Option Returns

Nonlinear martingale theory is used to form lower and upper price processes straddling a martingale. The martingale return is then modeled in terms of risk charges associated with the returns on the straddling lower and upper processes. The move to physically expected returns is made via the usual covariation risk factors. There are however two covariation terms, one for the lower and the other for the upper return. The expected returns are then seen to incorporate in addition to covariation considerations that address the martingale, the risk charges that assess out of martingale issues. The theory is tested on prices of options for 10 underliers over 170 days covering 4 maturities and 21 strikes. It is observed that the risk charges have positive significant coefficients while the upper covariation has a positive price with the lower covariation receiving a negative price