Emergence of Risk-Neutral Probabilities from a Game-Theoretic Origin
This chapter lays the geometric foundations for our approach to option pricing establishing a link between certain game-theoretic problems and risk-neutral probabilities. We show how these probabilities, in their abstract geometric representation, appear
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Emergence of Risk-Neutral Probabilities from a Game-Theoretic Origin
12.1 Geometric Risk-Neutral Probabilities and Their Extreme Points In this section we will introduce risk-neutral probabilities in an abstract geometric setting revealing their basic properties and describing their extreme points. This discussion belongs to convex analysis and is independent of either games or finance, but it will provide us with basic tools to use later. For a compact metric space E we denote by P(E) the set of probability laws on E and by C(E) the Banach space of bounded continuous functions on E. For our purpose here we will mostly need finite subsets E = {ξ1 , . . . , ξk } of Rd (in which case probability laws are given by the sets of positive numbers {p1 , . . . , pk } summing up to one). We will work in a bit more general setting anticipating further applications; more general compact sets pop in very naturally sometimes, especially when dealing with mixed strategies. For f ∈ C(E), μ ∈ P(E), the standard pairing is given by the integration or, in more probabilistic notation, by the expectation (where we will denote, with some abuse of notation, random and integration variables by the same letter): ( f , μ) =
E
f (x)μ (dx) = Eμ f (x) = Eμ f .
This pairing also extends to vector-valued functions f . The set P(E) is known to be a compact set in its weak topology [where μn → μ , as n → ∞, whenever ( f , μn ) → ( f , μ ) for any f ∈ C(E), as n → ∞]. Definition 12.1. Let us say that a probability law μ ∈ P(E) on E ⊂ Rd is risk neutral with respect to the origin or, more concisely, risk neutral if the origin is its barycenter, that is, E ξ μ (d ξ ) = 0. The set of all risk-neutral laws on E will be denoted by Prn (E).
P. Bernhard et al., The Interval Market Model in Mathematical Finance, Static & Dynamic Game Theory: Foundations & Applications, DOI 10.1007/978-0-8176-8388-7 12, © Springer Science+Business Media New York 2013
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12 Emergence of Risk-Neutral Probabilities
It will be convenient to work in a bit more general setting of probability laws with linear constraints. That is, for a compact subset E ⊂ Rn and a continuous mapping F : E → Rd let P(E; F) = { μ ∈ P(E) : (F, μ ) =
F(x)μ (dx) = 0}.
This is clearly a convex closed subset of P(E) (which may be empty of course), and Prn (E) = P(E; Id), where Id is the identical mapping of E. The key role for the analysis of risk-neutral probabilities belongs to the following two conditions for a subset of Rd . Definition 12.2. A set E ⊂ Rd is called weakly (resp. strongly) positively complete if there exists no ω ∈ Rd such that (ω , ξ ) > 0 [resp. (ω , ξ ) ≥ 0] for all ξ ∈ E. Geometrically, this means that E does not belong to any open (resp. closed) halfspace of Rd . Clearly E is strongly positively complete if for any ω ∈ Rd there exist vectors ξ1 , ξ2 ∈ E such that (ω , ξ1 ) > 0 and (ω , ξ2 ) < 0. If E ⊂ Rd is a compact convex set, then E is weakly positively complete if and only if it contains the origin. It is, moreover, strongly positively complete whenever
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