Hull White

Price-it®

Internal Quantitative Notes

Hull-White Interest Rate Model

May 11, 2012

General disclaimer

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b

Conventions

Boxed formulas are important:

dWQT= dWQ −B(t, T )σ(t)λdt

Highlighted formulas are effectively implemented in Price-It® :

A(t, T ) = B(t, T )(T − t) +1

2(V (0, T )− V (0, t))

c

Contents

General disclaimer b

Conventions c

1 Hull-White 1 Factor 11.1 Short rate diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Short rate conditional distribution . . . . . . . . . . . . . . . . . . 11.1.2 Monte-Carlo simulation . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Bond price formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Bond price diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Modelling in forward measure . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 Bank account . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.6 Analytical Pricing Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.6.1 Libor - Convexity Adjustment . . . . . . . . . . . . . . . . . . . . . 81.6.2 CMS - Convexity Adjustment . . . . . . . . . . . . . . . . . . . . . 91.6.3 Cap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.6.4 Swaption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.7 Excel test sheets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.7.1 MC Bank account & Libor . . . . . . . . . . . . . . . . . . . . . . 91.7.2 Analytical Libor & CMS . . . . . . . . . . . . . . . . . . . . . . . . 91.7.3 Analytical Cap & Swaption . . . . . . . . . . . . . . . . . . . . . . 9

1.8 Problems & Notices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.8.1 Bank account in forward measure . . . . . . . . . . . . . . . . . . . 9

2 Hull-White 2 Factors 10

3 Hull-White in IR-Equity hybrid 11

4 Hull-White in in IR-FX hybrid 12

d

1 Hull-White 1 Factor

1.1 Short rate diffusion

We consider the following extension of the Vasicek model by Hull and White [HW93]:

dr(t) = [θ(t)− ar(t)] dt+ σ(t)dW (t) (1.1)

in the risk neutral measure. The θ(t) chosen so as to exactly fit the term structure ofinterest rates currently observed in the market [BM07, §3.3.1],[AP10, §10.1.2.2] :

θ(t) =∂fM (0, t)

∂t+ afM (0, t) + y(t) (1.2)

y(t) =

∫ t

0σ(u)2e−2a(t−u)du (1.3)

where we denoting by fM (0, t) the market instantaneous forward rate at time 0 for thematurity t, i.e.,

fM (0, t) =∂PM (0, t)

∂t

with PM (0, t) the market discount factor.

1.1.1 Short rate conditional distribution

The equation (1.1) can be integrated to yield:

r(t) = r(s)e−a(t−s) +

∫ t

se−a(t−u)θ(u)du+

∫ t

sσ(u)e−a(t−u)dW (u)

= r(s)e−a(t−s) + α(t)− α(s)e−a(t−s) +

∫ t

sσ(u)e−a(t−u)dW (u) (1.4)

where:

α(t) = fM (0, t) +

∫ t

0e−a(t−u)y(u)du

= fM (0, t) +

∫ t

0

1

ae−a(t−u)

(1− e−a(t−u)

)σ(u)2du (1.5)

1

1.2 Bond price formula 2

1.1.2 Monte-Carlo simulation

Defining the process x(t) byx(t) =d r(t)− α(t) (1.6)

we immediately have that:

x(t) = x(s)e−a(t−s) +

∫ t

sσ(u)e−a(t−u)dW (u) (1.7)

x(0) = 0

In Price-It® we simulate the process x(t) defined in equation (1.6) and its diffusion isdescribed in equation (1.7). The volatility is parameterized as:

σ(t) =d e−µtΣ(t) (1.8)

with Σ(t) piece-wise constant. In Price-It® terminology, µ is called as mean reversionand Σ(t) as volatility. The process x(t) is exactly simulated.

1.2 Bond price formula

Note by P (t, T ) the price of a zero-coupon bond which pays 1 currency unit at maturityT . This can be calculated by an expectation in the risk neutral measure:

P (t, T ) = EQ[e−

∫ Tt r(u)du|Ft

](1.9)

Due to the Gaussian distribution of r(u) conditional on Ft,∫ Tt r(u)du is itself normally

distributed. From equation (1.4) we have:∫ T

tr(s)ds =

∫ T

tr(t)e−a(s−t)ds+

∫ T

t

(α(s)− α(t)e−a(s−t)

)ds

+

∫ T

t

∫ s

tσ(u)e−a(s−u)dW (u) ds (1.10)

We will calculate the mean and variance explicitly (always conditionally on Ft).

Random part

We re-write the martingale part in (1.10) on changing the integration order:∫ T

t

∫ s

tσ(u)e−a(s−u)dW (u) =

∫ T

tσ(u)

∫ T

ue−a(s−u)ds dW (u)

=

∫ T

tσ(u)

1

a

(1− e−a(T−u)

)dW (u)

=

∫ T

tσ(u)B(u, T )dW (u) (1.11)

Copyright © 2005-2012 Pricing Partners

1.2 Bond price formula 3

where we denote

B(u, T ) =d 1

a

(1− e−a(T−u)

)The variance V (t, T ) of

∫ Tt r(s)ds is:

V (t, T ) =d Var[∫ T

tr(s)ds|Ft

]=

∫ T

tσ2(u)B2(u, T )du (1.12)

Determinist part

The drift in (1.10) is:

E

[∫ T

tr(s)ds|Ft

]=

∫ T

tr(t)e−a(s−t)ds−

∫ T

tα(t)e−a(s−t)ds+

∫ T

tα(s)ds

= (r(t)− α(t))B(t, T ) +

∫ T

t

[fM (0, s) +

∫ s

0e−a(s−u)y(u)du

]ds

= (r(t)− α(t))B(t, T ) + lnPM (0, t)− lnPM (0, T ) +

∫ T

t

∫ s

0e−a(s−u)y(u)du ds

The last term in the previous equation can be showed to be:∫ T

t

∫ s

0e−a(s−u)y(u)du ds =

1

2(V (0, T )− V (0, t))

with V (0, t) the variance of∫ t0 r(s)ds as defined in the equation (1.12).

Bond price formula

With these above results and taking into account the fact that∫ t0 r(s)ds has Gaussian

distribution, we have the following expression for bond price:

Proposition 1. Knowing r(t) at time t, the bond price P (t, T ) is

P (t, T ) =PM (0, T )

PM (0, t)exp

{−1

2(V (0, T )− V (0, t)) +

1

2V (t, T )

}×

exp {−B(t, T )(r(t)− α(t))}

P (t, T ) =PM (0, T )

PM (0, t)exp

{1

2

∫ t

0σ2(u)

(B2(u, t)−B2(u, T )

)du

}×

exp {−B(t, T )x(t)} (1.13)


1.3 Bond price diffusion 4

The term in the first brakets could be written as:

1

2

∫ t

0

σ2(u)(B2(u, t)−B2(u, T )

)du = −1

2B2(t, T )

∫ t

0

σ2ue

−2a(t−u)du

−B(t, T )

∫ t

0

σ2ue

−a(t−u) 1

a

(1− e−a(t−u)

)du

Bond price in Monte-Carlo simulation

We see that the value of P (t, T ) at time t depends only on the short rate value r(t) at thattime. From equation (1.13), this price can be obtained once we know x(t) = r(t)− α(t)since every other terms are explicitly known from model parameters. However, rememberthat in our Monte-Carlo simulation, it is the process x(t) that we simulate, thus it is notnecessary to know α(t) to compute the bond price coherently.

In practice, we never need to know the short rate value r(t) but only quantities likethe bond price P (t, T ), which permit us later to have related quantities like the forwardLibor rates.

As an example suppose that in our product we need to simulate the Libor 3M at ntimes T1, T2, ..., Tn, the procedures are:

1. Simulate x(t) at the time points T1, T2, ..., Tn following the equation (1.7)

2. Calculate P (Ti, Ti + 3M) from x(Ti) as in the bond price equation (1.13)

3. Calculate Libor 3M as:

L(Ti, Ti + 3M) =1

δ

(1

P (Ti, Ti + 3M)− 1

)

1.3 Bond price diffusion

The bond price P (t, T ) viewed from date t = 0 is a random process and it is useful toknow its diffusion. From the equation (1.7) we see that P (t, T ) is a exponential function ofr(t) multiplied by a determinist term, thus the martigale part of its percentage variationis:

dP (t, T )

P (t, T )= ...−B(t, T )σ(t)dW (t)

The minus sign − is important and one should be careful to not miss it. This signifiesthe well-known fact that an increase in interest rate will cause bond price to fall.

Since we are in the risk-neutral measure and P (t, T ) is the price of a traded asset,its drift is r(t)dt, so we have:


1.4 Modelling in forward measure 5

Proposition 2. In the Q - risk neutral measure, the bond price diffuse as

dP (t, T )

P (t, T )= r(t)dt−B(t, T )σ(t)dWQ(t) (1.14)

Sometimes we need to know the diffusion in the QT ∗ - forward measure having P (t, T ∗)as numéraire (T ∗ may be different than T ). In this measure the market risk primium isjust the volatility of P (t, T ∗), so as:

dW T ∗= dWQ +B(t, T ∗)σ(t)dt (1.15)

Proposition 3. In the QT ∗ - forward neutral measure, the bond price diffuse as

dP (t, T )

P (t, T )= r(t)dt+B(t, T )B(t, T ∗)σ2(t)dt−B(t, T )σ(t)dW T ∗

(t) (1.16)

1.4 Modelling in forward measure

We consider now that the pricing is taken place in the QT ∗-forward measure. From (1.15)the diffusion of short rate is:

dr(t) = [θ(t)− ar(t)] dt+B(t, T̄ )σ2(t)dt+ σ(t)dW T ∗(t) (1.17)

One can simulate the short rate in this forward measure and apply the same formula forbond price as in proposition (1). The annexed process that we actually simulate x(t) iswritten in the forward measure as:

x(t) = x(s)e−a(t−s) +

∫ t

sB(u, T ∗)σ2(u)e−a(t−u)du+

∫ t

sσ(u)e−a(t−u)dW T ∗

(u) (1.18)

x(0) = 0

Suppose now we are living in the QT ∗-forward measure and the Brownian (or gaussiansgenerated in the program) in our simulation is W T ∗

(t). Suppose that we still simulatex(t) as before using the equation (1.7) instead of (1.18) so that we obtain a process x̃(t):

x̃(t) = x̃(s)e−a(t−s) +

∫ t

sσ(u)e−a(t−u)dW T ∗

(u) (1.19)

x̃(0) = 0

It is easy to see that we can recover the true x(t) from x̃(t) as:

x(t) = x̃(t) +

∫ t

0B(u, T ∗)σ2(u)e−a(t−u)du (1.20)

In Price-It®, the short rate is always simulated in the risk neutral measure, thus thecorrection term in (1.20) must be added to the bond price formula when we are pricing inthe forward measure. More explicity:


1.5 Bank account 6

Proposition 4. In the QT ∗-forward measure, suppose we still simulate the annexedprocess x(t) = r(t)− α(t) as in (1.18), the bond price is:

P (t, T ) =PM (0, T )

PM (0, t)exp

{1

2

∫ t

0σ2(u)

(B2(u, t)−B2(u, T )

)du

}× exp {−B(t, T )x(t)} × exp

{−B(t, T )

∫ t

0B(u, T ∗)σ2(u)e−a(t−u)du

}An additional term is added with respect to the equation (1.13). This expression could besimplified to give:

P (t, T ) =PM (0, T )

PM (0, t)exp

{−1

2B(t, T )[B(t, T )− 2B(t, T ∗)]

∫ t

0σ2(u)e−2a(t−u)du

}× exp {−B(t, T )x(t)} (1.21)

1.5 Bank account

The bank account is defined as

D(0, T ) =d exp

{∫ T

0r(s)ds

}which is the inverse of the discount factor, necessary for pricing in the risk neutral measure.From every date on the simulated path x(t0)), ..., x(tN ), we can reconstitute the bankaccount curve or the discount curve on these dates, i.e, D(0, t0), ..., D(0, tN ).

The price of a security which values S(T ) at date T and pays at T ∗ is:

Price = EQ[e−

∫ T∗0 r(s)dsS(T )

]= EQ

[e−

∫ T0 r(s)dse−

∫ T∗T r(s)dsS(T )

]= EQ

[e−

∫ T0 r(s)dsS(T )EQ

[e−

∫ T∗T r(s)ds|FT

]]The conditional expectation in the above equation is the forward bond price, we have:

Price = EQ[

1

D(0, T )S(T )P (T, T ∗)

](1.22)

The above equation (1.22) is at the heart of our simulation-pricing strategy. While T is areset date, i.e, the moment we determine the payoff’s value, it will be a simulation point.


1.5 Bank account 7

As stated above, the bank account curve will be constructed on all our simulation points sothat D(0, T ) will be known, while the forward bond price P (T, T ∗) is given by the equation(1.13)

Bank account curve constructionOver the time step [ti, ti+1], the advancement of x(t) on the schedule is:

x(ti+1) = E(x(ti+1)|x(ti)) +√

Var (x(ti+1)|x(ti))Zi (1.23)

and we recall from equation (1.7) that:

E(x(ti+1)|x(ti)) = x(ti)e−a(ti+1−ti)

Var (x(ti+1)|x(ti)) =

∫ ti+1

ti

σ2(u)e−2a(ti+1−u)du

What is the possible values of

I(ti, ti+1) =

∫ ti+1

ti

x(s)ds (1.24)

knowing that x(ti) and x(ti+1) are simulated?

Important point: If only x(ti) was known then the distribution of I(ti, ti+1),conditionally on x(ti), or Fti, is a Gaussian with mean and variance (see (1.12)):

E(I(ti, ti+1)|x(ti)) = x(ti)B(ti, ti+1)

Var (I(ti, ti+1)|x(ti)) =

∫ ti+1

ti

σ2(u)B2(u, ti+1)du

The value of I(ti, ti+1) is random and can be simulated using an other GaussianZ̃i independant of Zi:

I(ti, ti+1) = E(I(ti, ti+1)|x(ti)) +√

Var (I(ti, ti+1)|x(ti))Z̃i

Since we aslo have knowlegde on x(ti+1), the conditional distribution of I(ti, ti+1) ismodified and must take into account the mutual distribution of it and x(ti+1). Since con-ditionally on x(ti) both these distributions are Gaussian, their mutual distribution ischaracterized by their covariance.

The random part of x(ti+1) and I(ti, ti+1), conditionally on x(ti), are (see equations(1.7) and (1.11))

x(ti+1) = ...+

∫ ti+1

ti

σ(u)e−a(ti+1−u)du

I(ti, ti+1) = ...+

∫ ti+1

ti

σ(u)B(u, ti+1)du

thus their covariance is:


1.6 Analytical Pricing Formulas 8

Cov (I(ti, ti+1), x(ti+1)|x(ti)) =

∫ ti+1

ti

σ2(u)e−a(ti+1−u)B(u, ti+1)du

=

∫ ti+1

ti

σ2(u)1

ae−a(ti+1−u)

(1− e−a(ti+1−u)

)du (1.25)

1.6 Analytical Pricing Formulas

1.6.1 Libor - Convexity Adjustment

Suppose the Libor L(T, T + δ) is paid at T ∗ > TS , its forward price at T ∗ (with out thediscount P (0, T ∗) ) is:

L̃0 =1

δ

(ET

∗[

1

P (T, T + δ)

]− 1

)(1.26)

Note X(t) = P (t, T )/P (t, T + δ) a martingale in the QT+δ-forward measure, we willcompute its drift in the QT ∗-forward measure. We have the following relations from(1.15):

dW T ∗(t) = dWQ(t) +B(t, T ∗)σ(t)dt

dW T+δ(t) = dWQ(t) +B(t, T + δ)σ(t)dt

so that, from the martingale property in the QT+δ-forward measure:

dX(t)

X(t)= σX(t)dW T+δ(t)

= σX(t)(B(t, T + δ)−B(t, T ∗))σ(t)dt+ σX(t)dW T ∗(t)

A simple calculation using Ito’s lemma shows that:

σX(t) = σP (t,T )(t)− σP (t,T+δ)(t)

= −(B(t, T )−B(t, T + δ))σ(t)

and thus:dX(t)

X(t)= (B(t, T + δ)−B(t, T ))(B(t, T + δ)−B(t, T ∗))σ2(t)dt+ σX(t)dW T ∗

(t) (1.27)

We arrive at the convexity adjusted formula for Libor which is paid at T ∗:

1

δ

(exp

{∫ T

0(B(t, T + δ)−B(t, T ))(B(t, T + δ)−B(t, T ∗))σ2(t)dt

}P (0, T )

P (0, T + δ)− 1

)(1.28)


1.7 Excel test sheets 9

1.6.2 CMS - Convexity Adjustment

1.6.3 Cap

1.6.4 Swaption

1.7 Excel test sheets

1.7.1 MC Bank account & Libor

In this test we consider a Libor which pays in advance and in arrears. We price in using thecash numeraire (risk-neutral) and the bond price numeraire (forward neutral). This allowsus to verify the implementation of forward bond price P (t, T ) and of bank account D(0, t).

Test file: "HW-SingleAsset-BankAccount-Libor.xls"

1.7.2 Analytical Libor & CMS

In order to use the convexity correction, we must add ARR in the Libor keyword:

Libor(USD,StartDate, 3M,A/360, ARR)

Test file: "HW-SingleAsset-ConvexityCorrection.xlsx"

1.7.3 Analytical Cap & Swaption

1.8 Problems & Notices

1.8.1 Bank account in forward measure

In the pricing in the forward measure does not require the computation of bank account,we have not adapted the bank account curve construction for this case. This should bedone to prevent an eventual future bug.


2 Hull-White 2 Factors

10

3 Hull-White in IR-Equity hybrid

11

4 Hull-White in in IR-FX hybrid

12

Bibliography

[AP10] Leif Andersen and Vladimir Piterbarg. Interest Rate Modeling. Atlantic FinantialPress, 2010.

[BM07] Damiano Brigo and Fabio Mercurio. Interest Rate Models - Theory and Practicewith Smile, Inflation and Credit - 2nd Edition. Springer Finance, 2007.

[HW93] John Hull and Alan White. One factor interest rate models and the valuation ofinterest rate derivative securities. Journal of Financial and Quantitative Analysis,28(2), 1993.

13

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