Arbitrage Pricing Theory for Idiosyncratic Variance Factors

Abstract

We develop an arbitrage pricing theory framework extension to study the pricing of squared returns/volatilities. We analyze the interplay between factors at the return level and those in idiosyncratic variances. We confirm the presence of a common idiosyncratic variance factor, but do not find evidence that this represents a missing risk factor at the (linear) return level. Thereby, we consistently identify idiosyncratic returns. The price of the idiosyncratic variance factor identified by squared returns is small relative to the price of market variance risk. The quadratic pricing kernels induced by our model are in line with standard economic intuition.

Latent Factor Analysis in Short Panels

“…Assumption 1 allows for serial dependence in idiosyncratic errors in the form of martingale difference sequences, like individual GARCH and Stochastic Volatility (SV) processes, as well as weak cross-sectional dependence (see Assumption 2 below). It also accommodates common time-varying components in idiosyncratic volatilities by allowing different entries along the diagonal of V ε ; see Renault, Van Der Heijden and Werker (2022) for arbitrage pricing in such settings. 4 This paper focuses mainly on testing hypotheses on the number of latent factors k when T is fixed and n → ∞.…”

“…, where W is a T × n random matrix of scaled errors terms w i,t that are independent across i and uncorrelated across t, and Σ = (σ i,j ) As already remarked, the diagonal elements of V ε are the sample realizations of the common component driving the variance of the error terms at times t = 1, ..., T ; see Renault, Van Der Heijden and Werker (2022) for empirical evidence of a variance factor. A sphericity assumption cannot accommodate such a common time-varying component.…”