By Glenn Shafer, Vladimir Vovk
Offers a beginning for likelihood in accordance with video game idea instead of degree theory.* a robust philosophical process with sensible applications.* provides in-depth insurance of classical chance concept in addition to new concept.
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Offers a starting place for chance in accordance with video game idea instead of degree thought. * a robust philosophical procedure with useful purposes. * offers in-depth assurance of classical chance conception in addition to new conception.
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Extra resources for Probability and Finance: It's Only a Game!
2a = 31 __, bi , hl + b2 2 , , .. -< By combination with a sure thing 1 b2 Fig. 2 Contending on equal terms. 2. Suppose now that we propose to Paul that he receive b2 if he defeats Peter and bl if he loses to Peter, where bl < b2. What price can we fairly charge Paul for this opportunity? Pascal answered this question by recognizing that the payoff can be reproduced by combining two payoffs whose fair price is already established: 0 0 Paul gets bl no matter what happens. It is surely fair to charge Paul bl for this.
Borel shared PoincarC’s cautious attitude towards continuous probability [ 164, 245, 2701. For Borel, a concrete meaning could be given to the idea of choosing at random from a finite set and perhaps even from a denumerable set, but because of the nonconstructive character of the totality of real numbers, no meaning could be given to choosing at random a real number from an interval or a point from a region of the plane. But in 1909, Borel published a celebrated article that brought continuous measure theory into the very heart of probability.
As it turns out, binomial trees, although unrealistic as models of the market, can serve as computationally useful approximations to these widely used (although perhaps equally unrealistic, alas) continuous-time models. This point, first demonstrated in the late 1970s [66,67, 2561, has made binomial trees a standard topic in textbooks on option pricing. 5: GAME-THEORETIC PROBABILITY IN FINANCE Today’s price , Price at tomorrow noon $7 $8”,=‘ rS9d-r Tomorrow’s closing mice X $5 $0 . -$8 , $0 , ,, ..