Risk Neutral Pricing

A classical problem, coming up frequently in practical business, is the valuation of future cash flows which are somewhat risky. By the term “risky” we mean that the payment is not of a deterministic nature; rather there is some uncertainty on the amount of the future cash flows. Of course, in real life virtually everything happening in the future contains some elements of riskiness. As a first example let us think of an investment project, say, a company plans to build a new factory. A classical way to proceed is to calculate a net asset value. One tries to estimate the future cash flows generated by the project in the subsequent periods. In the present example they will initially be negative; this initial investment should be compensated by positive cash flows in later periods. Having fixed these estimates of the future cash flows for all periods, one calculates a net asset value by discounting these cash flows to the present date. But, of course, there is uncertainty involved in the estimation of the future cash flows and people doing these calculations are, of course, aware of that. The usual way to compensate for this uncertainty is to apply an interest rate which is higher than the riskless rate of return corresponding to the rate of return of government bonds. The spread between the riskless rate of return and the interest rate used for discounting the future cash flows in the calculation of the net asset value can be quite substantial in order to compensate for the riskiness. Only if the net asset value, obtained by discounting with a rather high rate of return, remains positive, the management of the company will engage in the investment project. Mathematically speaking, the above procedure may be described as follows: first one determines the expected values of the future cash flows and subsequently one discounts by using an elevated discount factor. However, there is no systematic way of mathematically approaching the question of how the degree of uncertainty in the determination of the expected values can be quantified, and in which way this should be taken into account to determine the spread between the interest rates. We now turn to a different approach which interchanges the rôles of taking expectations and discounting in taking the riskness of the cash flows into account. This approach is used in modern mathematical finance, in particular