Does risk aversion always increase option value? If so, explain why. If not, modify the example with different shapes to the payoff functions to provide an example where the risk neutral buyer would pay more. Let’s work out the option value provided by a flexible-fuel car in a numerical example. Let A1(x) = 1 – x be the payoff from a fossil-fuel-only car and A2(x) = x be the payoff from a bio-fuel-only car. The state of the world, x, reflects the relative importance of bio-fuels compared with fossil fuels over the car’s lifespan. Assume x is a random variable that is uniformly distributed between 0 and 1 (the simplest continuous random variable to work with here). The statistics section in Chapter 2 provides some detail on the uniform distribution, showing that the probability density function

(PDF) is f (x) = 1 in the special case when the uniform random variable ranges between 0 and 1.

risk neutrality. To make the calculations as easy as possible to start, suppose first that the car buyer is risk neutral, obtaining a utility level equal to the payoff level. Suppose the buyer is forced to choose a bio-fuel car. This provides an expected utility of

X xdx= 2 [ x = E[A] A(x)f(x) dx = ||