Pricing kernel factorization and recovery theorem

Recently, Ross proposed an idea, now known as the “Recovery Theorem,” that asserts that the real (physical) probability measure can be recovered from the market prices of derivatives. This work has generated a great deal of controversy in the finance literature. The purpose of this paper is to revisit the core idea of the recovery theorem and to examine its implications. In particular, issues concerning the so-called factorization of the pricing kernel will be examined from the viewpoint of the Flesaker–Hughston representation.

CONVERGENCE OF MINIMUM ENTROPY OPTION PRICES FOR WEAKLY CONVERGING INCOMPLETE MARKET MODELS

We study weak convergence of a sequence of (approximating) asset price models Sn to a limiting model S: both Sn and S are multi-dimensional asset price processes with some physical probability measures Pn resp. P, and a natural notion of process convergence is the weak convergence of the induced path probability measures, denoted by (Sn|Pn) resp. (S|P), on the abstract topological space of possible asset price trajectories. For the purpose of no-arbitrage pricing of options or more general derivatives on the model assets, there are two different aspects of this convergence: (i) convergence under the given physical probability measures, (Sn|Pn) → (S|P) and (ii) convergence under suitably chosen equivalent martingale measures (EMM) relevant for pricing derivatives, (Sn|Qn) → (S|Q). A simple example is the convergence of a sequence of discrete-time binomial models to the Black–Scholes model (geometric Brownian motion), where the model markets are complete and hence the choice of Qn resp. Q is unique. This example and the general case of complete limit markets (S|P) have been studied in [1]. In contrast we have several choices for Qn resp. Q when all the model markets are incomplete. A natural choice is the minimum entropy EMM [2, 3], defined as the (unique) EMM R minimizing the relative entropy H(R|P) to the physical measure P, among all EMMs. We prove the following: given that the approximating models converge under the physical measures, (Sn|Pn) → (S|P) — with some mild assumptions on P and on the minimum entropy EMMs Rn for Sn resp. R for S — entropy number convergence implies weak convergence of the minimum entropy option price processes: H(Rn|Pn) → H(R|P) implies (Sn|Rn) → (S|R). Several rigorous examples illustrate the result; cf. also [4].

Certainty and uncertainty in science: the subjectivistic concept of probability in physiology and medicine.

Most physiological scientists have restricted understanding of probability as relative frequency in a large collection (for example, of atoms). Most appropriate for the relatively circumscribed problems of the physical sciences, this understanding of probability as a physical property has conveyed the widespread impression that the "proper" statistical "method" can eliminate uncertainty by determining the "correct" frequency or frequency distribution. However, many relatively recent developments in the theory of probability and decision making deny such exalted statistical ability. Proponents of Bayes's subjectivist theory, for example, assert that probability is "degree of belief," a more tentative idea than relative frequency or physical probability, even though degree of belief assessment may utilize frequency information. In the subjectivist view, probability and statistics are means of expressing a consistent opinion (a probability) to handle uncertainty but never means to eliminate it. In the physiological sciences the contrast between the two views is critical, because problems dealt with are generally more complex than those of physics, requiring judgments and decisions. We illustrate this in testing the efficacy of penicillin by showing how the physical probability method of "hypothesis testing" may contribute to the erroneous idea that science consists of "verified truths" or "conclusive evidence" and how this impression is avoided in subjectivist probability analysis.