The impact of return nonnormality on exchange options

Abstract

The Margrabe formula is used extensively by theorists and practitioners not only on exchange options, but also on executive compensation schemes, real options, weather and commodity derivatives, etc. However, the crucial assumption of a bivariate normal distribution is not fully satisfied in almost all applications. The impact of nonnormality on exchange options is studied by using a bivariate Gram-Charlier approximation. For near-the-money exchange options, skewness and coskewness induce price corrections which are linear in moneyness, while kurtosis and cokurtosis induce quadratic price corrections. The nonnormality helps to explain the implied correlation smile observed in practice.

Valuation and applications of compound basket options

“…In the case of options and their portfolios, finding moments is not straightforward because they contain cumulative normal distribution functions. However, Li () and Li et al () provide the expected value of , where a and b are constants, and y follows a normal distribution. Moreover, Li, Zhou, and Deng () generalize this by allowing y to follow a multivariate normal distribution and b as a row vector in a multiasset spread option setting.…”

Using Four‐Moment Tail Risk to Examine Financial and Commodity Instrument Diversification

“…For example, while negative skewness occurs for equity markets, a number of studies document positive skewness for commodity futures returns (Gorton and Rouwenhorst, 2006; Lucey and Eastman, 2008), and studies of individual futures contracts generally report various degrees of leptokurtosis in the futures contracts (Stevenson and Bear, 1970; Hudson, Leuthold, and Sarassoro, 1987). The existence of skewness and excess kurtosis has led researchers to consider the role of higher moments in asset pricing and valuation (Kraus and Litzenberger, 1976; Harvey and Siddique, 2000), in examining hedge funds (Brooks and Kat, 2002; Bergh and Van Rensburg, 2008) and options (Li, 2008), and the pricing of futures contracts (Christie‐David and Chaudhry, 2001). However, skewness and kurtosis have not been employed in the analysis of the possible tail losses in diversified portfolios.…”