محاسبه ریسک نوسانات قیمت بازار در بازارهای کالایی انرژی

In this paper, we demonstrate the need for a negative market price of volatility risk to recover the difference between Black–Scholes [Black, F., Scholes, M., 1973. The pricing of options and corporate liabilities. Journal of Political Economy 81, 637–654]/Black [Black, F., 1976. Studies of stock price volatility changes. In: Proceedings of the 1976 Meetings of the Business and Economics Statistics Section, American Statistical Association, pp. 177–181] implied volatility and realized-term volatility. Initially, using quasi-Monte Carlo simulation, we demonstrate numerically that a negative market price of volatility risk is the key risk premium in explaining the disparity between risk-neutral and statistical volatility in both equity and commodity-energy markets. This is robust to multiple specifications that also incorporate jumps. Next, using futures and options data from natural gas, heating oil and crude oil contracts over a 10 year period, we estimate the volatility risk premium and demonstrate that the premium is negative and significant for all three commodities. Additionally, there appear distinct seasonality patterns for natural gas and heating oil, where winter/withdrawal months have higher volatility risk premiums. Computing such a negative market price of volatility risk highlights the importance of volatility risk in understanding priced volatility in these financial markets.

The importance of energy markets has increased with the development of futures and options markets, and with the always-important impact of energy on the economy. In this paper, we seek an in-depth understanding of priced volatility in the energy markets, as well as quantitatively displaying the mirror-image aspect of the energy and equity markets: Whereas prices and volatilities are negatively correlated in the equity markets, they tend to display a positive correlation in the energy markets. Because of this positive correlation, computing a negative market price of volatility risk in energy markets may imply a negative market price of commodity risk in the energy markets and consequently an upward-bias of energy futures contracts prices relative to expected spot prices. Financial research has made numerous advances in testing the sensitivity of a given data-generating process to changes in the instantaneous parameters that govern it. Currently there are a wealth of parametric models attempting to explain stock price movement. The most notable extensions to the Black and Scholes (1973) model are the inclusion of stochastic volatility à la Heston (1993), and the inclusion of jumps by Bates (1996) and others. Recent additions such as volatility jumps introduced by Duffie et al. (2000) and Bates’ crash risk are further advancements to the well-tested Black–Scholes model. A problem for current researchers is the ability to reconcile time-series and cross-sectional differences in spot and option prices by fitting a given underlying model to capture the distributions of both returns. The parameter of particular interest is the market price of volatility risk. While extensive research has focused on stochastic volatility models, there is conflicting information on the impact of the market price of volatility risk. Recently, findings in Bakshi and Kapadia, 2003, Coval and Shumway, 2001, Pan, 2002 and Doran, 2007 address the direction and magnitude of the market price of volatility risk, but with contradictory conclusions. Our hypothesis is that the market price of volatility risk is indelibly linked to the bias in Black–Scholes implied volatility. If the market price of volatility risk is significant and negative, this potentially explains the upward-bias observed in Black–Scholes implied volatility (henceforth, BSIV) as well as contributing to the Bates’ (1996) finding that out-of-the money (OTM) puts are expensive relative to other options (the so-called “volatility skew”). Additionally, Eraker, 2004 and Bates, 2000 have documented that selling options results in Sharpe ratios that are significantly higher than the Sharpe ratio of traditional equity portfolios. These results suggest large premiums for exposure to volatility risk, and lend justification for option traders tendency to be short options. Another strand of the literature has focused on the appropriate econometrics to estimate continuous-time models. Chernov and Ghysels (2000) use the Gallant and Tauchen (1998) EMM technique, Pan (2002) applied an IS-GMM framework, and Jones (2003) and Eraker (2004) have used Bayesian analysis to arrive at their estimates. These works have improved our understanding of equity market pricing dynamics and the risk premium within these markets by combining spot and options data on the S&P index. By comparison, the work in energy markets incorporating options is mostly unexplored. As Broadie et al. (2007) point out, it is very difficult to arrive at precise parameter estimates for multiple risk premia, especially when one is the volatility risk premium. Noting this issue, we attempt to estimate the volatility risk premium in energy markets by combining both implied and realized volatility in two-step estimation procedure. Using Black (1976) implied volatility (BIV) and implementing the relationship between expected volatility and instantaneous volatility as given in Aı¨t Sahalia (1996), we infer the instantaneous risk-neutral parameter estimates from the discrete-time analogue. The market price of volatility risk is then deduced via calibration from these instantaneous parameters and the difference between expected realized volatility and the actual realized volatility whilst avoiding the under-identification problem. The findings suggest a significant negative market price of volatility risk for three energy commodities – natural gas, crude oil and heating oil. Additionally, there appears to be a strong seasonal component to the volatility risk premium for natural gas and mild seasonality in heating oil. For robustness, in conjunctions with the volatility risk premium, the market price of risk and the correlation between the price and volatility innovations are estimated in the Hansen (1982) GMM framework, in a test for model mis-specification and stability of the parameter estimates. While we are unable to make a definitive statement on the commodity price risk premium, the volatility risk premium remained negative and significant using this alternative specification. The remainder of the paper is organized as follows. Section 2 will review the current findings on the market price of volatility risk. Section 3 will introduce the simulation and the findings for the various parametric-models tested. Section 4 will detail the estimation procedure and its application to the market price of volatility risk. Section 5 concludes.

As Jackwerth and Rubinstein (1996) pointed out, implied volatility is higher on average than realized volatility. In the realm of energy markets, we explain the difference between Black and Scholes (1973)/Black (1976) implied volatility and realized-term volatility by modeling and computing a negative market price of volatility risk. We use Monte Carlo simulation to demonstrate numerically that a negative market price of volatility risk is the key risk premium in explaining this disparity in both the equity and commodity- energy markets. Across several models, the market price of volatility risk is the critical parameter that generates the differences in risk-neutral (Black implied volatility) and real-world volatility (realized-term volatility). A negative market price of volatility risk translates to a BIV greater than realized volatility. Our results for energy markets mirror those found for equities by Coval and Shumway (2001) and Bakshi and Kapadia (2003). This has implications for the market price of risk for energy, which is more difficult to estimate (due to limited and more volatile data) than that of equities. By controlling such variables as correlation between the processes, underlying volatility, and jump size, inferences were made on the impact each parameter had on the difference between implied and realized volatility. Additionally, it was determined that kr was the key parameter, even in the presence of a jump-process, that determined this difference. This finding suggests that jump models, in energy or equities, fail to account for a negative market price of volatility risk are incomplete. The resulting empirical tests for kr revealed that the factor is negative and significant for all three energy commodities. In addition, there appears to be strong seasonality in volatility risk for natural gas and heating oil, while crude oil volatility risk is consistent across months. Arriving at these estimates required a two-step procedure, where the risk-neutral parameters were first estimated using implied volatility, and then calibrated in conjunction with the volatility risk premium to estimate the realized volatility. The results were robust to alternative specifications. The finding that kr is negative explains why option traders tend to be short options: They write options at the higher implied volatilities and reap the risk premium by delta-hedging their price exposures.