Gain, Loss and Asset Pricing

Antonio E. Bernardo and Olivier Ledoit


 In this paper we develop an approach to asset pricing in incomplete markets that gives the modeller the flexibility to control the tradeoff between the precision of equilibrium models and the credibility of no-arbitrage methods. We rule out the existence of investment opportunities that are very attractive to a benchmark investor. The key feature of our approach is the measure of attractiveness employed: the gain-loss ratio. The gain (loss) of a portfolio is the expectation, under a benchmark risk-adjusted probability measure, of the positive (negative) part of the portfolio's excess payoff. The benchmark risk-adjusted probability measure incorporates valuable prior information about investor preferences and portfolio holdings. A restriction on the maximum gain-loss ratio in the economy has a dual representation in terms of admissible pricing kernels: it is equivalent to a bound on the ratio of extreme deviations from the benchmark pricing kernel.

Price bounds are derived by computing all prices which do not permit the formation of portfolios with gain-loss ratios in excess of some prespecified level. We give an example where we bound the price of an option on a non-traded asset that is correlated with a traded asset. The resulting bounds lie strictly between the Black-Scholes price and the no-arbitrage bounds, and they are sharper when (i) the maximum allowable gain-loss ratio is lower, (ii) the correlation between the non-traded and traded asset is higher, and (iii) the volatility of the non-traded asset is lower. This has implications for pricing real options and executive stock options, and for performance evaluation of portfolio managers who use derivatives.

Journal of Political Economy, Volume 108, Number 1, February 2001, pages 144-172

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