Keeping risk alive

Risk premia in markets rise with the degree of discontinuity in price motion. This discontinuity is measured by the scale and speed coefficients of price movements embedded in the bilateral gamma process, which is modeled as the difference of two independent gamma processes. The bilateral gamma process is calibrated to both the time series data of daily returns and to option price data. Return distributions inferred from option prices are heavily influenced by the presence of risk premia. Strategies are proposed to unwind these effects with the goal of constructing return distributions that are more useful for investment analysis. Downward movements exhibit lower continuity than upward movements, both physically and under the risk-neutral measures. Moreover, risk-neutral movements are substantially less continuous than their physical counterparts for both upward and downward directions. Indices of discontinuity are developed for upward and downward movements, based on asymmetric variance-minimizing step sizes used in finite difference delta calculations.