We develop a simple commodity model to analyze (i) the effects of hedging with liquidity constraints, due to producers' inability to bear unlimited trading losses, (ii) the role of speculation in the process of risk allocation between consumers and producers, and (iii) the equilibrium implications of government price subsidies to the producers. We find that (1) liquidity constraints can cause futures prices to exhibit mean reversion, which then makes speculation profitable; (2) speculation tends to make futures price volatility an increasing function of futures price; and (3) government price subsidy, if actively hedged by the producers, serves to lower the futures risk premium and reduce futures volatility.

A martingale approach is used to characterize general equilibrium in the presence of portfolio insurance. Insurers sell to non-insurers in bad states, and general equilibrium requires that the risk premium rises to induce non-insurers to increase their holdings. We show that portfolio insurance increases price volatility, causes mean reversion in asset returns, raises the Sharpe ratio and volatility in bad states, and causes volatility to be correlated with volume. We also explain why out-of-the-money S&P 500 put options trade at a higher volatility than do in-the-money puts.

A martingale approach is used to characterize general equilibrium in the presence of portfolio insurance. Insurers sell to noninsurers in bad states, and general equilibrium requires that the risk premium rises to induce noninsurers to increase their holdings. We show that portfolio insurance increases price volatility, causes mean reversion in asset returns, raises the Sharpe ratio and volatility in bad states, and causes volatility to be correlated with volume. We also explain why out‐of‐the‐money S&P 500 put options trade at a higher volatility than do in‐the‐money puts.