1. law, govt and politics
2. government

Interest Rate Derivatives
Hedging & Trading
FI6002: Fixed Income Models
Barry Sheehan 0854867
Garry Lynch 0871117
Mark Moran 14067706
MSc. in Computational Finance
2015
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Title Page
University: University of Limerick
Program: MSc. in Computational Finance
Year of Submission: 2015
Authors: Barry Sheehan, Garry Lynch, Mark Moran
Title: Interest Rate Derivatives Hedging & Trading
Lecturer: Dr. Bernard Murphy
This assignment is solely the work of the authors and submitted in partial fulfilment of the
requirements of the MSc. in Computational Finance.
X
Barry Sheehan, Garry Lynch, Mark Moran
MSc. in Computational Finance Candidates
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Contents
Title Page ...................................................................................................................................ii
Table of Figures ........................................................................................................................ iv
Table of Tables .......................................................................................................................... v
A ‘Multi-Leg FRA’ Swap Key Rate Hedge & Yield - Spread Arbitrage Trading
................................................................................................................................. 1
Part A:
Strategy
1.1
Introduction to Swaps.................................................................................................. 1
1.2
Near Date Multi-Leg FRA Swap Key Rate Hedge ..................................................... 1
1.2.1
Introduction .......................................................................................................... 1
1.2.2
Swap Construction ............................................................................................... 2
1.2.3
Delta-Hedging of Near Dated Swap .................................................................... 5
1.3
2.0
Far-Dated Offsetting Par Swap Hedge ........................................................................ 7
Scenario Analysis /Yield-Spread Arbitrage Trading Strategy ...................................... 10
2.1
Near Dated Swap ....................................................................................................... 10
2.2
Far Dated Swap ......................................................................................................... 12
Part B:
Swaption Risk Management & Yield Spread Arbitrage Strategy ......................... 14
3.1
Introduction to Swaptions ......................................................................................... 14
3.2
Dual Key Rate Exposures ......................................................................................... 14
3.3
Method of Hedging and Key Hedging Insights ........................................................ 16
4.0
Swaption Yield Spread Arbitrage Trading Strategy: P&L Capabilities ....................... 19
4.1
Type of curve movement........................................................................................... 19
4.2
Hedged Swaption specific movements ..................................................................... 20
5.0 Swaption Yield Spread Arbitrage Trading Strategy: Implementation and Risk
Analysis.................................................................................................................................... 23
5.1
Swaption Construction and Key Rate Risk Exposure Analysis ................................ 24
5.2
Swap Hedging Strategy Implemented ....................................................................... 28
5.2.1
Payer Swap......................................................................................................... 28
5.2.2
Receiver Swap ................................................................................................... 29
5.3
6.0
Swaption Key Rate Hedge Portfolio Combined ....................................................... 30
Residual Exposure Strategy .......................................................................................... 34
Bibliography ............................................................................................................................ 37
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Table of Figures
Figure 1: Plain Vanilla 0x12M Payer Swap Creation ................................................................ 3
Figure 2: Key Rate Risk Exposures of Plain Vanilla 0x12M Payer Swap ................................ 4
Figure 3: Key Rate Risk Exposures of Plain Vanilla 0x12M Payer Swap as Forwards ............ 4
Figure 4: LIBOR 3 Month Reset Rates...................................................................................... 6
Figure 5: Plain Vanilla 1Y Payer Swap Hedged with 3 x 3m FRAs ......................................... 6
Figure 6: Key Rate Risks of Hedged Near Date Swap with FRAs ............................................ 6
Figure 7: Key Rate Risks of Hedged Swap Curve Instrument .................................................. 7
Figure 8: Dollar Delta Exposure of Hedged Near Date Swap with FRAs ................................. 7
Figure 9: Far Dated 1Yx4Y Par Swap Hedge ............................................................................ 8
Figure 10: Par Coupon Rate used for Far Dated Swap (Failry Priced) ..................................... 8
Figure 11: Key Rate Risk Exposure of the 1Yx4Y Receiver Swap .......................................... 9
Figure 12: Far Dated Swap Key Rate Risk Exposure ................................................................ 9
Figure 13: Mars Analysis for Far Date Offsetting Static Hedge ............................................. 10
Figure 14: Near Date 1 Year Delta-Hedged Payer Swap MARS Scenario P/L ...................... 11
Figure 15: Far Date 1Yx4Y Delta Hedged Swap .................................................................... 12
Figure 16: Swaption construction ............................................................................................ 17
Figure 17: Vanilla Swaption Risk Screens .............................................................................. 17
Figure 18: Swaption MARS Scenario 1bp analysis ................................................................. 18
Figure 19: MARS Scenario Analysis for vanilla Swaption ..................................................... 18
Figure 20: Zero Yield Curve (Short term decrease & Long term increase) ............................ 20
Figure 21: Zero Yield Curve (Short term decrease & larger long term decrease) ................... 21
Figure 22: Zero Yield Curve (Long term decrease)................................................................. 22
Figure 23: 1Y x 3Y Payer Swaption SWPM construction ...................................................... 24
Figure 24: Key Rate Risk Exposure of 1Y x 3Y Long Payer Swaption .................................. 25
Figure 25: Key Rate Risk Exposure to 1 year and 4 year swap rates ...................................... 25
Figure 26: Swaption MARS Scenario -1bp analysis ............................................................... 25
Figure 27: MARS Scenario Analysis for Vanilla Swaption .................................................... 27
Figure 28: 1yr Par Payer Swap Risk with Notional set to $1 .................................................. 28
Figure 29: 1 Year Pay Fixed Notional + DV01 ....................................................................... 29
Figure 30: 4 Year Receiver Swap Risk with Notional set to $1 .............................................. 29
Figure 31: 4 Year Receiver Swap Notional & DV01 .............................................................. 30
Figure 32: MARS Overview - 1Y x 3Y Payer Swaption, 1Y Payer Swap & 4Y Receiver
Swap ......................................................................................................................................... 30
Figure 33: Delta hedged Swaption ........................................................................................... 31
Figure 34: Residual Analysis of Delta-Hedged Portfolio ........................................................ 31
Figure 35: MARS Scenario Analysis - Delta-hedged Swaption .............................................. 33
Figure 36: 3M x 6M FRA SWPM construction....................................................................... 34
Figure 37: 3*6 Month FRA Reduces 6 Month Exposure ........................................................ 35
Figure 38: 9*12 Month FRA.................................................................................................... 35
Figure 39: 9*12 Month FRA reduces 1 Year Exposure .......................................................... 36
Figure 40: Final Swaption Portfolio Strategy DV01 ............................................................... 36
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Table of Tables
Table 1: Near Date Scenarios Profit/Loss ................................................................................ 11
Table 2: Far Date Delta-Hedged Swap Scenarios Profit/Loss ................................................. 13
Table 3: Scenario Analyses - P&L Breakdown of each portfolio component ......................... 33
Table 4: Residual Exposures of delta-hedged swaption and FRA National’s required ........... 34
v|Page
Part A:
A ‘Multi-Leg FRA’ Swap Key Rate Hedge & Yield - Spread
Arbitrage Trading Strategy
1.1 Introduction to Swaps
According to (Hull, 2015) , a swap can be defined as “An agreement to exchange cash flows
in the future according to a prearranged formula”. In the case of an interest rate swap, one
party agrees to exchange floating interest rate cash flows on a notional principal amount
usually determined by the London Inter-Bank Offer Rate (LIBOR) in exchange for a fixed
rate cash flow known as the par swap rate. Once the LIBOR agreement is entered into, it is
fixed for the term of the agreement i.e. “spot-starting” according to (Murphy, 2015), however
as in the case of this section, LIBOR rates for constant maturity loans change randomly with
the passage of time similar to movements in a stock from one day to the next, this is known
as LIBOR resetting. In the case of this project the LIBOR 3 month rate will be resetting
whilst the fixed swap rate remains constant for the duration of the swap. Interest rate swaps
have become a dominant force in the over-the-counter derivatives market representing 58.5%
of the total over-the-counter outstanding which as of June 2014 had a notional outstanding of
$691.5 Trillion according to (Bank for International Settlements, 2014).
1.2 Near Date Multi-Leg FRA Swap Key Rate Hedge
1.2.1 Introduction
A payer swap is implemented in this section whereby the holder pays the agreed fixed swap
rate and receives the floating rate on a notional of $79,780,657.89. At its inception, the payer
swap should have a market value of zero according to (Litzenberger, 1992) in order for the
swap to be priced “fairly” meaning that neither counterparty is required “to pay the other to
induce that party into the agreement” as per (Fifth Third Trading Center, n.d.). This economic
value of zero requires that the underlying bond positions of the swap are priced as what is
referred to as par value (
. According to (HedgeBook, 2013), the fixed
rate that neutralizes the value of the swap agreement is described as the equilibrium swap
rate. The equilibrium swap rate
seen in equation 1 below can be specified in terms of the
discrete forward-LIBOR rates
spanning the life of the swap where
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denotes the
present price of a discount bond (priced off the LIBOR-Swap Zero Curve) maturing on date
.
denotes the
day-count convention for the floating and fixed leg of the swaps.
∑
(
∑
The equilibrium swap rate
)(
(
)
(1)
)
can be determined by noting the current 3 month LIBOR rate
for the purposes of the 12 month payer swap in this section in addition to the further four 3
month forward rates corresponding to pre-determined LIBOR reset dates due to the forward
rate curve acting as a market participant proxy for the remaining future LIBOR rates payable
on the floating-leg of the swap as per (Murphy, 2015), meaning the equilibrium swap rate
(par rate) is the weighted average of the quarterly tenor LIBOR proxy forward rates of the
swap. The denominator in equation 1 represents the Present Value of a Basis Point (PVBP),
sensitivity of the fixed-leg to a 1 basis point increase in the fixed rate, the PVBP is also
referred to as the annuity discount factor according to (Murphy, 2015).
Given that the equilibrium swap rate
is determined by the weighted average of the
quarterly tenor LIBOR proxy forward rates of the swap and such rates act as proxies for the
LIBOR rates payable on the floating-leg of our payer swap, it would make sense that an
offsetting position in such forward rates would render the swap to be delta-hedged given that
the denominator it represents the sensitivity of the fixed leg to small changes similar to that of
equity delta-hedging. One such instrument enabling us to enter that offsetting position comes
in the form of a Forward Rate Agreement (FRA). The aim of this section is to delta-hedge our
1 year payer swap with a notional of $79,780,657.89 with three 3 month Forward Rate
Agreements thereby limiting the risk exposure of our swap to parallel shifts in the yield curve
as per (Lindquist, 2011).
1.2.2 Swap Construction
Using the (SWPM) function in Bloomberg Terminal, a 1 year payer swap was created (See
Figure 1). An equal notional amount of $79,780,657.89 was used for the fixed and floating
leg of the payer swap. The par coupon of 0.482633 was used as the fixed leg coupon in our
swap construction in order to ensure the swap was fairly priced. The market value in Figure 1
validates the swap construction with a market value of -$0.15 which in comparison to the
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notional amount is essentially zero as per the criteria of (Litzenberger, 1992). Note the PV01
and DV01 amounts in Figure 1. Due to the fact that the duration of the fixed leg is much
greater or “longer” than the floating leg and we have shorted the fixed rate in our payer swap
see equation 2 our net duration (Delta Dollar) DV01 value should be negative as is the case
in Figure 1. Additionally, because the PV01 value refers only to the fixed leg which we are
short combined with Bloomberg Terminals’ default -1 basis point PV01 convention
according to (Murphy, 2015) results in a positive PV01 (Basis Point Value) should be evident
see equation 3, this is the case in our swap creation as per Figure 1.
[
]
[
]
[
[
]
[
]
(2)
]
(3)
Figure 1: Plain Vanilla 0x12M Payer Swap Creation
Figure 2 and 3 highlight the key rate risks associated with the swap. These key rate risks
represent the delta dollar value of the fixed leg cash flows. The pay column in Figure 3
shows the practical use equation 1 whereby the total DV01 figure has now been stripped into
a series of Forward Rate Agreements which can be then used to offset the fixed-leg risk
exposure in a delta-hedge manner. Each DV01 value has been broken down for each fixed
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rate cash flow. Each FRA’s notional is approximately -$2000 and by receiving a 3mx6m, 6m
x9m and a 9mx12m Forward Rate Agreement we should be able to delta hedge our swap.
Note that the 0-3 month pay and receive values cancel each other out, this is due to the first
floating leg received being identical to a zero coupon bond whereby
in order for the swap to be priced at par and for the transaction the priced in a
fair manner as per (Litzenberger, 1992). Naturally, there is no need to use an FRA to hedge
something that is already been offset.
Figure 2: Key Rate Risk Exposures of Plain Vanilla 0x12M Payer Swap
Figure 3: Key Rate Risk Exposures of Plain Vanilla 0x12M Payer Swap as Forwards
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1.2.3 Delta-Hedging of Near Dated Swap
Due to the LIBOR 3 month floating rate being the reference floating rate in the 1 year swap,
the LIBOR 3 month reset rate was used as the rate in the FRA hedging legs, this is due to
LIBOR reset dates and rates being responsible for the individual floating-leg payments in the
swap since the contract inception as per (Murphy, 2015). Figure 4 shows the reset rates and
dates appropriate for our 3mx6m, 6mx9m and 9mx12m forward rate agreements with Figure
5 illustrating our FRA delta hedge with the appropriate reset rates. Using a notional of
$77,780,657.89 for each long receive fixed rate 3 month tenor FRAs spanning 3mx12m in
order to offset the identical swap notional the hedge was conducted according to (Murphy,
2015), if weights are equal you will have a low key rate risk exposure. The DV01 value
decreased dramatically when incorporating the FRA delta hedge diminishing from -6055 at
the inception of the swap (See Figure 1) to -1.91. The key rate risk exposures have also
dropped considerably from a min value of -1940 to -1.91 see Figure 6. The MARS delta
residual exposures in Figure 8 suggest that the hedge has been successful in virtually
eliminating the DV01 of the swap to almost zero, fluctuating between -$1.00 and $1.00 on a
notional of $77,780,657.89 = 0.00. The present value of the long receive fixed forward rate
agreements on the FRA settlement date can be denoted as:
PVFRAfixed = + [ RK – R(T,T+3) ] x T/360 x NP x DF(T+T)
(4)
RK denotes the FRA fixed rate which is set to equal the forward rate
which results in:
(5)
Should the LIBOR floating rate fall over the tenor of the FRA below the initial forward rate,
then by the T+3 settlement date the receiver will generate enough of a gain to offset the loss
in the value of the payer swap due to a parallel shift downwards in the LIBOR reference rate,
we are short the fixed rate in the payer swap and long the floating, therefore the receive fixed
FRA is an intuitive and practical approach to protecting capital from changes in the first
principal component and is an effective measure of hedging the near-dated swaps’ key rate
risk as evident by comparing Figure 3 and Figure 6.
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Figure 4: LIBOR 3 Month Reset Rates
Figure 5: Plain Vanilla 1Y Payer Swap Hedged with 3 x 3m FRAs
Figure 6: Key Rate Risks of Hedged Near Date Swap with FRAs
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Figure 7: Key Rate Risks of Hedged Swap Curve Instrument
Figure 8: Dollar Delta Exposure of Hedged Near Date Swap with FRAs
1.3 Far-Dated Offsetting Par Swap Hedge
A far dated 1Yx4Y receiver swap was created with a notional of $60,334,475.7 which is
calculated in Section B, this far dated swap was then hedged with an offsetting payer swap
with an equation notional amount in order to reduce our key rate exposure in a similar fashion
to section 1.3 and the constituents of the hedge can be seen in Figure 9 below, note that
similar to the near dated swap, the par coupon was used in the construction of the far dated
swap as seen in Figure 9 and 10 to ensure fair pricing upon inception. Notice the positive
DV01 value in Figure 10, this is down to being long the fixed bond in a receiver swap and
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thereby long the vast majority of the duration given that the duration of the fixed bond is
much longer than that of its floating counterpart. Figure 11 highlights the positive Key Rate
Exposure of the un-hedged far dated swap of +23570 with figure 12 exhibiting the success of
such an offsetting hedge with a hedged DV01 of +20.43 a reduction in exposure of 99.91%.
Figure 9: Far Dated 1Yx4Y Par Swap Hedge
Figure 10: Par Coupon Rate used for Far Dated Swap (Failry Priced)
[
]
[
]
[
[
]
[
]
(6)
]
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Figure 11: Key Rate Risk Exposure of the 1Yx4Y Receiver Swap
Figure 12: Far Dated Swap Key Rate Risk Exposure
Using the Risk Report feature in the Multi Asset Risk System in Bloomberg it became
apparent that the static offsetting hedge for the far dated swap reduced the residual exposures
significantly, see Figure 13. Utilizing multi-leg setup with an FRA hedging instrument for a
near date 1 year swap and an off-setting par payer swap hedging instrument for a far date
1Yx4Y receiver swap we were successfully able to delta-hedge against the Key Rate Risk
Exposures for both swaps as per our brief.
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Figure 13: Mars Analysis for Far Date Offsetting Static Hedge
2.0 Scenario Analysis /Yield-Spread Arbitrage Trading Strategy
2.1 Near Dated Swap
Whilst the near date 1 year payer swap is delta-hedged, it is not immune to gamma-like shifts
in the yield curve also known as 2nd and 3rd principal components similar to that of equity
delta-hedging. The 2nd principal component is a change in the slope of the yield curve and we
aim to profit from such a change. Figure 14 shows the MARS ran scenario test using the
following scenarios, the payoffs can be seen in Table 1:

Parallel Shift Down Scenario (-1 basis point)

Parallel Shift Up Scenario (+1 basis point)

Steepener Scenario ( Year 1 point on yield curve + 50 basis points)

Flattener Scenario ( Year 1 point on yield curve – 50 basis points)
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Figure 14: Near Date 1 Year Delta-Hedged Payer Swap MARS Scenario P/L
Near Date 1Y Payer Swap Delta-Hedged
Scenario
Profit/Loss
Parallel Up
$
Parallel Down
$
Steepner
$
Flattener
$
1.57
-2.28
-0.36
-0.36
Table 1: Near Date Scenarios Profit/Loss
Due to the implementation of the delta-hedge, one would expect a miniscule payoff with
regard to the 1st principle component i.e. a parallel shift up or down. Figure 14 and Table 1
illustrate this and validate the delta-hedging conducted on the near date swap using a multileg FRA hedging setup. It is interesting to note in Figure 14 and Table 1 that larger change in
the yield curve in the form of a steepener (1Y yield curve +50 basis points) and a flattener
(1Y yield curve -50 basis points) resulted in a smaller payoff than the parallel movements
despite the 2nd principal component being what we were looking to intentionally expose
ourselves to. Our intention was to profit from a steepener due to the fact that should rates
increase the fixed bond we are short in the payer swap should decrease significantly more
than the floating bond we are long given its much longer duration, thereby delivering a profit.
It seems as though in this instance we are somewhat neutral to slope changes in the yield
11 | P a g e
curve in addition to parallel movements despite our intention to profit from such a move. The
flattener loss would have been expected as decreasing rates would have outweighed the gains
from the fixed bond
but the magnitude of the losses were surprisingly small. Perhaps a
longer term near date swap would have yielded anticipated findings.
2.2 Far Dated Swap
Similarly to the near dated swap, out intentions were to expose the portfolio to non-parallel
shifts in the yield curve, in particular 2nd principal component slope change with an intention
of making a profit via the yield-spread. Figure 15 shows the MARS ran scenario test using
the following scenarios, the payoffs can be seen in table 2:

Parallel Down (-1 basis point)

Parallel Up (+ 1 basis point)

Steepener (4Y +50 basis points only)

Steepener (1Y -25 basis points, 4Y +25 basis points)

Flattener (4Y -50 basis points only)

Flattener (1Y +25 basis points, 4Y -25basis points)
Figure 15: Far Date 1Yx4Y Delta Hedged Swap
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Table 2: Far Date Delta-Hedged Swap Scenarios Profit/Loss
It is clear in the case of our far date 1Yx4Y delta-hedged swap that we have managed to gain
exposure to the changing slope 2nd principal component as we intended evident by the
payoffs in Table 2 and Figure 15. We are short the Libor-Swap yield-spread whereby a
steepening of the shorter term curve (1Y +25bps) in addition to a flattening of the longer
maturity curve (4Y -25bps) yielded a profit of $2193.49 and the flattening of the 4Y(-50bps)
yielded a profit of $4442.48. The differential between the two flatteners comes down to the
amplification of the present value from such a flattening event occurring in year 4 as per
(Murphy, 2015). Table 2 and Figure 15 also validated our delta-hedging strategy with a mere
$17.79 profit from a parallel movement downwards (1bps) and a $23.12 loss from a 1 basis
point increase in interest rates.
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Part B:
Swaption Risk Management & Yield Spread Arbitrage
Strategy
3.1 Introduction to Swaptions
Swaptions are options on interest rate swaps as they give the holder the right to enter into a
certain interest rate swap at a certain time in the future. There are two types’ swaptions, a
payer swaption and a receiver swaption. A payer swaption grants the holder the right but not
the obligation to enter into a swap agreement where the holder pays a fixed rate and receives
a floating rate. However a receiver swaption grants the holder the right but not the obligation
to enter into a swap agreement whereby they will receive a fixed rate and in turn they’ll pay a
floating rate. Our aim in part B of this project is to hedge a payer swaption using swaps,
which is similar to part A where swaps were hedged using FRA’s, and implement an
appropriate yield spread arbitrage trading strategy. The swaption used was a 1Y x 3Y
European payer swaption with a notional principle of $100 million dated from 04th May 2016
to 5th May 2019.
3.2 Dual Key Rate Exposures
The 1Y x 3Y payer swaption used for this assignment has a ‘dual’ key rate exposure which in
order to appropriately hedge must be fully understood. Analysing Black’s formula for valuing
a payer interest swaption underlying forward rate swap formula allows these ‘dual’ key rate
exposures can be rationalised.
[
]
[
(7)
]
√
√

PSt(T0,Tm,y(t,T0,Tm),K) denotes the current-date value of a payer swaption (i.e. a call
option on a forward-starting payer swap)

y(t,T0,Tm) denotes the current forward (par) swap rate (FPSR) on a payer swap
starting on date T0 > t and ending on date Tm

K denotes the strike level of the FPSR – if K = y(t,T0,Tm) we say that the swaption is
‘at-the-money-forward’, denoted as ATMF.
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
denotes the forward – swap ‘annuity factor’, the denominator term in the
following for imputing the FPSR (Murphy, 2015).
By examining equation 7 above one can ascertain that a payer swaption is essentially a call
option on the underlying forward par swap rate (y). Rearranging the above formula the
underlying forward par swap rate can be represented by the following expression, equation 8,
where the numerator term is the difference between the prices of two discount bonds. Similar
to how we are long S in a call option we are long the underlying forward swap rate. Hence if
the under lying forward rate (y) increases, similar to an underlying stock in a call option, the
swaption will increase in value.
(8)
In a payer swaption the investor is considered to be long a discount bond with expiry T0, in
this case a 1 year discount bond, and short a discount bond with expiry Tm, which in this case
is a 4 year discount bond. Thus the swaptions dual key rate exposure can be viewed in terms
of sensitivities to zero rates which affect the prices of the 2 discount bonds (P(t,T0) &
P(t,Tm)). By analysing these two key rate risk exposure we can see how the movement in
interest rates can affect the price of the swaption.
Firstly, with all others thing being equal (ceteris paribus) a decrease in the zero rates at
maturity, T0, will cause an increase in the value of 1 year discount bond P(t,T0) and hence a
ceteris paribus there will a rise in value of the payer swaption. This is the similar for the 4
year zero rate, however as we are short this position and the overall value of the payer
swaption decrease. Hence an increase in the 1 year zero rate would result in a fall in the
overall value of the payer swaption and an increase in the 4 year zero rate would result in a
rise in the value of the payer swaption. This intuitive understanding of what the dual key rate
exposures to the payer swaption are, along with how they affect the price of the swaption is
key to being able to fully understand how to implement an appropriate hedge. This analysis
can be reaffirmed using the 1Y x 3Y payer swaption with a 1 basis point drop in interest rates
in Bloomberg’s SWPM risk screen and MARS scenario screen as seen in figure 18 & 19
respectively. As can be seen the 1 year exposure increases the value of the swaption and the 4
year exposure decreases the value with a basis point drop and increases the value of the
swaption with a basis point increase. The net DV01 of the position is -18842.96. It is this figure
that we aim to hedge and get a close to zero as possible.
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3.3 Method of Hedging and Key Hedging Insights
In order to hedge the swaption we will use swaps that are sensitive to the same key rate
exposures as the swaption mentioned in the previous section. According to Hull (2012) in the
context of interest rates, delta risk is the risk associated with a shift in the zero rates curve.
However because there are so many ways in which the zero curve can change numerous
deltas can be calculated. Traders therefore prefer calculating the impact of small changes in
the quote for each of the instruments used in the construction of the Libor-Swap zero rates
curve. The Libor-Swap zero rates are determined by using a bootstrapping method. A
methodology used to ‘strip’ zero rates from the Libor market curve. Bootstrapping shows that
these zero rates are in fact directly correlated with the same-maturity Libor interest rates, and
hence practitioners view the key rate risk exposures as being to these market-quoted rates.
Which in turn, these key rates underline the dual (key) par swap rate exposures illustrated in
figure 17. The key advantage of using swaps to hedge key rate exposures of a swaption is that
swaps are sensitive to the same market quoted interest rates as swaptions, which is effectively
using instruments that have sensitivity to the swap rate rather than the zero rates. As the
swaption and the swap have sensitivity to the very same swap rate, it is an obvious choice to
use this as a prospective hedging instrument (Murphy, 2015).
These are some key hedging insights that practitioners use when hedging and are inferred
from the methodology used to “strip” zero rates from the LIBOR-Swap curve.
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Net DV01
Figure 16: Swaption construction
1Y +$4724
Dual Key rate risk exposures
4Y -$23592.26
Figure 17: Vanilla Swaption Risk Screens
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Figure 18: Swaption MARS Scenario 1bp analysis
Figure 19: MARS Scenario Analysis for vanilla Swaption
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4.0 Swaption Yield Spread Arbitrage Trading Strategy: P&L Capabilities
By delta hedging the payer swaption with swaps our portfolio is only hedged against small
parallel shirts in the yield curve. Movements in the yield curve can be typically categorised
into the following three risk factors which account for 95% or more of the variation in the
yield curve.

Level: are parallel shifts across the yield curve. This is when a one unit of that factor
movement in the yield curve. For example the 1yr rate increases by 1 bps and the 4yr
increases by 1 bps, and so on. As can be seen in figure 35 and table 3 the delta hedged
swaption is hedged against this type of small movement, incurring small gains and
losses for a +1 and -1 bps movement respectively.

Twist: Change in the slope curve (‘twist’). For example a ‘steepening’ or ‘flattening’
of the yield curve. The hedge does not protect against this and the position is fully
exposed to this type of movement. This is equivalent to Vega exposure on a delta
hedged call option.

Curvature: this risk factor is associated with the bending of the curve in the middle
region (Hull, 2012).
Therefore by delta hedging our swaption against small parallel shifts we deliberately exposed
to non-parallel movements in the slope of the yield curve in order to deliver a profit.
4.1 Type of curve movement

Steepener: this occurs when the gap between the yields of short-term rates and longterm rates increase causing a more positively sloped curve. The increase in this gap
indicates that yields on long-term rates are rising faster than yields on short-term
rates.

Flattener: this occurs when gap between the yields of short-term rates and long-term
rates decreases, reducing the slope of the curve causing it to appear flatter.
Both these types of movements allow for the potential of profit. If we take the view that the
curve will steepen then we can enter into a payer swaption. Inversely if we believe the curve
will flatten a put option on an interest rate should be purchased i.e. a receiver swaption.
19 | P a g e
4.2 Hedged Swaption specific movements
To analyse how movements in the yield curve can generate a profit for our yield spread
arbitrage trading strategy we must consider the swap valuation and payer swaption formula
below:
(8)
∑
[
[
(
∑
)
(
]
)
(9)
]
(10)
By considering the above formula changes in P(t,1Y) and P(t,4Y) in the swap valuation and
P(t,To) and P(t,Tm) in the payer swaption will have large effect on the profit generated
depending on the movements in rates.
The first yield curve movement in which a positive is generated is a decrease in the short
term rates, i.e. 1 yr. rates, while the long term rates, i.e. 4 yr. rates, increase. This has the
effect of steepening the yield curve as can be seen in figure 20 below.
Figure 20: Zero Yield Curve (Short term decrease & Long term increase)
20 | P a g e
The effects of the movement on the swaps and swaption values are summarised below:

1Y Payer Swap P&L. If the 1Y rate shifts downwards the 1Y swap will fall in value
as the payer swap has negative duration. The rationale behind this is that in a payer
swap you are effectively short the fixed bond and therefore once interest rates
decrease the bond value increases giving a loss on the swap.

4Y Payer Swap P&L. If the 4Y rate shifts upwards due to the receiver swap having a
positive duration the swap falls in value. As you are effectively long a fixed bond in a
receiver swap when interest rates rise, the bond value will in turn decrease.

1Y x 4Y Swaption P&L. The payer swaption increase in value can be explained by
using the payer swaption formula above. The decrease in the 1Y rate will increase the
value of the bond P(t,To) while the increase in the 4Y rate decreases the value of the
shorted bond P(t,Tm). Hence increasing the value of the underlying forward swap rate
and in turn increasing the value of the swaption. As can be seen from table 3 the
increase in value of the swaption is greater than the losses incurred by the 4Y
swaption which sequentially generates a profit.
A second yield curve movement that will generate a positive P&L is parallel shift plus a
change in the slope of the curve. This movement entails a decrease in the short term rates and
a greater decrease in the long term rates as seen in figure 21 below.
Figure 21: Zero Yield Curve (Short term decrease & larger long term decrease)
21 | P a g e
The effects of the movement on the swaps and swaption values are summarised below:

1Y Payer Swap P&L. If the 1Y rate shifts downwards the 1Y swap will fall in value
as the payer swap has negative duration. As mentioned in a payer swap are short the
fixed bond and therefore once interest rates decrease the bond value increases giving a
loss on the swap.

4Y Payer Swap P&L. If the 4Y rate shifts downwards to a greater extent than the 1Y
swap the receiver swap will increase in value much more than the decrease in the 1Y
bond. As you are effectively long a fixed bond in a receiver swap when interest rates
drop, the receiver swap will in turn increase.

1Y x 4Y Swaption P&L. The payer swaption decreases in value can be explained by
using the payer swaption formula above. The decrease in the 1Y rate will increase the
value of the bond P(t,To) while the increase in the 4Y rate increases the value of the
shorted bond P(t,Tm). Hence decreasing the value of the underlying forward swap rate
and in turn decreasing the value of the swaption. As can be seen from table 3 the
decrease in value of the swaption is greater than the losses incurred by the 4Y
swaption which sequentially generates a profit.
Another movement which will earn a positive P&L is a flattening of the yield curve whereby
the short term rates stay the same while the mid to long rates decrease as seen in figure 22
below.
Figure 22: Zero Yield Curve (Long term decrease)
22 | P a g e
The effects of the movement on the swaps and swaption values are summarised below:

1Y Payer Swap P&L. If the 1Y rate does not move there is no effect on the payer
swap value.

4Y Payer Swap P&L. If the 4Y rate decreases, due to the receiver swap having a
positive duration the swap rises in value. As you are effectively long a fixed bond in a
receiver swap when interest rates fall, the bond value will in turn increase.

1Y x 4Y Swaption P&L. The payer swaption decreases in value can be explained by
using the payer swaption formula. The decrease in the 4Y rate increases the value of
the shorted bond P(t,Tm). Since the 1Y bond P(t,To) stays constant the net effect is a
decrease in the forward swap rate thereby decreasing the value of the swap. As can be
seen from table 3 the decrease in value of the swaption is less than the gain main by
the 4Y swaption which sequentially generates a profit.
Although there are a number of movements in which a profit may be generated it is worth
noting there are movements whereby losses can be incurred such as, for example, when short
term rates increase and longer term rates increase to a greater extent. Another scenario which
a loss can be incurred is a ‘flattener’ at the short end of the curve.
5.0 Swaption Yield Spread Arbitrage Trading Strategy: Implementation
and Risk Analysis
In this section, the swaption yield curve spread arbitrage strategy portfolio will be
incrementally implemented, analyzing the DV01 exposure and vulnerabilities to yield curve
spread movements at each stage. Section 1 will show the implementation of the 1Y x 3Y
swaption and highlight the key rate risks and risk scenarios implicit to the interest rate
derivative. Section 2 will demonstrate the swap hedging strategy used to reduce the market
value exposure to small parallel shifts in the yield curve. Section 3 will describe how the use
of forward rate agreements (FRAs) was incorporated into the arbitrage portfolio to minimize
the residual exposure to short-term yield curve movements.
23 | P a g e
5.1 Swaption Construction and Key Rate Risk Exposure Analysis
Figure 23 summarizes the key features with regards the long payer swaption being exploited
to create the yield spread arbitrage trading strategy. This highlights the interest rate
derivatives negative exposure to small parallel yield curve movements in terms of market
value, with a DV01 of -18,842.96. This implies that if there is a -1bp parallel shift in the yield
curve, the market value of the swaption would decrease by -$18,842.96. The swaption
provides the investor the right but not the obligation to enter into a 3 year long payer swap on
05/06/2016 at 1.721377% fixed rate in exchange for the floating rate.
Figure 23: 1Y x 3Y Payer Swaption SWPM construction
The key rate risk exposure of the swaption was investigated using the SWPM Risk
functionality on the Bloomberg terminal and summarised in Figure 24. Figure 25 indicates
that the 1Y x 3Y long payer swaption is positively exposed to the one year swap rate and
negatively exposed to the 4 year swap rate.
This feature can be explained by the fact that the long payer swaption holder is indirectly
long the 1 year discount bond price and short the 4 year discount bond price:
(12)
Where
is the “underlying” forward par swap rate, and the numerator values
denote the 1 year and 4 year discount bonds respectively. Since the payer swaption provides
the right to enter into a payer swap in 1 year, the market value of the swaption increases when
the underlying payer swap increases via upward shifts in the zero rates curve.
24 | P a g e
Figure 24: Key Rate Risk Exposure of 1Y x 3Y Long Payer Swaption
Figure 25: Key Rate Risk Exposure to 1 year and 4 year swap rates
To further illustrate the interest rate derivatives positive sensitivity to short term yield shifts
and negative sensitivity to the 4 year swap rate, the Multi-Asset Risk platform in Bloomberg
was leveraged to perform analytics to validate the ‘dual-risk’ exposure to the 1 year and 4
year swap rates and to investigate the swaptions behaviour under various yield curve
scenarios.
Figure 26: Swaption MARS Scenario -1bp analysis
25 | P a g e
Validating figures 24 and 25, figure 26 demonstrates a significant negative sensitivity to the 4
year swap rate and upward exposure over the O/N to 1 year maturity spectrum to a -1bp
parallel shift. The cumulative intuition stemming from these results implies that, in order to
hedge the interest rate derivatives exposure to small parallel movements, a 1 year payer swap
and a 4 year receiver swap should be included into the portfolio. The negative duration of the
1 year payer swap determines that small downward parallel shifts would create a fall in
market value or P&L. Thereby, through the manipulation of the notional of the 1Y payer
swap, the swaption would be immunised in terms of DV01. Additionally, through the positive
DV01 feature of the 4 year receiver swap, the swaptions negative exposure to the 4 year swap
rate may be hedged.
In order to illustrate the potential market scenarios whereby the 1Y x 3Y payer swaption
would generate profit or loss, the Bloomberg MARS user-defined scenario functionality was
utilized and provided in Figure 27. This highlights positive profitability characteristic of the
1Y x 3Y payer swaptions’ market value with regard to both upward parallel shifts in the yield
curve and the steepening of the yield curve. This can be explained via the greater sensitivity
of the swaption to the 4 year swap rate, generating greater profit through upward yield curve
movements at the 4 year mark.
In terms of small ( 1bp) parallel movements in the yield curve, the swaption has greater
positive exposure to upward shifts than negative exposure downward shifts. This is due to the
excess profit generated by the short 4Y discount bond beyond the loss impacted by the long
1Y discount bond.
The steepening and flattening of the zero curve by 50bps at the 4Y mark only generates a
greater magnitude of profit and loss respectively, when compared to the tilt imposed at the
1Y and 4Y maturities. This highlights the interest rate derivatives substantial sensitivity to the
absolute value change to the 4Y zero rate, which will be addressed via a swap hedging
strategy in Section 2.
26 | P a g e
Figure 27: MARS Scenario Analysis for Vanilla Swaption
27 | P a g e
5.2 Swap Hedging Strategy Implemented
It is comprehensively identified and illustrated in Section 1 that the 1Y x 3Y payer swaption
has ‘dual rate’ exposure to the 1 year swap rate and the 4 year swap rate. Swaps, as identified
in Section A, are fixed income financial assets which are also analogously exposed to
movements in the swap rates. These simultaneous key rate risks can be utilized to In order to
hedge these key rate sensitivities and thereby reduce DV01 exposure, a 1Y payer swap and a
4Y receiver swap will be added to the portfolio.
5.2.1 Payer Swap
To hedge the positive key rate risk exposure to the 1 year swap rate, a 1 year payer par swap
is included into the portfolio. The gain expected from a -1bp parallel shift in the yield curve
at year 1 is identified in Figure xx to be +$6063.33. Since a 1-year maturity payer swap will
fall in value for this particular interest rate decrease, a long position will be demonstrated in
the portfolio.
The notional principal amount required to immunize the portfolio to the 1 year par swap rate
was calculated by the change in value of the payer swaption divided by the corresponding
change in value of the payer swap. Therefore, using the 1Y KRR exposure from the SWPM
Risk screen of the swaption (+$6063.33) and the Risk figure generated from the par payer
swap (see Figure 28), the notional was calculated as follows:
(13)
Figure 28: 1yr Par Payer Swap Risk with Notional set to $1
Figure 29 shows the final swap manager tool for the 1Y payer swap, illustrating that the
notional amount calculated in Equation 13 combined with utilizing the par coupon, generates
a negligible costing 1 year payer swap with DV01 exposure of -6055.00. This swap will be
added to the portfolio to hedge against the positive +6063.33 DV01 sensitivity of the payer
swaption to the 1 year swap rate.
28 | P a g e
Figure 29: 1 Year Pay Fixed Notional + DV01
5.2.2 Receiver Swap
In order to hedge the significant second key rate risk sensitivity to the 4 year swap rate, a 4
year receiver par swap is required i.e. short 4Y payer swap. Figure 25 indicated that a small
negative parallel shift in the yield curve at year 4 maturity generates a $23,590.75 loss for the
1Y x 3Y payer swaption. The 4-year receiver swap will gain in value for a small decrease in
the yield curve due to its positive duration, providing an off-setting hedge for the swaptions
second key rate risk.
The notional amount was calculated using the 4Y KRR exposure (-$23,590.75) and the Risk
figure generated from the 4 year par receiver swap provided Figure 21, which was generated
by setting the notional value to $1 and coupon to the par coupon.
(14)
Figure 30: 4 Year Receiver Swap Risk with Notional set to $1
Figure 31 indicates that the notional amount generated in Equation 14 combined with the par
coupon, generates an approximately zero cost 1 year payer swap with DV01 exposure of
+23,571.77. Its inclusion in the portfolio will off-set the $23,590.72 negative exposure of the
1Y x 3Y payer swaption to the 4 year swap rate.
29 | P a g e
Figure 31: 4 Year Receiver Swap Notional & DV01
5.3 Swaption Key Rate Hedge Portfolio Combined
The resultant DV01 gained via the combined portfolio of the 1Y x 4Y Payer Swaption and
dual key rate risk off-setting 1 Year Payer Swap and 4 Year Receiver Swap was analyzed on
the MARS platform. Figure 32 indicates a DV01 value of -1302.10 residual exposure
underlying the combined portfolio, a 92.03% reduction from the vanilla swaption DV01 of
18,842.96.
Figure 32: MARS Overview - 1Y x 3Y Payer Swaption, 1Y Payer Swap & 4Y Receiver Swap
The risk analysis functionality of MARS provides the P&L breakdown of each constituent
component of the portfolio at defined maturities corresponding to a -1bp parallel shift in the
yield curve as illustrated in Figure 33. This function provides investors with greater ability to
investigate such residual exposures over the SWPM Risk tool as it provides a granular
30 | P a g e
breakdown of the component exposures over the different terms. Figure 33 highlights that the
1 year payer swap generates 6 month and 12 month exposure in excess of that required by the
payer swaption, resulting in residual exposures that will be addressed later in the report.
Figure 33: Delta hedged Swaption
The time series of net residual exposures of the Swaption Key Rate Hedge portfolio are
replicated in Figure 34, exemplifying a clear need to add a 3M x 6M Forward Rate
Agreement (FRA) and a 9M x 12M (FRA) to hedge these sensitivities.
Figure 34: Residual Analysis of Delta-Hedged Portfolio
Bloomberg’s MARS user-defined scenario framework was utilized to quantitatively analyse
the resultant constituent P&Ls of the Swaption Key Rate Hedge portfolio given a
comprehensive set of underlying yield curve movements. Figure 34 summarises the effect of
6 separate interest rate scenarios at a portfolio level, while Table 1 analyses the P&Ls
generated by each scenario by at a granular level.
It is noted that the swaption has been adequately delta-hedged, with 1bp parallel upward and
downward shifts in the yield curve provoking +$1,510.54 and -$1,558.32 respectively. This
31 | P a g e
portfolio idiosyncrasy has been identified in the the residual analysis as over-exposure
implicit to the 1 year payer swap, a result which is further validated in columns 1 and 2 in
Table 1.
Given a non-parallel steepening of the yield curve at the 4 year tenor only, the hedged
portfolio achieves a $242,423 profit, resulting from the profit generated by the Payer
Swaption in excess of the the loss inflicted on the portfolio by the 4Y Receiver Swap. The 1Y
Payer Swap component is not affected by this steepening effect of the yield curve at the later
maturity.
The ‘steepening’ effect of the yield curve was investigated in terms of widening the spread
between the two key rate risk maturities, by reducing the 1 year yield by 25bp and increasing
the 4 year yield by 25bp. This market scenario generates $70,111 profit within the portfolio,
explictily through the excess profit generated by the payer swaption over the long dated 4
year receiver swap. Notably, the 1Y payer swap is not effected by this non-parallel slope
movement, implying that the fixed income financial asset is insensitive to these forms of
curve movements since the initial -25bp shift initiates at its matutity date.
Conversely, the impact of the contraction of the yield curve spread was analysed in the final
two scenarios in a reverse manner to that of the ‘Steepener’ market scenarios. It is seen in
Table 1 that the 4 year receiver swap generates surplus profit over resulting losses observed
in the 1Y x 3Y payer swaption. Again, the negligable effects of the ‘flattening’ yield curve is
demonstrated by the 1Y payer swap in an analogous fashion to the ‘steepening’ scenarios
provided above.
Through the analysis of the resultant P&Ls generated from the comprensive battery of user
defined scenarios on the Bloomberg MARS platform, it is clear that the Swaption Key Rate
Hedge portfolio is satisfactorily delta-hedged and positively exposed to both upward and
downward non-parallel yield curve movements. As mentioned in section 4.2 movements such
as a steepener, which is the decrease in short term rates and the increase in the long term
rates, a profit should be generated.
32 | P a g e
Figure 35: MARS Scenario Analysis - Delta-hedged Swaption
Portfolio
Component
Swaption
1Y Payer Swap
4Y Receiver Swap
Total P&L
Parallel Down
(-1bps)
$
$
$
$
-22,731
-6,050
27,223
-1,558
Parallel Up
(+1bps)
$
$
$
$
15,350
6,059
-19,899
1,511
Steepener
(4Y +50bps Only)
$
$
$
$
1,412,948
5
-1,170,531
242,423
Steepener
(1Y -25bps 4Y
+25bps)
$
654,556
$
5
$
-584,450
$
70,111
Flattener
Flattener
(4Y -50bps Only) (1Y +25bps 4Y -25bps)
$
$
$
$
-862,522
5
1,185,947
323,430
$
$
$
$
-521,954
5
593,789
71,841
Table 3: Scenario Analyses - P&L Breakdown of each portfolio component
33 | P a g e
6.0 Residual Exposure Strategy
As an additional feature to our Swaption Key Rate Hedge portfolio, a Residual Exposure
Strategy is imposed with the objective of reducing the combined portfolios sensitivity to
near-dated surplus risk to small yield curve movements. The aggregate effect stemming from
the inclusion of this strategy is to reduce the total portfolio’s DV01 value.
Residual exposures to 6 month and 12 month small movements in the yield curve are
identified in section 5.3, indicating that the appropriate measure is to include a 3M x 6M
FRA and 9M x 12M FRA into the portfolio.
The residual exposures identified in Section 2.3 are summarised in Table 2 and the Notional’s
are calculated using the formula:
(
)
(4)
Setting the coupon to the par, as per Figure 36, the Market value of the 3M x 6M FRA
reduces to approximately zero.
Table 4: Residual Exposures of delta-hedged swaption and FRA National’s required
Figure 36: 3M x 6M FRA SWPM construction
34 | P a g e
Figure 37: 3M x 6M FRA Reduces 6 Month Exposure
The inclusion of the short 3M x 6M FRA into the Swaption Key Rate Hedge portfolio
demonstrates a dynamic hedge to the 6 month residual exposure as seen in Figure 37.
The 9M x 12M FRA was constructed similarly, setting the coupon to the Market Rate and
calculating the notional amount required to off-set the residual exposure to the 12 month rate
using Equation 4 as seen in Figure 38.
Figure 38: 9*12 Month FRA
35 | P a g e
Figure 39: 9M x 12M FRA reduces 1 Year Exposure
By including the short 3M x 6M FRA into the combined portfolio, the 12 month residual
exposure is considerably reduced as demonstrated in Figure 39. As a result of implementing
the FRA residual exposure reduction strategy, the overall DV01 of the Swaption Key Rate
Hedge portfolio has reduced from -1302.10 to -247.55, a 98.7% reduction from the initial
vanilla swaption DV01 of 18,842.96.
Figure 40: Final Swaption Portfolio Strategy DV01
36 | P a g e
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