1 factor vs.2 factor gaussian model for zero coupon bond pricing final

1 Factor vs. 2Factor GaussianModel for Zero Coupon BondPricingA Computational Approach

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2Gaussian Model for Zero Coupon Bond Pricing

Topics Covered

1. Gaussian Models : 2 Factor vs. 1 Factor – A degree of fre edom

2. Setting up PDE & Computational Approach in C++

3. Appendix

1. Gaussian Integral

2. Instantaneous vs. Terminal Correlation considerations in term structuremodelling

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1Gaussian Models : 2 Factor vs. 1

Factor – A degree of freedom

Financial Algorithms™


One Factor vs. Two Factor : Term Structure of Zero Rates

These days interest rates tend to hover in negative zone; well that really increases the

complexity of pricing various instruments and derivatives. One of the complex case is the

pricing of zero coupon bond.

In HW set up short rate follows the dynamics :

�� Where �� a Wiener process, & � � a deterministic function of time. Solution of this SDE is :

� � � �� 1 � ��

��

The short rate has a normal distribution, with mean

� �� 1 � ��& variance

�� 2� 1 � ��

we see that the bigger the value of a, the faster � � tends to its limit distribution

Zero Coupon Bond Pricing in Negative Interest Rates Scenario



Negative Rates : Hull-White model compatibility

Negatives Rates

» Since the Hull-White model implies that the short rate has a normal distribution, this short

rate could technically take every value of real line ℝ, and a fortiori negatives values. In fact,

we can even compute the probability of it relatively easily:

� � � � 0 � ɸ � � � � !��

» If the short rate in the market are very low (near 0), volatility tends to be also very low.

» In the One-Factor Hull-White model, this scenario would return a bigger mean reversion and

a smaller θ � .

» From the expression of � � (see previous slide), we see that the bigger the mean reversion,

the lower the variance or � � .

– Real world scenario : we have seen recently that banks trading CHF (Swiss Franc) exchange a

negative overnight rate. USD LIBOR short rate were in the negative zone for couple of months during

GFC.



Issues with one factor model

» The short rate and its distributional properties suffice to characterize the yield curve, as we

have the relation :

� �, $ � � �� % & ' (')*

and the fact that with all the bond prices, we can reconstruct the yield curve.

» However, a not so robust short rate model would lead to a poor representation of the yield

curve and its evolution.

» One factor models such as the classic Hull-White gives 100% correlated LIBOR rates & that

is an issue in real world scenario.

» In reality, as we often see the yield curve steepening (short term LIBOR rates get lower, long

term LIBOR rates get higher).

» A one-factor model only allows us to make parallel moves of the yield curve.



The motivation for multiple factor model

» As we have seen that the One-Factor Hull-White model is a model where the rates tends to

reach a limit mean given by θ�� at a certain pace, given by the mean reversion �.

» The function θ�� is deterministic, but an intuitive way would be to add to it a stochastic

component u�� with a mean reversion b, lower than �.

» Therefore, after a certain time, our short rate model tends to be a one-factor stochastic

function which itself tends to be a deterministic function. We thus introduce a correlation

factor ρ between the Wiener processes driving the dynamics of r�� and u�� . » This prompts us to study the Two-Factor Hull-White model :

�� / � ��/�� 0� � �� Where ��/�� = ρ��

» This is actually equivalent to the two additive factor Gaussian Model. (see next slide)



Two additive factor Gaussian model

» The two factor Hull-White model is defined such that it assumes the short rate evolves in

the risk-adjusted measure according to :

with �1/, 1� a two dimensional Brownian motion such that �1/�� 1�� = ρ��, ��, a, b, /, � are positive constants, and −1 ≤ ρ ≤ 1.

» The deterministic function � � is chosen to fit the current term structure of interest rates.

» Now, let us define a new stochastic process where

χ/ � � � � �� 4�5 & χ� � � ��

5�4» Now, differentiating χcarefully, (function needs two step transformation if your

differentiation procedure is correct), both the stochastic process can be rewritten as :

� � � 6 � � χ/ � � χ� �Where 6 � � �� % ��7 ��8 �

� �7 & � � � (9 �(� � �6 �

�� / � �1/ � &� 0 � �� 0� � �� 1� � &� 0 � ��



The Gaussian Equations under Hull-White interest rate setup

» Zero coupon bond: Two factor Gaussian

model :

� � 6 � � ;/ �;�Where :

<;= ��ƙ=;= �� =�?=&

�?/ @ �?� � A��

» Zero coupon bond: One factor Gaussian

model :

� � 6 � � ;/Where :

<;= ��ƙ=;= �� =�?=with 6 � a deterministic function of time such

that 6 0 � �� (�ƙ= i.e. a , b from previous

slides)

Thus, the Hull-White Two-Factor Model is equivalent to a “Two-Additive-Factor Gaussian

Model”. This equivalence is very useful as it is more easy to interpret the different

parameters of the Hull-White model and their influence on the price and volatility structures,

the shape of the Gaussian model allows us to easier calculations for the prices of bonds

and derivatives

The Two-Factor Hull-White Model : Pricing and Calibration of Interest Rates Derivatives by Arnaud Blanchard (Page 21)



2 Setting up PDE & Computational

Approach in C++



Setting up grids for Crank-Nicolson Central Space Scheme for One Factor Model

PDE : Crank-Nicolson in C++

» Coordinate transformation concentrates grid points at x=0 (for concentration near

spot values)

Region[yMin<=y<=yMax && 0<=tau<=TMax, GenCoordins[{y},tau]];

ExCoordins[{{x},t}];

x==xMax Sinh[y]/Sinh[yMax];

t==tau;

» Default grid parameters (note that it is determined by sigma & kappa )

xMax==Max[6 Sqrt[sigma^2/(2 kappa)*(1-Exp[-2 kappa TMax])], 0.05];

yMax==GridParam; yMin==-GridParam;

GetorCreateParam[{GridParam}]; GridParam==1;



One Factor Partial Differential Equation with initial & boundary conditions

» One-factor Gaussian model PDE for bond

B�7, � B� �12� B�;

B;� � C; B 7, ;B; � φ7 � ;7 � 0

When [

der[V,t] + 1/2 sigma^2 der[V,{x,2}] - kappa x der[V,x] - (phi + x) V == 0;

];

When[Boundary, AutomaticBC; OneSidedDifference[1]];

When[max[tau], V==1];



One Factor Calibration & Output

» Calibrate time dependent E F to initial zero curve

E F � FG∆�I J K FG∆�I

J � F�∆�I J K F�∆�I

J∆�I � 5J

J & LM �N �O �MF )

phi== β + (1/2) a^2;

β ==((t + delta[tau]/2) InterpLin [ZCurve, ZDates, t+ delta[tau]/2] -

(t - delta[tau]/2) InterpLin [ZCurve,ZDates, t- delta[tau]/2]) / delta[tau];

a==(sigma/kappa) (1-Exp[-kappa t]);

ReadArray[{ZDates,ZCurve}, {ic,0,nC}, "ZeroCurve.txt"];

» Output[V, x==0, "ZeroBond.txt"];



Setting up grids for Crank-Nicolson Central Space Scheme for Two Factor Model

PDE : Crank-Nicolson in C++

» Coordinate transformation concentrates grid points at x1=x2=0 (for concentration

near spot values)

Region[yMin<=y1<=yMax && yMin<=y2<=yMax && 0<=tau<=TMax,

GenCoordins[{y1,y2},tau]];

ExCoordins[{{x1,x2},t}];

x1==x1Max Sinh[y1]/Sinh[yMax];

x2==x2Max Sinh[y2]/Sinh[yMax]; t==tau;

» Default grid parameters (note that it is determined by sigma & kappa for each

factor)

x1Max==Max[6 Sqrt[sigma1^2/(2 kappa1)*(1-Exp[-2 kappa1 TMax])], 0.1];

x2Max==Max[6 Sqrt[sigma2^2/(2 kappa2)*(1-Exp[-2 kappa2 TMax])], 0.1];

yMax==3; yMin==-yMax;



Two Factor Partial Differential Equation with initial & boundary conditions

» Two factor Gaussian model PDE for bond

B�7, � B� �12 =�

B=�;=B;=�

� A/�/�B 7, ;/

B;/B 7, ;�

B;�� C=;= B 7, ;=

B;=� φ7 � ;/7 � ;�7 � 0

When[

der[V,t] + 1/2 sigma1^2 der[V,{x1,2}]

+ 1/2 sigma2^2 der[V,{x2,2}]

+ rho sigma1 sigma2 der[der[V,x1],x2]

- kappa1 x1 der[V,x1]

- kappa2 x2 der[V,x2]

- (phi + x1 + x2) V == 0;

];

When[Boundary, AutomaticBC2];

When[max[tau], V==1];



Two Factor Calibration & Output

» Calibrate time dependent E F to initial zero curve

E F � FG∆�I J K FG∆�I

J � F�∆�I J K F�∆�I

J∆�I � �PQG�QQG�R�P�Q

J & STUT

�N � O �UTF

phi == ((t + delta[tau]/2) InterpLin[ZCurve,ZDates,t+delta[tau]/2] -

(t - delta[tau]/2) InterpLin[ZCurve,ZDates,t-delta[tau]/2]) / delta[tau] +

(1/2) (a1^2 + a2^2 + 2 rho a1 a2);

a1 == (sigma1/kappa1) (1-Exp[-kappa1 t]);

a2 == (sigma2/kappa2) (1-Exp[-kappa2 t]);

ReadArray[{ZDates,ZCurve}, {izc,0,nZC}, "ZCurve.txt"];

» Output[V, x1==0&&x2==0, "ZeroBond.txt"];



3 Appendix



Gaussian Integral

» Since % ��VQ�; is an improper and indefinite integral – it can not be solved unless we

transform this into % ��VQ �;GW�W .

» Again, solving it on Cartesian coordinates is cumbersome (via Laplace); thus easier way to

solve this by doubling it and transforming into polar coordinates using �� axis.

» The results after taking a square root of the integral will be X .

» Similarly, for the functions like ��YQQyou will get the results like

/�Z ��Y

QQ i.e. PDF of

standard normal distribution.

» And, for the function �� Y[\ QQ]Q you will get PDF

/S �Z �� Y[\ Q

Q]Q

» It is important to keep in mind the difference between the domain of a family of densities and

the parameters of the family. Different values of the parameters describe different

distributions of different random variables on the same sample space (the same set of all

possible values of the variable).



Instantaneous vs. Terminal Correlation considerations in term structure modelling

» The notion of correlation in the term structure modelling should be described in a more

formal way. For calibrating LIBOR market rates, instantaneous correlation is modelled.

However, for pricing correlation-sensitive products, terminal correlation is used.

» The instantaneous correlation of the LIBOR rates L(t, Ti−1, Ti) and L(t, Tj−1, Tj ) is given by

the correlation of the increments of the Brownian motions :

ρij(u) := instantaneous correlation between variables with indices i,j a moment u.

» Given instantaneous correlation and instantaneous volatility, terminal correlation can be

computed :

term_ρij(T) = [∫0TσI(u)σj(u)ρij(u)du] / √(vivj)

where: vi = ∫0Tσi(u)2du

» If instantaneous volatilities are not constant, they have a significant impact on terminal

correlation and can produce terminal de-correlation, even in the case of perfect

instantaneous correlation. Prices of correlation-sensitive products depend on terminal

correlation, and thus instantaneous correlation and instantaneous volatility.



Instantaneous vs. Terminal Correlation considerations in term structure modelling cont.’

» It is important to note that there is no instrument that is sensitive solely to instantaneous

correlation. Therefore, estimating correlation from a product that is sensitive to multiple

factors is not straight-forward and can lead to ambiguous results.

» Empirical evidences show that a plot of correlation versus parameter indices will be

sigmoidal in shape since positive correlation remains high (close to+1) for adjacent

forward rates and then jump to values close to zero(0) between short and long maturity

rates.

» A remedy to this is to use decorrelation; decorrelation is important for market rate

calibration as it is actually observed in the market for different forward rates.

» Consequently to get the best fit to the market data, what matters most is the calibration

of terminal correlation, which in turn requires fitting both the instantaneous correlation

and instantaneous volatilities.

» Typically if one estimates instantaneous correlation historically from time series of zero-

coupon rates at a given set of maturities, and then approximates by lower rank matrix

(called parameterization). This allows the users to reduce the correlation bias from the

various points of the yield curve during calibration process.



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1 factor vs.2 factor gaussian model for zero coupon bond pricing final

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