Derivatives Lecture 2: Hedging with Futures
1) The purpose of hedging is to reduce risk—reduce the probability and magnitude
of a negative payoff. Using futures (or forwards) the cost of this is potential gains
beyond the rate locked in at hedge inception. Thus, the net effect is to compress
returns about an expected value and reduce return volatility.
Spot
2) Hedging cedes potential gains as well as limiting potential losses. If a party
chooses to hedge, they won’t know until expiration whether the hedge position
made or lost money. However, what a hedge will do with certainty is limit
variance of a portfolio.
3) Hedge Types:
-
Short Hedge: Sell a future contract—used to hedge a long position on an asset
(you’re concerned that the price may fall/interest rates may rise)
 You’re long an asset and wish to fix its future value
 You plan sell an asset/commodity at a future date and wish to fix
its price
 Have issued floating rate debt and want to limit interest rate risk,
or plan to borrow in the future
-
Long Hedge: Buy a future contract—hedge a short position on an asset.
 You’re short an asset and want to limit the risk of an increase in
price
 You plan to purchase an asset in the future and want to fix it’s
value today
4) Profits from Hedging—Conventional Thinking (and where the books are in error):
-
Short Hedge:  S  S t  S 0    f t  f 0  . Long the spot and short the futures.
Pay S0 at time 0, and Sell for St at time t. Short futures at 0, and buy at t.
-
Long Hedge:  L   f t  f 0   S t  S 0  . Short the spot and long the futures. Sell
S0 at time 0, and Buy for St at time t. Long futures at 0, and sell at t.
-
The problem is that the above implies the change in futures prices are intended to
counter the change in the value of the spot at inception. That is conceptually
incorrect.
5) Example of How the Hedge is Supposed to Work (In theory)
-
Short Hedge: Long 100 shares, S0 = $100, and f0 = $105.
 We are concerned the price ↓, so we sell a futures contract.
 In theory, if St = $90, the futures price ft = $95.
 S  St  S0    ft  f 0   $90  $100  $95  $105  0 .
-
Long Hedge: Short 100 shares, S0 = $100, and f0 = $105.
 We are concerned that the price ↑, and buy a futures to hedge.
 Again, in theory, if St = $110, we expect ft = $115.
 L   ft  f 0   St  S0   115  105  110  100  0 .
6) The Problem
-
Futures prices move with the deviation of the spot from the forward rate, not
the change in the spot from its original value.
-
Example: Suppose that S0 = 100, f0 = 105, and that at time t, St = 110.
Future versus Spot
Basis narrowing at
constant rate
$110
$5
Future
$105
$10
Note: The change in the future
price is equivalent to the deviation
of the spot from its original
forward price—which equals f0
because we are holding the future
to maturity.
Expected Path of Spot/Future
Spot
$100
Actual Path of Spot/Future
Maturity
-
Since the futures price must converge on the spot (St) at maturity, the future will
increase only by $5 and fails to counteract the change in value of the spot ($10).
The Actual Hedge Results will be:
If St  110  ft  110 :
Short Hedge:  S  St  S0    ft  f 0   $110  $100  $110  $105  5
Long Hedge:  L   ft  f 0   St  S0   $110  $105  $110  $100  5
If St  90  ft  90 :
Short Hedge:  S  St  S0    ft  f 0   $90  $100  $90  $105  5
Long Hedge:  L   ft  f 0   St  S0   $90  $105  $90  $100  5
The hedge will not lock in the current spot S0—Profits are never zero.
Benchmarking against the spot incorrectly states the gains/losses of the hedge.
7) True Measure of Hedging Profits
-
Short Hedge:  S  St  F0    ft  f 0  . Long the spot and short the futures.
-
Long Hedge:  L   f t  f 0   St  F0  . Short the spot and long the futures.
The difference between this and the conventional method of measuring profits or
losses is that the hedging objective is not to fix the price of the underlying asset at
the spot rate at time 0, but at the time t Forward price of the asset at time 0.
8) Real World Example
- Assume that we have a long position of 1000 ounces of gold. The February 10
spot is $385.50 and the futures price for the June contract is $390.10
-
Since each futures contract controls 100 ounces, we sell 10 contracts.
Selling at Maturity: If the position is held to expiration, then fT  ST .
Spot
June 20 Forward
June 20 Future
Net Gains/Losses
Feb 10
385.50
390.10
390.10
June 20
380.00
Gains/Losses
- 10.10
Gain/Loss ($)
- 10,100
380.00
+ 10.10
+ 0.00
+10,100
0
Selling prior to maturity: May 01, Forward price is 388.33
Spot
May 01 Forward
June 20 Future
Net Gains/Losses
Feb 10
385.50
388.33
390.10
May 01
387.00
Gains/Losses
- 1.33
Gain/Loss ($)
- 1,330
389.10
+ 1.00
- 0.33
+ 1,000
- 330
We sell the Gold at $387.00 on May 01 and profit $1.00 on the futures position: your net
is $388.00. This is much more closely aligned to the forward as a target ($388.33), than
the original spot ($385.50).
$/P
390.10
Future (expected path)
Future (actual path)
389.10
Gain on sale of future $1.00
388.33
Loss on Spot -$1.33
387.00
Spot (expected path)
Spot (actual path)
May 01
Jun 20
Conventional Measure: Suppose we use S0 as our objective (wrong objective):
Selling at Maturity: If the position is held to expiration, then fT  ST .
Spot
June 20 Forward
June 20 Future
Net Gains/Losses
Feb 10
385.50
June 20
380.00
Gains/Losses
- 5.50
Gain/Loss ($)
- 5,500
390.10
380.00
+ 10.10
+ 4.60
+10,100
+ 4,600
Gains/Losses
+ 4.50
Gain/Loss ($)
+ 4,500
+ .10
+ 4.60
+ 100
+ 4,600
Gains/Losses
+ 14.50
Gain/Loss ($)
- 5,500
+
+10,100
+ 4,600
Suppose the spot exceeds the forward: St = 390.00
Feb 10
June 20
385.50
390.00
Spot
June 20 Forward
390.10
390.00
June 20 Future
Net Gains/Losses
Or: St = 400.00
Feb 10
June 20
385.50
400.00
Spot
June 20 Forward
390.10
400.00
June 20 Future
Net Gains/Losses
9.90
4.60
No matter, the total value of our portfolio on June 20 is $385.50 + $4.60 = $390.10.
We get the forward price.
How about if we are short 1000 ounces of Gold at $385.50 and buy 10 futures contracts:
Spot
June 20 Forward
June 20 Future
Net Gains/Losses
Feb 10
385.50
June 20
380.00
Gains/Losses
+ 5.50
Gain/Loss ($)
+ 5,500
390.10
380.00
- 10.10
- 4.60
- 10,100
- 4,600
Gains/Losses
- 4.50
Gain/Loss ($)
- 4,500
-
100
- 4,600
Suppose the spot exceeds the forward: St = 390.00
Feb 10
June 20
385.50
390.00
Spot
June 20 Forward
390.10
390.00
June 20 Future
Net Gains/Losses
Or: St = 400.00
Feb 10
June 20
385.50
400.00
Spot
June 20 Forward
390.10
400.00
June 20 Future
Net Gains/Losses
.10
4.60
Gains/Losses
- 14.50
Gain/Loss ($)
+ 5,500
+
-
- 10,100
- 4,600
9.90
4.60
No matter, the total value of our portfolio is -$385.50 - $4.60 = -$390.10. We are short
gold at the forward rate.
You might say then why hedge the short position?
If the spot goes to $400.00, you will owe 1000 ounces of Gold at $400.00. The hedge
limits your risk to $390.00, but does not lock in the spot price—it locks in the forward.
9) Currency Example using the Forward as the objective:
-
Today is January 2. You are a middle man. You have an agreement to sell
equipment to a US firm for $330,000. Your Mexican supplier will sell you the
equipment at that time for 3,060,000P.
-
The current spot is .1000$/P, and the April 20 futures price is .097656$/P.
-
Peso Contracts are available with a value of 500,000P per contract
Selling at Maturity: If the position is held to expiration, then FT  ST .
Jan 2 ($/P)
Spot
Apr 20 Forward
Apr 20 Future
Net Gain/Loss
.097656 $/P
.097656$/P
Apr 20 ($/P)
.097116 P/$
Gains/Losses ($/P)
+.000540 $/P
Gain/Loss ($)
+1,652
.097116 P/$
- .000540 $/P
- 1,620
+ 32
Selling prior to maturity: April 02, Forward price is .098039$/P
Jan 2 ($/P)
Spot
Apr 2 Forward
Apr 20 Future
Net Gain/Loss
.098039 $/P
.097656 $/P
Apr 2 ($/P)
.097116 $/P
Gains/Losses ($/P)
+.000923 $/P
Gain/Loss ($)
+2,825
.096590 $/P
- .001066$/P
- 3,198
- 373
Spot: .000923P / $  3,060,000 P  $2,825
Future: .001066$ / P  500,000 P  6  $3,198
There are two problems:
- We are not fully hedged: The obligation is 3,060,000 Pesos, but we can only buy a round
number of contracts (6). So we under-hedged by 60,000 pesos.
-
Basis risk: The change in the futures prices (.001066 $/P) did not exact equal the deviation of the
spot from the forward rate (.000923 $/P).
Basis risk occurs because the asset/liability being hedged has a different maturity than the future
being used to hedge it. So the change in value in response to the spot is somewhat faster/slower.
$/P
spot
Anticipated Path
Realized Path
.098039
.097656
Gain from the Relative
Depreciation of spot of
.000923$/P
.097116
future
Loss on future of .001066$/P
.096590
Apr 2
Apr 20
10) Rolling Futures Contracts
-
Futures contracts have a maximum maturity of 9 or 12 months.
-
“Rolling” one contract into another is a method of synthetically creating a longer
maturity contract.
-
To roll a contract, you simply hold it to (or near) expiration, then close the
position, and take the same position on another longer contract:
Long: You buy a 1-year contract, hold it to expiration, sell it, and purchase the new contract.
Short: You sell a contract, at expiration you buy one to close, and sell the new contract.
 Close the position on the desired date.
Example:
-
Today is Jan 2, 2013. You owe 3,060,000P due April 3, 2015 (2 years, 91 days)
The spot is .10 $/P, and forward rate for April 03, 2015 is .0836375 $/P.
The longest futures contract available is Jan 17, 2014. You buy 6 contracts.
Jan 2 ($/P)
Spot
Apr 03, 2015 Forward
Jan 17, 2014 Future
Net Gain/Loss
.083638 $/P
.092083 $/P
(Buy Jan 16 contract)
Spot
Apr 03, 2015 Forward
Jan 16, 2015 Future
Net Gain/Loss
Jan 17 ($/P)
(Buy Apr 17 contract)
Spot
Apr 03, 2015 Forward
Apr 17, 2015 Future
Total Futures
Net Gain/Loss
Jan 16 ($/P)
.083638 $/P
.085917 $/P
.083638 $/P
.084571 $/P
Jan 17, 2014 ($/P)
.09300 P/$
Gains/Losses ($/P)
.09300 P/$
+.000917 $/P
Jan 16, 2015 ($/P)
.08600 P/$
Gains/Losses ($/P)
.08600 P/$
+.000083 $/P
Apr 03, 2015 ($/P)
.084500 P/$
Gains/Losses ($/P)
- .000862 $/P
.084400 P/$
- .000171 $/P
+.000829 $/P
Gain/Loss ($)
Gain/Loss ($)
Gain/Loss ($)
- 2,638
+ 2,487
- 151
Note, the forward rate for April 17, 2015 will have changed each year as the forecast on
the $/P spot on that date has changed. However, we use the original forward as our target
because that was the rate we wished to lock in.
11) Interpolating the Forward Rate
-
In theory, we can simply compute the Forward price using Ft  St  (1  r )T  t or in
continuous time: Ft  St  e rC T  t 
T
-
1  r$

r  r T t 
For exchange rates: F$ / P  S$ / f 
 or Ft  St  e $ f
1

r
f 

r T  t 
$
Note the latter comes about because: e
e
r f t  t 
e
r$  r f T t 
-
The problem is that we may not have the correct risk-free rates, they embed a
premium, but you would like a more direct market-based estimation of the correct
forward rate.
-
The forward can be interpolated from the two surrounding futures prices:
Suppose the April 20 future is .097656 $/P. That is also a forward rate on the
Peso for April 20. We wish to compute the April 2 Forward.
March 16 contract has a price of .098402 $/P.
Apr 02  Mar16
Apr 20  Mar16
17
FAPR03  .098402 P$  .097656 P$  .098402 P$ 
 .098040 P$
35
FAPR03  f MAR16   f APR20  f MAR20  
Our interpolated forward would be .098040 $/P, whereas the actual is .098039.
Pretty close.
Note: the forward and futures prices were computed using risk-free rates of 1.6%
for the US and 10% for Mexico.
 1.10
1.0161.10
1.0161.10
F$ / P  Mar16  .10 P$ 1.016
F$ / P  Apr02  .10 P$
F$ / P  Apr 20  .10 P$
74
 .098402
365
91
365
109
 .098039
365
 .097656
12) Basis and Basis Risk
-
Basis is the difference between the Spot and Futures price at any point in time.
 At time 0: b0  S 0  f 0
 At time t: bt  S t  f t
-
The profits/losses from a hedge can be written as the change in basis.
 Short Hedge: bt  b0
 Long Hedge: b0  bt
-
Basis Risk is the uncertainty surrounding the Change in Basis—the movement of
spot relative to the futures price. There is no Basis Risk if:
 The position is held to maturity, and
 The future is on the same asset as the one being hedged
13) Computing Gains/Losses from Change in Basis—Gold futures example (8)
b0  S0  f 0  $385.50  $390.10  $4.60
If the position is held to expiration, fT  ST  380.00 and bT  ST  fT  0 .
 S  bt  b0  0   $4.60  $4.60
Spot
June 20 Forward
June 20 Future
Basis
Feb 10
385.50
June 20
380.00
Gains/Losses
- 5.50
Gain/Loss ($)
- 5,500
390.10
(b0)
- 4.60
380.00
(bT)
0.00
+ 10.10
(b0 – bT)
+ 4.60
+10,100
+ 4,600
Computing gains/losses using a Change in Basis is just a different way of Accounting for
them. The results are the same.
If we close the hedge early, calculating Gains/Losses from a Change in Basis is still no
different from calculating the total gains/losses from the change in the spot:
Spot
May 01 Forward
June 20 Future
Basis
Feb 10
385.50
May 01
387.00
Gains/Losses
+ 1.50
Gain/Loss ($)
+ 1,500
390.10
(b0)
( 4.60)
389.10
(bT)
( 2.10)
+ 1.00
(b0 – bT)
+ 2.50
+ 1,000
+ 2,500