1. Optimal Hedging for the Real World
Risk Management
Prof. Ali Nejadmalayeri,
a.k.a. “Dr N”

2. Basis Risk
Suppose that futures price for maturity T is given by Ft(T). Now, the basis is Bt = Ft – St. The basis risk, volatility of basis over time, then has determined by:
In perfect markets, the basis risk is due to interest rates, convenience yield, payout, and storage costs uncertainties.
In imperfect markets, the basis risk is due to interest rates, convenience yield, payout, and storage costs uncertainties. But additionally there is a pricing error due to the fact that futures pricing is not valid anymore.

3. Hedge Ratio with Basis Risk
Suppose that futures price for maturity T is given by Ft(T). Now, the basis is Bt = Ft – St. In general, the optimal hedge ratio, h, is:
Having considered the basis risk, then the optimal hedge ratio, h, is:

4. Variance of hedged position = (1 – h)2 [Variance of Spot]
In theory, a hedged position should not have any volatility, but in reality, because of the basis risk, the hedged position value fluctuates over time, and:
Variance of hedged position = (1 – h)2 [Variance of Spot]
+ h2 [Variance of Basis]

5. Factors affecting hedging
Availability of underlying asset:
In absence appropriate underlying, a “Cross-Hedge” can be implemented. For instance, airliners use heating oil or gasoline contacts to hedge jet fuel risk.
Liquidity of markets:
We usually assume that setting up of a hedge position is matter of exercising correct trades. However, execution of these orders requires liquidity. Large orders can move the price in direction detrimental to success of the hedge.
Market impact, open interest, rollover risk, etc.

6. Imperfect Divisibility
Since contracts have standard size, finding the exact number of contracts needed is not always possible. The number of contracts has to be rounded. This then give rise to a number of problems:
Rounding error can make hedging sub-optimal
Rounding up or down can have significantly different results.
Hedge ratios change over time, so initial numbers also have to consider possibility of future adjustments.