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Unformatted text preview: A Note on Zerolevel Pricing and the HARA Utility Function 1 S.M. Guu 2 and J.N. Wang 3 Abstract. In the literature, zerolevel pricing method has been proposed to pro vide a unique price for a nonmarketable new asset. Its disadvantage from the viewpoint of robust pricing theory is dependent on investor’s utility function and initial wealth. In some situations, the zerolevel price is universal in a sense of inde pendent of utility function and initial wealth. In this note, we show that only one parameter of the HARA (hyperbolic absolute risk aversion) utility function affects the zerolevel price of a new asset. This implies that, if this parameter is fixed, zerolevel prices are identical for all individuals with the HARA utility functions and the different levels of initial wealth. Key Words. Asset pricing, zerolevel pricing, hyperbolic absolute risk aver sion utility, maximization. 1. Introduction Pricing a new asset is a fundamental problem of finance. One commonly seen method of pricing a new asset is by use of the noarbitrage principle. In many cases, however, the noarbitrage principle produces only a range for the price. In order to yield a unique price for a new asset, zerolevel pricing method has been proposed in the literature (Refs. 14). Zerolevel pricing method determines a price so that an investor will include this new asset in his or her portfolio at a zero level. It turns out that under reasonable conditions, zerolevel pricing has very nice properties such as linearity and free of arbitrage. Being dependent on investor’s utility function and initial wealth, however, is its disadvantage from the viewpoint of robust pricing theory. Luenberger (Ref. 2) proposed the notion of universality for the zerolevel price. Precisely, the zerolevel price is universal if it is the same for all utility functions in some class and all positive wealth levels. Luenberger (Ref. 2) showed that universal zerolevel prices exist in the following four situations: the new asset is statistically independent of all marketed 1 This research is partially supported under the grant of NSC 952221E155049. 2 Department of Business Administration, Yuan Ze University, Taoyuan, Taiwan, R.O.C. Email: ies mguu@saturn.yzu.edu.tw. 3 Graduate School of Management, Yuan Ze University, Taoyuan, Taiwan, R.O.C. assets and there is a riskfree asset; all assets are normally distributed; the singleperiod market is partially complete; all individuals have quadratic utility functions. For easy reference, we follow the settings in Luenberger’s paper (Ref. 2). Consider a single period model with n risky assets and one riskfree asset in the market. Let the price for riskfree asset be 1 and the corresponding prices of n risky assets be denoted by p 1 ,p 2 ,...,p n at time t = 0. The gross return of the riskfree asset is R and let ˜ R denote the payoff of the riskfree asset at time 1. Payoffs of n risky assets at time 1 are denoted by d 1 ,d 2 , ··· ,d...
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 Spring '09
 Mar,Lee

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