Hedging with Forward &
Futures
Risk Management
Prof. Ali Nejadmalayeri,
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 ΔSi = Si – Si-1, then expected change is:
T
E ST 1   S  1 T  Si
i 1
• The volatility of the change is :
1 T
2
Si  S 
Var ST 1  

T  1 i 1
Square Root Volatility
• 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
• Value of hedged position is the sum of
Cash Position + Gain from Hedge
• 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
• Value 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 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:
St  const.  h  Ft  error
– Simply put then, the hedge ratio is:
CovSt , Ft 
h
Var Ft 
Hedge with Random Basis
• Relationship between cash position
and futures price is only
approximately deterministic: