A Stochastic Control Approach to the Pricing of Options

Abstract

We construct a stochastic control model of a portfolio in which the investor may invest in stock, call options on the stock, or risk free bonds. He may also borrow funds. We assume that the investor is attempting to maximize expected utility of total wealth at expiration of the options where the utility function is given as wealth to the power γ, 0 ≤ γ ≤ 1. This includes both the risk averse and risk neutral cases. We allow the borrowing interest rate, R, to be distinct from the lending interest rate, r. We make two fundamental assumptions: (i) the investor cannot achieve infinite expected utility of wealth in finite time (“stable markets”) and (ii) the market will set the option price to minimize the investor's maximal expected utility of wealth, i.e., options are priced using a minimax strategy. The market is stable only when R ≥ r. We prove in the case R > r and γ < 1 that the minimax (European) call option price, F(t, S), is given as the solution of the Black-Scholes equation Ft + ½ σ2S2FSS + δ S FS − δ F = 0, where δ = r when r ≥ α + σ2(γ − 1); δ = R when α + σ2(γ − 1) ≥ R; and δ = α + σ2(γ − 1) if R ≥ α + σ2(γ − 1) ≥ r. The parameter α is the stock return drift and σ is the stock return diffusion. If R = r, then the call option price must, by stable markets, always be the Black-Scholes price with rate r. In the risk neutral (γ = 1) case we must have R ≥ α ≥ r for a stable market and the option price given by the Black-Scholes equation but with δ = α. Explicit solutions are presented and the optimal portfolios and returns are also derived. Comparable results are obtained for American call options which can be exercised at any time up to and including the expiration date. In this case, the option pricing function is shown to be the solution of an obstacle problem.