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Unformatted text preview: A Note on Zero-level Pricing and the HARA Utility Function 1 S.-M. Guu 2 and J.-N. Wang 3 Abstract. In the literature, zero-level pricing method has been proposed to pro- vide a unique price for a non-marketable 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 zero-level 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 zero-level price of a new asset. This implies that, if this parameter is fixed, zero-level prices are identical for all individuals with the HARA utility functions and the different levels of initial wealth. Key Words. Asset pricing, zero-level 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 no-arbitrage principle. In many cases, however, the no-arbitrage principle produces only a range for the price. In order to yield a unique price for a new asset, zero-level pricing method has been proposed in the literature (Refs. 1-4). Zero-level 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, zero-level 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 zero-level price. Precisely, the zero-level 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 zero-level 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 95-2221-E-155-049. 2 Department of Business Administration, Yuan Ze University, Taoyuan, Taiwan, R.O.C. Email: ies- email@example.com. 3 Graduate School of Management, Yuan Ze University, Taoyuan, Taiwan, R.O.C. assets and there is a risk-free asset; all assets are normally distributed; the single-period 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 risk-free asset in the market. Let the price for risk-free 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 risk-free asset is R and let ˜ R denote the payoff of the risk-free 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