Closed-form Transformations from Risk-neutral to Real-world Distributions

Abstract

Risk-neutral (RN) and real-world (RW) densities are derived from option prices and risk assumptions, and are compared with densities obtained from historical time series. Two parametric methods that adjust from RN to RW densities are investigated, firstly a CRRA risk aversion transformation and secondly a statistical calibration.Both risk transformations are estimated using likelihood techniques, for two flexible but tractable density families. Results for the FTSE-100 index show that densities derived from option prices have more explanatory power than historical time series.Furthermore, the pricing kernel between RN & RW densities may be more regular than previously reported and a more reasonable risk aversion function is estimated.

Density forecast comparisons for stock prices, obtained from high‐frequency returns and daily option prices

“…This contrasts with the index results of Shackleton et al (2010), which favor option-based methods for 2-and 4-week horizons but high-frequency methods for forecasts 1-day ahead. We note that option-based density forecasts generally outperform historical forecasts obtained from daily returns for a 1-month horizon, according to Liu et al (2007); Høg and Tsiaras (2011);and Trujillo-Barrera et al (2012). Furthermore, Yun (2014) shows that "options & returns" models outperform "only returns" models for all horizons from 1 day to 4 weeks.…”

“…There are several ways to rank density forecasts, and we will use the standard log-likelihood criterion previously employed by Bao, Lee, and Saltoğlu (2007); Liu et al (2007); and Shackleton et al (2010). For a given horizon h, assume method m gives densities f m;t x ð Þ at times i; ::: ; j for the asset price at times i þ h; ::: ; j þ h. Our goal is to find the method which maximizes the out-of-sample log-likelihood of observed asset prices, which for method m is given by…”

“…1 Chang, Christoffersen, Jacobs, and Vainberg (2012) show option-implied betas have significant predictive power compared with historical betas, while Kempf, Korn, and Sassning (2015) show that option prices can be used to enhance portfolio optimization. Liu et al (2007) provide the first evidence that option prices provide density forecasts that are superior to forecasts which rely solely on historical prices.…”

“…Previously, equity market comparisons have been made by Liu et al (2007) for FTSE 100 futures, Shackleton, Taylor, and Yu (2010) for S & P 500 futures, and Yun (2014) for the S&P-500 index. Comparisons for other asset classes include Høg and Tsiaras (2011) for crude oil futures and Trujillo-Barrera, Garcia, and Mallory (2012) for lean hog futures, while related studies of Euribor interest rate futures are Gutiérrez and Vincent-Humphreys (2012); Vergote and Gutiérrez (2012); and Ivanova and Gutiérrez (2014).…”

“…Economic theory motivates pricing kernel transformations using power and/or exponential utility functions (Bakshi, Kapadia, & Madan, 2003;Bliss & Panigirtzoglou, 2004;Liu, Shackleton, Taylor, & Xu, 2007) and the hyperbolic absolute risk aversion (HARA) function (Kang & Kim, 2006). Liu et al (2007) use both utility and statistical calibration transformations, and they find that a statistical, parametric calibration gives a higher log-likelihood for observed outcomes than a utility transformation. Shackleton et al (2010) compare parametric and nonparametric econometric transformations, obtaining the best diagnostic test results for the latter.…”