# The implied Sharpe ratio

## TL;DR

The paper introduces the implied Sharpe ratio, a new measure to compare European options' investment attractiveness in incomplete markets, aiding risk-averse investors in portfolio decisions.

## Contribution

It proposes the implied Sharpe ratio as a novel metric for evaluating European options' utility contribution in incomplete markets.

## Key findings

- Implied Sharpe ratio helps identify the most beneficial options for portfolios.
- The measure varies with investor risk-aversion, influencing investment choices.
- Numerical studies demonstrate its effectiveness in comparing options.

## Abstract

In an incomplete market, including liquidly-traded European options in an investment portfolio could potentially improve the expected terminal utility for a risk-averse investor. However, unlike the Sharpe ratio, which provides a concise measure of the relative investment attractiveness of different underlying risky assets, there is no such measure available to help investors choose among the different European options. We introduce a new concept -- the implied Sharpe ratio -- which allows investors to make such a comparison in an incomplete financial market. Specifically, when comparing various European options, it is the option with the highest implied Sharpe ratio that, if included in an investor's portfolio, will improve his expected utility the most. Through the method of Taylor series expansion of the state-dependent coefficients in a nonlinear partial differential equation, we also establish the behaviour of the implied Sharpe ratio with respect to an investor's risk-aversion parameter. In a series of numerical studies, we compare the investment attractiveness of different European options by studying their implied Sharpe ratio.