Tn 2 Futures Forwards Swaps 2011

8/2/2019 Tn 2 Futures Forwards Swaps 2011

1/74

Futures, Forwards, and Swaps

Hui Chen

MIT Sloan

15.437, Fall 2011

c Hui Chen (MIT Sloan) Futures, Forwards, and Swaps 15.437, TN 2 1 / 74


2/74

Forward Contracts

Outline

1 Forward ContractsThe Forward PriceThe Value of a Forward Contract

2 Futures ContractsHedging with FuturesThe Futures Price

3 Swaps

Swaps versus ForwardsThe Swap RateOther Swaps



3/74

Forward Contracts

No Arbitrage

Arbitrage Opportunity

An arbitrage opportunity is a trading strategy that either

1 Costs nothing today and yields a positive profit in the future; or

2 Yields a positive profit today, and zero cash in the future.

In well functioning markets, no arbitrage opportunities may arise

If there were, arbitrageurs would take massive positions to profit from them,equilibrating the market

The value of derivative securities, including forwards, futures, swaps andoptions, are determined by assuming that no arbitrage opportunities exist

No arbitrage = No market frictions



4/74

Forward Contracts

No Arbitrage

The Law of One Price

Securities with identical payoffs must have the same price.

Otherwise, an arbitrage opportunity arises

Buy Low / Sell High

Buy the security with low price and sell the one with high price

At maturity the arbitrageur is hedged, as the cash flows from the twopositions are the same and exactly offset each other

Q: Is the payoff of a security the same as its cash flows?



5/74

Forward Contracts

Forward Contracts

Forward contract

A Forward contract is an agreement between two counterparties to buy or sella prespecified amount of goods or securities at a prespecified future date Tfor a prespecified price F.

It does not cost anything to enter into a forward contract at time 0: thecontract is an agreement to exchange goods (or securities) for money inthe future, and not today

The prespecified price F is set to ensure that the value of the forwardcontract is zero for both counterparties at the inception of the contract


F d C t t


6/74

Forward Contracts

Forward Contracts

Let Mt be the price of one unit of the good or security at t, and N be thesize of the contract (# of units)

The Profit/Loss (P/L) at the maturity T are

P/L counterparty long the forward = N (MT F)

P/L counterparty short the forward = N (FMT)

For instance, the party long the forward agrees to buy at T a security for

F when its value is MTIf MT > F, it makes a profit (it pays only F instead of MT)If MT < F, it makes a loss (it pays F instead of the cheaper MT)


Forward Contracts


7/74

Forward Contracts

Example

A US firm has sold a piece of equipment to a German client and now ithas a receivable of EUR 5 million in T = 6 months

Let Mt = USD/EUR exchange rate at t. Assume the current rateM0 = 1.2673, the (c.c.) US interest rate is r$ = 5% and the Euro interestrate is re = 3%

Dollar payoff at T = 5 mil MT

Exchange rate risk: Euro can depreciate versus the dollar (MT decline)


Forward Contracts


8/74

Forward Contracts

Example (cont.)

Hedging strategy: enter into a forward contract with a bank to exchange

euros for dollars at T = 6 months at an exchange rate F, say F = 1.28,decided today

The firm is short the Euro forward

Dollar P/L of forward contract at T = 5 mil (FMT)

Total payoff at T

= payoff from original position T + payoff of forward contract at T

= (NMT) + N (FMT)

= 5 mil 1.28 = $6.4 mil

The hedge helps lock in the dollar payoff at T to be N F, regardless ofthe exchange rate movement


Forward Contracts


9/74

Forward Contracts

Forward Contracts: The Payoff Diagram

Euro

PayoffOriginal Exposure

Euro

Payoff Forward Position

Hedged Position

Euro


Forward Contracts The Forward Price


10/74


The Forward Price

How is the prespecified exchange rate F = 1.28 USD/EUR decided?

We will examine this question in 3 different ways:

1 How much should the bank charge to be able to break even?

2 What if the forward exchange rate in the market is higher (lower) thanF = 1.28?

3 An arbitrage argument based on the covered interest rate parity




11/74

1. A Synthetic Forward

The bank can hedge its commitment to pay dollars for euros at the fixed rate F usingthe following steps:

At time 0:

1 Borrow the present value of 5 mil euros at the rate re = 3% for T = 6 mthsBorrow

ereT 5 mil

= e4.925 mil

2 Exchange this amount into dollars todayIf M0 = 1.2673 = the bank gets: e4.925milM0 = $6.242mil

3

Invest this amount in dollar deposit at rate r$ = 5% for T = 6 mths= Today the bank nets 0

At time T:1 Pay back the 5 mil euro loan2 Proceeds from the dollar investment: $6.242 mil er$T = $6.4 mil

Effective exchange rate: F =

$6.4 mil

e5 mil = 1.28 $/e

Consolidate with the forward contract payoff:

Receive 5 mil euros from the US firm the exact amount needed to payback the euro loanGive the $6.4 mil proceeds from dollar deposit to the US firm

The bank has effectively created a short position in the synthetic forward!c Hui Chen (MIT Sloan) Futures, Forwards, and Swaps 15.437, TN 2 11 / 74



12/74

Diagram of Banks Hedging Strategy

0 = today T = 6 month

1. Firms and Bank agree

Bank will buy 5 mil euro (forward)at T for F=1.28 $/Euro

2. (a) Bank borrows Euros4.925 m = PV(5 m) at r

e=3%

(b) Exchange them into dollars at

M0 = 1.2673 $/Euro(c) Invest proceeds of $ 6.242 mil

at r$

= 5%

1. Firms gives bank 5 mil Euros

2. (a) Bank uses Firms payment toclose 5 mil Euro loan

(b) Bank liquidates $ investment

$6.242 e5%

= $6.4 mil

(c) Gives the proceeds to the firm

Exchange Rate for firm = $6.4mil / Eur 5 mil

= 1.28 $/Eur= quoted F at 0




13/74

Forward Price

We can now generalize the forward exchange rate formula using thepayoff of the synthetic forward:

Payoff at T =

ereT 5 milM0

er$T

= M0 e(r$re)T 5 mil

= Effective exchange rate the bank can promise is

F = M0 e(r$re)T

F = Forward rate.Since F is determined at time 0 for an exchange of euros for dollars at T, wedenote it as

F0,T




14/74

2. What if the markets are not in line?Suppose: F0,T < M0 e

(r$re)T

Arbitrageur: Buy low Sell high

But what is low, and what is high? (actual vs. synthetic forward)

At time 0:

1 Enter forward contract with maturity T to buy N euros at forward rate F0,T

2 Borrow N ereT euros; exchange them into N ereT M0 dollars; andinvest N ereT M0 dollars at the dollar rate r$

= At t = 0 arbitrageur evens out

At time T:

1 Receive N euros from forward contract, and pay $F0,T N for them2 Pay back the euro loan of N euros (N ereT accrued with interest)

3 Receive proceeds

N ereT M0 er$T from dollar investment

= Net Dollar Profit at T = N M0 e(r$re)T F0,T > 0




15/74

The reverse case

What if F0,T > M0 e(r$re)T? What is low and what is high now?

At time 0:

1 Enter forward contract with maturity T to sell N euros at forward rate F0,T

2 Borrow N ereT M0 dollars; exchange them into N ereT euros; and

invest N ereT euros at euro rate re until T

= At t = 0 arbitrageur evens outAt time T:

1 Receive $F0,T N from forward sale of euros in exchange of N euros

2 Position is covered by the proceeds from the euro investment, as N ereT eurosat 0 grow up to N by T

3 Pay back dollar loan of N ereT M0 er$T= Dollar Profit at T = N

F0,T M0 e

(r$re)T> 0

Q: How large should N be?




16/74

3. Yet another arbitrage argument

Consider a US investor who has $100 mil to invest for T = 6 months

There are two strategies:

A. Invest in 6-months U.S. Treasury bills at the rate r$ = 5% yielding:

Payoff of strategy A at T = $100mil er$T

B. (a) Exchange the $100 mil into euros at the current rate 1/M0 =EUR/USD(b) Invest the proceeds in 6-month Euro Treasuries(c) Enter into a forward contract today to sell 100

M0 ereT mil euros at F0,T

Payoff of strategy B at T = $100milM0

ereT F0,T

What are the differences between the two strategies?




17/74

Deriving the forward price one more time (cont.)

Both strategies final payoffs are known in certainty at time 0. Thus, wemust have

Payoff of strategy A at T = Payoff of strategy B at T

or, substituting

$100mil er$T =$100mil

M0 ereT F0,T

Solving for F0,T gives

F0,T = M0 e(r$re)T




18/74

Actual Data: April 4 2007 (Financial Times)

DOLLAR SPOT FORWARD AGAINST THE DOLLAR

Closing Change Bid/offer Days mid One month Three months One year J.P. Morgan

mid-point on day spread high low Rate %PA Rate %PA Rate %PA Index

EuropeCzech Rep. (Koruna) 20.8984 -0.0132 844 - 123 21.0040 20.8844 20.8478 2.9 20.7499 2.8 20.4229 2.3 -Denmark (DKr) 5.5734 0.0004 726 - 741 5.5944 5.5708 5.5673 1.3 5.5562 1.2 5.5265 0.8 110.00Hungary (Forint) 183.947 0.3426 867 - 028 184.720 183.590 184.357 -2.7 185.117 -2.5 188.057 -2.2 -Norway (NKr) 6.1194 0.0046 172 - 215 6.1372 6.1120 6.1138 1.1 6.1049 0.9 6.1006 0.3 112.60Poland (Zloty) 2.8765 -0.0009 746 - 783 2.8878 2.8746 2.8734 1.3 2.8681 1.2 2.8594 0.6 -Russia (Rouble) 25.9694 -0.0042 669 - 719 26.0270 25.9620 25.9662 0.2 25.9603 0.1 25.9865 -0.1 -Slovakia (Koruna) 25.0013 0.2019 761 - 265 25.2560 24.8350 24.9695 1.5 24.8948 1.7 24.6513 1.4 -Sweden (SKr) 6.9687 -0.0387 667 - 706 7.0301 6.9603 6.9570 2.0 6.9347 1.9 6.8702 1.4 101.80Switzerland (SFr) 1.2181 0.0002 178 - 184 1.2232 1.2174 1.2150 3.1 1.2089 3.0 1.1870 2.6 107.10Turkey (Lira) 1.3730 - 715 - 745 1.3790 1.3680 1.3889 -13.9 1.4182 -13.2 1.5600 -13.6 -UK (0.5058)* () 1.9772 -0.0014 770 - 774 1.9775 1.9717 1.9770 0.1 1.9761 0.2 1.9658 0.6 103.50Euro (0.7477)* (Euro) 1.3375 0.0002 373 - 376 1.3380 1.3319 1.3391 -1.4 1.3421 -1.4 1.3506 -1.0 128.80SDR - 0.6605 -Americas

Argentina (Peso) 3.1013 - 000 - 025 3.1025 3.1000 3.0978 1.4 3.1023 -0.1 3.1663 -2.1 -Brazil (R$) 2.0365 -0.0021 355 - 375 2.0400 2.0310 2.0460 -5.6 2.0610 -4.8 2.1465 -5.4 -Canada (C$) 1.1575 0.0009 572 - 577 1.1593 1.1544 1.1564 1.1 1.1544 1.1 1.1473 0.9 123.90Mexico (New Peso) 10.9998 0.0160 975 - 020 11.0025 10.9714 11.0160 -1.8 11.0520 -1.9 11.2448 -2.2 81.90Peru (New Sol) 3.1815 0.0010 810 - 820 3.1830 3.1800 3.1795 0.8 3.1765 0.6 3.1795 0.1 -USA ($) - - - - - - - - - - - 88.10Pacific/Middle East/Africa Australia (A$) 1.2231 -0.0056 228 - 234 1.2398 1.2204 - - - - - - 125.40Hong Kong (HK$) 7.8161 0.0009 159 - 163 7.8171 7.8147 7.8097 1.0 7.7929 1.2 7.7434 0.9 92.70India (Rs) 43.0700 0.0050 600 - 800 43.1350 42.8450 43.3225 -7.0 43.7050 -5.9 44.8423 -4.1 -Indonesia (Rupiah) 9116.00 21.0000 200 - 000 9121.00 9098.00 9116.00 - 9116.00 - 9116.00 - -Iran (Rial) 9244.00 -1.0000 800 - 000 -Israel (Shk) 4.1310 -0.0290 280 - 340 4.1580 4.1240 4.1275 1.0 4.1201 1.1 4.0968 0.8 -Japan (Y) 118.615 -0.1700 600 - 630 119.080 118.550 118.139 4.8 117.231 4.7 113.528 4.3 80.20Kuwait (Dinar) 0.2893 -0.0001 892 - 893 0.2894 0.2892 0.2892 0.1 0.2893 - 0.2894 -0.1 -Malaysia (M$) 3.4570 -0.0035 545 - 595 3.4600 3.4545 3.4516 1.9 3.4415 1.8 3.3985 1.7 -New Zealand (NZ$) 1.3865 -0.0013 860 - 870 1.3953 1.3841 - - - - - - 137.20Philippines (Peso) 48.0500 -0.0750 400 - 600 48.2000 47.8700 48.0492 - 48.0563 -0.1 48.0615 - -Saudi Arabia (SR) 3.7504 - 499 - 509 3.7509 3.7499 3.7489 0.5 3.7472 0.3 3.7435 0.2 -Singapore (S$) 1.5150 -0.0038 147 - 153 1.5205 1.5137 1.5119 2.5 1.5062 2.3 1.4824 2.2 105.70South Africa (R) 7.1718 -0.0168 667 - 768 7.1962 7.1460 7.1942 -3.8 7.2421 -3.9 7.4743 -4.2 -South Korea (Won) 936.500 -0.2500 000 - 000 937.000 935.500 935.850 0.8 934.600 0.8 930.300 0.7 113.00Taiwan (T$) 33.0900 -0.0310 850 - 950 33.1360 33.0640 33.0550 1.3 32.9300 1.9 32.3500 2.2 87.20Thailand (Bt) 34.9450 -0.0200 100 - 800 34.9800 31.9000 34.9215 0.8 34.8600 1.0 34.5550 1.1 -U A E (Dirham) 3.6723 0.0001 721 - 724 3.6724 3.6721 3.6715 0.2 3.6700 0.2 3.6688 0.1 -

*The closing mid-point rates for the Euro and are shown in brackets. The other figures in both rows are in the reciprocal form in line with market convention. Official rate set byMalaysian government. The WM/Reuters rate for the valuation of capital assets is 3.80 MYR/USD. Bid/offer spreads in the Dollar Spot table show only the last three decimal places.J.P. Morgan nominal indices: Base average 2000 = 100. Bid, offer, mid spot rates and forward rates in both this and tha pound table are derived from the WM/REUTERS 4pm (Londontime) CLOSING SPOT and FORWARD RATE services. Some values are rounded by the F.T.

Apr 4




19/74

Actual Data: April 4 2007 (cont.)

The formula for forward price: F0,T = M0 e(r$re)T. Does it work?

Interest rates on April 4 2007 were in the following table

MARKET RATES

*Libor rates come from BBA (see www.bba.org.uk) and are fied at 11am UK time. Other data sources:

US $, Euro & Cds: dealers; SDR int rate: IMF; EONIA: ECB; EURONIA & SONIA: WMBA.

Over Change One Three Si One

night Day Week Month month months months year

US$ Libor* 5.32125 +0.015 +0.010 +0.023 5.32000 5.35000 5.33563 5.23656Euro Libor* 3.85250 +0.003 +0.028 +0.284 3.86538 3.94300 4.06863 4.20350

Libor* 5.30625 -0.011 -0.033 +0.050 5.51000 5.63688 5.74625 5.88875Swiss Fr Libor* 2.15000 +0.010 -0.023 +0.130 2.20917 2.30000 2.41417 2.57000 Yen Libor* 0.62375 -0.010 -0.156 +0.045 0.64063 0.66063 0.69625 0.78375Canada Libor* 4.25000 -0.002 - - 4.26000 4.27583 4.29000 4.32000Euro Euribor - - - - 3.86 3.94 4.07 4.20Sterling CDs - - - - 5.46 5.59 5.69 5.83US$ CDs - - - - 5.27 5.28 5.28 5.18Euro CDs - - - - 3.845 3.920 4.035 4.150US o`night repo 5.25 -0.010 +0.005 +0.020Fed Funds eff 5.20 -0.050 -0.050 -0.020US 3m Bills 4.91 -0.015 -0.020 -0.060SDR int rate 4.09 - - -EONIA 3.84 - +0.020 +0.290EURONIA 3.8369 +0.0047 +0.0240 +0.2849SONIA 5.2673 - -0.0296 +0.0404

Apr 4

Over One One Three Si Onenight Week month months months year

Interbank 5.30-5.22 5.44-5.35 5.51-5.43 5.64-5.56 5.74-5.66 5.89-5.81

source: Financial Times www.ft.comc Hui Chen (MIT Sloan) Futures, Forwards, and Swaps 15.437, TN 2 19 / 74



20/74

Actual Data: April 4 2007 (cont.)

1 month US and EURO Libor rates: R$ = 5.32%, Re = 3.86538%

The day-count convention for Libor is Actual/360The continuous compounded interest rate arer$ = log(1 + R$ 30/360) 12 = 5.3082%re = log(1 + Re 30/360) 12 = 3.8592%

Since M0 = 1.3375, we obtain that for T = 1/12

F0,T = M0 e(r$re)1/12 = 1.3391

which coincides with market forward exchange rate

The same steps for T = 3/12 yield r$ = 5.3145% and re = 3.9237%, and

thusF0,T = M0 e

(r$re)3/12 = 1.3422

which (almost) coincides with the market forward exchange rate

Q: Why are we using data from 2007?




21/74

The Determinant of the Forward Price

The forward price only depends on the interest rate differential of the twocountries, i.e., (r$ re).

If r$ > re, then F0,T > M0 and if r$ < re, then F0,T < M0.

Forecast of future exchange rates

There is no more information in the forward price about future exchange ratesthan there is in the interest rate differential.

12/30/1998 12/30/1999 12/29/2000 12/31/2001 12/31/2002 12/31/2003 12/30/2004 12/30/2005 1/1/2007 1/1/2008

3

2

1

0

1

2

3

Percent

Forward Spread: (Ft,T

Mt)

Interest Rate Differential: (r$r

e)


Forward Contracts The Value of a Forward Contract


22/74

The Value of a Forward Contract

The Forward contract costs nothing to enter. So its initial value is 0. Atmaturity T, its value is equal to the payoff (e.g., FMT for a shortposition). But what is its value in between 0 and T?

Lets go back to the initial example:Recall that a US company was hedging a 5 million Euro receivable through aforward contract with F0,T = 1.28.Assume that at time t = 3 months the Euro appreciated from M0 = 1.2673 toMt = 1.29If the US firm wants to cancel the contract with the bank, how much does ithave to pay?

The US firm can enter into the reverse forward contract with the bank

Dollar payoff at T of reverse forward contract = 5 mil (MT Ft,T)

Now (at t = 3 mths):Ft,T = Mt e

(r$re)(Tt) = 1.29 e(0.050.03)0.25 = 1.296 $/e

We now see that the reverse contract neutralizes the former one




23/74

The Value of a Forward Contract (cont.)

Payoff at T from forward + reverse forward = 5m (F0,

T MT) + 5m (MT Ft,

T)= 5m (F0,T Ft,T)

= 5m (1.28 1.296) = $80, 000

= The US firm will have to pay the bank at T exactly $80, 000.

The Present Value of $80, 000 is the value of the original forward contract to theUS firm:

ft,T = er$(Tt) (F0,T Ft,T) 5m = $79, 006.2

Since it costs $79,006.2 to close the position, the value of the forward contract tothe firm must equal this amount. Viceversa, the value to the bank must be

$79,006.2

The time-t value of a forward contract to sellone euro at a prespecified price K at timeT is always given by

ft,T = er$(Tt) (K Ft,T)




24/74

The Value of a Forward Contract (cont.)

0 = today T = 6 month

M0

= 1.263 USD/EUR

F0,T

= 1.28 USD/EUR

Firm enters into a reverseforward to buy 5 mil eurosat new forward rateF

t,T

=1.296

t = 3 month

Mt= 1.29 USD/EUR

Ft,T

= 1.296 USD/EUR

Firm enters into a forwardto sell 5 mil euros at

F0,T=1.28

P/L = 5 mil (1.28 MT

)

P/L = 5 mil (MT

1.296)

Total = 5 mil (1.28 1.296)= $80,000

Value at t = PV(80,000)= 79,006.2




25/74

Forward Price and the Value of a Forward Contract

To summarize, we have found:

1 The Forward (Delivery) Price the price decided at time 0 to buy / sellgoods (Euros) in the future is given by

F0,T = M0 e(r$re)T

2 The value of an existing forward contract to deliver goods (Euros) atpredetermined price K (set at some time in the past) is equal to the profit(cost) of closing the contract:

ft,T

= [K Ft,T

] er$(Tt)

Q: This is the value of a forward contract to sell. What is the value of aforward contract to buy? (i.e. a long forward contract)




26/74

Forward Price and the Value of a Forward Contract

Important Reminder

The value of a forward contract at inception t = 0 is zero.

The Forward delivery price K = F0,T, which makes f0,T = 0.

No exchange of money when two counterparties enter into a forwardcontract

In fact, they agree to exchange money only in the future, depending onMT



F d C f h i i


27/74

Forward Contracts for other securities

The earlier derivation was obtained for currency forwardsSimilar derivations hold for other securities, such as stocks, bonds, and

commoditiesHere is a list of formulas. A good exercise is to go over the steps again

Security Forward Price

Currency (e.g. dollar vs euro) F0,T = M0 e(r$re)T

Stock: no dividend F0,

T = S0

erT

Stock: dividend yield q F0,T = S0 e(rq)T

Stock: known Dividend D at T1 < T F0,T =

S0 D erT1

erT

Commodity: Storage cost U. No convenienc yield F0,T = (S0 + PV(U)) erT

Commodity: % storage cost u. No convenienc yield F0,T = S0 e(r+u)T

Commodity: % storage cost u, convenient yield y F0,T = S0 e(r+uy)T

Bond: Zero Coupon with Maturity T

> T F0,

T =

Z(0,T)

Z(0,

T)Bond: with Semi-annual Coupon c, Maturity TnPayment dates T1, ...,Tn and with Tm = T

F0,T =

ni=m+1

c2

Z(0,Ti)Z(0,T)

+ Z(0,Tn)Z(0,T)

In all cases, the value of a forward contract (to sell) at time t> 0 (afterinitiation) is

ft,T = er(Tt)

[F0,T Ft,T]c Hui Chen (MIT Sloan) Futures, Forwards, and Swaps 15.437, TN 2 27 / 74



28/74

The Forward Price of a Stock with KnownDividend Payment

Consider the a stock with price S0 which pays a known dividend at timeT

1< T.

The pricing formula F0,T = (S0 PV(D)) erT

Arbitrage argument. What if

F0,T > (S0 PV(D)) erT?

Arbitrageur:At time 0:

(a) Short forward F0,T;(b) borrow $(S0 PV(D)) with maturity T and $PV(D) with maturity T1 ;(c) use total $S0 to buy stock.

At time T1Receive dividend D from stock. Use it to replay PV(D) loan at T1.

At time T:

(a) Receive F0,T from sale of stock (which is covered, because of (c) above);

(b) repay the loan (S0 PV(D))erT

Payoff at T = F0,T (S0 PV(D))erT > 0



S C


29/74

The Forward Price of a Stock with Constant Dividend Yield

Dividend Yield = Stocks payoff per unit of stock price = D/St

Continuously compounded dividend yield q= Total dividend in a small interval [t, t + dt] is Dt = q St dt

Forward price: F0,T = S0 e(rq)T

What if F0,T > S0 e(rq)T?

1 Short Forward at F0,T2 Borrow S0 e

qT and buy N0 = eqT < 1 shares

3 For every t reinvest the dividends in the stock

Change in number of shares in a small interval dt:Nt+dt Nt =

NtDtSt

= Nt q dt

Total number of shares between 0 and T: NT = N0 eqT = eqT eqT = 1

The arbitrageur has exactly the right amount of shares to cover the shortforward position

Payoff at T = F0,T S0 e(rq)T > 0



Th F d P i f C diti


30/74

The Forward Price of Commodities

Can we apply the same formula for commodities?

What if the commodity is held by economic agents for consumption

purposes as well?Stocks, bonds and currencies are held for investment reasonsMost commodities are held by economic agents for an economic purpose,such as heating houses, or feeding hungry peopleWhat may go wrong in the no arbitrage strategy?



C d I t t R t P it Vi l ti d i th 2007 2009


31/74

Covered Interest Rate Parity Violation during the 2007 - 2009

Financial Crisis

Sometimes, arbitrageurs fail to keep markets together

The 2007 - 2009 provides an interesting example

Define the discrepancy between market and theoretical forward rate as:

Basis = Traded Forward Rate Theoretical Forward Rate= Ftradedt,t+T Ft,t+T

where T = maturity (e.g. 3 month, 6 months etc.), and recall

Ft,t+T = Mt e(r$re) T

The latter relation is also called Covered Interest Rate Parity (CIP)

If Basis above is not close to zero, we say that there is a violation of thecovered interest rate parity



CIP Vi l ti d i th 2007 2009 Fi i l C i i E /D ll


32/74

CIP Violation during the 2007 - 2009 Financial Crisis: Euro/Dollar

6/1/2007 10/2/2007 1/31/2008 6/2/2008 10/1/2008 1/30/2009 6/2/2009 10/1/2009 2/1/2010 6/2/20101

0.5

0

0.5

1

1.5

2

Percentag

e

Panel B. Financial Crisis

1 year Fwd

6 month Fwd

12/30/199812/30/199912/29/200012/31/200112/31/200212/31/200312/30/200412/30/2005 1/1/20071

0.5

0

0.5

1

1.5

2

Percentage

Panel A. PreCrisis

1 year Fwd

6 month Fwd



CIP Violation during the 2007 2009 Financial Crisis: UKP/Dollar


33/74

CIP Violation during the 2007 - 2009 Financial Crisis: UKP/Dollar

6/1/2007 10/2/2007 1/31/2008 6/2/2008 10 /1/2008 1/30/2009 6/2/2009 10/1/2009 2/1/2010 6/2/20103

2

1

0

1

2

3

4

Percentag

e

Panel B. Financial Crisis

1 year Fwd

6 month Fwd

6/15/1990 6/16/1992 6/16/1994 6/17/1996 6/17/1998 6/16/2000 6/18/2002 6/17/2004 6/19/20064

2

0

2

4

Percentage

Panel A. PreCrisis

1 year Fwd

6 month Fwd



Some Practical Considerations Before You Take on the


34/74

Some Practical Considerations Before You Take on the

Arbitrage ...

Bid-ask spreads eat into arbitrage profits

At t, suppose we have Ftradedt,t+T > Ft,t+T1 Enter forward contract with maturity T to sell N euros at forward rate Ftraded

t,T

2 Borrow N ere(Tt) M0 dollars at rate r$ and use it to buy N ere(Tt)

euros3 Invest N ere(Tt) euros at euro rate re until T

= At t arbitrageur evens out

At T ...Bid or ask? Ftraded,bidt,T < F

traded,askt,T , M

bid0 < M

ask0 , r

lende < r

borrowe , r

lend$ < r

borrow$

Mark-to-market

In reality, if spot prices move against you, you will be required to post morecollateral (to limit default risk) margin callsWhat if you dont have any capital left?



Why may arbitrage fail?


35/74


1. Lack of borrowing capacity

Most arbitrage trades require long-short strategies and leverage to increasethe return on capital.If borrowing is difficult due to market disruptions or increase in risk aversionof lenders (e.g. banks), it may not be worthwhile to engage in the arbitrage

strategy.

2. Funding risk

An arbitrage strategy must be held until maturity T to payoff. Thus, it mayimply large outflows if spreads increase before T. Lenders may ask for morecollateral (margin calls) while holding the position.If an arbitrageur thinks it cannot hold the position until maturity, it wont takethe trade.





36/74


3. Lack of capital or fear of withdrawals

Many hedge funds experience large outflows if trading position lose money.

A possible increase in spreads generates mark-to-market losses, which can

require the fund to close the position before arbitrage pays off.

Snowball effect: As some funds must close the positions, the spread increases,generating mark-to-market losses to other funds.

Lock-up provisions mitigate the problem, but competition across hedge fundsmay make them hard to implement



Why Did CIP Fail during the Crisis?


37/74

Why Did CIP Fail during the Crisis?

Holding US Treasuries has its own convenience yield when everyoneneeds cash collateral

During the crisis, from the graph, we had the basis being positive. That is:

Ftradedt,t+m > Ft,t+m = Mt e(r$re)m

Recall that in this case, an arbitrage trade requires the following:(a) short forward;(b) borrow dollars (or sell US Treasuries);(c) change them into Euro;(d) invest in Euro (or buy Euro bonds)

But point (b) failed during the crisis, as:1 Market-wide increase in credit risk concerns impaired the ability of any

financial institution to borrow;2 Holding safe dollars (US Treasuries) is valuable during a financial crisis for

liquidity managementUS Treasuries are the only collateral accepted for short-term lendingtransactions.It is very valuable to hold on to them for future cash management



Summary


38/74

Summary

What is a forward contract?

Understand the different arguments for forward prices

Synthetic forwardCovered interest rate parity

Value of a forward contract (not always 0)

Forward price for stocks, bonds, commodity


Futures Contracts

Outline


39/74

Outline


2 Futures Contracts

Hedging with FuturesThe Futures Price

3 SwapsSwaps versus ForwardsThe Swap RateOther Swaps


Futures Contracts

Futures Contracts


40/74

Futures Contracts

Futures contracts are similar to forward contracts, as they are agreementsbetween two counterparties to exchange a pre-specified amount of good(e.g. corn or Euros) at a prespecified time for a predetermined price

The are three notable differences between forwards and futures

1 The contracts are traded on an exchange (e.g. CME)2 The contracts are standardized (size, maturity)

3 Profits and losses are marked to market (daily settlement)

Standardization is important for improving liquidity

but it introduces basis risk: mismatch between futures contract size andmaturity versus actual needs

Mark-to-market is important to decrease default risk

At the end of each trading day, profits and losses accrue to the account ofthe traderIf trading losses become too large, exchange issues a margin call: the tradermust post additional collateral or the position is immediately closed


Futures Contracts

Futures Contracts


41/74

Futures Contracts

Futures contracts are available on numerous goods and financialsecurities.

Commodities, such as

Corn, Wheat, Soybean, etc.Cocoa, Coffee, Orange Juice,Gold, Silver etc.Crude Oil, Natural Gas etc.

Currencies, such as

USD/EUR, USD/UKP, USD/Australian Dollar, Euro/Australian Dollar etc.

Equities, such as

S&P 500, NASDAQ 100, RUSSELL 2000, S&P 500/Citigroup Value, etc.

Interest rates, such as

Eurodollar, 30 year T-Bond, 10-year T-Note etc.

Weather, such asHeating Degree in Atlanta, Chicago, New York, Las Vegas etc.

Energy, such as Electricity etc.

For more information, go to CME website: http://www.cmegroup.com/


Futures Contracts

Futures Contracts


42/74

utu es Co t acts

Futures price Ft,T = delivery price at which trader who is short the futuresagrees at t to sell the underlying commodity or securities at T to the

trader who is long the futures

As futures price moves, profits and losses accrue to both counterpartiesThe party Long the futures who agrees to buy the good at maturity gainswhen price increases, and his/her daily P/L is given by

Daily P/L = Contract Size Ft,T Ft1,TThe party Short the futures who agrees to sell the good at maturity gainswhen price decreases, and the P/L is the opposite

Settlement and Delivery:

Physical vs cash settlementFutures are referred to by their delivery months, specified by the exchangeOnly during these months delivery of the good can occur (typically, the exactperiod within the month is specified as well)

In this class we typically assume delivery occurs on a specific date, for simplicity

Delivery must occur on a given location, also specified by the contract


Futures Contracts

Futures Contracts


43/74

Convergence Property: At maturity T we must haveF

T,T= M

T.

Otherwise, there is an immediate arbitrage at TIf FT,T > MT = (a) short the futures; (b) buy underlying; (c) deliver.If FT,T < MT = (a) long the futures; (b) sell underlying (if you have it);(c) buy it back from futures.

Margin LimitsBoth counterparties must set up an amount of collateral with the exchange,called initial marginAs P/L accrue on the account, the margin may be eroded over time

If the amount falls below a maintenance margin, the exchange issuesa margin call

The trader must post additional collateral up to the initial margin to keep theposition open, otherwise it is closedFor instance, on CME Euro FX Futures: Initial Margin = $2,995; MaintenanceMargin = $1,700.


Futures Contracts

Mark to Market: Example


44/74

p

On December 12, you bought 10 contracts of December 15 gold futures onthe CME, at a price of $500/oz

Contract size 100 troy ounces. Agreed to buy 1,000 ounces of gold (10

contracts) on December 15

No money changed hands initially

As of August 24, 2011, the benchmark 100-troy-ounce gold contract hasinitial margin of $9,450 and maintenance margin of $7,000


Futures Contracts

Mark to Market: Example (cont.)


45/74

p ( )


Futures Contracts

Futures Contracts: 09/06/2011


46/74

Open Sett Change High Low Est.vol OpenintSep6

Sources:*NYBOT;Sterling100,000andYen:100,000.CME:Australian$:A$100,000,Canadian$:C$100,000,Euro:125,000;MexicanPeso:500,000,SwissFranc:SFr125,000;Yen:12,5m($per100);Sterling:62,500.CMEvolume,high&lowforpit&electronictradingatsettlement.Contract sshownarebasedonthevolumestradedin2004.

-Sterling* Sep 0.8805 0.8763 - 0.8812 0.8757 29 6,330

-Yen* Sep - 108.4500 - - - - 6,006

$-Can$ Sep 1.0138 1.0097 -0.0057 1.0145 1.0030 137,874 90,814

$-Euro Sep 1.4162 1.3986 -0.0198 1.4283 1.3969 579,419 177,942

$-Euro Dec 1.4152 1.3980 -0.0191 1.4277 1.3962 14,775 8,247

$-SwFranc Sep 1.2685 1.1612 -0.1078 1.2792 1.1595 77,291 41,895$-Yen Sep 1.3012 1.2882 -0.0150 1.3041 1.2872 185,633 127,491

$-Yen Dec 1.3025 1.2900 -0.0145 1.3056 1.2891 4,685 2,776

$-Sterling Dec 1.6148 1.5919 -0.0270 1.6188 1.5905 4,121 2,776

$-Aust$ Sep 1.0584 1.0471 -0.0148 1.0611 1.0460 189,371 123,369

$-MexPeso Nov - 79300 -550.00 - - - -

CURRENCY FUTURES

(Source: Financial Times)


Futures Contracts

CME Euro FX Futures


47/74

6/13/2005 9/13/2005 12/14/20053/16/2006 6/16/2006 9/18/2006 12/19/2006 3/21/2007 6/21/2007 9/21/20071.15

1.2

1.25

1.3

1.35

1.4

EURO FX

CME Mar 07 Futures

CME Jun 07 Futures

CME Sep 07 Futures


Futures Contracts

Contango and Backwardation


48/74

For commodity futures, the following jargons are often used:

Contango: Futures prices increase with maturity

Backwardation: Futures prices decrease with maturity

Another definition is one that adjusts for the time-value of money:

Contango: F0,T > S0erT

Backwardation: F0,T < S0erT


Futures Contracts

Contango and Backwardation


49/74


Futures Contracts Hedging with Futures

Example: Hedging with Futures


50/74

Consider earlier example: The US firm could hedge its Euro exposurewith futures contracts instead of forwards

CME Euro FX Futures have size of 125,000 Euro and expire on Mar, Jun,Sep and Dec.

Suppose that T = Mar 2007 and F0,T = 1.28US firm can short k = 5 mil/125,000 = 40 futures contracts

Every day t,(P/L)t = k 125,000 (Ft1,T Ft,T) = 5 mil (Ft1,T Ft,T)At maturity T:

Payoff at T = (P/L)1 + (P/L)2 + + (P/L)T

= 5m (F0,T F1,T) + 5m (F1,T F2,T) + + 5m (FT1,T FT,T)= 5m (F0,T FT,T)= 5m (F0,T MT)

Q: Are we forgetting anything?c Hui Chen (MIT Sloan) Futures, Forwards, and Swaps 15.437, TN 2 50 / 74


Hedging with Futures


51/74

Caveat: The total payoff from using a constant number of contracts kevery period is actually not exactly equivalent to the one of forwardcontract

Because of mark-to-market, trading profits and losses accrue over timetothe hedger

The correct statement of the payoff at T is in fact:

Payoff at T = (P/L)1 er$(Tdt) + (P/L)2 e

r$(T2dt) +

+ (P/L)T1 er$dt + (P/L)T

where dt = 1/365 = 1 day (in annual units)



Hedging with Futures (cont.)


52/74

To obtain the forward contracts payoff, we must tailor the hedge andchoose carefully the number of contracts kt per period as follows (recall

5 mil125,000 = 40):

k0 = 40 er$(Tdt)

k1 = 40 er$(T2 dt)

ki = 40 er$(Ti dt)

kT1 = 40

which yields the payoff sequence

Payoff at T = 5m er$(Tdt) (F0,T F1,T) e

r$(Tdt)+

+

5m er$(T2dt) (F1,T F2,T) er$(T2dt) + + 5m (FT1,T FT,T)= 5m (F0,T F1,T) + 5m (F1,T F2,T) + + 5m (FT1,T FT,T)= 5 mil (

F0,T MT)



Speculating with Futures


53/74

Futures are also excellent vehicles to speculate on a view about thevariation in the underlying

Example: Let today be Friday Jan 1, 2010. The Euro/Dollar exchangerate is M0 = 1.4326

A speculator believes that the Euro/Dollar rate will increase over the

weekendConsider two strategies:

1 Funded Speculative Position:

Buy 125,000 Euros for 125, 000 1.4326 = $179, 075

2 Unfunded Speculative Position through Futures:

Go long 1 Futures contract at the CME. For example, the March 10 Futures priceon Jan 1, 2010 was Ft,T = 1.4334Post initial margin $2,995



Speculating with Futures (cont.)


54/74

On Monday, Jan 4, 2010 (t), the exchange rate was Mt = 1.4413 and the Mar 10Futures price was Ft,T = 1.441

1 Funded Speculative Position: The position is125,000 1.4413 = $180162.5

Profit = $180162.5 $179, 075 = $1087.5

Return on Investment =$1087.5

$179,075= 0.607%

2

Unfunded Speculative Position through Futures:Profit = 125, 000 (1.441 1.4334) = $950

Return on Investment =$950

$2, 995= 31.72%

Futures provide a much higher profit (or loss) as a percentage of investment

The margin can be cash equivalent (e.g. T-bills) which earns interestA futures position is equivalent to a levered investment in underlying security

(a) Borrow $179, 075 = 125,000 1.4326 minus the safety margin (haircut)$2, 995

(b) Invest in Euros (as before).The return on investment is then $1087.5/$2,995=36.3%



55/74

Futures Contracts The Futures Price

Summary


56/74

What are the differences (pros and cons) between futures and forwards?

How does mark-to-market work?Gaining leverage with futures


Swaps

Outline


57/74


2 Futures Contracts

Hedging with FuturesThe Futures Price

3 SwapsSwaps versus Forwards

The Swap RateOther Swaps


Swaps Swaps versus Forwards

Swaps


58/74

Back to the example, assume the US firm is due to receive the 5 mileuros in 5 equal installments, every 6 months

The US company can enter into five forward (or futures) contracts, andhedge each single installment as a stand-alone cash flow

M0 = 1.2673, r$ = 5%, re = 3% imply the following forward rates

Forward Rates

Maturity T 0.5 1 1.5 2 2.5Forward Rate F0,T 1.28 1.2929 1.3059 1.3190 1.3323

For each cash flow, same arguments as above hold



Swaps


59/74

Alternatively, the US firm can also enter into an FX swap:

An FX swap is a contractual agreement between two counterparties toexchange prespecified amounts of money denominated in differentcurrencies.

For instance, the swap contract between the US firm and a bank may bespecified as follows:

1 US firm pays Bank 1 mil euros on the following dates T = .5, 1, ..., 2.5;2 Bank pays US firm 1 mil K (where K is the swap rate, say K = 1.306)

dollars on the same dates.

What is the net cash flow for the firm from the swap at any payment date?

We need to express cash flows in the same currencyEvery T, the firm receives 1 mil K dollars, and must pay 1 mil MT dollars

Net $ CF at T = 1 mil (K MT)



Example: Hedging with Swaps


60/74

US FirmGerman Firm

Bank(Swap Dealer)

N Mt

N

M tK N = (1.306 $/Eur) N

= $ 1.306 Mil

N = Notional = 1 Mil EurosM

t= $/Eur Exchange Rate

t=0.5, 1, 1.5,...,2.5


Swaps The Swap Rate

Swap Rate


61/74

How is the swap rate K = 1.306 determined?

The Swap Rate K is chosen at time 0 so that the value of the swap is equalto zero= no exchange of money at inception but only in the future.

How can we determine the value of a swap?

Lets use the same methodology we used to value forwards, that is, lets findhow much would the firm pay to get out of the position.

No arbitrage condition: Suppose the firm wants to close the swapexposure by using a sequence of forwards

The payoff for every T = 0.5,...,2.5 is

Payoff at T of swap + reverse forward = 1 mil (K MT) + 1 mil (MT F0,

T)= 1 mil (K F0,T)


Swaps The Swap Rate

Swap Rate


62/74

The Present Value of these sequence of net payments is

PV(swap + reverse forwards cash flows) = er$0.5 1 mil (K F0,0.5) +

+ er$1 1 mil (K F0,1) + ...+

+ ... + er$2.5 1 mil (K F0,2.5)

No arbitrage = At time 0, PV(swap + reverse forwards cash flows) = 0Why? K and F0,T are chosen so that it costs nothing to enter into a swap orforward.If PV of these cash flows = 0, infinite profits are available.

We obtain one equation with one unknown K0 = er$0.5 1 mil (K F0,0.5) + e

r$1 1 mil (K F0,1) + ... +

+... + er$2.5 1 mil (K F0,2.5)


Swaps The Swap Rate

Swap Rate


63/74

The solution to the equation is a weighted average of forward prices:

Currency Swap Rate = K = w0.5F0.5 + w1F0,1 + ... + w2.5F0,2.5

The weights wT are given by the relative time value of money acrossmaturities

wT =er$T

er$0.5 + er$1 + ... + er$2.5

We obtain an alternative (equivalent) formulation by substituting theforward prices F0,T = M0 e

(r$re)T:

Currency Swap Rate = K = M0

ere0.5 + ere1 + ... + ere2.5

er$0.5 + er$1 + ... + er$2.5

The FX Swap rate equals the current exchange rate multiplied by the ratio ofthe relative borrowing costs in the two currencies.


Swaps The Swap Rate

Hedging with Swaps versus Forwards

Th ff fil f h f f d d i


64/74

The payoff profile from the sequence of forwards and one swap isdifferent:

1

The sequence of forwards imply the US firms gets less money early on, andmore later on (from $ 1.28 mil to $1.3323 mil).2 The swap implies the firm gets a constant amount $1.306 mil every six

months.

0 0.5 1 1.5 2 2.5 31.25

1.26

1.27

1.28

1.29

1.3

1.31

1.32

1.33

1.34

1.35

time

miliondollars

Payoff from Hedging Strategies

sequence of forwards

swap

Both strategies perfectly hedge the exposure, as the exchange rate risk iseliminated and both payoff profiles are known at 0.


Swaps The Swap Rate

The Value of the Swap Contract after Initiation


65/74

What is the value of a swap contract after initiation?

Let today be t, K be the swap rate of the existing swap, and T1, T2, ...., Tn be

the remaining payment periods.What would the firm have to pay to get out of the position?As before, using a sequence of forwards to get out of the position gives

Vswapt = PV of swap + reverse forward cash flows

= er$(T1t) (K Ft,T1

) + er$(T2t) (K Ft,T2

) + ... + er$(Tnt) (K Ft,Tn)

Obtaining the formula

Vswapt =n

i=1

er$(Tit) (K Ft,Ti)

Substituting the value of Ft,Ti = Mt e(r$re)(Tit) we obtain the equivalent

formula:

Vswap

t = K

n

i=1

er$(Tit)

Mt

n

i=1

ere(Tit)


Swaps The Swap Rate

Plain Vanilla FX Swaps


66/74

In the example above I assumed the counterparties swap fixed paymentsin the two currencies each period

The US Firm pays EUR 1 Mil to the Bank every six months and at the sametime the Bank pays $1.306 mil to the US Firm

The plain vanilla FX (or currency) swap is slightly different, as the twocounterparties not only exchange a series of fixed cash flows over time(coupons), but also a (larger) notional amount at both the initiation andmaturity of the contract

The primary use of plain vanillaFX swaps is to allow firms to borrow in anyforeign currency and hedge the foreign exchange risk

Because of this, it is important for the firm to also hedge the principalamount of the loan, which must be paid at maturity of the bond


Swaps The Swap Rate

Plain Vanilla FX Swaps - Example


67/74

A US firm issues 100 million Euro-denominated 5 year note with coupon c = 4%

If the firms revenue is mainly in US dollars, then the firm is subject to

exchange rate riskEvery six month, the firm must pay e2 mil = 100 mil 2%. In addition, atT = 5, the firm must pay back the e100 mil principal

A plain vanilla FX swap that hedges this exchange rate risk works as follows:

(a) At initiation (time 0), the two counterparties exchange the principal at the

current exchange rate: US firms gives e100 mil to the Bank; Bank gives USfirm $126.73 million (current exchange rate M0 = 1.2673$/EUS)

This first exchange of principals ensures the firm receives the USD funding itwanted. Why not raise debt in dollars directly?

(b) After initiation,(i) US Firm receives from Bank e2 mil every six months, plus e100 mil at T;

(ii) US Firm pays the bank C/2 126.73 mil dollars every six month, plus $126.73mil at T.

What is Swap Rate C?

It is the value of the coupon rate that makes the value of the swap equal tozero at inception.


Swaps The Swap Rate

Plain Vanilla FX Swaps - ExampleWhat is the value of the swap to the US firm?


68/74

In the swap contract, the firm is long a Euro-denominated 4% coupon bond withe100 mil principal, and short a C% dollar-denominated bond with $126.73 mil

principal

Let Be4%(t,T) and B$C(t,T) be the value of the two bonds (in their respective

currencies). The subscripts indicate the coupon rate.

Suppose the (c.c.) rates are constant across maturities at re = 4% and r$ = 6%

Be4%(0,T) = 100 2%ere0.5 + ere1 + + ere5 + ere5 = e99, 819, 335

B$C(0,T) = 126.73

C%

2

er$0

.5 + er$1 + + er$5

+ er$5

The value of the swap to the US firm in US dollars at t is then

Vswapt = Mt Be4%(t,T) B$C(t,T)

At time t = 0, look for the value C that makes the value of the swap equal to zero:

Vswap0 = 0 = M0 Be4%(0,T) = B

$C(0,T)

The coupon rate that ensures VSwap = 0 is C = 6.04845%


Swaps Other Swaps

Other Swaps


69/74

The number of swaps that exist in the market is very large. Some popularexamples:

1 Interest rate swaps

One party pays a fixed coupon and the other party pays a floating rate.Plain vanilla IR swaps have floating rate given by LIBOR

2 Energy swaps

One party pays a fixed amount and the other party pays a floating amountlinked to some energy index, such as oil prices, gas prices, etc.

3 Basis swaps

Both parties make floating payments, but linked to different indices, such asLIBOR versus Treasuries


Swaps Other Swaps

Total Return Swaps


70/74

Two parties both make floating payments, linked to the total return of differentsecurities

Example

One party (Bank A) pays the total return of a junk-rated corporate bond andthe other (Bank B) pays the total return of treasury plus a spread

Why bother?It allows one party (Bank B) to derive the economic benefit of owning anasset without putting that asset on its balance sheet, and allows the other(Bank A, which does retain the risky asset on its balance sheet) to buyprotection against loss in its value

Regulatory arbitrage: to gain leverage (Bank B), exploit funding costadvantage (Bank A), circumvent regulation (e.g., Bank B may not beallowed hold junk bonds), etc.


Swaps Other Swaps

Volatility and Variance Swaps

Volatility swap


71/74

o at ty s ap

An agreement to exchange the realized volatility between time 0 and time T

for a prespecified fixed volatility with both being multiplied by a prespecifiedprincipal(realized Kvol) N

realized realized volatility of an asset (say S&P500) quoted in annualterms

Kvol annualized volatility delivery price (say 30%)

N principal per annualized volatility point (say $250,000/(volatility point))

To compute realized volatility, need to specify1 the source and observation frequency of stock or index prices (e.g., daily

closing prices of S&P500)2 the annualization factor (e.g., 260 business days convention)3 whether to substract sample mean from each return, or to assume a zero

mean (theoretically preferred)

Variance swap: similar but replace volatility with variancec Hui Chen (MIT Sloan) Futures, Forwards, and Swaps 15.437, TN 2 71 / 74

Swaps Other Swaps

Why Volatility Swaps?


72/74

Advantage: provide pure exposure to volatility alone (unlike, say a stock

option, whose volatility exposure is contaminated by its stock-pricedependence)

Who can use volatility swaps?

Directional trading of volatility levels

Example

Long a volatility swap if you foresee a rapid rise in political and financial turmoil afterthe Greek crisis

Trading the spread between realized and implied volatility levels (more later)Hedging implicit volatility exposure embedded in businesses

Example

Portfolio managers who are judged against a benchmark have tracking error that mayincrease in periods of higher volatility


Swaps Other Swaps

Inflation Swaps

Z C I fl ti S


73/74

Zero-Coupon Inflation Swap

A bilateral contract that enables an investor to secure an inflation-protectedreturn with respect to an inflation index. The inflation buyer (a.k.a. the inflationreceiver) pays a predetermined fixed rate, and in return receives from theinflation seller (a.k.a. the inflation payer) inflation-linked payment(s).

Example of a US CPI zero-coupon inflation swapNotional: $100,000,000Index: US CPI-NSA (non-revised)Source: First publication by BLS as shown on Bloomberg CPURNSATrade date: 10-Feb-2005Start date: 12-Feb-2005

End date: 12-Feb-2010First fixing: 190.775Fixed leg: (1 + 2.75%)5 1

Inflation leg: I(12Feb10)I(12Feb05) 1


Swaps Other Swaps

Summary


74/74

How are swaps related to forwards?

How to determine the swap rate?

Popular swap products: plain vanilla currency swap, interest rate swap,total return swap, inflation swap ...


Documents

Tn 2 Futures Forwards Swaps 2011