By Hans Follmer, Alexander Schied
ISBN-10: 3110171198
ISBN-13: 9783110171198
This e-book is an advent to monetary arithmetic for mathematicians. it really is meant either for graduate scholars with a undeniable history in chance concept in addition to for pro mathematicians in and academia. not like many textbooks on mathematical finance, merely discrete-time stochastic types are thought of. This environment has the virtue that the textual content can focus from the start on standard difficulties that are recommended through monetary purposes. furthermore, yes rules, corresponding to the final incompleteness of practical industry types, turn into therefore extra obvious and visual. nonetheless, all types are in accordance with normal chance areas, and so the textual content captures the interaction among likelihood thought and practical research that is commonplace for contemporary mathematical finance. the 1st a part of the booklet features a learn of monetary investments in a static one-period industry version. right here, an investor faces intrinsic possibility and uncertainty, which can't be hedged away. The instruments provided to accommodate this example variety from the classical thought of anticipated application till the more moderen improvement of measures of hazard. within the moment a part of the publication, the assumption of dynamic hedging and arbitrage-free pricing of contingent claims is constructed in a multi-period framework. Such industry versions tend to be incomplete, and specific concentration is given to tools combining the dynamic hedging of a dicy place with the instruments of assessing threat and uncertainty as offered partially. Contents: Mathematical finance in a single interval: Arbitrage conception. anticipated application. optimum investments. Measures of danger Dynamic Arbitrage idea: Dynamic hedging of contingent claims. American contingent claims. not obligatory decomposition and super-hedging. effective hedging in incomplete markets. Minimizing the hedging errors. Hedging below constraints References. Index
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Additional resources for Stochastic finance: an introduction in discrete time
Example text
This is a bounded Donsker class since F = {f1 f2 : f1 ∈ F1 , f2 ∈ F2 }, where Fj = {1{yj ≤ t}, t ∈ R} is a bounded Donsker class for j = 1, 2. We thus obtain consistency in probability of the bootstrap. 7, since F is bounded by 1. 4 The Functional Delta Method √ X for Suppose Xn is a sequence of random variables with n(Xn − θ) some θ ∈ Rp , and the function φ : Rp → Rq√has a derivative φ′ (θ) at θ. The standard delta method now tells us that n(φ(Xn ) − φ(θ)) φ′ (θ)X. d. data involve maps from empirical processes to spaces of functions, and hence cannot be handled by the standard delta method.
Estimation under the model P is more taxing than estimation under any parametric submodel P0 = {Pθ : θ ∈ Θ0 } ⊂ P, where Θ0 is finite dimensional. Thus the information for estimation under model P is worse than the information under any parametric submodel P0 . If the information for the regular estimator Tn is equal to the minimum of the information over all efficient estimators for all parametric submodels P0 , then Tn is semiparametric efficient. For semiparametric models, this minimizer is the best possible, since the only models with more information are parametric models.
Another interesting application—to “large p, small n” asymptotics for 32 2. 5 of Part II, although we will not address the theoretical derivation of the KMT construction. d. data are the U-processes. The mth order empirical U-process measure Un,m is defined, for a measurable function f : X m → R and a sample of obervations X1 , . . , Xn on X , as n m −1 f (Xi1 , . . ,im )∈In,m where In,m is the set of all m-tuples of integers (i1 , . . , im ) satisfying 1 ≤ i1 < · · · < im ≤ n. , Un,1 = Pn .
Stochastic finance: an introduction in discrete time by Hans Follmer, Alexander Schied
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