10 - Multifrequency Jump-Diffusions

Abstract

This chapter develops a continuous-time asset pricing economy with endogenous multifrequency jumps in stock prices. Multifractal diffusions are exogenously specified and have continuous sample paths. The price process is endogenized by considering a continuous-time version of the exchange economy. Dividends and consumption follow continuous diffusions, but their drift rates and volatilities can undergo discrete Markov switches. These regime changes in fundamentals trigger endogenous jumps in asset prices, consistent with the well-known property that discrete arrivals of information about cash flows and discount rates can cause price discontinuities in equilibrium. When dividends follow a continuous-time MSM, the construction generates a new class of stochastic processes called multifrequency jump-diffusions. The stock price displays endogenous jumps of heterogeneous frequencies, and the largest discontinuities are triggered by the most persistent volatility shocks. The model thus produces many small jumps and fewer large jumps. The equilibrium model contributes to the process by endogenizing the heterogeneity of jump sizes and the association between jump-size and frequency. The limiting behavior of the economy is investigated when the number of volatility components goes to infinity. A gap is thus bridged between exogenously specified jump-diffusions and discrete-time equilibrium models of volatility feedback.