‫תמחור חוזים עתידיים וחוזה החלף‬
Stock Index Futures
The most active contract is the S&P500 futures contract traded
on the CME, where the contract notional is defined as $250
times the index level.
If we actually invested in the S&P500 index, our rate of return
would be higher than the index, because we would receive the
cash dividends.
The pricing formula is derived by the no-arbitrage argument,
using a strategy composed of buying the Index , selling a futures
contract, and borrowing. such that the net investment is zero
Strategy
Cash Flow Today
Cash Flow at the
End of the Period
S0
 S0 (1  r ) τ
 S0
ST  D
Sell One futures contract
0
F0  ST
Net Position
0
F0  S0 (1  r ) τ  D
Borrow S 0
Buy the index for S 0
F0  S0 (1  r ) τ  D
0  F0  S0 (1  r )  D
τ
D
F0  S0 (1  r )  S0
 S0 (1  r  d ) 
S0

If we have annualized and continuing compounded dividend and
interest:
F0  S0 e
 r  d τ
Numerical Example
Suppose the NISE Index closed at 342. If dividend yield is 2%
and the current risk-free interest rate is 4%, what is the
equilibrium value of a six-month futures contract on the NYSE
Index?
F0  S0 (1  r  d) τ 
342  (1  0.04  0.02)1/ 2  $345.4
Assume that the futures contract is traded at $347, show
arbitrage strategy!
Strategy
Borrow S 0
Buy the index for S 0
Cash Flow Today
Cash Flow at the
End of the Period
342
 342  (1.04)0.5  348.8
 342
ST  0.01 342  ST  3.42
Sell One futures contract
0
Net Position
0
347  ST
1.6  347  345.4
Currency Futures
Currency futures contracts are used by firms having exposure
to foreign exchange risk.
For example, a Israeli firm sell its goods in US and therefore
receives USD in exchange for its product.
To minimize the effect of FX risk on the value of the product
sold, the firm may enter into a futures contract to sell USD in the
future with predetermined NIS/$ exchange rate.
Strategy
Cash Flow Today
Borrow 1$
Lend S 0 NIS in Israel
Buy futures position to buy
(1  rF ) τ $
Net Position
Cash Flow at the
End of the Period
S0
 ST (1  rF ) τ
 S0
S0 (1  rL ) τ
0
(1  rF ) τ (ST  F0 )
0
S0 (1  rL ) τ  F0 (1  rF ) τ
τ
0  S0 (1  rL )  F0 (1  rF )
τ
τ
 1  rL 
  S0 (1  rL  rF ) τ
F0  S0 
 1  rF 
If we have annualized and continuing compounded interest:
F0  S0 e
 rL  rF τ
Numerical Example
Suppose you are an arbitrage trader in the Swiss franc foreign
exchange rate. You observe the following information:
$0.65
$0.64
S0 
, F0 
, rL  3%, rF  6%, τ  1 / 2
SwF
SwF
Are these prices in equilibrium? How will you profit if they
are not?
The equilibrium futures price should be:
1
2
 1  rL 
 1.03 


F0  S0 
 0.65  


 1.06 
 1  rF 
1/ 2
 $0.6407
Thus, the current future price is lower than the equilibrium
price.
Strategy
Cash Flow Today
Cash Flow at the End of
the Period
Borrow 1SwF
$0.65
 ST  (1.06)1/ 2
Lend $0.65 Dollars in US
 $0.65
$0.65  (1.03)1/ 2
Buy futures position to buy
(1.06)1/ 2 SwF
0
Net Position
0 $0.65  (1.03)1/ 2  $0.64  (1.06)1/ 2  $0.0007
(1.06)1/ 2  (ST  $0.64)
Numerical Example
Assume that the British pound Des 2004 futures contract
settled at $1.6664/£ and Mar 2005 contract settled at $1.6604/£
What is the implied interest rate difference between the pound
and dollar?
 1  rL 

FDes 2004  S0 
 1  rF 
τ1
FMer 2004  1  rL 

 
FDes 2004  1  rF 
 1  rL 

FMer 2005  S0 
 1  rF 
τ 2  τ1
τ2
τ 2  τ1  3 / 12  1 / 4
1/ 4
1.6604  1  rL 

 
1.6664  1  rF 
1/ 4
 1  rL 


 1  rF 
 0.9857  (1  rL  rF )1/ 4
rL  rF  1.43%
Commodity Futures
To price commodity futures, we need to consider storage costs
and insurance costs.
The pricing formula is derived by using a strategy composed of
buying the asset , selling a futures contract, and borrowing.
Strategy
Cash Flow Today
Buy the asset at price S 0
 S0
Borrow S 0
S0
Sell a futures contract on
the asset
0
Net Position
0
Cash Flow at the End of
the Period
ST  C
 S0 (1  r ) τ
F0  ST
F0  S0 (1  r ) τ  C
F0  S0 (1  r ) τ  C
Numerical Example
Assume that the spot price of gold is $340 per ounce and the
one year futures price is $357. If the risk-free interest is 3%,
what is the implied storage cost for gold in percent?
F0  S0 (1  r ) τ  C
357  340 1.03  C
C  $6.8
C 6.8

 2%
S0 340
Swap Contracts
Swap contracts are OTC agreements to exchange a series of
cash flow according to some pre-specified terms.
The underlying asset can be :
an interest rate, an exchange rate, an equity, a commodity price
or any other index.
The most common swap contracts are: an Interest Rate
Swap (IRS), a Foreign Exchange Swap (FES) and a Credit
Default Swap (CDS)
Interest Rate Swap
Consider the case of a firm that has issued long term bonds
with total par value of $10M at a fixed interest rate of 8%.
However, it can change the nature of its obligation from fixed
rate to floating rate by entering a swap agreement to pay a
floating rate and to receive a fixed rate.
A swap with notional principle of $10M that exchanges
LIBOR for an 8% fixed rate:
$800K ↔ $10M * rLIBOR
Suppose that the swap is for three years and the LIBOR rates
turns out to be 7%, 8% and 9% in the next three years
$900K
$800K
$700K
Floating rate payments
Fixed rate payments
$800K
LIBOR
7%
$800K
8%
$800K
9%
IRS - Pricing
A swap contract can be viewed as a portfolio of forward
transactions, but instead of each transaction being priced
independently, on forward price is applied to all of the
transactions.
The Yield and the Forward Curve
year
Forward curve (%)
Ft-1,t
Yield Curve (%)
yt
1
7
7
2
9
8
3
11.03
9
F* – Fixed rate
yt, is the appropriate yield from the yield curve for discounting
dollars cash flows.
F3
F1
F2



2
3
(1  y1 ) (1  y 2 )
(1  y 3 )
 1

1
1

F 


2
3 
(1  y 3 ) 
 (1  y1 ) (1  y 2 )
*
.07
.09
.1103



2
3
(1  .07) (1  .08) (1  .09)
F*
F*
F*
*



F
 8.88%
2
3
(1  .07) (1  .08) (1  .09)
IRS – Quotations
Swaps are quoted in terms of spreads relative to the yield of
similar-maturity Treasury notes.
For instance, a dealer quote 10 years swap rates as 31/35bp
against LIBOR.
If the current note yield is 7%:
The dealer is willing to pay 7%+0.31%=7.31% against
receiving LIBOR and to receive 7%+0.35%= 7.35% against
paying LIBOR.
Interest Rate Swap – Motivation
Consider two firms, A and B that can raise funds either at
fixed or floating rates, $100M over 10 years. A want to raise
floating and B want to raise fixed.
Cost of Capital Comparison
Firm
Fixed (%)
Floating (%)
A
10
LIBOR+0.3
B
11.2
LIBOR+1
Interest Rate Swap – Motivation
Firm A has an absolute advantage in both markets
However, it has a comparative advantage in raising fixed
If both will directly issue funds in their desired market, the total
cost: LIBOR+0.3% (for A) + 11.2% (for B) = LIBOR + 11.5%
If they will raise funds where each has a comparative advantage,
the total cost: 10% (for A) + LIBOR+ 1% (for B) = LIBOR + 11%.
Thus, the gain to both firms from entering a swap is:
11.5%-11%= 0.5%.
A swap that splits the benefit equally between the two parties:
Swap to firm A
Firm A issues fixed debt at 10% and enters a swap whereby it
promises to pay LIBOR+0.05% in exchange to receiving 10% fixed
payments, which will offset the required debt payments.
Operation
Fixed
Issue debt
Pay 10%
Enter swap
Receive 10%
Floating
Pay LIBOR+0.05%
Net
Pay LIBOR+0.05%
Direct cost
Pay LIBOR+0.3%
Saving
0.25%
A swap that splits the benefit equally between the two parties:
Swap to firm B
Firm B issues floating debt at LIBOR+1% and enters a swap
whereby it promises to pay 10% fixed payments in exchange to
receiving LIBOR+0.05%, which is less than the direct cost by
0.25%
Operation
Floating
Fixed
Issue debt
Pay LIBOR+1%
Pay 10%
Enter swap
Receive LIBOR+0.05%
Net
Pay 10.95%
Direct cost
Pay 11.2%
Saving
0.25%
Foreign Exchange Swap
Foreign Exchange Swaps are agreements between to parties
to exchange currencies according to a pre-determined formula.
FES enable the firm to quickly and cheaply hedge its
currency exposure.
For Instants, a U.S. firm sell its goods in UK and therefore
receives British pound in exchange for its product.
To minimize the effect of FX risk on the value of the
product sold, the firm may enter into a swap contract to sell
British pound in the future with predetermined $/£ exchange
rate.
Foreign Exchange Swap
A U.S. firm has a 3 years contract of selling goods to UK
firm for £100M each year. The U.S. firm can enter to a FES
whereby it promises to pay £100M in exchange to receiving
$X.
The current exchange rate is: $1.8/£
The term structure of US and UK interest rate
year
1
US (%)
2
UK (%)
4
2
3
4
3
3.5
4
The Forward rates:
2
 1.03 
F2  1.8  
  1.766
 1.04 
1.02
F1  1.8 
 1.76
1.04
3
 1.035 
F3  1.8  
  1.774
 1.04 
£100M
$176M
Therefore,
£100M
£100M
$176.6M
$177.4M
Alternatively, we can calculate a constant rate of F* dollars per
pound to be exchanged each year:
F3
F1
F2
F*
F*
F*





2
3
2
1  y1 1  y 2  1  y 3  1  y1 1  y 2  1  y 3 3
where y1, y2 and y3 are the appropriate yields from the yield
curve for discounting dollars cash flows.
1.76
1.766
1.774
F*
F*
F*





2
3
2
1  .02 1  .03 1  .035 1  .02 1  .03 1  .0353
F*  1.7665
In this case the swap agreement will be:
£100M
$176.65M
£100M
$176.65M
£100M
$176.65M
Credit Default Swap
In a credit default swap contract, a protection buyer pays a
premium to the protection seller in exchange of payment if
credit event – default - occurs.
Buyer Periodic Payment
Seller
Contingent Payment
The contingent payment is triggered by a Credit Event on the
underlying credit
Investing in a risky bond is equivalent to investing in a riskfree bond plus selling a credit default.
Numerical Example
A protection buyer enters a 1-year CDS on a notional of
$100M worth of 10-year bonds issued by XYZ. The swap
entails an annual payment of 50bp.
At the beginning of the year, the buyer pays $500K to the
protection seller.
At the end of the year, XYZ defaults on this bond, which
now traded at 40% of the notional value (Recovery Rate)
The seller has to pay $60M (Loss Given Default).