Model Specification and Risk Premia: Evidence from Futures Options

Abstract

This paper examines model specification issues and estimates diffusive and jump risk premia using S&P futures option prices from 1987 to 2003. We first develop a time series test to detect the presence of jumps in volatility, and find strong evidence in support of their presence. Next, using the cross section of option prices, we find strong evidence for jumps in prices and modest evidence for jumps in volatility based on model fit. The evidence points toward economically and statistically significant jump risk premia, which are important for understanding option returns.THERE ARE TWO CENTRAL, RELATED, issues in empirical option pricing. The first issue is model specification, which comprises identifying and modeling the factors that jointly determine returns and option prices. Recent empirical work on index options identifies factors such as stochastic volatility, jumps in prices, and jumps in volatility. The second issue is quantifying the risk premia associated with the jump and diffusive factors using a model that passes reasonable specification hurdles.The results in the literature regarding these issues are mixed. For example, tests using option data disagree over the importance of jumps in prices:

Inferring Volatility Dynamics and Risk Premia from the S&P 500 and VIX Markets

“…, Todorov and Tauchen (2011) and Jacod and Todorov (2010) find striking evidence for co-jumps in S&P 500 returns and in the VIX. See also Eraker (2004), Broadie et al (2007), Cont and Kokholm (2013). Bates (1996), Pan (2002) and Eraker (2004) argue in favor of using state-dependent jumps in returns, which is intuitively appealing, as jumps tend to occur more frequently when volatility increases.…”

“…10 Parameters obtained when calibrating to daily options prices are not stable over time, as explained in Broadie et al (2007) and Lindström et al (2008).…”

“…2 Time-consistent estimation methods have been previously used to calibrate models to index returns and options. See, e.g., Pan (2002), Eraker (2004), Broadie et al (2007), Christoffersen et al (2010), Johannes et al (2009) and Duan and Yeh (2011). However, as underlined in Ferriani and Pastorello (2012), most papers filtering information from option prices rely on one option per day or a limited set of options.…”

Saddlepoint approximations for affine jump-diffusion models

“…7 However, empirical values found in the literature stay small enough for the LR method to produce small relative errors. More specifically, as Broadie et al [11] summarize in their paper, Eraker et al [33], Andersen et al [2], Chernov et al [17] and Eraker [32] report 4.07%, 1.95%, 0.7% and 6.63% for σ J , respectively. Broadie et al [11] report σ J between 9% and 10% when a risk premium for σ J is assumed to exist.…”

Non-parametric estimation of forecast distributions in non-Gaussian, non-linear state space models

“…The product of the elements p (y t+1 jy 1:t ) in (21), for t = 1; 2; :::; T 1, along with the marginal distribution p (y 1 ) in (15), de…nes the likelihood function in (4). Motivated by the prior belief that the true unknown distribution of is a smooth function that declines in the tails, the logarithm of this likelihood function is penalized accordingly.…”