Fixed NumeriX I n c o m e M o d e l s : I m p l e m e n t a t i o nN o t e s

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NumeriX

R

Fixed Income Models: Implementation Notes

NumeriX Quantitative Research

March 6, 2007

Abstract

This document describes the quantitative models implemented in NumeriXR _analytics

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NumeriX is the recognized independent leader in pricing and risk ana-lytics for fixed income, credit, foreign exchange, hybrids, cross currency, inflation rate and equity derivatives. NumeriX has a financial engineer-ing and quantitative team composed largely of Ph.D.’s on the same scale as the very largest of financial institutions. More than 200 clients across 25 countries rely on NumeriX for speed and accuracy in valuing their structured products and derivatives. Trading and risk platform vendors leverage NumeriX analytics to gain a time-to-market advantage by em-bedding the power of NumeriX into their systems. Founded in 1996, the company is privately held and has offices in New York, Chicago, Santa Fe, Toronto, London, Paris, Singapore and Tokyo. For more information visit

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for ExcelR

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1 Short rate IR models

1.1 Black model

Black’s model is a basic model of interest rates in which either forward or swap interest rates are treated as lognormal stochastic variables, (see, e.g., [2] or [3]). The model gives access to very rapid analytics for a limited set of instruments, including caps and European swaptions.

1.2 Hull-White 1-factor

The Hull-White model has a normal short rate process

dr(t) = (θ(t)−λ(t)r(t))dt+ σ(t)dW(t) (1)

where:

r(t) is the short rate

λ(t) is the mean reversion

θ(t) is a deterministic function introduced to match the initial discount curve

σ(t) is the short rate volatility

dW(t) is Brownian motion (in risk neutral measure)

This model is implemented using several methods, including PDE Green’s function-based backward induction, trinomial trees, and forward simula-tions.

The zero coupon bond (ZCB) dynamics are determined from the short rate r(t),

PT(t) = Et[e−

RT

t r(u)du] (2)

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1.3 Hull-White 2- and 3-factor

For multifactor Hull-White, the short rate is modeled as a sum of several Markov processes xi(t) and an additional deterministic function α(t) used to match the original discount curve,

r(t) = d

X

i=1

xi(t) +α(t). (3)

Each auxiliary process obeys the Ornstein-Uhlenbeck equation,

dxi(t) = −λixi(t)dt+σi(t)dWi(t). (4)

The Brownian motions are correlated,

hdWi(t)dWj(t)i = ρijdt. (5)

Implemented are the cases of dimensionality d = 2,3. Similar to the one factor case, all integral functionals of the short rate are normally dis-tributed, and transition probabilities have a Gaussian kernel, which allows for an efficient Green’s function-based implementation.

1.4 Black-Karasinski and Black-Derman-Toy models

The Black-Karasinski (BK) model is a one-factor interest rate model where the logarithm of the short rate follows a standard Ornstein-Uhlenbeck process in the risk-neutral measure. If

x(t) = lnr(t) (6)

then

dx(t) = (θ(t)−λx(t))dt+σ(t)dW(t). (7)

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1.5 Spot-Skew model

The spot-skew model is a one-factor interest rate model that interpolates between Hull-White and Black-Karasinski. It is implemented as a PDE-based backward induction. The short rate r(t) is related to an auxiliary stochastic variable x(t) and a skew parameter β by

r(x;β) = 1

β[exp(βx)−1] +β. (8)

The variable x(t) follows a standard Ornstein-Uhlenbeck process in the risk-neutral measure:

dx(t) = (θ(t)−λx(t))dt+σ(t)dW(t). (9)

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2 BGM and other market models

Direct modeling of market observables such as LIBOR or swap rates nat-urally leads to the class of market models (MM). A standard formulation is due to Brace, Gatarek, and Musiela (BGM model, [1]).

2.1 Definition of market models and volatility structure

A market model is defined for a set of indexes {In(t)}. For LIBOR mar-ket models (LMM) the indexes are forward LIBOR rates while for swap market models (SMM) they are swap rates. The set of indexes should be sufficiently rich to express the forward discount factors on a given set of dates, Ti. That is, for any Ti it should be possible to derive zero bond prices P(Ti, Tj) observed on Ti and maturing at a later time Tj ≥ Ti. A diffusion SDE is imposed on each index In(t),

dIn(t) = · · ·+σn(I, t)·dW, (10)

where, in the most general case, the volatility function σn(I, t) can depend on time and on the entire set of the indexes. The Brownian motion W

can be multi-dimensional, in which case σn(I, t) is a vector and dot is the inner vector product. A measure-dependent drift term is understood by the notation “ . . . ” . For the classic BGM model the volatilities linearly depend on the indexes, and the index evolution is lognormal.

Thus a definition of an F-factor LMM includes

• a set of maturities Tn, n = 0, . . . , N such that 0 < T0 < T1 < · · · < TN

• initial forward LIBOR rates Ln(0)

• a volatility vector function σn(t, I) = {σn,0(t, I)· · ·σn,F−1(t, I)}, n = 0, . . . , N −1

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Tn+1

Ln(t) = 1

δn





P(t, Tn)

P(t, Tn+1)

−1



, (11)

where P(t, Tn) is the time t price of a zero coupon bond with maturity

Tn. We restrict ourselves to two specific LMM models: classic BGM and shifted BGM.

In the classic BGM model the forward LIBOR rates satisfy

dLn(t) = · · ·+Ln(t)λn(t)·dW(t) (12)

where vector notations are used for W(t) = {W0(t),· · ·, WF−1(t)} and

λn(t) = {λn,0(t),· · ·, λn,F−1(t)}. The Brownian increments are indepen-dent, E[dWidWj] = δijdt.

The shifted BGM model introduces an additional set of shift parameters,

sn, subject to the conditions

−Ln(0) < sn < 1

δn

. (13)

The SDE reads

dLn(t) = · · ·+ (Ln(t) +sn)λn(t)·dW(t). (14)

The BGM volatility has a large number of free functions, which makes it convenient to use simplifying assumptions regarding the volatility struc-ture. Popular volatility decompositions are (see [2] for other examples):

• λn,f(t) =σ1,f(t)σ2,f(Tn −t)

(available in NumeriX for direct model construction). The first term corresponds to a common time-dependent component while the second term is time-homogeneous.

• λn,f(t) =γn(t)Nn,f with |Nn(t)| = 1

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forward curve at time t has F degrees of freedom, each one defined by a global factor Xf(t) = R0tdWf(s)σf(s). This form guarantees time-independent correlations between forward LIBOR rates, Cnm = Nn · Nm.

2.2 Implementation of market models: numeraire and

discretiza-tion

The drift in the SDEs (12–14) depend on the measure which is fixed by a choice of a numeraire. We use a measure associated with the so-called rolling spot (RS) numeraire

N(t) = 1

P(T0, T1)

· · · 1

P(Ti−1, Ti)

P(t, Ti+1)

P(Ti, Ti+1)

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for Tn ≤ t ≤Tn+1. This numeraire is defined on a discrete set of maturities

Ti.

The SDEs for the BGM and the shifted BGM forward LIBOR rates in the RS measure have the following form:

dLn(t) = Ln(t)



 

n

X

j=η(t)

δj Lj(t) 1 +δjLj(t)

λn(t)·λj(t)





 dt+Ln(t)λn(t)·dW(t)

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dLn(t) = (Ln(t)+sn)



 

n

X

j=η(t)

δj (Lj(t) +sj) 1 +δj Lj(t)

λn(t)·λj(t)





dt+(Ln(t)+sn)λn(t)·dW(t)

(17) where η(t) = n+ 1 for Tn < t ≤ Tn+1.

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For a given time grid tk one has the following exact equation for the LIBOR evolution:

Ln(tk+1) = Ln(tk) exp

Z t_k₊₁

tk

dn(t)dt+

Z t_k₊₁

ki

λn(t)·dW(t)− 1 2

Z t_i₊_k

tk

|λn(t)|2dt

!

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Z t_k₊₁

tk

dt dn(t) = n

X

i=η(t)

Z t_k₊₁

tk

dt λi(t)·λn(t)

δiLi(t) 1 +δiLi(t)

. (19)

Note that we consider the volatility λn(t) to be step-constant on the dis-cretization dates tk, i.e., λn(t) =λn(tk) for tk ≤ t < tk+1.

Now the goal is to express the exponent in the simulation iteration scheme (18) as function of the Brownian increment ∆Wk = W(tk+1) −

W(tk) and time increment ∆tk = tk+1 −tk. The simulation of the second and the third terms is straightforward with

Z t_k₊₁

ki

λn(t)· dW(t) = λn(tk) ∆Wk (20)

and

Z t_i₊_k

tk

|λn(t)|2dt = |λn(tk)|2∆tk. (21)

The simulation of the drift term is non-trivial. We rewrite it as

Z t_k₊₁

tk

dt dn(t) = n

X

i=η(t)

λi(tk)·λn(tk)

Z t_k₊₁

tk

dt δiLi(t)

1 +δiLi(t)

. (22)

Each element in the sum is discretized up to the second order in ∆tk:

Z t_k₊₁

tk

dt δiLi(t)

1 +δiLi(t)

' δiLi(tk) 1 +δiLi(tk)

∆tk

+ δiLi(tk) 2 (1 +δiLi(tk))2





∆Wk ·λi(tk) + ∆tk

i−1

X

j=η(t)

δjLj(tk) 1 +δjLj(tk)

λj(tk)·λi(tk)



 ∆tk.

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2.3 Interpolation and dynamics of rates

The simulated BGM model permits direct calculation of instruments with fixing and payment dates falling on the model time grid because all ratios

P(Ti, Tn)/P(Ti, Tn+1) are known. To evaluate an instrument dependent or zero bond values P(t, T) for t, T 6= Ti interpolation is required.

Bond ratios

Mn(t) =

P(t, Tn)

P(t, Tn+1)

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satisfy a model-dependent SDE

dMn(t) =· · ·+Mn(t)αn(t)·dW (25)

with the diffusion term

αn(t) =

δnLn(t) 1 +δnLn(t)

λn(t) (26)

for the BGM model and

αn(t) =

δn(Ln(t) + sn) 1 +δnLn(t)

λn(t) (27)

for the shifted BGM model.

We use an arbitrage-free lognormal interpolation of the bonds which is done in two steps. The interpolation in maturity time T is given by

P(t, T) = P(t, Tn)

Tn+1−T

Tn+1−Tn P(t, T n+1)

T−Tn Tn+1−Tn

× exp

(₁

2

T −Tn

Tn+1 −Tn

Tn+1−T

Tn+1 −Tn

Z t

0 dτ|αn(τ)| 2

)

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for Tn < T < Tn+1. The interpolation in t is done according to

P(t, Tn) = P(Ti, Ti+1)

Ti+1−t

Ti+1−Ti

× exp

(

−1 2

t−Ti

Ti+1−Ti

Ti+1 −t

Ti+1 −Ti

Z T_i

0 dτ|αi(τ)| 2

)

× 1

Mi+1(t)Mi+2(t)· · ·Mn−1(t)

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for Ti < t ≤Ti+1.

Given interpolated bonds P(t, T) one can compute any financial rates (indexes) and their dynamics. We introduce a generic rate R(t) as a ratio of two linear combinations of bonds

R(t) =

P

iaiP(t, ti)

P

ja0j P(t, t0j)

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for any times ti, t0i and coefficients ai, a0i. For example, the forward LIBOR is

L(t) = 1

T0 −T 



P(t, T)

P(t, T0) −1



 =

P(t, T)−P(t, T0)

(T0 −T)P(t, T0) (31) and the forward CMS rate is

CM S(t) = _P_nP(t, t0)−P(t, tn) j=1(tj −tj−1)P(t, tj)

. (32)

The SDE for the rate R(t) has the form

dR(t) = · · ·+

P

j a0jP(t, t0j)

P

iaidP(t, ti)−PiaiP(t, ti)Pj a0jdP(t, t0j)

P

ja0jP(t, t0j)

2

(33) or

dR(t) =· · ·+ R(t)_P 1 iaiP(t, ti)



 X

i

aidP(t, ti)−R(t)

X

j

a0_j dP(t, t0_j)



.

(34) The instantaneous lognormal volatility σR(t) of the process for R(t)

dR(t) = · · · −R(t)σR(t)·dW(t) (35)

is given by

σR(t) =

1

P

iaiP(t, ti)



 X

i

aiP(t, ti)σ(t, ti)−R(t)

X

j

a0_j P(t, t0_j)σ(t, t0_j)



.

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An instantaneous correlation at the origin between two rates R and R0

(used in the calibration) can be easily calculated using the bracket process

hdR(t)dR0(t)i = R(t)R0(t)(σR(t)·σR0(t))dt.

Thus,

Corr(R(0), R0(0)) = q (σR(0)·σR0(0))

(σR(0)·σR(0)) (σR0(0)·σ_R0(0))

. (37)

Note that the rate log-volatilities depend on the initial yield curve and vector LMM volatilities λf(0, Tn) at origin.

2.4 Calibration and pricing using market models

The goal of the calibration is to come up with model parameters such that the constructed model

• fits the calibration input,

• has smooth volatility parameters, and

• remains attractive and effective for further numerical implementation; for example, it has a minimal number of factors and LIBOR rates to simulate.

The Market Model calibration strategy can include the following steps:

• Given a calibration input choose the model LIBOR dates Tn (or use a direct input for Tn).

• Choose the volatility decomposition form (or some other practical volatility characteristic).

• Choose a European swaptions analytical formula.

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Calibration input. The calibration input includes market information on option prices, correlation, and frozen model parameters, such as the num-ber of factors, and value of maturities and shifts.

• Market information

One can specify one market input type from the list below

– arbitrary set of swaptions/caplets

– arbitrary set of swaptions/caplets and correlations between arbi-trary forward LIBOR rates

– arbitrary set of swaptions/caplets and correlations between arbi-trary rates, such as CMS, etc.

The option input consists of a vector of options (including target prices) and their weights (optional).

The correlation input can have several forms

1. forward LIBOR correlations matrix – correlation matrix C_ij0

– correlation dates: start dates τ_i(st) and end dates τ_i(end)

A forward LIBOR with start τ_i(st) and τ_i(end) has an instantaneous market correlation at the origin equal to C_ij0 with a forward LIBOR

with start τ_j(st) and τ_j(end).

2. general rates correlations matrix – correlation matrix C_ij0

– rates vector: Ri

A rate Ri and rate Rj has an instantaneous market correlation at the origin equal to C_ij0 .

3. general rates correlations vector – correlation vector C_i0

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A rate Ri and rate R0_i has an instantaneous market correlation at the origin equal to C_i0.

If the correlation dates coincide with the BGM maturities one can use exact formulas for model correlations, otherwise a volatility interpola-tion should be used.

• Fixed model parameters are:

– Number of factors

– Maturities (optional)

– Shifts (optional)

One can fix in this way shifts sn for the shifted BGM model and calibrate only volatilities.

Calibration output. The calibration output for LMM includes

• Maturity dates Tn

• Volatilities

• Additional parameters, including shifts sn for the shifted BGM.

Choice of maturities. The BGM maturities Tn are ordered exercise dates

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two underlying rates. Otherwise, if one chooses 1Y coupon date intervals, the model will have the rates (3M LIBOR and short CMS rate) almost perfectly correlated due to the interpolation.

Volatility structure choice. The user can choose between two volatility

structures:

• general volatility

a complete matrix λn,f(t) of volatility curves

• constant in time (flat) correlations form

λn,f(t) =γn(t)Nn,f.

The time-dependent functions are assumed to be step-constant between exercise dates, in other words, the curves nodes are the options exercise dates. There is one exception from this rule: for a 1-factor model the nodes are only those exercise dates corresponding to more than one option.

Pricing with the model. Pricing non-path dependent callable instruments

requires the availability of zero coupon bonds P(t, T) and conditional ex-pectations. The conditional expectation is calculated using Monte Carlo (MC) least-square minimization (Longstaff-Schwartz type) for some set of states {si(t)}, i.e.,

E[X |Ft] →E[X |σ(s1(t), s2(t),· · ·)] = f(s1(t), s2(t),· · ·). (38)

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References

[1] A. Brace, D. Gatarek, and M. Musiela, “The Market Model of Interest Rate Dynamics,” Math. Finance 7, 127–155 (1997).

[2] D. Brigo and F. Mercurio, “Interest Rate Models: Theory and Prac-tice,” Springer-Verlag (2001).

Fixed NumeriX I n c o m e M o d e l s : I m p l e m e n t a t i o nN o t e s

NumeriX

Fixed Income Models: Implementation Notes

Contents

1

Short rate IR models

2

BGM and other market models

References