1. Introduction to Derivatives

2. 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

3. 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

4. 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.

5. 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

6. Profit or Loss on Long and Short Forward Contract
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

7. Futures Contract
futures contracts are differ from forward contracts as follows:
Futures are traded in organized exchanges in contrast to forwards, which are traded on OTC market.
Standardization – Futures contracts are offered with a limited choice of expiration dates and trade in fixed contract size.
Clearinghouse – After each transaction, the clearinghouse interpose itself between the buyer and the seller, ensuring the performance of the contract.

8. Cash flows to Buyer and Seller of Cotton 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.
Cash flows to Buyer and Seller of Cotton Futures Contracts
26/11
90.25
25/11
93.75
24/11
90.75
Closing Price
(cents per pound)
Buyer pays 3.5 cents per pound
Buyer receives 3 cents per pound
Purchases cotton futures at cents per pound
Buyer
Seller receives 3.5 cents per pound
Seller pays 3 cents per pound
Sells cotton futures at cents per pound
Seller

9. 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.

10. Futures Contracts
The main categories of forward/futures contracts are:
Currency
Commodity
Stock Index
Bond

11. 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.

12. 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

13. If we have annualized and continuing compounded dividend and interest:
Cash Flow at the End of the Period
Cash Flow Today
Strategy
Borrow
Buy the index for
Sell One futures contract
Net Position
If we have annualized and continuing compounded dividend and interest:

14. 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?
Assume that the futures contract is traded at $347, show arbitrage strategy!

15. Cash Flow at the End of the Period
Cash Flow Today
Strategy
Borrow
Buy the index for
Sell One futures contract
Net Position

16. 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.

17. If we have annualized and continuing compounded interest:
Cash Flow at the End of the Period
Cash Flow Today
Strategy
Borrow 1 £
Lend Dollars in the US
Buy futures position to buy
Net Position
If we have annualized and continuing compounded interest:

18. 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:

19. Thus, the current future price is lower than the equilibrium price.
Cash Flow at the End of the Period
Cash Flow Today
Strategy
Borrow 1 £
Lend $ Dollars in US
Buy futures position to buy
Net Position

20. 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?

22. 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.
Cash Flow at the End of the Period
Cash Flow Today
Strategy
Buy the asset at price
Borrow
Sell a futures contract on the asset
Net Position

23. 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?

24. 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)

25. 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

26. $800K
$700K
$900K
Fixed rate payments
Floating rate payments
LIBOR % % %

27. The Yield and the Forward Curve
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
Yield Curve (%) yt
Forward curve (%)
Ft-1,t
year 7 1 8 9 2
11.03 3

28. F* – Fixed rate
yt, is the appropriate yield from the yield curve for discounting dollars cash flows.

29. 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.

30. 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
Floating (%)
Fixed (%)
Firm
LIBOR+0.3 10 A
LIBOR+1
11.2 B

31. 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) % (for B) = LIBOR %
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%.

32. 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.
Floating
Fixed
Operation
Pay 10%
Issue debt
Pay LIBOR+0.05%
Receive 10%
Enter swap
Net
Pay LIBOR+0.3%
Direct cost
0.25%
Saving

33. 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%
Fixed
Floating
Operation
Pay 10%
Pay LIBOR+1%
Issue debt
Receive LIBOR+0.05%
Enter swap
Pay 10.95%
Net
Pay 11.2%
Direct cost
0.25%
Saving

34. 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.

35. 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
UK (%)
US (%)
year 4 2 1 3
3.5

36. The Forward rates:
Therefore,
£100M
$176M
$176.6M
$177.4M

37. Alternatively, we can calculate a constant rate of F
Alternatively, we can calculate a constant rate of F* dollars per pound to be exchanged each year:
where y1, y2 and y3 are the appropriate yields from the yield curve for discounting dollars cash flows.

38. In this case the swap agreement will be:

39. Buyer Periodic Payment Seller
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.

40. 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).