RISK MANAGEMENT

APPLICATIONS
OF SWAPS & SWAPTIONS

Professor Kian-Guan Lim

Singapore Management University
KEY POINTS HEADLINED IN RED

Copyright Prof Kian-Guan Lim SMU

Using interest rate swap to convert variable
funding cost to fixed funding cost
 Firm issues $100 million 2-year FRN with semi-annual coupons
reset to LIBOR + 2%. Lm is LIBOR that will be realized in m-months.
0 6 12 18 24 months

100m ½ (L0+2) ½ (L6+2) ½ (L12 +2) ½ (L18 +2)+100m

 To eliminate variable interest rate risk or funding cost, firm enters

into an IRS as 8% fixed-rate payer on notional of $100 million.
4% 4% 4% 4%

½ L0 ½ L6 ½ L12 ½ L18

IRS dealer pays firm LIBOR

 The net funding cost to the Firm is fixed 10% p.a. for 2 years.

Copyright Prof Kian-Guan Lim SMU

Using interest rate swap to convert fixed
funding cost to variable funding cost
 Firm issues $100 million 2-year Notes with semi-annual coupons
fixed at 10%. Lm is LIBOR that will be realized in m-months.
0 6 12 18 24 months

100m 5% 5% 5% 5%

 To take a view on falling interest rates or funding cost, firm enters

into an IRS as 7% fixed-rate receiver on notional of $100 million.
½ L0 ½ L6 ½ L12 ½ L18

3.5% 3.5% 3.5% 3.5%

IRS dealer pays firm fixed 7%

 The net funding cost to the Firm is fixed LIBOR + 3% p.a. for 2
years. Firm gains if LIBOR falls below 7%.
Copyright Prof Kian-Guan Lim SMU
Using interest rate swaps to adjust bond
portfolio duration
 An IRS is a capital market instrument with its
own modified duration Dm measure.
 Need to define Dm carefully here.
Dm = - (∆ B/B) ÷ ∆ y
or % change in market value of bond per % change in the bond yield.
Dm > 0 for long position in bond since dB/dy < 0.
However, the use of signed measure for duration allows for short position in
bond to have negative duration if we employ B>0 for long position, and B<0
for short position.
The latter is convenient since we can effectively reduce duration or
sensitivity to zero by shorting the long position in bond.

Copyright Prof Kian-Guan Lim SMU

Value of floating rate note/bond marks to par
at each reset time
 Market value VFL of a FRN paying LIBOR is
100% of par at initiation and at each reset.
Realized LIBOR rates
L0 L1 L2 L3 L4 LN+100
accruals a0

t*
at*

Pays L0xa0 L1xa1 L2xa2 L3xa3 LN-1 xaN-1

Suppose at this reset time, VFL < 100, then

At non-reset time t*, VFL = arbitrageur could buy FRN. At the same time, sell
100(1+L0xa0)/(1+Lt* xat* ) $100 1-reset period FRN. At t=2, again sell $100
where 0<t*<a0, at* = a0 – (use this to redeem the previous sold FRN). At t=2,
t*, and Lt* is the new use the N-period FRN L1xa1 to pay for the FR
discount rate at t*. Here, interest in the short 1-period position. Net gain at
VFL needs not be 100. t=1 is $100 – VFL .
Copyright Prof Kian-Guan Lim SMU
Modified duration of a FRN

 Since the price of a N-period FRN converges to 100 at each reset, the
Macaulay duration of such a FRN behaves like the duration of a 1-
period FRN. Modified duration is Macaulay duration ÷ (1+y x a) or
approximately 1-period. V is full price ≅ Quoted price 100(1+L xa ) /
FL t t*
(1+L xa )+accrued interest 100L (a -a )
 In general, dVFL /dLt* = d{100(1+Ltxat)/(1+L t* xat* )}/dLt*
t* t* t t t*

≅ - at* VFL , or Dm ≅ at* that is between 0 and a. Choose ½ a or half times

the reset interval as an approximation.
 Suppose each reset is 6 months e.g. a semi-annual floating rate coupon
of a 1 yr or 2 yr or in general N yr bond. Then a long position in this FRN
has an approximate duration of ½ x 0.5 or 0.25
 Suppose each reset is 3 months for a FRN, then approximate duration
is ½ x 0.25 = 0.125

Copyright Prof Kian-Guan Lim SMU

Modified duration of an IRS
 A fixed payer position in an IRS is equivalent to short the fixed leg
and long the floating leg.
 In a 2-year semi-annual reset IRS, suppose the short fixed leg bond
with price B has a duration of Dm= - 0.8.
 The long FRN has Dm ≅ 0.25.
 Then a fixed payer position in the swap has dollar duration 0.25 x
100 – 0.8 x B.
 Divide this by 100 to obtain the swap duration of
0.25 – 0.8 x B/100
Thus, the swap duration is sensitivity defined with respect to the
notional principal.
 In the above, suppose B=90. Then the swap duration is Dsw = 0.25 –
0.72 = - 0.47

Copyright Prof Kian-Guan Lim SMU

Amount of IRS to use to adjust
bond portfolio duration
 Firm has par value $200 million bonds with B=$180 m,
and modified duration 6.5. It wants to bring total duration
(thus cash-flow risk) down to 3.5 in the wake of
anticipated interest rate hikes.
 Firm considers a 5-year fixed payer IRS with annual
reset. Suppose the swap duration is
– 4.5. Let the desired notional principal be P.
 Total dollar duration = $180m x 6.5 + P x (-4.5)
 This equals the target dollar duration = $180m x 3.5
 180m x 6.5 – 4.5 P = 180m x 3.5 ⇒ P = $120m

Copyright Prof Kian-Guan Lim SMU

Use of IRS to make risky arbitrage
profit in structured notes
 Example of Inverse Floater: Firm issues inverse FRN that pays 8% -
LIBOR with maximum at 8% and minimum at 0%. Firm buys fixed rate
note at 5%. Firm enters into a IRS paying LIBOR and receiving 4% fixed.
All instruments have same notional and same resets and maturity.

Inflow (+) Inverse Fixed Rate IRS NET

Outflow (-) Floater Note
LIBOR = L - (8% - L) + 5% - L + 4% 1%

Risk when 0% + 5% - 6% -1%

L>8%, e.g.
L=10%

Copyright Prof Kian-Guan Lim SMU

Using equity swap to convert exposure
to equity market risk into funding cost
 Securities dealer carries a large inventory of market shares
approximating the S&P 500 index
 The dealer firm funds the holding by bank loan
 It is more concerned about meeting the funding cost, and does not
wish to be exposed to market risk, since it makes its earnings from
buying and selling via the spread, and not from market movements.
 It enters into an Equity Swap with an Insurance Company that
prefers some equity returns.
 Dealer pays %∆ in S&P 500 index to Insurance Firm and receives
in exchange LIBOR
 Fixed tenor of N years with pre-specified reset times
 Note that the Dealer firm does not need to liquidate its share
holdings

Copyright Prof Kian-Guan Lim SMU

Other types of equity swaps

 Returns on single name swapped into returns on an

index portfolio is an equity swap to reduce concentration
risk
 Returns on a domestic portfolio index swapped into an
international portfolio index is an equity swap to diversify
internationally without physically involved in the stock
transactions
 Returns on bonds index swapped into small caps funds
index is an equity swap to allow a funds manager to
reduce exposures to bonds and increase exposures to
small cap equities as part of asset re-allocation

Copyright Prof Kian-Guan Lim SMU

Equity swap has special problems
 In an ∆ indexi to ∆ indexj swap, sometimes
both changes can be in opposite
directions so that one party pays on both
legs to the other party
 In an equity swap exchanging ∆ index risk
with interest cost, the index may not track
the portfolio exactly, resulting in tracking
error risk

Copyright Prof Kian-Guan Lim SMU

Using interest rate swaptions

 To obtain the option to convert payments on

future loans from floating to fixed (using payer
swaption: enter a swap as fixed rate payer) or
fixed to floating (using receiver swaption: enter a
swap as fixed rate receiver)
 To terminate an existing swap by exercising a
swaption
 Add or remove a call feature to a bond

Copyright Prof Kian-Guan Lim SMU

Selling a receiver swaption to convert a
callable bond into a non-callable bond
 Callable bond issuer has the call option
 Call option will be exercised by issuer when interest rate falls and bond price
reaches the call value. Issuer will exercise to stop paying the higher fixed
rate when interest rate has fallen too low. By calling, the issuer will refinance
at a cheaper rate.
 If issuer believes that the interest rate will not fall low enough, it may decide
to cash-in and sell the call option feature.
 It can do this by selling a receiver (or put) swaption (issuer pays fixed when
swaption is exercised by investor)
 When interest rate is high, investor will not exercise swaption.
 When interest rate becomes too low, investor will exercise swaption to
receive fixed and pay low floating. The issuer calls the bond. But its total
exposure now includes this swaption that is exercised into a swap in which
the issuer now pays high fixed rate and receives low floating. This is as if
the bond was not called and the issuer continues to pay the high bond rate.

Copyright Prof Kian-Guan Lim SMU

Buying a receiver swaption to convert a non-
callable bond into a callable bond
 Non-callable bond issuer anticipates falling interest rate and wants to
have a call option on the bond
 Call option will be exercised by issuer when interest rate falls and
bond price reaches the call value. Issuer will exercise to stop paying
the higher fixed rate when interest rate has fallen too low. By calling,
the issuer will refinance at a cheaper rate.
 Issuer can have this option by buying a receiver (or put) swaption
(issuer receives fixed when it exercises the swaption)
 When interest rate is low, issuer will exercise the swaption to receive
fixed and pay low floating. The issuer continues to pay on the non-
callable bond. However, the net effect is that the issuer effectively
pays the lower floating, and it is as if it had called the bond. The swap
tenor must equal the remaining tenor of the non-callable bond.

Copyright Prof Kian-Guan Lim SMU

Black’s Model for European Eurodollar futures
Options at CME
 Not appropriate to assume that the futures price is lognormal since for
short-term instruments, their value converges to par. Assume instead the
yield is lognormally distributed.
 Yield yt = 100 – index price Ft ; volatility of yt or of (100-Ft) is σ y2. Call is in-
the-money when index price Ft is higher, or yield (100-FT) is lower than a
strike yield (100-X). Thus, it is like a put –> exercise call when yT is lower
than 100-X.

c=P 0 ,T[  1 0 0 -X N-d2 −  1 0 0 -F0  N -d1 ]

ln[ 1 0 0 -F0  /1 0 0 -X]σ 2y T /2
d 1= ;   d 2 =d 1−σ y  T
σ y T
The European Eurodollar futures put option price is:
p = P(0, T)[(100 - F0 ) N(d1 ) − (100 - X ) N(d 2 )]

Copyright Prof Kian-Guan Lim SMU

Caps and Floors
 A cap is a portfolio of call options or caplets on LIBOR.

 Payoff at time tk+1 is N δ k max(Rk+1-K, 0) where N is the notional

principal, δ k =tk+1-tk is the reset interval between last reset time tk
and present reset time tk+1.

K is the cap rate, and Rk+1 is the current LIBOR rate at time tk+1

 A floor is a portfolio of put options on LIBOR. Payoff at time tk+1 is

N δ k max(K – Rk+1, 0)

 Each cap or floor has a maturity of T years with 2xT semi-

annual resets (or some with 4xT quarterly resets)

Copyright Prof Kian-Guan Lim SMU

 Black’s Model for Caps
 Each caplet is a call option on a future LIBOR rate with the payoff
occurring in arrears
 Using Black’s model we assume that the interest rate underlying
each caplet is lognormal
 The value of a caplet, for period (tk, tk+ 1) is

Nδ k P(0, t k +1 )[Fk N(d1 ) − KN(d2 )]

ln(Fk /K) + σ 2k t k /2
where d1 = and d 2 = d1 - σ t k
σk t k

where Fk = forward interest rate for (tk, tk+1 )

σ k = forward rate volatility
N = notional principal
K = cap rate
δ k = tK+1 – tK

Copyright Prof Kian-Guan Lim SMU

Theoretical Justification for Cap Model

Using a forward risk neutral probabilit y measure

with respect to zero - coupon bond maturing at time t k +1 ,
the option price is
P(0, t k +1 ) N E k +1[max(R k − K,0)]
Since E k +1[R k ] = Fk
where Fk is the expected future rate.

Copyright Prof Kian-Guan Lim SMU

Swaptions
 A swaption or swap option is a contract that gives
the holder the right, not obligation, to exercise the
option into an interest rate swap in the future
before the swap option expiry.
 A payer swaption provides the holder the right to
pay a specified fixed rate and receive floating
LIBOR when the interest rate swap is exercised.
 A receiver swaption provides the holder the right to
receive a specified fixed rate and pay a floating
LIBOR when the interest rate swap is exercised.

Copyright Prof Kian-Guan Lim SMU

Black’s Model for European Swaption
 Assume that the swap rate is lognormal
i.e. dS = σ S dW
 Consider a payer swaption which gives the holder
the right to exercise the swap and pay fixed rate K
on a n-year swap starting at time T. The payoff on
each swap payment date is
N
max(s T −principal,
where N is themnotional K, 0) m is the
payment frequency and sT is market n-year swap
rate at time T.

Copyright Prof Kian-Guan Lim SMU

Black’s Model for European Swaption

The Black model value of the swaption is

N × A × [s 0 N(d1 ) − K N(d 2 )]
ln s0 /Kσ 2 T /2
w h e re  d 1= ;  d 2=d 1−σ  T
σ T

where s0 is the forward swap rate

σ is the swap rate volatility
tk is the time from today until the kth swap payment
mn
1
and A=
m k =1 ∑
P(0, t k )

Copyright Prof Kian-Guan Lim SMU

Swaptions and Bonds
 An interest rate swap is essentially an exchange of
a fixed-rate bond for a floating-rate bond.
 Therefore, a payer swaption is an option for a
holder to exchange a fixed-rate bond and receive a
floating-rate bond.
 Therefore, a receiver swaption is an option for a
holder to exchange a floating-rate bond and receive
a fixed-rate bond.

Copyright Prof Kian-Guan Lim SMU

Swaptions and Bond Options
 When a swaption is exercised into a swap. At the start of
this swap, the floating-rate bond is at par value of 100%.
 Therefore a payer swaption (where the exercised swap is
for holder to pay fixed rate and receive floating rate) is a
put option on a bond having similar reset frequency,
maturity and fixed coupon rate, with a strike price of par
100%.
 A receiver swaption (where the exercised swap is for
holder to receive fixed rate and pay floating rate) is a call
option on a bond having similar reset frequency, maturity
and fixed coupon rate, with a strike price of par 100%.

Copyright Prof Kian-Guan Lim SMU