Total risk aversion and the pricing of options

The objective of this paper is to derive the price of a call option in a stable market for amarket maker who is totally risk averse in the sense that he has a utility function with infinite index of absolute risk aversion. We prove that in the case when the borrowing rateR is the same as the lending rater the option price must be the Black—Scholes price with rateR = r. In the more interesting caseR > r, we prove that two option pricing functions are necessary and sufficient for making a market in the option. These two functions are the Black—choles prices with rater orR. Which price to use at each time will depend on the circumstances (buying or selling) and the capitalization of the market maker. We determine which price to use at each time and the optimal riskless portfolio.Partially supported by a grant, No. AFOSR-86-0202, from the Air Force Office for Scientific Research and a grant from Loyola University of Chicago.Partially supported by Grant No. AFOSR-86-0202 from the Air Force Office for Scientific Research, a grant from the National Science Foundation, and a grant from Loyola University of Chicago.