introduction to derivatives. derivatives– overview and definitions a derivative instrument is...
TRANSCRIPT
Introduction to Derivatives
Derivatives– Overview and Definitions
A derivative instrument is defined as a private contract
whose value is derived from some underlying asset price,
reference rate or Index.
A derivative instrument is a contract between two parties –
buyer and seller - who agree to exchange some asset for cash
at some future date/s, at a predetermined price.
The main categories of derivatives are:
Futures and Forward contracts
Swap contracts
option contracts
Futures, Forward and Swap contracts are distinctly different
from option contracts:
With an Options contract the buyer has the right to buy or
sell some asset in the future.
With Futures, forward and swap contracts the buyer is
obligated to buy some asset in the future
Forward Contracts
Forward contract is a private agreement to exchange a given asset against cash at a fixed point in the future, at a predetermined price.
The terms of the contract are: Underlying Asset, quantity, or number of units or shares, date to delivery, and price at which the exchange will be done.
The seller of the contract has to deliver the asset while the buyer has a commitment to receive the asset.
Thus: the seller of the contract is in a short position, while the buyer is in a long position.
Denoting
T – time to delivery, also called the maturity date
t – current time
– T-t: time to maturity
St – current spot price of the underlying asset
F – forward price of the asset to delivery at T
Vt – current value of the contract
n - quantity, or number of units in contract
The notional amount, also called the principal value is defined as the amount nF to pay at maturity
The value of the forward contract at expiration, for one unit of
the underlying asset is,
VT = ST - F
Profit or Loss on Long and Short Forward Contract
Profit or Loss
ST
F
Sell Forward
Profit or Loss
ST
F
Buy Forward
Futures Contract
futures contracts are differ from forward contracts as follows:
1. Futures are traded in organized exchanges in contrast to
forwards, which are traded on OTC market.
2. Standardization – Futures contracts are offered with a
limited choice of expiration dates and trade in fixed contract
size.
3. Clearinghouse – After each transaction, the clearinghouse
interpose itself between the buyer and the seller, ensuring the
performance of the contract.
Futures Contracts
Marking to Market – Futures are marked to market on a daily
basis which involves cash settlement of the gains and the
losses on the contract every day.
Closing Price
(cents per pound)
24/11
90.75
25/11
93.75
26/11
90.25
BuyerPurchases cotton futures at 90.75 cents per pound
Buyer receives 3 cents per pound
Buyer pays 3.5 cents per pound
SellerSells cotton futures at 90.75 cents per
pound
Seller pays 3 cents per pound
Seller receives 3.5 cents per
pound
Cash flows to Buyer and Seller of Cotton Futures Contracts
Margin Requirements
To provide some guarantee of the contract’s performance,
initial margin are required by the clearinghouse for both buyer
and seller.
The initial margin is the monies placed with the clearing house
when the trade is initially executed.
When the minimum margin level is reached, the investor have
to post more margin.
In case he/she fails to meet the margin call, the broker has the
right to liquidate the position.
Futures Contracts
The main categories of forward/futures contracts are:
Currency
Commodity
Stock Index
Bond
Valuing Futures Contracts
Generally forward contracts are established so their initial
value is zero.
This is achieved by setting the forward price F so there will
be no arbitrage relationship between the spot and the futures
market.
No-arbitrage is a situation where economically equivalent
portfolio have the same price.
Stock Index Futures
The most active contract is the S&P500 futures contract traded
on the CME, where the contract notional is defined as $250
times the index level.
If we actually invested in the S&P500 index, our rate of return
would be higher than the index, because we would receive the
cash dividends.
The pricing formula is derived by the no-arbitrage argument,
using a strategy composed of buying the Index , selling a futures
contract, and borrowing. such that the net investment is zero
StrategyCash Flow TodayCash Flow at the End of the Period
Borrow
Buy the index for
Sell One futures contract
Net Position
0S
0S0S
0S
0
τ0 )r1(S
DST
T0 SF
0 D)r1(SF τ00
D)r1(SF0 τ00 D)r1(SF τ
00
If we have annualized and continuing compounded dividend and interest:
τdr00 eSF
)dr1(SS
DS)r1(SF 0
0000
Numerical Example
Suppose the NYSE Index closed at 342. If dividend yield is 2%
and the current risk-free interest rate is 4%, what is the
equilibrium value of a six-month futures contract on the NYSE
Index?
4.345$)02.004.01(342
)dr1(SF2/1
τ00
Assume that the futures contract is traded at $347, show arbitrage strategy!
StrategyCash Flow TodayCash Flow at the End of the Period
Borrow
Buy the index for
Sell One futures contract
Net Position
0S
342
342
0
8.348)04.1(342 5.0
42.3S34201.0S TT
TS347
0
0S
4.3453476.1
Currency Futures
Currency futures contracts are used by firms having exposure
to foreign exchange risk.
For example, 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 futures contract to sell British
pound in the future with predetermined $/£ exchange rate.
StrategyCash Flow TodayCash Flow at the End of the Period
Borrow 1 £
Lend Dollars in the US
Buy futures position to buy
£
Net Position
0S
0S
0S
0
τFT )r1(S
τL0 )r1(S
)FS()r1( 0Tτ
F
0 τF0
τL0 )r1(F)r1(S
If we have annualized and continuing compounded interest:
τrr00
FLeSF
τF )r1(
τF0
τL0 )r1(F)r1(S0 τ
FL0
τ
F
L00 )rr1(S
r1
r1SF
Numerical Example
Suppose you are an arbitrage trader in the Swiss franc foreign
exchange rate. You observe the following information:
Are these prices in equilibrium? How will you profit if they
are not?
The equilibrium futures price should be:
2/1τ%,6r%,3r,SwF
64.0$F,
SwF
65.0$S FL00
6407.0$06.1
03.165.0
r1
r1SF
2/12
1
F
L00
Thus, the current future price is lower than the equilibrium price.
StrategyCash Flow TodayCash Flow at the End of the Period
Borrow 1 £
Lend $0.65 Dollars in US
Buy futures position to buy
£
Net Position
2/1)06.1(
65.0$
65.0$
0
0
2/1T )06.1(S
2/1)03.1(65.0$
)64.0$S()06.1( T2/1
0007.0$)06.1(64.0$)03.1(65.0$ 2/12/1
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? 1τ
F
L02004Des r1
r1SF
2τ
F
L02005Mer r1
r1SF
12 ττ
F
L
2004Des
2004Mer
r1
r1
F
F
4/112/3ττ 12
4/1
F
L
r1
r1
6664.1
6604.1
4/1FL
4/1
F
L )rr1(9857.0r1
r1
%43.1rr FL
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.
StrategyCash Flow TodayCash Flow at the End of the Period
Buy the asset at price
Borrow
Sell a futures contract on the asset
Net Position
0S
0S
0S
0S
0
CST
τ0 )r1(S
T0 SF
0 C)r1(SF τ00
C)r1(SF τ00
Numerical Example
Assume that the spot price of gold is $650 per ounce and the
one year futures price is $678. If the risk-free interest is 3%,
what is the implied storage cost for gold in percent?
5.8$C
C03.1650678
C)r1(SF τ00
%3.1650
5.8
S
C
0
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
$800K $800K$800K
$700K
$800K
$900K
Fixed rate payments
Floating rate payments
LIBOR 7% 8% 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
yearForward curve (%)
Ft-1,t
Yield Curve (%)
yt
177
298
311.039
33
221
*
33
3,22
2
2,1
1
1,0
)y1(
1
)y1(
1
)y1(
1F
)y1(
F
)y1(
F
)y1(
F
F* – Fixed rate
yt, is the appropriate yield from the yield curve for discounting dollars cash flows.
%88.8F)09.1(
F
)08.1(
F
)07.1(
F
)09.1(
1103.
)08.1(
09.
)07.1(
07.
*3
*
2
**
32
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
FirmFixed (%)Floating (%)
A10LIBOR+0.3
B11.2LIBOR+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 FixedFloating
Issue debtPay 10%
Enter swapReceive 10%Pay LIBOR+0.05%
NetPay LIBOR+0.05%
Direct cost Pay LIBOR+0.3%
Saving0.25%
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 FloatingFixed
Issue debtPay LIBOR+1%Pay 10%
Enter swapReceive LIBOR+0.05%
NetPay 10.95%
Direct cost Pay 11.2%
Saving0.25%
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
yearUS (%)UK (%)
124
234
33.54
The Forward rates:
76.1
04.1
02.18.1F1 766.1
04.1
03.18.1F
2
2
774.104.1
035.18.1F
3
3
Therefore,
£100M £100M £100M
$176M $176.6M $177.4M
Alternatively, we can calculate a constant rate of F* dollars per
pound to be exchanged each year:
33
*
22
*
1
*
33
32
2
2
1
1
y1
F
y1
F
y1
F
y1
F
y1
F
y1
F
where y1, y2 and y3 are the appropriate yields from the yield
curve for discounting dollars cash flows.
3
*
2
**
32 035.1
F
03.1
F
02.1
F
035.1
774.1
03.1
766.1
02.1
76.1
7665.1F*
In this case the swap agreement will be:
£100M £100M £100M
$176.65M $176.65M $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 risk-
free 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).