🤖 AI Summary
This study aims to unify the theoretical framework of probability measures in asset pricing by addressing how market prices can be represented through measure changes. It systematically traces the conceptual evolution from state prices and risk-neutral measures to stochastic discount factors (pricing kernels), emphasizing that asset pricing fundamentally relies on equivalent measures—adjusted via discounting, numéraire normalization, or utility weighting—rather than the original physical probability measure. The work innovatively incorporates data-driven information such as textual content, attention metrics, and sentiment into the measure transformation process, thereby extending the learning paradigm for pricing kernels in incomplete markets. By integrating stochastic discount factors, Radon–Nikodym derivatives, Girsanov transformations, implied densities, and machine learning techniques, the paper constructs a cohesive framework that bridges classical asset pricing theory with modern empirical methodologies, offering both theoretical grounding and practical guidance.
📝 Abstract
This review summarizes the historical development of probability measures in asset pricing, from early mathematical finance and state price theory to risk-neutral valuation, martingale measures, forward measures, stochastic discount factors, incomplete-market measure selection, benchmark pricing, robust and nonlinear pricing, and modern data-driven probability transformations. The central theme is that asset pricing is not merely an exercise in estimating physical probabilities. Instead, pricing theory constructs, transforms, or selects probability measures so that market prices can be represented as expectations after discounting, numeraire normalization, marginal utility weighting, entropy penalization, calibration, or information conditioning. The paper emphasizes landmark contributions including Bachelier's probabilistic model of speculation, Arrow-Debreu state-contingent claims, Black-Scholes-Merton option pricing, Harrison-Kreps and Harrison-Pliska's martingale formalization, Delbaen and Schachermayer's fundamental theorem, Breeden-Litzenberger implied state price densities, change of numeraire methods, Hansen-Jagannathan stochastic discount factor restrictions, Cochrane's SDF synthesis, and recent empirical and machine learning work on learned pricing kernels. Text-, attention-, and sentiment-based probability transformations are treated as recent information-adjusted forecasting extensions that complement, rather than replace, martingale, numeraire, SDF, and incomplete-market frameworks. The paper also collects key formulas for state prices, stochastic discount factors, Radon-Nikodym densities, Girsanov changes of measure, risk-neutral valuation, forward measures, implied densities, coherent risk measures, benchmark pricing, learned SDFs, and information-adjusted forecasting.