Risk Appetite and Endogenous Risk

Jon Danielsson, Hyun Song Shin and Jean-Pierre Zigrand
July 2009

Market volatility reflects traders' actions, while their actions depend on perceptions of risk. Equilibrium volatility is the fixed point of the mapping that takes perceived risk to actual risk. We solve for equilibrium stochastic volatility in a dynamic setting where risk-neutral traders operate under Value-at-Risk constraints. We derive a closed form solution for the stochastic volatility function in the benchmark model with a single risky asset. Even though the underlying fundamental risks remain constant, the resulting dynamics generate stochastic volatility through traders' reactions in equilibrium. Volatilities, expected returns and Sharpe ratios are shown to be countercyclical. If the purpose of financial regulation is to shield the financial system from collapse, then basing regulation on individually optimal risk management may not be enough.