A0067927 amit sinha_thesis

Thesis to get the degree of Master of Science in
Financial Engineering

Pricing and Exposure measurement
of IR derivatives:A Short rate model
approach
A0067927

SINHA Amit Kumar

25/11/2011

National University of Singapore
Risk Management Institute

Contents

Abstract 1

1 Theory and Implementation Notes 3
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Quantitative measure of Counter-Party Exposure . . . . . . . . . . . 4
1.3 Interest rate derivatives and their analytical pricing . . . . . . . . . . 9
1.4 Short Rate Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.5 Simulation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2 Model Implementation and Results 23
2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Bootstrapping yield curve . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3 Calibrating CIR Model . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.4 Calibrating Hull-White 1 Factor Model . . . . . . . . . . . . . . . . . 29
2.5 Exposure Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3 Appendix 41
3.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2 Market Data and Matlab Code . . . . . . . . . . . . . . . . . . . . . 41

Nomenclature 47

i

Abstract
Pricing and Exposure measurement of IR derivatives:A Short rate model approach
Short rate models have been very popular and still in use for term-structure mod-
elling, pricing and hedging interest rate derivatives. Under short rate model, both
equillibrium model and no-arbitrage models have been actively used for derivative
pricing and risk management. Potential Future Exposure ,however, is a relatively
new entrant. PFE modelling can be used effectively to calculate exposure at dif-
ferent level: trade level, portfolio level,counteryparty level. We compare the results
obtained from these two models and show that PFE evolution for a longer term is
very much model dependent and initial.
Here we use one tradtional term-structure approach model: Cox–Ingersoll–Ross(CIR)
model and no-arbitrage model: Hull-White 1-factor model to evolve interest rate,
price interest rate swaps and calulated exposure metrics. We assume pure interest
rate risk to reduce correlation measurement. Term structure modelling: Market In-
terest Rate data(US) is used to construct an initial discount curve. Term-structure
modelling/fitting is performed based on this initial Zero coupon yield curve. CIR
model parameters are estimated based on intial DF curve. HW1F model is cali-
brated based to interest rate caps. Pricing Methodology: Pricing of Interest rate
swap is performed under deterministic model, assuming expected forward rate can be
calculated based on inital discount curve. For interest rate options, pricing model
is also a short model based approach. Market simulation modelling: Short rate
models with monte-carlo, which are also risk neutral models, are used to simulate
future IR risk variables( short rate,instanenous forward rate,etc) . PFE measure-
ment: Based on the simulated future risk variables ,MTM of contract/portfolio is
calculated at for number of simulations and at a particular monitoring frequency.
Netting/non-netting effects amd effect of diversification in a IR portfolio is studied.
In conclusion we demonstrate the effect of netting and diversification in a port-
folio.Effect of downward and upard sloping yield curve on exposure measurement.
Also, as shown in different literature a comparison of HW1F and CIR model.

1

1 Theory and Implementation Notes
This section contains some of the theory relevant to this thesis as well as some
specific implementation notes.

1.1 Motivation
Since Barings in 1995 to Lehman, AIG in 2008, finance domain has experienced
many disasters in last two decades. Other than incurring great losses and economic
slowdown these incidents have also forced us to think about better ways to manage
risk. Credit crisis of 2008 was particularly important in bringing risk management
at the heart of financial institutions.Particularly, managing risk in OTC instruments
has become very crucial.Value at Risk(VaR) has been adopted and recommended
widely as a measure to check market risk. Another component of financial risk is
counter-party risk. Profit and loss over an OTC contract becomes even more dicey
and asymmetric in in case of counter-party default. A lot of effort has recently gone
into account for the counter-party credit risk in OTC market. It turns out that
a methodology for pricing and measuring future exposure of derivatives contract
and their portfolio forms most important component of Counter-party credit risk
measurement. We will use a dynamic term structure modeling and pricing coupled
with Monte-carlo simulation technique to capture Price and exposure of derivatives
contract/portfolio.In rest of the thesis , we will discuss methodology to capture
portfolio/contract exposure of interest rate derivatives.
Counter-party exposure measurement serves as very effective risk management tool
to manage credit limits on a given counter-party. Just like VaR, it can give a
simple and easy to explain representation of exposure to a counter-party to be put
in front of the management board or regulatory bodies. This can be used top decide
different policies w.r.t to lending and amount of credit limit to each counter-party.
A good counter-party can worsen and a bad counter-party can improve on its credit
worthiness.These exposures can also be updated and monitored from time to time.
Management or regulatory bodies can set a cap on credit line to a particular counter-
party and limit the losses in case there is a actual counter-party default. Firms can
use other mitigation methods like netting and collateral to reduce loss if counter-
party defaults. These effects can also be incorporated in Exposure measurement
and effects of these measures can be demonstrated. Therefore a metric to measure
counter-party exposure also helps in implementing exposure aggregation provisions.

3

Chapter 1 Theory and Implementation Notes

Future pricing and exposure measurement can serve as input to other quantitative
measures like Counter-party value adjustment (CVA).These measures have becomes
exceedingly important in present times and they are part of internal rating based
model under Basel II accord.
.

1.2 Quantitative measure of Counter-Party Exposure

Exposure can be aggregated at diﬀerent hierarchies:
1. Trade/Contract level: Considering exposure of each trade separately.
2. Counterparty level:Considering exposure against each counterparty along with
credit risk mitigation techniques
3. Portfolio level: Considering exposure of portfolio as a whole against all counter-
parties.
Asymmetric nature of Exposure:As in other circumstances , the contracts at the time
of default are settled based on MTM values.If a defaulted counter-party cannot pay
the MTM value then its a loss to the party (+MTM ). If the party owes MTM to a
defaulted counter-party, the party is still obliged to make payments (-MTM).
A simple illustration:We consider a single IR swap with current MTM value as
MTM(t). So exposure is deﬁned as

E(t) = max(M T M (t)i , 0)

Credit risk mitigation techniques:
ISDA Master agreement and Credit support annex outline several agreement that
counterpaties can have in a OTC contract to limit exposure in time of counter-party
default.
• Netting agreement: It is a legal contract that allows parties to aggregate
positions in case of default. Agreement could be made to decide on the trades
that could be netted together. Typically in a portfolio, there would be some
netting and some non-netting. Exposure with Netting:

E(t) = max( M T M (t), 0)i

A combination of netting and non-netting sets:

E(t) = max( M T M (t), 0)i + max(M T M (t)i , 0)

4 4


Without Netting

5


With Netting

• Collateral and margin agreement: It is one of the most commonly used risk
mitigation method. Collateral agreement is a legally binding agreement where
the counter-party has to provide securities when it is ’out of the money’. Collat-
eral is posted if the MTM is more than a per-determined threshold. Exposure
with collateral can be modeled as :

E(t) = M ax( M T Mi − C(t))

Trade level counter-party credit exposure metric

We will start with a ﬁgure to represent diﬀerent exposure related quantities.

6 6


All the risk measures can be approximated either analytically or by simulation of
market risk factors. As we shall see later, we have used Monte-carlo simulation based
approach to calculate there exposure related metrics. Simulation based approach
provides us extra ﬂexibility in terms of evolving risk factor such as interest rate like
any popular practitioners model( like HW1F,HJM,CIR, etc).

7


Value of these exposure metrics depend on two most important factors:
1. Evolution of risk factor in future time and underlying risk factor volatility.This
factor is more of a model dependent phenomenon. For example- A mean
reversion model will control the evolution of risk factor in longer term by
models rate of mean reversion and long term rate.
2. Secondly the nature of the contract also plays an important role. In some
contracts the exposure increases with time as they may only have a single or
concentrated cash-flow towards the maturity of the deal. Others may have
uniform payment at a particular frequency and thus, exposure gets amortized
as we reach towards the maturity of the deal.
• Current Exposure: Current exposure is the value of contract lost if the the
counter-party defaults now. Current exposure for any contract is typically
same as the MTM value. At the start of a par contract, current exposure is
usually zero.
• Expected Exposure: It is the average value of counterparty exposure at any
given time before the maximum tenor in the portfolio.

EE(t) = AV ERAGE[M ax( Vi (t), 0)]

• PFE or Potential future exposure : It is the exposure that will not be exceeeded
with certain confidence interval a.

P F E(t) = inf {X(t) : P (E(t) ≥ X(t)) ≤ 1 − α}

PFE reminds very closely of Var for except there are certain differences:
– VaR will typically be calculated at one fixed horizon, where as PFE is
evaluated at different points(t) before the maturity of a contract. Com-
bination of all these PFE(t) gives us a PFE profile.
– VaR is a loss where as PFE represents a gain in the contract.
– VaR is a measure of Market Risk where as PFE is a measure of Counter-
party credit risk.
• Maximum PFE: It is maximum value of PFE through time ,through all sce-
narios at t given confidence interval.

maxP F E(α) = max[EE(ωi , tk )]

where w(i),tk represent each path and each time step respectively.
• Expected Positive Exposure: It is the weighted average of Expected Exposure
over time.

EP E = Average(EE(t))

8 8

1.3 Interest rate derivatives and their analytical pricing

• Effective Expected Exposure: It is a non decreasing function of Expected
Exposure through time.

EEE = M ax(EEi , EEEi−1 )

• Effective EPE:Time weighted average of EEE.
1
Ef f EP E = EEE(ti ) t
t

It is important to understand that bottle-neck in Expsoure measurement is the com-
bination of pricing methodology,factor simulation modeling and MTM calculation
at each time step for number of simulation. Our work mainly will focus on these
lines. Generating different exposure metric is quite simple once MTM simulations
through time is available.

Exposure at Default(EAD):

As part of regulatory capital calculation under Basel II , EAD is used as a parameter
to calculate economic capital. EAD calculation under internal rating based model
is based on Effective EPE methodology.

EAD = a x Effective EPE

where a lies between 1.2 to 1.4.

1.3 Interest rate derivatives and their analytical
pricing
Our portfolio will consist of IR instruments so, its worthwhile to understand basic
nature and popular pricing methods. In this section we will touch upon IR swaps
and IR options. We will also discuss procedure to construct discount curve given
set of liquid market instruments. This procedure will be utilized in our analysis.

IR Swap:

We will discuss the most vanilla form of Interest rate swaps which a fixed-float same
curency swap instrument.

9


A swap has two payment legs:
1. Fixed Leg:fixed payer pays fixed rate through out the swap maturity on pre-
determined coupon payment dates. In the figure above, Fixed leg payers pays
two coupons of size C/2 at 6m and 1y respectively. Floating leg payer receives
these fixed leg payments.
2. Floating Leg: Floating leg payer pays floating leg which a benchmark rate
(like LIBOR) every floating leg coupon payment date. A vanilla swap has
floating leg which is reset in advance which means that floating rate paid at
3m is being set at 0m (figure above).
Price of a receiver IRS is given by :

P rice(IRS) = P V (F ixedleg) − P V (F loatleg)

IR Options:

We will discuss European calls and puts on interest rates. These calls(or puts) are
on a liquid benchmark which is usually of short maturity. There are also other kind
of options on interest rate derivatives like swaptions ,but we are not delving in to
them for the sake of being focused.
The most basic building block for a Interest rate option is a caplet or a floorlet.
Caplet can be defined as a ’single’ option to buy a floating rate at a particular strike
rate. For instance- A Caplet can be an option on 6M LIBOR to buy this particular
benchmark at ,say, 5%. Lets suppose today is t and 6M LIBOR will be resetting on
T and end of period will be T+6M
Therefore the payoff from caplet at the end of T+6M will be :

M ax(L6M (T,T +6M ) − 5%) ∗ DCF (T, T + 6M )

10 10


Price of a caplet ,therefore , will be simple the discounted expected value of above
payoﬀ.

P rice = Z(t, T + 6M ) ∗ E[M ax(L6M (T,T +6M ) − 5%) ∗ DCF (T, T + 6M )]

Cap then is just a series of caplets with same strike rate.Market practice is to price
each caplet with Black’s formula (to be discussed soon) and add them to get cap
price. As we will see, Cap prices are usually quotes in in terms of volatility implied
by Black’s formula.

Bootstrapping DF curve:
In order to price swap , we need to discount each coupon payment of each leg to
its present value. One way to calculate initial discount curve is by bootstrapping
discount factors from actively traded interest rate instruments:
1. Money market instruments like interbank lending rates (example-LIBOR rates)
which can available up to one year maturity.
2. Euro dollar futures: Some of these instruments are very liquid. Instruments
with high liquidity have very lower liquidity.
3. Swaps: We get market quoted swap rates for a large tenor (up to 30y).
Below we describe procedure to boot strap DF from MM instruments and Swap:

First part is to build DF curve using cash rates quoted in the market

Discount factor Z(T)of the cash rates can be calculated using simple present value
formula:
1
Z(T ) = ((1+r(T )a(0,T ))

Where a is the Day count factor as per the money market day count basis. Discount
factors for rates up to 9M cash deposit rates can be calculated using above relation-
ship. Day count fraction basis used is Act/360. We can deﬁne DF as of today as 1,
in that case DF at spot date(t+2) will not be 1.
The discount factors calculated for all the maturities staring 3m will be adjusted
using overnight(o/n) and tom-next rate(t/n). First we calculate overnight discount
factor as :

11


1
((1+.29
= 0.999998

Next, we use the tomorrow next rate to calculate the discount factor for the spot
date. The tomorrow next rate is a forward rate between trade day plus one business
day to trade date plus two business days. Therefore, the discount factor for the spot
date is:
1
((1+.302
= 0.99998

Discount factors from Swap rates:

In order to extract DF from swap rate, we consider following par swap rate, S(t).
We also assume that Libor Swaps are quoted mid-market, semi-annual swap rates
and pay the floating 3-month Libor rate.
The fixed leg of a swap is calculated by adding up the PVs of all future cash flows:
nf ixed
P V F IXED = S(t) (J=1) αj Z(0, Tj )
For the Floating leg:
nf loat
P V F LOAT = (J=1) Lj αj Z(0, Tj )
Where Lj is the LIBOR forward rate for settlement at Tj-1 .
Lj = F (Tj , Tj − 1)
1 Z(0,T(j−1) )
Lj = αj
( Z(0,Tj ) − 1)
Using above relationship in floating PV calculation we get:
(nf loat) 1 Z(0,T(j−1) )
P V F LOAT = (J=1) αj ( Z(0,Tj ) − 1)αj Z(0, Tj )
(nf loat) Z(0,T(j−1) )
P V F LOAT = (J=1) )( Z(0,Tj ) − 1)Z(0, Tj )
(nf loat)
P V F LOAT = (J=1) (Z(0, T(j−1) ) − Z(0, Tj ))
P V F LOAT = 1 − Z(0, T(nF loat) )
For a par swap, P VF LOAT IN G = P VF IXED
Therefore we can put them together as:
(nf ixed)
S(tn ) (J=1) αj Z(0, Tj ) = 1 − Z(0, T(nF loat) )
After rearranging for nth period DF and assuming nf ixed = nf loat we get,
(n−1)
(1−S(tn ) (J=1)
αZ(0,Tj ))
Z(0, Tn ) = (1+αn S(tn ))

We can use the expression above to calculate the discount factor associated with the
last swap coupon payment. In order to use this expression to calculate 1YR DF, we

12 12


need 1 YR swap rate and discount factor associated with fixed leg of the underlying
par swap.
(6M )
(1−S(t1Y ) (J=1)
aZ(0,Tj ))
Z(0, T1Y ) = (1+a6M −12M S(t!Y ))
)
Here assumption is that the underlying swap is quoted semi-annually with Act/360
For Swap rate 1Y=0.5180%, DF 1Y is
Z(0, T1Y ) = 0.99477

Interpolation:

It is also worth noticing that Swap are not quoted for all tenors, therefore it usually
comes down to a good interpolation method to back out DF at in-between points.
We can either interpolate between rates or we can interpolate between DF. It is
usually better to interpolate between DF because DF curve is monotonically de-
creasing with tenor so it is usually provides smoother and better approximation for
missing DF points. Zero coupon curve on the other hand may have different shapes
depending on the market forces it may exhibit a uniformly increasing then flatten-
ing curve or a hump shaped curve of a decreasing curve. This makes interpolation
between rates less accurate. Once DF is calculating rates can be backed out based
on formulas we discussed already.
For example-
The 2-year semiannual par-swap rate is quoted as 0.770%.
In order to calculate Z(2Y ) we need to use following relationship:
(18M )
(1−S(t2Y ) (J=1)
aZ(0,Tj ))
Z(0, T2Y ) = (1+a18M −24M S(t2Y ))
)
Above is an equation with two unknowns: Z(0, 18m), Z(0, 2Y R).We order to esti-
mate Z(0, 2Y ) ,we have to use some interpolation method. We use linear interpola-
tion method in discount factors:
1
Z(0, 18m) = 2 (Z(0, 1Y R + Z(0, 2Y R))
Using above equation is our standard discount factor calculation formula we obtain
Z(0, 2Y R) as: 0.98450

Forward LIBOR:

Putting it all together we have Discount factor curve from trade date to 5Y for
standard tenor points. These discount factors can be used to generate forward rate
curve for valuation of swaps in our problem statement Forward rate between any
two dates T1 and T2 from the discount curve for a given trade/ spot date can be
calculated as:
1 Z(0,T1 )
L1X2 = a1X2 ( Z(0,T2 ) − 1)

13


DF Curve:

Putting all these relationships in a tabular form we get following discount factors:

Below is the DF curve as obtained from the market instruments upto 5Y Data:

This Market data used here is just for illustration purpose. In order to obtain results
for the thesis more data points have been used. Results will be produced in the next
section.
After boot-strapping with the full market data we will also ﬁt a cubic slpine using
matlab sub-routine so that we can get a DF for any tenor.

14 14


Pricing a vanilla swap using deterministic model:
Deterministic pricing of swap has the same underlying principle as bootstrapping.
We say that Expected forward LIBOR rate can be calculated from present yield
curve.

E(F (t, T, S) = 1
( Z(t,T )
αT XS Z(t,S)
− 1)

Where E= expected value
We will define following convention:
LongSwap = F ixedleg − F loatingleg(Receivef ixed, payf loating)
ShortSwap = F loatingleg − F ixedleg((Receivef ixed, payf loating)
Value of swap is given by :

P rice(IRS) = P V (F ixedleg) − P V (F loatleg)

The fixed leg of a swap is calculated by adding up the PVs of all future cash flows:
nf ixed
P V F IXED = S(t) (J=1) αj Z(0, Tj ) x Notional
Where S(t) is the swap rate
For the Floating leg:
nf loat
P V F LOAT = (J=1) Lj αj Z(0, Tj ) x Notional
Where Lj is the LIBOR forward rate for settlement at T(j−1) .
L1X2 = 1
( Z(0,T1 )
α1x2 Z(0,T2 )
− 1)

Pricing a IR Cap using Black model:
Under Black’s model we assume that forward libor rate/forward interest rate is log-
normally distributed. Under BS framework it becomes intuitive to have closed form
solution like this for each caplet:

N∗ t ∗ Z(t, T + t) ∗ [f (t0 , T, T + t)N (d1) − K ∗ M (d2)]

Where
ln(f (t0 ,T,T + t)) σ 2 ∗T
K
+ 2
d1 = √
σ∗ T

15


ln(f (t0 ,T,T + t)) σ 2 ∗T √
K
− 2
d2 = √ = d1 − σ ∗ T
σ∗ T

Please note the discounting and volatility scaling used in the above formula. Caplet
is discounted for the full period and volatility is scaled only till the point of obser-
vation.
Therefore price of a European cap is simply the summation of caplets using above
formula. Formula ﬂoorlets put can be obtained in the similar way.

1.4 Short Rate Models
Though short rate models are not the only and most advanced models that currently
in use, We will use two short rates models( CIR and HW1F) because of two main
reasons:
1. They are fast: We can restore to market model frame work like HJM or BGM
but calibration process is too time consuming. The idea of measuring exposure
is not get the most accurate pricing ,but to get fairly accurate results based on
a market accepted model. Typically the size of portfolio to be examined will
have thousand of contracts and will be become very computationally expensive
if we resort to very complex modeling methodology.
2. They are still in use: short rate models have performed reasonably well and are
still being used. They have consistently provided accurate pricing of vanilla
swaps and options.
Short rate: it is an instantaneous interest rate r(t) applicable for an investment
between period tand t + dt .
For a bond maturing at Time T , price can be written in terms of short rate :

´T
P (t, T ) = E[e− t r(t)dt
]

Where E is the expected value under risk neutral world.

CIR Interest Rate Model
CIR model belongs a family of processes given by:

dr = a(t)[θ(t) − r(t)]dt + σ(t)r(t)β dW (t)

16 16


For Cox-Ingersoll-Ross process β = .5, a square root process, This ensures that the
short rate never goes negative. Sigma in CIR process will change in proportion the
the square of short rate. We will estimate a and θbased on discount curve generated
using boot strapping method and cubic spline DF generation.
For Estimation , We will compute zero coupon prices for current market data using
following closeed form formulas:

P (t, T ) = A(t, T )e−B(t,T )r

Where r is short rate. Short rate assumed in the overnight rate for current market
data.

φ1 eφ2 (T −t)
A(t, T ) = ( )φ3
φ2 (eφ1 (T −t) − 1) + φ1

eφ1 (T −t)
B(t, T ) = ( )
φ2 (eφ1 (T −t) − 1) + φ1

√
φ1 = a2 + 2σ 2

a + φ2
φ2 =
2

2aθ
φ3 =
σ2

Estimation of CIR parameters

1. While we already constructed a DF curve we can use observed Discount factors
and unit zero coupon bonds. Then we can compare the ZC DF curve with the
model implied bond prices for different tenors( of quoted instruments) . Then
we can minimize the sum of the square of the differences between these two
prices to get best fit.
2. Second approach can be to get zero coupon yield curve and then try to fit
model paramters by minimizing sqaured errors.

17


John C hull and Rebonato in two independent research works have suggested cali-
bration to prices not zero coupon yield. However, Tuckman suggested that pricing
to rates does away the weightage problem associated with bonds of differnet matu-
rities. The problem stems from the fact that in shorter term interest rate are not
very senstive compared to longer term.We, however, tried both method and show
results.

Hull and White 1-Factor Interest Rate Model
One problem with the CIR model is that it doesnt exactly fit the term-structure.
If the drift in basic SDE is coverted to a time varying function, it turns out that
it can excatly fir the term-structure of current interest rates.drift term θmade sure
that the dynamic term-structure modeling is consistent with initial term structure.
That is to say that DF curve evolved from will ensure a no-arbitrage situation in a
risk neutral world.
Second model that we have used is a Hull-While 1-factor model.
SDE for a HW1F model is as follows:

dr = [θ(t) − αr]dt + σdz

Where
∂f (0, t) σ2
θ(t) = + αf (0, t) + (1 − e−2αt )
∂t 2α

Discount Curve in terms of HW1F can be calculated as:

P (t, T ) = A(t, T )e−B(t,T )r(t)

where
1
B(t, T ) = (1 − e−α(T −t) )
α

DF (0, s) ∂lnP (0, t) 1
lnA(t, T ) = ln − B(t, T ) − 3 σ 2 (e−αT − e−αt )2 (e2αt − 1)
DF (0, T ) ∂t 4α

r(t) = shortrate

18 18


Caplet pricing using HW1F model

P ayof fcaplet = τ max(R − K, 0)
Above is the payoﬀ of a caplet at time tk+1 with reset at tk .
payoﬀ at tk

1
P ayof fcaplet = τ max(R − K, 0)
(1 + τ R)

1 1
P ayof fcaplet = max( − , 0)(1 + τ K)
(1 + τ K) (1 + τ R)

Therefore caplet price with strike K and Notional L is same as a european put
options with strike = (1+ 1 τ K) and (1 + τ K) notional
Therefore price of a caplet under HW1F has a analytical form dependent on model
parameters :

caplet(t, T, T + T ) = KD(t, T )N (−d2 ) − D(t, T + T )N (−d1 )

Where
ln D(t,T + ) )
KD(t,T
T
σP
d1 = +
σP 2

d2 = d1 − σP

σ2
σP = (1 − e−2α(T −t) )(1 − e−α(T +∆T ) )2
2α3

D = DiscountF actor
1
K=
(1 + T Rk )

D(t, T + T ) = D(t, T + T ) ∗ (1 + T Rk )

19


Calibration Procedure

HW1F model parameters are computed by calibrating it to market quoted cap prices.
We will use market quotes cap volatility to back-out Black implied Cap prices.
Assuming market has N liquid cap volatility and hence N cap prices, Calibation is
done by minimizing following expresssion:

minσ,α = (modeli (σ, α) − marketi (σ, α))2

If we don’t want to fit cap volatilities we can directly fit the σ, α to zero coupon
curve formula given in the above section. However, We have done the calibration to
ATM cap volatlities to get the comparison between the two models.
Cap volatilty as obtained from the market data is assumed to be flat volatility (not
spot volatility). Same volatility is applied to all caplets in a cap.

1.5 Simulation Model
After calibrating our two models to market data and obtaining the model parame-
ters, We simulate future market data. This can be very slow and needs judgement
to decide between
• Risk factors
• Time steps
• Number of paths
In our case , we have one source of randomness(short rate), we will simulate risk
factors at each coupon payment date. If there are more than one instrument in a
portfolio we will simulate risk factors at smallest time interval up to longest maturity.

CIR model Simulation

r(t + t) = r(t) + α(θ − r(t)) t + σ r(t) t (0, 1)

where

∼ N (0, 1)

For each simulation step,r(t) is evolved at every time step upto the maturity of the
contract. Based on r(t) at each time step, Discount curve is obtained. DF curve
is then used to calculate the exposure of IR instruments/portfolio. For short rate
volatility ,we use the same model parameters as the initial calibration.

20 20

1.5 Simulation Model

Above discretization of r(t) can produce negative values ,therefore we will use a
diﬀerent implementation to simulate CIR short rate.
Milstein approach for :4αθ > σ 2

√
r(t) + αθ t + σ r(t) t (0, 1) + 1 σ 2 ( (0, 1)2 − 1)
4
r(t + t) =
1+α

else

r(t + t) = r(t) + α(θ − max(r(t), 0)) t + σ max(r(t), 0) t (0, 1)

HW1F model Simulation

r(t + t) = r(t) + (θ(t) − αr(t)) t + σ t (0, 1)

where

∼ N (0, 1)

∂f (0, t) σ2
θ(t) = + αf (0, t) + (1 − e−2αt )
∂t 2α

As we can see, drift term θ(t) evolves with time,and is a function initial forward
curve. f(0,t) is same as intial ZCYC that we have already bootstrapped and splined
in section 1.3. Thereforef (0, t) and ∂f∂t can be easily obtained from the initial
(0,t)

curve.
Same as CIR ,for each simulation step r(t) is evolved at every time step upto the
maturity of the contract. Based on r(t) at each time step, Discount curve is obtained.
DF curve is then used to calculate the exposure of IR instruments/portfolio. For
short rate volatility ,we use the same model parameters as the initial calibration.
Please note that exposure is always calculated under r.n measure.

Pricing Model

Once risk factors are simulated instruments are priced as:
1. IR swap: Deterministic model
2. IR Caps:Caplet pricing using HW1F model

21


Collateral Model

For Collateral modeling, we have assumed a simple collateral construction.
We assume that there are no minimum transfer amount and collateral is posted
immediately.Also in event of default ,there is a immidiate settle vis-a-vis collateral.
Under these simpliﬁcations,

C(t) = M ax(M T M (t) − CTcounterparty )

Where

C(t) = collateral

CTcounterparty = CollateralT hreshold

Under this condition, expsoure of the portfolio is :

Ecollaternal = min(E(t), CTcounterparty )

MTM Matrix

By simulating the value of portfolio through time steps and number of paths we
obtains a matrix of portfolio MTM.
For example, for 12 timesteps and 1000 simulation paths we have a 12x1000 MTM
matrix. These represent the value of the exposure at each time step calculated using
an anlytical formula( as described under Pricing model). From the MTM matrix dif-
fernet exosure related metrics can be calculated.(Potential future exposure,Expected
Exposure ,etc.)

1.6 Summary
In this section we outlined theory and methodology used to build the pricing and
exposure measurement model. In the next section , we will go through each of these
steps in detail with actual market data and results. We will also present an analysis
and comparative study based on choice of model,instrument,eﬀects of netting and
aggregation on the portfolio.

22 22

2 Model Implementation and Results

2.1 Overview
In this part, we will go through each step of model implementation and show results
based on our sample market data. Underlying theory and implementational nitty
grittys have been explained in previous section. Here we will support claims made
in previous section and take note of any deviation. Plots and parameters presented
here can be regenerated with matlab programs and market data attached in the
appendix.

2.2 Bootstrapping yield curve
Market data date used for Boot Strapping Interest Rates is of 21/10/2005.
Below is the discount curve as obtained by boot-strapping cash rates(o/n to 1Y)
and Swap rates(2Y to 50Y). In-between interpolation is done in matlab using ppval
matlab function.

23

Chapter 2 Model Implementation and Results

Below is the Zero coupon yield curve as obtained from DF curve (for points af-
ter Money market rates ) and by boot-strapping cash rates(o/n to 1Y) and Swap
rates(2Y to 50Y). In-between interpolation is done in matlab using ppval matlab
function.

2.3 Calibrating CIR Model

CIR model is calibrated to the given DF curve at each circular point(each quoted
instrument) in the Discount curve of section 2.2.

Model params calibrated values description
a 0.4415 rate of mean reversion
θ 0.0519 long-term rate
σ 0.0233 volatility
r 0.0378 shortrate(observed o/n)

24 24


Calibration comparison with full term structure(O/N to 50Y):

Using full term structure, ﬁt for short term maturity is not very good as long term
maturity.

Since we are going analyse swap instruments only upto ﬁve years, It’s worth trying
to estimate CIR model paramters to a shorts Zero coupon curve. However the

25


optimization of three parameters does not fare very well when the tenor is shortened.
It throws a negative value for the volatility parameter.

Constant volatility parameter

To avoid this problem, we ﬁx the volatility to 3Y cap ATM volatility as obtained
from the cap surface.

Model params calibrated values description note
a 2.1007 rate of mean reversion from estimation
θ 0.0483 long-term rate from estimation
σ 0.1940 volatility 3YR ATM CAP VOL

Estimation in zero rates by minimizing error in model rate and
zero coupon rate:

Model params calibrated values description note
a 4.2936 rate of mean reversion from estimation
θ 0.04747 long-term rate from estimation
σ 0.1940 volatility 3YR ATM CAP VOL

26 26


As we can observe , estimation by minimizing squared error in interest rates provides
the best ﬁt. We will use model parametera backed-out by this method.

Risk factor simulation

Based on estimated parameters , we can simulate risk factors at diﬀerent interval.
In the plot below, 1000 paths are simulated at each 3m period upto 3years( 12 time
steps)

27


As expected we observe a faster mean reversion (a=4.2936)

If we just change the mean reversion rate to 0.5 ,we obtain a much smoother picture.

28 28

2.4 Calibrating Hull-White 1 Factor Model

2.4 Calibrating Hull-White 1 Factor Model

We calibrate Hull white 1-factor model to intertest rate caps such that it minimizes
the squared difference between model cap prices and market cap prices.

By calibrating to cap prices upto 10Y we obtain

α = 0.0023

and

σ = 0.0084

Below is the plot comparing Model Cap prices with Market cap prices as calculated
using Market quoted volatility. Please note that we could have done optimiztion for
three parameters (α, σ, r)where r is the short rate. But the optimization would run
into problem of negative short rate.

A check on Yield curve fitting: Hull-White 1F model is a no-arbitrage model. That
implies that it should fit the today’s yield curve . Here try to compare the model
implied Discount curve with market discount curve:

29


Please note that the calibration is doneto cap prices upto 10Y. In the above plot,
upto 10Y point the ﬁt is reasonable ,after which model implied DF drops faster and
touching almost zero as 50Y point.

Risk factor simulation

Based on estimated parameters , we can simulate risk factors at diﬀerent interval.
In the plot below, 1000 paths are simulated at each 3m period upto 3years( 12 time
steps)

30 30

2.5 Exposure Measurement


Swap simulation

All results are for a 3Y reciever vanilla swap , swpa rate= 8.4% . Swap is actually
a par swap with 8.4% determined by setting P V (f loat) = P V (f ixed).

CIR model calibration to partial term-struture( up to 5Y):

MTM proﬁle:

31


PFE Proﬁle(95 %tile):

32 32


Exposure can be extracted from the MTM profile as Exposure = M AX(M T Mt , 0)

CIR model calibration to full term-struture:

MTM profile:

PFE Profile(95 %tile)

33


HW model calibration to CAP volatilities:

MTM proﬁle:

PFE proﬁle(95%tile)

34 34


A comparison of PFEs using HW1F and CIR model:

As we can see in the above plot, CIR model produces higher PFE proﬁle than the
HW1F model. This can be explained on the basis of standard deviation component
√
of CIR process. In the CIR process ,standard deviation of r is varies with r.

35


Higher Short rate will increases the volatity of change in short rate. This accounts
for higher dispersion of MTM values under CIR model.

Exposure Metrics

A brief recap of exposure formulas:
•
EE(t) = AV ERAGE[M ax( M T Mi (t), 0)]

•
P F E(t) = inf {X(t) : P (E(t) ≥ X(t)) ≤ 1 − α}

•
maxP F E(α) = max[EE(ωi , tk )]

•
EP E = Average(EE(t))

•
EEE = M ax(EEi , EEEi−1 )

A comparison of diﬀerent exposure metrics. Plot below is generated with HW model
calibrated to CAP volatilities:

36 36


Legend:
green:Expected Exposure
blue:Potenrial Future Exposure
red:maxPFE
magenta:Eﬀective Expected Exposure
’+’: Expected Postive Exposure
All results from this point onwards are based on CIR model with model parameters
estimated to full term structure.

Eﬀect of Netting:

Portfolio of two swaps :
1. Payer swap unit notional , 6M payment frequency
2. Reciever swap unit notional ,3M payment frequency
Without Netting:
Exposure=Max(MTM_3M,0)+Max(-MTM_6M,0)

With Netting:
Exposure=Max(MTM_3M-MTM_6M,0)

37


Effect of Collateral:

Collateral threshold: We assume a simple collateral model. Everytime the MTM
exposure exceeds 8% of notional, counterparty posts collateral:

C(t) = max(M T Mt − T hreshold, 0)

Exposure(t) = max(M T Mt − C(t), 0)

f rom above two equations :

Exposurecollaterlized = min(Exposure(t), T hreshold)

We fixed threshold arbitratily to show to effect of collateral:

38 38


39

3 Appendix

3.1 References
1. Counterparty Credit Risk: The New Challenge for Global Financial Markets"
by Jon Gregory 2010
2. Measuring and marking counterparty risk by Eduardo Canabarro Head of
Credit Quantitative Risk Modeling, Goldman Sachs, Darrell Duﬃe Professor,
Stanford University Graduate School of Business 2008
3. Implementing Derivatives Models by Les Clewlow and Chris Strickland 2001
4. Options, Futures, and. Other Derivatives, Seventh Edition John C. Hull 2010
5. Interest Rate Models by Damiano Brigo and Fabio Mercurio 2004
6. A guide to modelling counterparty credit risk by pykhtin and Zhu 2007

3.2 Market Data and Matlab Code

Market Data :
Market Data used is of 21/10/2005 as obtained from bloomberg:
• BBA LIBOR US O/N to 12 Months
• United States Dollar USD Fixed Float Swap 1Y to 30Y
• Caps - Black ATM Volatility BBIR Mid 10/21/11
Market Data is has been submitted in csv format( as read by the matlab code).

Matlab Codes :
There are close to 20 matlab ﬁles and they are attached as a part of thesis submission.
However to make sure that results could be reproduced , main function that calls
all other sub-routines is submitted here:
%This Procedure Generates the short rate paths for the Interest Rate

41

Chapter 3 Appendix

%Swap(reciever swap)
%PFE Calculation using the Cox, Ingersoll Ross model(CIR) and Hull-White 1-
factor model.
function pfe_irswap(model,iscalibrated,pathstr,nsim)
[ini_DF_curve ini_fwd_curve difnumdate cap_data]=DF_capvol(pathstr);
swap_details=csvread([pathstr ’swap.csv’],1);
if iscalibrated==0
model_calibration(ini_DF_curve,ini_fwd_curve,difnumdate,cap_data,model);
end
%load model params
%swap trade details
Notional=swap_details(1);
maturity=swap_details(2);
freq=swap_details(3);
DCF=1/freq;
delta_t=DCF;
tenor=zeros(freq*maturity,1);
%swap payment schedule
for i=1:size(tenor,1)
tenor(i)=i*DCF;
end
%nsim=1000;
shortrate_paths=ones(nsim,size(tenor,1));
dfcurve_paths=ones(nsim,size(tenor,1),size(tenor,1));
MTM_paths=zeros(nsim,size(tenor,1));
PFE_proﬁle=zeros(size(tenor,1)+1);
MTM_paths1=zeros(nsim,size(tenor,1));
PFE_proﬁle1=zeros(size(tenor,1)+1);
if strcmp(model,’HW’)
model_params=load_params(model,pathstr);
alpha=model_params(1);
sigma=model_params(2);

42 42


r_0=ppval(ini_fwd_curve,0);
for I=1:nsim
shortrate_paths(I,:)=HW_short_rate(tenor,alpha,sigma,r_0,ini_fwd_curve,delta_t);
last=size(tenor,1);
for J=1:size(tenor,1)
if(J==1)
dfcurve_paths(I,J,:)=shortrate_paths(I,1:last);
else
dfcurve_paths(I,J,:)=[shortrate_paths(I,1:last) zeros(size(tenor,1)-last,1)’];
end
%Change the maturity and ZCYC used in the calculation
Rates=squeeze(dfcurve_paths(I,J,:));
%for short rate
DF=Givenshortrate_getDF_HW(Rates(1:last),tenor(1:last),model_params,ini_DF_curve);
if(J==1)
Swaprate=SwapRate(DF,DCF);
end
MTM_paths(I,J)=SwapValue(Notional,Swaprate,DF,DCF);
% end
last=size(tenor,1)-J;
end
end
for I=1:1:size(tenor,1)
if I~=size(tenor,1)+1
PFE_proﬁle(I)=prctile(MTM_paths(:,I),95);
end
end
elseif strcmp(model,’CIR’)
model_params=load_params(model,pathstr);
k=model_params(1);
theta=model_params(2);

43

Chapter 3 Appendix

sigma=cap_data(3,2);
r_0=ppval(ini_fwd_curve,0);
for I=1:nsim
shortrate_paths(I,:)=CIR_short_rate(tenor,k,theta,sigma,r_0,delta_t);
last=size(tenor,1);
for J=1:size(tenor,1)
if(J==1)
dfcurve_paths(I,J,:)=shortrate_paths(I,1:last);
else
dfcurve_paths(I,J,:)=[shortrate_paths(I,1:last) zeros(size(tenor,1)-last,1)’];
end
%Change the maturity and ZCYC used in the calculation
Rates=squeeze(dfcurve_paths(I,J,:));
%for short rate
DF=Givenshortrate_getDF(Rates(1:last),tenor(1:last),model_params,sigma);
if(J==1)
Swaprate=SwapRate(DF,DCF);
end
MTM_paths1(I,J)=SwapValue(Notional,Swaprate,DF,DCF);
% end
last=size(tenor,1)-J;
end
end
for I=1:1:size(tenor,1)
if I~=size(tenor,1)+1
PFE_proﬁle1(I)=prctile(MTM_paths(:,I),95);
end
end
save(’PFE_data.mat’,’MTM_paths1’,
else
error(’Invalid Model: Please enter HW or CIR as Model’);

44 44


end
% for I=1:1:size(tenor,1)
% if I~=size(tenor,1)+1
% PFE_proﬁle(I)=prctile(MTM_paths(:,I),95);
% end
%
% end
end

45

Nomenclature
DF Discount Factor

IR Interest Rate

MTM Mark to Market

OTC Over the counter

PFE Potential Future Exposure

SDE Stochastic Diﬀerential Equation

VaR Value at Risk

47

A0067927 amit sinha_thesis

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