Narrow Results

This paper deals with the problem of determining the correct risk measure for options in a Black–Scholes (BS) framework when time is discrete. For the purposes of hedging or testing simple asset pricing relationships previous papers used the "local", i.e., the continuous-time, BS beta as the measure of option risk even over discrete time intervals. We derive a closed-form solution for option betas over discrete return periods where we distinguish between "covariance betas" and "asset pricing betas". Both types of betas involve only simple Black–Scholes option prices and are thus easy to compute. However, the theoretical properties of these discrete betas are fundamentally different from those of local betas. We also analyze the impact of the return interval on two performance measures, the Sharpe ratio and the Treynor measure. The dependence of both measures on the return interval is economically significant, especially for OTM options.

In this chapter, we look at how estimation is performed on an important finance valuation model to address the problem of risk pricing and also to apply the model to evaluate investment performances.

A key foundation of finance is the tradeoff between risk and return. Assets returning higher ex-ante (or conditional expected) return should also bear higher ex-ante risk, and vice versa. Finance theory and models seek to establish the exact relationship between this return and risk. One of the earliest finance models is the Sharpe-Lintner capital asset pricing model (CAPM) that links expected return to only systematic risk. The unsystematic or residual risk can be diversified away and is, therefore, not priced. Systematic risk in the form of market risk is priced. For any asset, this market risk is reflected in the sensitivity of the asset return to the market factor movement. Higher sensitivity implies a higher beta or systematic risk. When we refer to risk, it is usually quantified in terms of standard deviation instead of variance since the former is expressed in terms of the same unit as the underlying variable. We shall begin our study of CAPM with the statistical market model.