Option Valuation and the Price of Risk

A valuation model is presented for options on stocks for which BlackScholes arbitrage does not entirely eliminate risk. The price dynamics of a portfolio of options and the underlying security is quanti ed by requiring that the excess reward-to-risk ratio of the portfolio be identical to that of the underlying stock: excess return risk portfolio = excess return risk 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 di erent valuations. For in nitesimal 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-themoney call. The present model is discussed in the context of previous work, and the e ect of nonlinearity on the valuation of a portfolio of derivative securities is considered.