swap pricing after the lehman collapse & continous...
TRANSCRIPT
Swap pricing after the Lehman
collapse &
Continous rate models Christian Heggen, Nordea Markets
LIBOR
How to calculate LIBOR payments
22,472.88360
91035,0000.000.10
360
actLNotR
LIBOR is quoted on an act/360 – basis. I.e. One
assumes 360 days in a year, but interest rates are paid
on actual days…
1, 3, 6, 9 eller 12 mnd
2 days
Interest rate
period LIBOR fixing
IR are paid
Basic swap pricing
%0.2%991.1
360
1820185.01
360
911
360
91017.01
),(),(1),(),(1),(),(1 313132322121
x
x
ttttLttttLttttL
• You have bought a house and you have been granted a mortgage in
a bank
• When you go to collect the mortgage and sign the papers, you are
given the opportunity to choose between 3 mths or 6mhts fixed rate
in the first period
• 3mths: 1.70
• 6mths: 1.85
• What do you choose? (and how should you think?)
Forward rates in USD and NOK
-
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
4.50
5.00
0 1.25 2.5 3.75 5 6.25 7.5 8.75 10 11.25 12.5 13.75
NOK USD
Loan in combination with an IRS
Company
Loan
Loan
agreement
IRS
Fixed rate
3 m Libor
3 m Libor
+ cm
Company pay 3mth Libor + cm (loan)
Company pay Fixed rate (IRS)
Company receive 3mth Libor (IRS)
Company pay Fixed rate + cm
IRS in combination with a bond issue
Company
Bond
Loan
agreement
IRS
6m LIBOR+cm
Fixed coupon
Fixed
coupon
Company pay Coupon (Bond)
Company pay 6mL+cm (IRS)
Company receive Coupon (IRS)
Company pay 6mL + cm
Cross Currency swap
• Start (spot) • Maturity
Company Company
X
(EUR)
X*S
(USD)
S=FX spot rate
X=Notional
X
(EUR)
X*S
(USD)
• During the term
Company
EUR
3M
LIBOR
+/- B
USD
3M
LIBOR
EUR
3M
LIBOR
+/- B
USD
3M
LIBOR
B is the price of the currency swap
• Company can fund itself in USD but needs EUR
• Company delivers USD to Nordea in exchange of EUR in a currency swap
• Company pays EURIBOR to Nordea and recives LIBOR in currency swap
• At maturity the notionals are reversed at initial exchange rate
• A currency swap is basically an exchange of loans in different currencies
Credit Support Annex (CSA)
• A CSA is an agreement that regulates the maximum credit exposure between
two derivative counterparties
• If the mtm of the portfolio party A has against party B is negative, then party B
has implicitely granted party A a loan
• Under banking regulations, party B must then hold capital against credit losses
in case party A defaults before it has paid off the mtm of the derivatives
• In order to minimize the capital usage, party A can post collateral to party B in
order to reduce the credit risk party B bears
• How the collateral is calculated and posted is regulated in a CSA agreement
• Parameters in the agreement:
• Threshold
• Frequency of posting collateral
Agenda
The fundamental assumptions used when pricing up interest rate derivatives have
changed dramatically over the last years resulting in new methods for constructing
rate curves. This presentation describes the changes that have happened in the
market and the new interest rate curves needed to cope with the new market
situation.
• Brief historic overview → What happened to the market?
• How must curves be changed to adapt the new market conditions?
• Curve design in Nordea, before and after the crisis, focus on basis swaps.
Basic swap pricing
),(),(1),(),(1),(),(1 313132322121 ttttLttttLttttL
• Forward rates can be calculated using bootstrapping
• Discounting on forward rates
)(1)(),(),(1
11 m
m
i
iiiflii tDtDttttL
Standard yield curves, pre crisis
• It was plausible to construct one yield curve for pricing of all derivatives. Swap rates had next to
no dependency on the floating rate index. The reason for this was that the credit and liquidity
premium on 6m rates over 3m rates was negligible.
• The various indices were assumed to be risk-free. Forecasting of any index and discounting
could be done on the same curve which represented the risk-free “time value of money”.
• A yield curve could therefore be constructed from a number of liquid different instruments
although these were linked to different indices. Example: deposits of various lengths, FRAs /
FUTs (typically 3m) and swaps (typically 6m).
• The value of collateral agreements (CSAs) were disregarded.
2.0
2.5
3.0
3.5
4.0
4.5
0 5 10 15 20 25 30
Tenor
Sw
ap
ra
te
EONIA 3M 6M
Example of pre-crisis EUR rates
(02-01-2007)
Crisis Start!
),(),(1),(),(1),(),(1 313132322121 ttttLttttLttttL
• Banks realize that they can loose money when lending to each other,
even the biggest players may go bust (Lehman Brothers).
• The longer the deposit time, the higher the credit add-on.
)(1)(),(),(1
11 m
m
i
iiiflii tDtDttttL
The development of the 3v6 basis
0.0
1.0
2.0
3.0
4.0
5.0
6.0
Jan-04 Jan-05 Jan-06 Jan-07 Jan-08 Jan-09 Jan-10
%
0
10
20
30
40
50
60
Bp
3m EURIBOR 6m EURIBOR 3m3m FRA 3m6m basis (ra)
Pre credit crisis
Lehmann collapse
),()),((1),(),(1),(),(1 313132322121 ttBSttLttttLttttL
Post crisis, the standard yield curves break down
• Swaps linked against different indices trade with a significant basis, indicating that:
• Multiple curves representing different indices are needed.
• All instruments on one curve must be linked to the same index in order for the curve to be
consistent.
• Forecasting and discounting cannot always be done on the same curve.
• The value of CSAs are now being taken into account in derivatives pricing.
Example of post-crisis EUR rates
(06-09-2010)
0.5
1.0
1.5
2.0
2.5
3.0
0 5 10 15 20 25 30
Tenor
Sw
ap
ra
te
EONIA 1M 3M 6M
Four major changes due to the crisis
1. Forward curves are split up
2. Discount- and forward-curves separate
3. Discounting depends on the collateral
4. Forward rates are “OIS-based”
Crisis change 1 - forward curves are split up
• After the crisis trades against different indices
have different prices where they used to be the
same, e.g. the 3m vs. 6m Euribor spread used to
be zero’ish.
• Any pricing setup must incorporate this such that
the forecasting of future fixings is correct, i.e.
forward rate curves must be tenor dependent.
Before Crisis
All Euribor indices
on one forward
curve
After Crisis
Separate forward
curves for different
Euribor indices
Crisis change 2 – discount- and forward-curves separate
• In order to avoid arbitrage, all trades must be discounted on the
curve that most correctly reflect the time value of money.
• This means that forecasting and discounting are different for some
trades. A curve setup that allows for different forecast and
discounting curves is necessary: Two-Curves.
• The correct discount curve to use for non-collateralized trades is the
funding curve of any bank, but for practical reasons a Libor curve is
often used.
Before Crisis
One Euribor curve
After Crisis
Forecasting on one curve
– discounting on another.
Crisis change 3a – discounting depends on the collateral
• A collateral agreements will produce extra cash-flows
associated with a trade.
• The extra cash-flows can be seen as an “ad-on-trade”
to the actual trade.
• The extra cash-flows basically change the funding
costs involved. If the collateral earns the overnight rate
the trade should therefore be discounted on a
overnight rate.
Trade
Collateral
agreement
Part A Part B
cash
NPV
cash
Interest
With collateral
Trade Part A Part B
cash
NPV
Without collateral
Funding
source
Interest
cash
𝐷𝐹(1 + 𝑂𝐼𝑆 ∆𝑇) = 1
𝐷𝐹 =1
1 + 𝑂𝐼𝑆 ∆𝑇
Alternative argument:
One EUR received in the future. If
the trade is fully collateralized, the
collateral must be equal to the NPV.
This grown at the OIS rate must be
equal to the future cash.
OIS swaps
• An Overnight Indexed Swap “OIS” is a swap where the floating leg
compounds daily rates.
m
i
i
o
k
k
n
j
jijfixn tDdrtDttO1 11
1 )(1)1(1)(),(
Crisis change 3b – collateral discounting depends on currency
• The CSAs typically specify that collateral can
be posted in different currencies.
• Due to the cross currency basis spreads, the
value of the collateral depends on the
currency of the collateral.
• The correct rate to discount with is the
overnight rate plus the OIS basis spread (the
basis spread for a currency basis swap where
the floating legs exchanged are linked to the
OIS index).
• The OIS basis spread can be calculated from
“ordinary” basis swap discount curves
combined with OIS forward curves.
• A basis swap will always have at least one leg
where the collateral is in a different currency
than the payment currency.
Market
Trade
Collateral
agreement
Part A Part B
cash (EUR)
NPV (EUR)
cash USD
Interest (USD)
EUR trade with USD collateral
Pay cash USD
receive cash EUR
Receive USD interest
Pay EUR interest + basis spread
Basis swap
Crisis change 4 – Forward rates are “OIS-based”
• The interbank trades are almost always collateralized.
• Therefore, the observed market is collateralized.
• There can only be one correct forecast of the forward rate
for any given period, namely the implied forward rate
constructed with OIS discounting.
• When curves are constructed from swaps, which are OIS
discounted, the implied forward rates are different from
what they would be if the cash flows were discounted on
for example an Euribor curve.
Standard curve construction
• Choose a number of market observed instruments.
• Choose the basic object of the curve / the object in
which you want to do the interpolation – e.g. zero
coupon rates or forward rates. R(t).
• For each market observed instrument choose a time,
typically the latest payout of the instrument Tj .
• Choose an interpolation method – e.g. flat, linear, spline.
• Adjust the rates R(Tj) such that all instruments are
priced at par. There is one unique solution as long as all
market instruments contain different and non-
overlapping information and the set of {Ti} are chosen
wisely.
• The solution can be found either by bootstrapping or
global solving.
R
t
R(T1)
R(T2)
R(T3)
R(T4) R(T5)
Typical EUR curve, pre crisis
ON
TN
1W
2W
3W
1M
2M
3M
Mar2011F
Jun2011F
Sep2011F
Dec2011F
Mar2012F
Jun2012F
Sep2012F
Dec2012F
3Y
4Y
5Y
6Y
7Y
8Y
9Y
10Y
11Y
12Y
13Y
14Y
15Y
16Y
17Y
18Y
19Y
20Y
21Y
22Y
23Y
24Y
25Y
26Y
27Y
28Y
29Y
30Y
35Y
40Y
50Y
Deposits
1M, one month deposit
FRAs/FUTs
Jun2011F: June starting IMM-future
Swaps
10Y: 10Y swap using "correct" conventions
• Before the crisis, a standard yield curve could be
constructed from a number of different instruments
although these were linked to different indices.
Example: deposits of various lengths, FRAs / FUTs
(typically 3m) and swaps (typically 6m).
Curve construction with multiple curves
• For the forward and discount curve choose
instruments, interpolation object, interpolation
method and dates for curve points.
• Start by finding the solution for the discount curve.
This is done in the exact same way as the standard
curve.
• More than one discount curve may be constructed
to price trades under different collateral
agreements.
• Continue by finding the solution for the forward
curve. This is done by adjusting the points on the
forward curve until the instruments on the curve is
priced at par using the discount factors from the
discount curve.
• For advanced curve setups there may be multiple
forward curves that depend on each other. In such
cases all forward curves must be build together.
1d Fwd
1m Fwd
3m Fwd
6m Fwd
1y Fwd
OIS Disc
(1) Construct OIS disc curve.
(2) Use the OIS disc factors when
constructing all forward curves.
EUR
"build as if the build-date was jan 1 2012"
OIS Disc Curve LIBOR Disc Curve Fwd Curves-Build from OIS Disc
BASE
Explanations / Examples 1d 3m 1d 1m 3m 6m 1y OIS Basis swaps Libor Basis swaps
Deposits ON ON ON RFR0M:1M RFR0M:3M RFR0M:6M RFR0M:12M
1M, one month deposit TN TN TN
FRAs/FUTs 1W 1W 1W
RFR0M:3M: 3M fixing 2w 2w 2w
Jun2011F: June starting IMM-future 3w 3w 3w
Swaps 1M 1M 1M
10Y: 10Y swap using "correct" conventions 2M 2M 2M 2M
Float spreads 3M 3M 3M 3M Mar2012F Mar2012:6M:3M Mar2012:12M:3M 3M 3M
4M 4M 4M
Fixed spreads 5M 5M 5M
Dec2011:6M:3M: IMM-dated spread 3v6, 6M long 6M 6M 6M 6M Jun2012F Jun2012:6M:3M Jun2012:12M:3M 6M 6M
5Y:6M: spot starting spread, 5Y long. 7M 7M 7M
Basis swaps 8M 8M 8M
9M 9M 9M 9M Sep2012F Sep2012:6M:3M Sep2012:12M:3M 9M 9M
Fx swaps 10M 10M 10M
11M 11M 11M
1Y 1Y 1Y 1Y Dec2012F Dec2012:6M:3M Dec2012:12M:3M 1Y 1Y
Mar2013F Mar2013:6M:3M Mar2013:12M:3M
18M 18M 18M 18M Jun2013F Jun2013:6M:3M Jun2013:12M:3M
Sep2013F Sep2013:6M:3M Sep2013:12M:3M
2Y 2Y 2Y 2Y Dec2013F Dec2013:6M:3M 2Y 2Y
Mar2014F Mar2014:6M:3M
Jun2014F
3Y 3Y 3Y 3Y:6M 3Y:6M 3Y 3Y:6M 3Y 3Y
4Y 4Y 4Y 4Y:6M 4Y:6M 4Y 4Y:6M 4Y 4Y
5Y 5Y 5Y 5Y:6M 5Y:6M 5Y 5Y:6M 5Y 5Y
6Y 6Y 6Y 6Y:6M 6Y:6M 6Y 6Y:6M 6Y 6Y
7Y 7Y 7Y 7Y:6M 7Y:6M 7Y 7Y:6M 7Y 7Y
8Y 8Y 8Y 8Y:6M 8Y:6M 8Y 8Y:6M 8Y 8Y
9Y 9Y 9Y 9Y:6M 9Y:6M 9Y 9Y:6M 9Y 9Y
10Y 10Y 10Y 10Y:6M 10Y:6M 10Y 10Y:6M 10Y 10Y
11Y 11Y 11Y 11Y:6M 11Y:6M 11Y 11Y:6M
12Y 12Y 12Y 12Y:6M 12Y:6M 12Y 12Y:6M 12Y 12Y
13Y 13Y 13Y 13Y:6M 13Y:6M 13Y 13Y:6M
14Y 14Y 14Y 14Y:6M 14Y:6M 14Y 14Y:6M
15Y 15Y 15Y 15Y:6M 15Y:6M 15Y 15Y:6M 15Y 15Y
16Y 16Y 16Y 16Y:6M 16Y:6M 16Y 16Y:6M
17Y 17Y 17Y 17Y:6M 17Y:6M 17Y 17Y:6M
18Y 18Y 18Y 18Y:6M 18Y:6M 18Y 18Y:6M
19Y 19Y 19Y 19Y:6M 19Y:6M 19Y 19Y:6M
20Y 20Y 20Y 20Y:6M 20Y:6M 20Y 20Y:6M 20Y 20Y
21Y 21Y 21Y 21Y:6M 21Y:6M 21Y 21Y:6M
22Y 22Y 22Y 22Y:6M 22Y:6M 22Y 22Y:6M
23Y 23Y 23Y 23Y:6M 23Y:6M 23Y 23Y:6M
24Y 24Y 24Y 24Y:6M 24Y:6M 24Y 24Y:6M
25Y 25Y 25Y 25Y:6M 25Y:6M 25Y 25Y:6M
26Y 26Y 26Y 26Y:6M 26Y:6M 26Y 26Y:6M
27Y 27Y 27Y 27Y:6M 27Y:6M 27Y 27Y:6M
28Y 28Y 28Y 28Y:6M 28Y:6M 28Y 28Y:6M
29Y 29Y 29Y 29Y:6M 29Y:6M 29Y 29Y:6M
30Y 30Y 30Y 30Y:6M 30Y:6M 30Y 30Y:6M 30Y 30Y
35Y 35Y 35Y 35Y:6M 35Y:6M 35Y 35Y:6M
40Y 40Y 40Y 40Y:6M 40Y:6M 40Y 40Y:6M
50Y 50Y 50Y 50Y:6M 50Y:6M 50Y 50Y:6M
Nordea EUR curve, post crisis
• Multiple forward curves
with consistent
instruments.
• Two discount curves
depending for pricing of
trades with and without
collateral.
• Continous time interest rate models
• Short rate models
• Hull-White
• Libor Market Models
Introduction
• In Norway the general perception is that floating rate mortgages is the best in the
long run with the least risk
• In USA (and Denmark) the general perception is that fixed rate mortgages is the
best in the long run
• Why?
• One possible explanation is that fixed rate mortgages in USA can always be
refinanced at par
• Fixed rate mortgages in Norway have to be refinanced over par if rates have fallen
• How can banks in USA and Denmark offer fixed rate mortgages where the
borrower always can refinance at par?
Cap
1. fix 2.fix 3.fix 4.fix 5.fix 6.fix 7.fix 8.fix 1
1,5
2
2,5
3
3,5
4
4,5
5
No No Yes Yes No No No Yes Exercise
Option?
Cap:3,5 %
Cap - pricing
n
i
iiiii dNKdNFTDNotCap1
211,0
• Market practice is to price caps and floors with Black 76
• Hence market prices on caps and floors are quoted in volatility terms with the
assumption that one is using Black 76 to calculate the actual price
• All other parameters are assumed to be exogenous variables
• All intuition around option prices is gained and communicated with the
assumption that you are in a Black & Scholes world
• The price of a cap is therefore
• Where F is the forward rate starting at time i, K is the strike, D is the 0-coupon
bond maturing at time i+1 and delta is the time fraction for period i
Swaptions and Bermudas
• A Swaption is an option where the owner has the right but not the obligation to
pay fixed rate (payer swaption, call) or receive fixed rate (receiver swaption, put)
in an IRS at the maturity of the option period
• The market practice is to price swaptions using Black 76
• Note: A cap is the sum of options on forward rates while a swaption is an option
on the sum of forward rates
• Price of cap >= Price of swpation
• Why?
• In an American swaption the holder has the right to exercise the option any time
before expiry
n
i
ii dKNdNFTPHPayer1
2101,0
Swaptions and Bermudas
• In a Bermuda option, the holder has the right to exercise the option on specific
dates before expiry.
• The tenor on the underlying swap becomes shorter for each exercise
• Hence a 5 yrs Bermudan swaption is the sum of exercises:
• 0->5yrs swap, 1->4yrs swap, 2->3yrs swap, 3->2yrs swap, 4->1yr swap
• BUT each exercise contingent upon the latter not being exercised
• How to price such a swaption?
• Black & Scholes world?
• Sum of the values of each exercise
Short rate models – Hull White
• Models the short rate and the whole yield curve is decribed in terms of the
instantanious short rate
• The Hull White model is an Ohrnstein Uhlenbeck process
• It has a drift along the initial yield curve and a mean reversion back to the initial yield
curve at a rate a
• Zero coupon bonds (or discount factors) are described analytically as:
• Where:
tttt dWdtardr
trTtBeTtATtD
,,,
14
1,
,0
,0lnln 222
3
atataT
tt eeeta
FTtBtD
TDA
a
eTtB
tTa )(1),(
at
t ea
taFtFt
22
12
,0,0
Short rate models – Hull White
Dynamics in a H-W model
0.00%
0.50%
1.00%
1.50%
2.00%
2.50%
3.00%
3.50%
4.00%
4.50%
0.2
5
0.5
0.7
5 1
1.2
5
1.5
1.7
5 2
2.2
5
2.5
2.7
5 3
3.2
5
3.5
3.7
5 4
4.2
5
4.5
4.7
5 5
Initial forward curve One path for r Remaining forward curve
Short rate models – Hull White
• The Hull White model can be solved analytically for caps (and swpations)
• This makes it easy to calibrate the model to market observed option prices
• When the model is calibrated to the market, one can price more complex
structures
• Note:
• Cannot calibrate to swaptions and caps simultaniously
• May be large errors on some caplets due to low number of parameters
],1,[1
1 iii
n
i
i
Pii hNTtDKhNTtDNotCap
ii
ii
P TTBa
aT,
2
2exp11
1
2),0(
),0(ln
1 1
i
P
i
i
i
P
iTDK
TDNoth
Short rate models – Hull White
• Pros
• Markov property – possible to implement in tree or PDE
• Ensures fast and accurate pricing and hedging
• Fast Monte Carlo simulation
• Easy to implement and easy to calibrate due to closed form solutions for swaptions
and caps/floors
• Cons
• Parameters not intuitive as caps/floors and swaptions priced with B & S
• Not possible to calibrate to caps/floors and swaptions simultaniously
• No yield curve dynamics
• Not possible to obtain forward volatilites or correlations
• Normal distribution of the short rate gives a positive probability of negative rates
LIBOR market model (BGM)
• Since a and σ in the HW model have a difficult intuitive interpretation, market
practitioners have been asking for a new and more intutitive model
• The market has been pricing caps with B&S for years:
• 𝐶𝑎𝑝𝑙𝑒𝑡 = 𝐸 exp (− 𝑟𝑡𝑑𝑡)𝑇
0∆ 𝐿 𝑆, 𝑇 − 𝐾 + =D 0, 𝑇 𝐸𝑄 ∆ 𝐹 𝑆; 𝑆, 𝑇 − 𝐾 +
• With a lognormal distribution for F. The dynamics of F(t;S,T) under the measure
associated with D(t,T) is chosen as:
• Where v is the instantanious volatility and W is a standard Brownian motion
under the Q(T) measure.
Q
tdWTStvFTStdF ),;(),;(
LIBOR market model (BGM)
• Let B be the discretely balanced bank account when investing 1 at time 0
• Choose B as numeraire and use Girsanov’s theorem starting from the dynamics
𝑑𝐹𝑘 = 𝜎𝑘𝐹𝑘𝑑𝑊𝑘 under 𝑄𝑘 to obtain the dynamics under B
• The measure 𝑄𝐵 associated with B is called the spot LIBOR measure. Then we
have the following:
k
tj
B
kkk
jj
jjkjj
kkk tdWtFtdttF
tFttFttdF
)(
,)()()(
)(1
)()(
Libor Market Model (BGM)
• The LMM for F’s allows for:
• Immidiate and intuitive calibration of caplets (better than any short rate models)
• Easy calibration to swaptions through algebraic approximation (better than most short
rate models)
• Can virtually calibrate a high number of market products exactly or with a precision
impossible to short rate models
• Clear correlation parameters since these are instantatious correlations of forward rates
• Powerful diagnostics: can check future volatility and terminal correlations strucutres
• Can be used for Monte Carlo simulation
• High dimensionality (many F’s are evolving jointly)
• Unknown joint distribution of the F’s (although each is lognormal under its canonical
measure)
• Difficult to use with PDEs or trees, but «recent» Monte Carlo approaches such as
Least Square MC make trees and PDEs less necessary
How do we choose between models?
• The picture below shows yield curves from 2006 until today in NOK swaps
• We see three different shifts in the yield curves
• Parallell (absolute level)
• Tilt (upward or downward sloping)
• Curvature (convex or concave)
1
2
3
4
5
6
7
1 2 3 4 5 6 7 8 9 10
Bermudan swaption
• What are the underlying dynamics?
• The model of choice needs to be able to capture the dynamics of the 1->4 swap, the 2-
>3 swap …. For each exercise.
• Calibration to swaptions
• No exposure to cap prices
• Exposure to curvature or tilt in yield curve?
• A Berumdan is an option with an early exercise feature
• Easiest to price in a tree or with finite difference grid
• Points to a short – rate model
• Hull – White or the 2-factor HW model would be the prefered model
Autocap
• In a 5 year cap with 3mth LIBOR as underlying, there are 20 caplets
• 20 individual options to price
• Often a cap might seem expensive
• In a 5 year auto cap against 3mth LIBOR, the holder can choose how many
caplets he wants to buy. A caplet is automatically exercised if it is ITM at any
fixing (exercise) date.
• If LIBOR rises above the strike immidiately and stays there throughout the tenor of the
cap, the holder will exercise all caplets after the 10 first fixings and he will have no
protection against higher LIBOR for the last 2,5 years
• Clearly the price of the auto cap is less or equal to a regular cap
• But which model should one use to price an auto cap?
Auto cap
• Underlying dynamics?
• Only against LIBOR -> Calibration to caps
• No simultanious yield curve dynamics at exercise times
• Automatic exercise and start from time 0
• Forward simulation -> Monte Carlo
• Could use both LMM and HW
• HW is sufficient as there is no yield curve dynamics in the product
• Mean reversion parameter might be an issue
Knock – In swap
• A knock – in swap is an interest swap that comes to live if LIBOR breaches a
barrier
• Example: Tenor 5 years. If 3mth LIBOR breaks 3,50% then it knocks into a swap at
3,75% for the remaining maturity
• Product for liability managers who are exposed to rising LIBOR
• Which model to price and risk manage this product?
• Dynamics:
• Clearly exposure to LIBOR which calls for calibration to caps
• Exposure to correlation between LIBOR and the swap you get knocked into
• Automatic knock – in -> Forward simulation -> Monte Carlo model
• Exposure to different points on the yield curve simultaniously calls for a multi
factor model
Rate expectations in a historical perspective
46
0
1
2
3
4
5
6
7
04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22
USD LIBOR 3m
30y – 10y CMS spread range accrual
• The CMS Spread Range Accrual was originally an invstor product designed to
give investors a substantial pick-up in yield by taking a view on the shape of the
yield-curve. The product was usually structured as a note (bond).
• The payoff in the product was: 𝑃𝑎𝑦𝑜𝑓𝑓 = 10% ∙𝑛
𝑁∙30
360
• Where n=number of days CMS30-CMS10 > 0 and N is total number of days in period
• In many cases, the bank had the options to call the bond every year
• Which model should we choose?
• Need for a model which models the whole yield curve simultaniously
• One option to price every day in the structure (often 10 years) -> need for a fast model
• Preferably a model solved in a tree due to speed
• LMM first choice for pricing but very slow
• Multi dimensional short rate model a possibility but very difficult to estimate
parameters
• Model the spread directly
Nordea set – up
• Model development
• Nordea employs 10 model developers to continiously develop new models and
improve implementation
• Recent focus has been to adjust models for multi – curve set up
• E.g. Is the volatility on products priced against 3m LIBOR different from 6m
LIBOR?
• Model set – up
• All models are programmed in C++
• In-house scripting language for programming of pay – off structures and cash flows
gives great flexibility to price almost any option or swap
• Excel is used as GUI for developers and strucutrers
• All models added as functions in Excel
• Nordea has bought a 3rd party system and replaced all models with its own, but
kept the database and other functionalities
Linear Rate Linear rate instruments,
vanilla swaps etc.
Linear Inflation linear inflation
instruments, e.g.
inflation swaps.
Linear Credit
linear credit, e.g.
cds, correlation
credit products
cdo’s, n to default,
etc.
Stochastic Volatility
European options with
one underlying. E.g.
swaptions, caps, CMS’s,
arrears swaps, quantos,
…
Linear
Equity linear equity
instruments.
BGM model
Path dependent
interest rate
options, e.g.
bermudans,
ratchet, range
accruals, …
Cheyette model
Monte-Carlo and
finite difference
models. Path
dependent interest
rate options.
Equity model
Monte-Carlo jump
diffusion model for
European and
path dependent
equity
instruments.
Thank you!
Nordea Markets is the name of the Markets departments of Nordea Bank Norge ASA, Nordea Bank AB (publ), Nordea Bank Finland Plc and Nordea Bank Danmark A/S.
The information provided herein is intended for background information only and for the sole use of the intended recipient. The views and other information provided herein are the
current views of Nordea Markets as of the date of this document and are subject to change without notice. This notice is not an exhaustive description of the described product or the
risks related to it, and it should not be relied on as such, nor is it a substitute for the judgement of the recipient.
The information provided herein is not intended to constitute and does not constitute investment advice nor is the information intended as an offer or solicitation for the purchase or
sale of any financial instrument. The information contained herein has no regard to the specific investment objectives, the financial situation or particular needs of any particular
recipient. Relevant and specific professional advice should always be obtained before making any investment or credit decision. It is important to note that past performance is not
indicative of future results.
Nordea Markets is not and does not purport to be an adviser as to legal, taxation, accounting or regulatory matters in any jurisdiction.
This document may not be reproduced, distributed or published for any purpose without the prior written consent from Nordea Markets.