# Gain, Loss and Asset Pricing

## Antonio E. Bernardo and Olivier Ledoit

### Abstract

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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