套利數(shù)學(xué)

內(nèi)容概要

in 1973 f. black and m. scholes published their pathbreaking paper [bs 73]on option pricing. the key idea -- attributed to r. melton in a footnote of the black-scholes paper -- is the use of trading in continuous time and the notion of arbitrage. the simple and economically very convincing ''principle of no-arbitrage" allows one to derive, in certain mathematical models of financial markets (such as the samuelson model, [s 65], nowadays also referred to as the "black-scholes" model, based on geometric brownian motion), unique prices for options and other contingent claims.this remarkable achievement by f. black, m. scholes and r. merton had a profound effect on financial markets and it shifted the paradigm of deal-ing with financial risks towards the use of quite sophisticated mathematical models.

書籍目錄

part i a guided tour to arbitrage theory   1 the story in a nutshell     1.1 arbitrage     1.2 an easy model of a financial market     1.3 pricing by no-arbitrage     1.4 variations of the example     1.5 martingale measures     1.6 the fundamental theorem of asset pricing   2 models of financial markets on finite probability spaces     2.1 description of the model     2.2 no-arbitrage and the fundamental theorem of asset pricing     2.3 equivalence of single-period with multiperiod arbitrage     2.4 pricing by no-arbitrage     2.5 change of numeraire     2.6 kramkov's optional decomposition theorem   3 utility maximisation on finite probability spaces     3.1 the complete case     3.2 the incomplete case     3.3 the binomial and the trinomial model   4 bachelier and black-scholes     4.1 introduction to continuous time models     4.2 models in continuous time     4.3 bachelier's model     4.4 the black-scholes model   5 the kreps-yan theorem     5.1 a general framework     5.2 no free lunch   6 the dalang-morton-willinger theorem     6.1 statement of the theorem     6.2 the predictable range     6.3 the selection principle     6.4 the closedness of the cone c     6.5 proof of the dalang-morton-willinger theorem for t＝1     6.6 a utility-based proof of the dmw theorem for t＝1     6.7 proof of the dalang-morton-willinger theorem for t≥1 by induction on t     6.8 proof of the closedness of k in the case t≥1     6.9 proof of the closedness of c in the case t≥1 under the (na) condition     6.10 proof of the dalang-morton-willinger theorem for t＞1 using the closedness of c     6.11 interpretation of the l∞-bound in the dmw theorem   7 a primer in stochastic integration     7.1 the set-up     7.2 introductory on stochastic processes     7.3 strategies, semi-martingales and stochastic integration   8 arbitrage theory in continuous time: an overview     8.1 notation and preliminaries     8.2 the crucial lemma     8.3 sigma-martingales and the non-locally bounded case part ii the original papers   9 a general version of the fundamental theorem of asset pricing (1994)     9.1 introduction     9.2 definitions and preliminary results     9.3 no free lunch with vanishing risk     9.4 proof of the main theorem     9.5 the set of representing measures     9.6 no free lunch with bounded risk     9.7 simple integrands     9.8 appendix: some measure theoretical lemmas   10 a simple counter-example to several problems in the theory of asset pricing (1998)     10.1 introduction and known results     10.2 construction of the example     10.3 incomplete markets   11 the no-arbitrage property under a change of numeraire (1995)     11.1 introduction     11.2 basic theorems     11.3 duality relation     11.4 hedging and change of numeraire   12 the existence of absolutely continuous local martingale measures (1995)     12.1 introduction     12.2 the predictable radon-nikodym derivative     12.3 the no-arbitrage property and immediate arbitrage     12.4 the existence of an absolutely continuous local martingale measure   13 the banach space of workable contingent claims in arbitrage theory (1997)     13.1 introduction     13.2 maximal admissible contingent claims     13.3 the banach space generated by maximal contingent claims     13.4 some results on the topology of     13.5 the value of maximal admissible contingent claims on the set me     13.6 the space under a numeraire change     13.7 the closure of g∞ and related problems   14 the fundamental theorem of asset pricing for unbounded stochastic processes (1998)     14.1 introduction     14.2 sigma-martingales     14.3 one-period processes     14.4 the general rd-valued case     14.5 duality results and maximal elements   15 a compactness principle for bounded sequences of martingales with applications (1999)     15.1 introduction     15.2 notations and preliminaries     15.3 an example     15.4 a substitute of compactness for bounded subsets of h1     15.4.1 proof of theorem 15.a     15.4.2 proof of theorem 15.c     15.4.3 proof of theorem 15.b     15.4.4 a proof of m. yor's theorem     15.4.5 proof of theorem 15.d     15.5 application part iii bibliography   references

章節(jié)摘錄

　　Let us turn back to the no-arbitrage theory developed in Chap. 2 to raise againthe question： what can we deduce from applying the no-arbitrage principlewith respect to pricing and hedging of derivative securities？　　While we obtained satisfactory and mathematically rigorous answers tothese questions in the case of a finite underlying probability space inChap. 2， we saw in Chap. 4， that the basic examples for this theory， theBachelier and the BlackScholes model， do not fit into this easy setting， asthey involve Brownian motion.　　In Chap. 4 we overcame this difficulty either by using well-known resultsfrom stochastic analysis （e.g.， the martingale representation Theorem 4.2.1for the Brownian filtration）， or by appealing to the faith of the reader， thatthe results obtained in the finite case also carry over —— mutatis mutandis ——to more general situations， as we did when applying the change of num~rairetheorem to the calculation of the Black-Scholes model.

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