By Hansjörg Albrecher, Walter Schachermayer, Wolfgang J. Runggaldier
This e-book is a suite of state of the art surveys on numerous themes in mathematical finance, with an emphasis on contemporary modelling and computational ways. the amount is expounded 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 info for Advanced Financial Modelling (Radon Series on Computational and Applied Mathematics)
How would the situation be, if we would take a step back and reduce instead of increase the number of risky assets? Suppose we start from a trivial initial market without risky assets S where only the riskless numeraire asset with price 1 is tradable. 5) of Qngd (1) yields Qngd (1) = P ngd . 1) for some Q ∈ P ngd , does not permit trading opportunities that are ‘too good’. 8) for X ∈ L2 (P ). That is, ρ(X) = π u (X; Qngd (1)) is of the same ‘good-deal’-type as π u (X) = π u (X; Qngd (S)) but defined with respect to 1 instead of S .
S.. α 5 Good-deal valuation and hedging via BSDEs In this section, we obtain a dynamic description for the good-deal valuation bounds, that arise from no-good-deal restrictions on optimal expected growth rates. The valuation bounds are given by the solutions to standard non-linear backward SDEs, whose generator satisfies a Lipschitz condition. Moreover, we develop a corresponding notion of hedging and show that also the hedging strategy is described by a BSDE. 8, let us define for X ∈ L2 (P ) ⊃ L∞ πtu (X) := ess sup EtQ [X] , Q∈Qngd t ≤ T¯ .
1) Using an L2 space for X fits conveniently with the present BSDE setting. 1) induces a mapping L2 (P ) → L2 (P, Ft ). -4. -6. 1). 1. Let Q ∼ P with density process dQ dP |F =: D = E( λdW ) for a predictable and bounded process λ. 2) for X ∈ L2 (P ). Moreover Y is a Q-martingale and W λ := W − λdt is a Q Brownian motion, satisfying Yt = EtQ [X] = Y0 + t 0 Z dW λ , t ≤ T¯ . e. Z·W is a P -BMO martingale. Proof. 2) are standard and it has a unique solution (Y, Z) in ST2¯ × HT2¯ . By application of Itˆo’s formula, the process DY is seen to be a local P -martingale.
Advanced Financial Modelling (Radon Series on Computational and Applied Mathematics) by Hansjörg Albrecher, Walter Schachermayer, Wolfgang J. Runggaldier