Financial jeopardy

Learning the pre-limited liability value process of equity claims and its relationship to the stock price is an answer to the financial jeopardy question when observed option prices are the answer being given by the market. Constant dollar equity holder values, prior to the imposition of limited liability, are the signed conditional expectations of the integral of discounted net residual equity claims through all time. The stock is modelled as a limited liability claim imputing positive dividend flows to shareholders in certain circumstances coupled with a call option written on the integral of all discounted net residual equity claims. The underlying signed value has a known characteristic function when revenues and expenses are modelled as independent gamma processes. The stock price is a positive function of this signed underlying value, given by the solution of a partial integro differential equation. Options on the stock are then options on this function of the signed underlying value and are solved for using its density obtained by Fourier inversion of the characteristic function. The calibration of model parameters, the imputed dividend function and the terminal call strike is conducted on option prices at a single maturity for four underliers, AMZN, SPY, GOOG and JNJ. In all these cases it is observed that risk neutrally up moves arrive more frequently and are generally smaller while down moves are less frequent and are larger. The terminal option strikes were in the money for SPY and JNJ, and out of the money for AMZN and GOOG.