**I**n Chapter 6, we developed option price relations in the absence of costless arbitrage opportunities. While the no-arbitrage price relations have useful applications, they provide only bounds on prices, not exact option values. In this chapter, we develop valuation equations for European-style options and show how the equations can be used for risk measurement. We also describe how to estimate of the parameters of the valuation equation.

An option, like any other security, can be valued as the present value of its expected cash flows. For a European-style call option, the expected cash flow is at the option's expiration and equals the expected difference between the underlying asset price and the exercise price conditional upon the asset price being greater than the exercise price. Thus the call's expected cash flow depends on, among other things, the expected risk-adjusted rate of price appreciation on the underlying asset between now and expiration. Once the call's expected terminal value is established, it must be discounted to the present. The discount rate applied to the expected terminal option value is the expected risk-adjusted rate of return for the option. The problem with this “traditional” approach to valuation is that it is difficult, if not impossible, to estimate precisely the expected risk-adjusted return parameters.

A major theoretical breakthrough occurred in 1973, with the publication of research papers by Black ...

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