A Continuous-Time Conditional Factor Model for Returns

Summary

The document sets out a conditional factor model for the excess returns of multiple assets. At each observation time, an asset’s return is represented by a time-varying intercept, exposure to a set of common tradable factors, and an idiosyncratic noise term. It then gives a continuous-time analogue for log prices, driven by time-varying drift, factor movements, and Brownian noise.

The motivating question is how to infer and simulate this diffusion from observed log prices and factors, with particular interest in the paths of the drift and factor loadings. The text supplies the model equations and identifies the observations available, but does not present an estimation procedure, simulation algorithm, or empirical results. It also leaves important modeling choices unspecified, including how the coefficient paths evolve and what assumptions govern the noise and factor processes. Consequently, it frames a statistical modeling problem rather than establishing that a particular inference method is efficient or suitable.

Key ideas

• Asset excess returns are modeled using time-varying intercepts, common factor exposures, and residual noise.
• A continuous-time formulation describes log prices through drift, factor increments, and Brownian disturbances.
• The stated goal is to infer and simulate the process from discrete observations of prices and factors.
• The document does not specify coefficient dynamics or provide an estimation or simulation method.

Tags

# Stochastic Differential equation: CAPM


# Stochastic Differential equation: CAPM












Let $R=(R_1, \dots ,R_M)$′ denote a vector of excess returns of M assets observed at $n$ time points, $0<t_1<t_2< \cdots <t_n<T$, within a time span $T>0$.

We wish to explain the returns through a set of $J$ common tradeable factors, $f=(f_1 \dots ,f_J)$′ which are observed at the same time points.

We assume the following conditional factor model explains the returns of stock $k$, $(k=1,…,M)$ at times $t_i$, $(i = 1, ..., n)$: \begin{align} R_{k,i} = \alpha_{k}(t_i) + \beta_{k}(t_i)′f_i + ω_{kk}(t_i)z_{k,i}. \end{align}

Where $R_{k,i}$ and $f_i$ are the observed returns and factors at time $t_i$.

This can be rewritten in matrix notation as: \begin{align} R_{i} = \alpha(t_i) + \beta(t_i)′f_i + \Sigma^{1/2}(t_i)z_{i}. \end{align}

In the paper on Testing conditional factor models they propose a continuous-time stochastic differential equation version of the above discrete time factor model to do theoretical analysis \begin{align} \mathrm{d}s(t) = \alpha(t)\mathrm{d}t + \beta(t)\mathrm{d}F(t) + \Sigma^{1/2}(t)\mathrm{d}B(t) \end{align} Where $s(t) = \log S(t)$ are the $M$ log prices, $F(t)$ are the $J$ factors and $B(t)$ is an $M$-dimensional Brownian motion.

We have observed the $s(t)$ and $F(t)$ at the $t_i$'s

My question is whether there are any efficient ways to do inference and possibly simulation of the diffusion process?

My main interest lies in sample paths of $\alpha(t)$ and $\beta(t)$.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.