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Derivatives & Risk Management

Lecture 4:

a) Swaps

b) Options: properties and non-parametric bounds

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Nature of Swaps

• A swap is an agreement to exchange cash flows at specified future times according to certain specified rules

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Example: a “Plain Vanilla” Interest Rate Swap

• An agreement by “Company B” to receive 6-month LIBOR & pay a fixed rate of 5% per annum every 6 months for 3 years on a notional principal of $100 million

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---------Millions of Dollars---------

LIBOR FLOATING FIXED Net

Date Rate Cash FlowCash FlowCash Flow

Mar.1, 1998 4.2%

Sept. 1, 1998 4.8% +2.10 –2.50 –0.40

Mar.1, 1999 5.3% +2.40 –2.50 –0.10

Sept. 1, 1999 5.5% +2.65 –2.50 +0.15

Mar.1, 2000 5.6% +2.75 –2.50 +0.25

Sept. 1, 2000 5.9% +2.80 –2.50 +0.30

Mar.1, 2001 6.4% +2.95 –2.50 +0.45

Cash Flows to Company B

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Typical Uses for an Interest Rate Swap

• Converting a liability from

– fixed rate to floating rate

– floating rate to fixed rate

• Converting an investment from

– fixed rate to floating rate

– floating rate to fixed rate

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A and B Transform a Liability

A B

LIBOR

5%

LIBOR+0.8%

5.2%

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Financial Institution is Involved

A F.I. B

LIBOR LIBORLIBOR+0.8%

4.985% 5.015%

5.2%

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A and B Transform an Asset

A B

LIBOR

5%

LIBOR-0.25%

4.7%

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Financial Institution is Involved

A F.I. B

LIBOR LIBOR

4.7%

5.015%4.985%

LIBOR-0.25%

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The Comparative Advantage Argument

• Company A wants to borrow floating• Company B wants to borrow fixed

Fixed Floating

Company A 10.00% 6-month LIBOR + 0.30%

Company B 11.20% 6-month LIBOR + 1.00%

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The Swap

A B

LIBOR

LIBOR+1%

9.95%

10%

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The Swap when a Financial Institution is Involved

A F.I. B

10%

LIBOR LIBOR

LIBOR+1%

9.93% 9.97%

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Criticism of the Comparative Advantage Argument

• The 10.0% and 11.2% rates available to A and B in fixed rate markets are 5-year

• The LIBOR+0.3% and LIBOR+1% rates available in the floating rate market are six-month rates

• B’s fixed rate depends on the spread above LIBOR it borrows at in the future i.e. it is fixed only as long as its creditworthiness stays the same

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Alternatives

• Information asymmetry

• Flexible and liquid instruments

• Tax and regulatory arbitrage

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Valuation of an Interest Rate Swap

• Interest rate swaps can be valued as the difference between the value of a fixed-rate bond & the value of a floating-rate bond

flfix BBV

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Valuation in Terms of Bonds

• The fixed rate bond is valued in the usual way

• The floating rate bond is valued by noting that it is worth par immediately after the next payment date

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Valuation as bonds

n

i

trtrfix

nnii QekeB1

11* trfl ekQB

K* is the floating rate know from at the beginning of the period

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Example

• A financial institution pays 6 month LIBOR and receives 8% (semi-annually) on $100 million notional principal.

• The FI has sold a floater and bought a fixed rate bond

• remaining life 1.25 years

• market rates for 3, 9, 15 months to go are 10%, 10.5% and 11%

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Example II

• The 6 month LIBOR when the swap was set up 3 months ago was 10.2%.

1.5

2102.0

1

100100 *

*

k

k

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Example III

51.1021.105

238.98$

10444

12

310.0

12

1511.0

12

9105.0

12

310.0

eB

million

eeeB

fl

fix

272.451.102238.98 V

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Forward Rate Agreement

• A forward rate agreement (FRA) is an agreement that a certain rate will apply to a certain principal during a certain future time period

• An FRA is equivalent to an agreement where interest at a predetermined rate, RK is exchanged for interest at the market rate

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Forward Rate Agreementcontinued (Page 100)

• Capital R is the rate measured with compounding rate reflecting maturity, i.e. if the T2 – T1 is three months the rate is compounded quarterly etc.

• The agreed cash flows are:

• T1: - L

• T2: L [1+ Rk (T2-T1)]

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FRA

• Note if Rf = Rk the FRA is worth 0. Why?• To value the FRA, we can compare now two payments

at time T2:• One that pays Rk and one that pays Rf

• Note: we are not assuming anything more than no arbitrage

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2

-r 2k 2 1 f 2 1

-r 2k f 2 1

L 1+ R (T -T ) -L 1+ R (T -T ) e =

L R -R (T -T )e

T

T

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Valuing future cash flows

2r

1r fr

Hypothetical cash flowCash

settlement

2 1fr T TXe X

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Alternative Valuation of Interest Rate Swap: portfolio of FRA

• Swaps can be valued as a portfolio of forward rate agreements (FRAs)

• Each exchange of payments in an interest rate swap can be analyzed as an FRA

• The relevant interest rates are the fixed for one leg, and the forward associated with the period to be valued for the other leg

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Swaps as FRA’s

• Suppose an interest rate swap promises fixed rate payments C and receives floating payments P1

fl at regular intervals

• We have seen that this could be valued as a portfolio of bonds

• What about valuating it as a package of FRA’s?

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Swaps as FRA’s II

flP3flP2

flP1flP4

CCC C

t1T 4T3T2T

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Swaps as FRA’s III

• The floating rate payment is computed based on the prevailing spot rate at T1

CP fl 2

• Consider the second exchange of payments (the first is known)

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Swaps as FRA’s IV

2,12

~100 RP fl

cC 100

t1T 2T

1r

2r

1,2R

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Swaps as FRA’s V

• So we want to compute the PV of

cR 2,1

~100

• Which can be written as

cR 1100~

1100 2,1

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Swaps V

• The value of the fixed part of this payment is obvious

1221100 TTrfix ecv

• The value of the floating part less so because it involves a random interest rate

2,1

~1100 RPVv fl

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Swaps as FRA’s VI

• We know that at T2, the floating rate payment will be worth

2,1,

~1100

2Rv Tfl

• And thus T1, it must be worth

100~1

~1100

2,1

2,1, 1

R

Rv Tfl

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Swaps as FRA’s VII

• And thus at time t it must be worth

11100 Trfl ev

• Recall that by no arbitrage

• So that

2 2 1 1 2 1fr T rT r T T

1 1 2 2 2 1frT r T r T T

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Swaps as FRA’s VIII

• Hence

2 2 2 1

2 1 2 2

100

100

f

f

r T r T Tfl

r T T r T

v e

e e

Changing compounding convention

2 2100 1 r TfR e

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Swaps as FRA

2 2 2 2

2 2

1,2100

100 1 100 1

100

r T r TF

r TF

PV R c

R e c e

R c e

So to value a fixed for floating exchange, compute the present value of the exchange between the forward rate and the fixed rate

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An Example of a Currency Swap

An agreement to pay 11% on a sterling principal of £10,000,000 & receive 8% on a US$ principal of $15,000,000 every year for 5 years

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Exchange of Principal

• In an interest rate swap the principal is not exchanged

• In a currency swap the principal is exchanged at the beginning &the end of the swap

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The Cash Flows

Years

Dollars Pounds$

------millions------

0 –15.00 +10.001 +1.20 –1.102 +1.20 –1.10

3 +1.20 –1.104 +1.20 –1.10

5 +16.20 -11.10

£

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Typical Uses of a Currency Swap

• Conversion from a liability in one currency to a liability in another currency

• Conversion from an investment in one currency to an investment in another currency

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Comparative Advantage Arguments for Currency Swaps

• Company A wants to borrow AUD• Company B wants to borrow USD

USD AUD

Company A 5.0% 12.6%

Company B 7.0% 13.0%

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Valuation of Currency Swaps

• Like interest rate swaps, currency swaps can be valued either as the difference between 2 bonds or as a portfolio of forward contracts

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Swaps & Forwards(continued)

• The value of the swap is the sum of the values of the forward contracts underlying the swap

• Swaps are normally “at the money” initially– This means that it costs NOTHING to

enter into a swap– It does NOT mean that each forward

contract underlying a swap is “at the money” initially

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Credit Risk

• A swap is worth zero to a company initially• At a future time its value is liable to be

either positive or negative• The company has credit risk exposure

only when its value is positive