A new volatility dependent pricing kernel in commodity market Presented by Minhao Cai Joint with Weidong Tian UNC CHARLOTTE

What we are trying to do? 1. Try to construct a new volatility dependent pricing kernel 2. Try to use this pricing kernel to explain several stylized findings in commodity market 3. Empirically check some new stylized findings based on our pricing kernel

Literature Review-pricing kernel 1. Consumption based approach 2. Risk neutral probability approach Widely used in asset pricing literatures (Black-Schole ( JPE 1973)) Feature: independent of consumption and a representatives utility function In short of economic intuition

Literature Review-pricing kernel 3. Fill the gap between consumption based approach and risk neutral approach Brennan (JF1979) and Rubinstein (BJE 1979): find that a power utility function is a necessary condition for the existence of the risk neutral probability measure in Black Schole in a discrete time model. Bick (JFQA 1987) extends their result in a continuous time equilibrium model.

Literature Review-pricing kernel 4. The stochastic volatility model Christoffersen, Heston and Jacobs (2011) document a volatility dependent pricing kernel in equity market. The new pricing kernel can explain: (1) negative variance premium (2) the U shaped relationship between the pricing kernel and the stock return (3) fat tails (4) over reaction of long term option to changes in short-term variance

Several stylized findings in commodity market 1. The negative volatility risk premium and implied volatility puzzle Trolle and Schwartz ( JD 2010) Doran and Rong ( JBF 2008) Trolle and Schwartz ( RFS 2009) Hughen ( JFM 2010)

Several stylized findings in commodity market 2. The U shape relationship between the pricing kernel and the underlying asset return Christoffersen, Heston and Jacobs (2011) Bakshi Maden and Panayotov (JFE 2010) 3. The V shape or U shape between the volatility of futures price and the lagged slope of forward curve Kogan, Livdan and Yaron ( JF 2009)

Model S(t) is crude oil spot price. y(t,T) is the time t instantaneous forward cost of carry at time T.

The new volatility dependent pricing kernel

Check whether slope of forward curve can predict volatility of futures price Where and

The empirical result

The new volatility dependent pricing kernel

Try to sign the parameters 1. a<0 because the marginal utility function is a decreasing function of the return of underlying assets. 2. g>0, when v(t) increase, we anticipate the hedging needs will increase in time of uncertainty. 3. rho_{13}<0 and rho_{23}<0 Trolle and Schwartz (2008) and Hughen (2010)

Main Findings 1. The V shape between the volatility of futures price and the slope of forward curve.

Main Findings 2. The sign of risk premium

Main Findings 3. The pricing kernel is a non-monotone function of underlying asset return, Slope ratio of forward curve,Futures price and volatility component. 4. The pricing kernel is a decreasing function of the correlation between the underlying asset price return and the slope ratio of forward curve.

Main Findings

Future studies Focus on empirical checking the possible stylized facts: (1)Pricing kernel is a inverse U shape function of volatility (or deceases as variance of volatility increases) (2) Pricing kernel is a inverse U shape function of slope ratio of forward curve (or deceases as variance of slope ratio of forward curve increases) (3) The pricing kernel is a decreasing function of the correlation between the underlying asset price return and the slope ratio of forward curve.