Download PDF by Hansjörg Albrecher, Walter Schachermayer, Wolfgang J.: Advanced Financial Modelling (Radon Series on Computational

By Hansjörg Albrecher, Walter Schachermayer, Wolfgang J. Runggaldier

ISBN-10: 3110213133

ISBN-13: 9783110213133

This ebook is a suite of state of the art surveys on a variety of subject matters in mathematical finance, with an emphasis on contemporary modelling and computational techniques. the quantity is said to a 'Special Semester on Stochastics with Emphasis on Finance' that happened from September to December 2008 on the Johann Radon Institute for Computational and utilized arithmetic of the Austrian Academy of Sciences in Linz, Austria.

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Additional resources for Advanced Financial Modelling (Radon Series on Computational and Applied Mathematics)

Example text

Recursiveness) For any stopping times T ≤ τ ≤ T¯, it holds that πTu (X) = πTu (πτu (X)) . 3. (Stopping-time consistency) For stopping times T ≤ τ ≤ T¯, the inequality πτu (X 1 ) ≥ πτu (X 2 ) implies πTu (X 1 ) ≥ πTu (X 2 ). 4. (Supermartingale property) (πtu (X)) is a supermartingale under any Q ∈ S . 5. (Coherent risk measure) For any stopping time T ≤ T¯ and mT , αT , λT ∈ L∞ (FT ) with 0 ≤ αT ≤ 1, λT ≥ 0, the mapping X → πTu (X) satisfies the properties: • • • • monotonicity: X 1 ≥ X 2 implies πTu (X 1 ) ≥ πTu (X 2 ) translation invariance: πTu (X + mT ) = πTu (X) + mT convexity: πTu (αT X 1 + (1 − αT )X 2 ) ≤ αT πTu (X 1 ) + (1 − αT )πTu (X 2 ) positive homogeneity: πTu (λT X) = λT πTu (X) 6.

1. 14) minimises the a priori risk measure ρt of the residual risk simultaneously for all t ≤ T¯. Being coherent, ρ is a monetary risk measure (see [2]) and the good deal bound can be interpreted as the minimal capital required to make the position acceptable, after optimal hedging according to φ∗ . 2. By the relation π (X) = −π u (−X) between lower and upper good-deal bounds, the result also yields the lower good-deal bound and the corresponding hedging strategy. 4) is non-linear, upper and lower bounds as well as the respective hedging strategies are different, in general.

By the definition of Qngd , since | − ξ + η|2 = |ξ|2 + |η|2 ≤ h2 ⊥ with equality holding for η¯ := h2 − |ξt |2 Π⊥ t (Z)/|Πt (Z)|. 5). 1, we have Yt = Et [X] for η = η Q and Ytη¯ = EtQ [X]. 6). Dynamic good-deal hedging We are going to show in this section, that a hedging strategy which minimises a suitable dynamic coherent risk measure is naturally linked to the good-deal valuation bounds. To this end, let P ngd denote the set of those equivalent probability measures Q whose density process is of the form E( λdW ) for a predictable bounded process λ with |λ| ≤ h, that is P ngd := Q∼P dQ dP F =E λdW for λ predictable with |λ| ≤ h .

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Advanced Financial Modelling (Radon Series on Computational and Applied Mathematics) by Hansjörg Albrecher, Walter Schachermayer, Wolfgang J. Runggaldier


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