiod models. (English) Zbl 1107.91059

Roorda, Berend; Schumacher, J. M.; Engwerda, Jacob

Math. Finance 15, No. 4, 589-612 (2005).

Summary: The framework of coherent risk measures has been introduced by P. Artzner et al. [Math. Finance 9, 203–228 (1999; Zbl 0980.91042)] in a single-period setting. Here, we investigate a similar framework in a multiperiod context. We add an axiom of dynamic consistency to the standard coherence axioms, and obtain a representation theorem in terms of collections of multiperiod probability measures that satisfy a certain product property. This theorem is similar to results obtained by L. Epstein and M. Schneider [J. Econ. Theory 113, No. 1, 1–31 (2003; Zbl 1107.91360)] and T. Wang [J. Econ. Theory 108, No. 2, 286–321 (2003; Zbl 1040.91033)] in a different axiomatic framework. We then apply our representation result to the pricing of derivatives in incomplete markets, extending results by P. Carr, H. Geman, and D. B. Madan [J. Financial Econ. 32, 131–167 (2001)] to the multiperiod case. We present recursive formulas for the computation of price bounds and corresponding optimal hedges. When no shortselling constraints are present, we obtain a recursive formula for price bounds in terms of martingale measures.

MSC:

91B30	Risk theory, insurance (MSC2010)
91G20	Derivative securities (option pricing, hedging, etc.)
60G44	Martingales with continuous parameter

Keywords:

dynamic consistency; robustness; option pricing; incomplete markets

Citations:

Zbl 0980.91042; Zbl 1107.91360; Zbl 1040.91033

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