Option Valuation and the Price of Risk

A valuation model is presented for options on stocks for which Black- Scholes arbitrage does not entirely eliminate risk. The price dynamics of a portfolio of options and the underlying security is quantified by requiring that the excess reward-to-risk ratio of the portfolio be identical to that of the stock. The nonlinear evolution equation for the portfolio value is homogeneous of degree one. A representative distribution is obtained from recent stock-history time series; numerical solutions for European calls are usually close to the Black- Scholes values, but naked and covered calls have different valuations. For infinitesimal time steps and a lognormal stock-price distribution, the evolution equation reduces to the Black-Scholes form. An analytically tractable non-lognormal distribution is analyzed near option expiration, and a formula expressing the deviation from the lognormal case is obtained for an out-of-the-money call. The present model is discussed in the context of previous work, and the effect of nonlinearity on the valuation of a portfolio of derivative securities is considered.