Hedging with Forward & Futures Risk Management Prof. Ali Nejadmalayeri, Dr N a.k.a. “Dr N”

Measuring Statistics Suppose we have T observations of past changes and we need to forecast change and volatility in T+1. Let’s say change is ΔS i = S i – S i-1, then expected change is: The volatility of the change is :

Square Root Volatility i.i.d.If a series of random variables are identically, independently distributed, i.i.d., with volatility per period of σ, the volatility of the series of random variables over N periods is σ√ N

Hedging with no Basis Risk Cash Position + Gain from HedgeValue of hedged position is the sum of Cash Position + Gain from Hedge 1.65  Vol(ΔPV of Hedged Position)One-day VaR of hedged position is 1.65  Vol(ΔPV of Hedged Position) In perfect hedging, i.e., making the expected value change zero, then requires correct Hedge Ratio. As the forward price changes, the hedge ratio changes. To change the hedge due to marked to market is Tailing Hedge.

Hedging with Basis Risk Payoff of Cash Position + Payoff of HedgeValue of hedged position is the sum of Payoff of Cash Position + Payoff of Hedge In any date Basis is the difference between spot and forward price. The Basis Risk is when the basis is not deterministic. Volatility-minimizing hedge ratio  Exposure to the risk factorVolatility-minimizing hedge is Volatility-minimizing hedge ratio  Exposure to the risk factor

Hedge with Basis Risk Relationship between cash position and futures price is deterministic:

Hedge with Basis Risk Hedge Size Hedge Position

Hedging with Random Basis When basis is random, then an approximate linear relationship between spot and futures is needed to figure out how changes in the spot and changes in the futures are linked with each other. –We need to a run a regression: –Simply put then, the hedge ratio is:

Hedge with Random Basis Relationship between cash position and futures price is only approximately deterministic: