Derivatives & Risk
Management
Lecture 4:
a) Swaps
b) Options: properties and nonparametric bounds
1
Nature of Swaps
• A swap is an agreement to exchange
cash flows at specified future times
according to certain specified rules
2
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
3
Cash Flows to Company B
---------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
4
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
5
A and B Transform a Liability
5%
5.2%
A
B
LIBOR+0.8%
LIBOR
6
Financial Institution is Involved
4.985%
5.015%
5.2%
A
F.I.
B
LIBOR+0.8%
LIBOR
LIBOR
7
A and B Transform an Asset
5%
4.7%
A
B
LIBOR-0.25%
LIBOR
8
Financial Institution is Involved
4.985%
5.015%
4.7%
A
F.I.
B
LIBOR-0.25%
LIBOR
LIBOR
9
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%
10
The Swap
9.95%
10%
A
B
LIBOR+1%
LIBOR
11
The Swap when a Financial
Institution is Involved
9.93%
9.97%
10%
A
F.I.
B
LIBOR+1%
LIBOR
LIBOR
12
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
13
Alternatives
• Information asymmetry
• Flexible and liquid instruments
• Tax and regulatory arbitrage
14
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
V  B fix  B fl
15
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
16
Valuation as bonds
n
B fix   ke
 ri ti
 Qe
 rn t n
i 1


B fl  Q  k e
*
 r1t1
K* is the floating rate know from at the
beginning of the period
17
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%
18
Example II
• The 6 month LIBOR when the swap was
set up 3 months ago was 10.2%.

100 k 
100 
k
*
0.102
1
2
*
 5.1
19
Example III
B fix  4e
3
 0.10
12
 4e
9
 0.105
12
 104e
15
 0.11
12
 $98.238 m illion
B fl  105.1e
3
 0.10
12
 102.51
V  98.238  102 .51  4.272
20
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
21
Forward Rate Agreement
continued (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)]
22
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
L 1+ R
k
(T2 -T1 ) -L 1+ R f (T2 -T1 )  e
-r2T 2
=
L  R k -R f  (T2 -T1 )e-r2T 2
23
Valuing future cash flows
r1
rf
r2
Xe
 rf T2 T1 
Cash
settlement
X
Hypothetical
cash flow
24
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
25
Swaps as FRA’s
• Suppose an interest rate swap promises
fixed rate payments C and receives
floating payments P1fl 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?
26
Swaps as FRA’s II
C
P1
t
fl
T1
P2
T2
C
C
C
fl
P3
T3
fl
P4
fl
T4
27
Swaps as FRA’s III
• Consider the second exchange of
payments (the first is known)
P
2
fl
C

• The floating rate payment is computed
based on the prevailing spot rate at T1
28
Swaps as FRA’s IV
r1
C  100  c
R1,2
P2
r2
t
T1
fl
~
 100  R1, 2
T2
29
Swaps as FRA’s V
• So we want to compute the PV of

~
100 R1, 2  c

• Which can be written as


~
100 1  R1, 2  100 1  c 
30
Swaps V
• The value of the fixed part of this payment is
obvious
v fix  1001  ce
 r2 T2 T1 
• The value of the floating part less so because
it involves a random interest rate
v fl


~
 PV 1001  R1, 2

31
Swaps as FRA’s VI
• We know that at T2, the floating rate
payment will be worth
v fl ,T2

~
 100 1  R1, 2
• And thus T1, it must be worth
v fl ,T1



~
100 1  R1, 2

 100
~
1  R1, 2


32
Swaps as FRA’s VII
• And thus at time t it
must be worth
 r1T1
v fl  100e
• Recall that by no arbitrage
r2T2  rT
1 1  rf T2  T1 
• So that
rT
1 1  r2T2  rf T2  T1 
33
Swaps as FRA’s VIII
• Hence
v fl  100e

 r2T2  r f  T2 T1 
 100e
r f  T2 T1 
e
 r2T2
Changing compounding convention
 100 1  R f  e
 r2T2
34
Swaps as FRA
PV 100  R1,2  c


 100 1  RF  e
 100 1  c  e

 r2T2
 100  RF  c  e
 r2T2
 r2T2
So to value a fixed for floating exchange,
compute the present value of the exchange
between the forward rate and the fixed rate
35
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
36
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
37
The Cash Flows
Dollars Pounds
$
£
Years ------millions-----0
–15.00 +10.00
+1.20 –1.10
1
2
+1.20 –1.10
3
+1.20 –1.10
4
+1.20 –1.10
5
+16.20 -11.10
38
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
39
Comparative Advantage Arguments
for Currency Swaps
• Company A wants to borrow AUD
• Company B wants to borrow USD
USD
Company A
AUD
5.0% 12.6%
Company B 7.0% 13.0%
40
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
41
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
42
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
43