Elements of Hull-White ModelYan Zeng

Version 1.0, last revised on 2011-12-13.

Abstract

In this note, we summarize the elements of Hull-White model.

Contents1 One-factor Hull-White model 2

1.1 Dynamics of short rate rt and state variable Xt under risk-neutral measure . . . . . . . . . . 21.2 Pricing formula of zero coupon bond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Joint density of (∫ t

0Xsds,Xt) and value of EQ

[e−

∫ t0Xsds

∣∣∣Xt

]under risk-neutral measure . . 4

1.4 Pricing formula of European contingent claim . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 Dynamics of short rate rt and state variable Xt under forward measure . . . . . . . . . . . . 51.6 Pricing formula of caplet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.6.1 Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.6.2 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.7 Calibration to caps market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.8 Pricing formulas of Asian swaps/caps/floors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.8.1 Price of Asian floater . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.8.2 Price of Asian caplet/flooret . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.9 Pricing cross-currency European contingent claims under HW1F . . . . . . . . . . . . . . . . 121.9.1 Dynamics of foreign rate under domestic money market account measure . . . . . . . 131.9.2 Pricing of European contingent claims . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.9.3 Joint density of (

∫ t

0Xd

s ds,Xdt , X

et , X

ft ) and value of Ed

[e−

∫ t0Xd

s ds∣∣∣Xd

t , Xet , X

ft

]. . . 15

1.9.4 Pricing of cross-currency swaption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.10 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.10.1 Implementation of ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.10.2 Implementation of νh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.10.3 Implementation of νH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2 Two-factor Hull-White model 202.1 Dynamics of x1(t) and x2(t) under risk-neutral measure . . . . . . . . . . . . . . . . . . . . . 202.2 Dynamics of x1(t) and x2(t) under T -forward measure . . . . . . . . . . . . . . . . . . . . . . 212.3 Dynamics of

∫ T

tx1(u)du and

∫ T

tx2(u)du under risk-neutral measure . . . . . . . . . . . . . . 22

2.4 Derivation of zero-coupon bond price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

A Summary of Girsanov’s Theorem for continuous semimartingale 23

B Lognormal approximation of sum of lognormals 24

C Characteristic function of a Gaussian random vector 25

1

1 One-factor Hull-White model1.1 Dynamics of short rate rt and state variable Xt under risk-neutral measure

Under the assumption of one-factor Hull-White model, the short rate process under the risk-neutralmeasure Q (the martingale measure associated with money market account numeraire) follows the dynamics

drt = (bt − κrt)dt+ σtdWt,

where κ is a constant, bt and σt are deterministic functions of t, and W is a standard Brownian motion.Solving the SDE gives rt = e−κtr0+e−κt

∫ t

0eκsbsds+e−κt

∫ t

0eκsσsdWs. Setting θt = e−κtr0+e−κt

∫ t

0eκsbsds

and Xt = e−κt∫ t

0eκsσsdWs. Then θt is a deterministic function of t and Xt is Gaussian process with mean

0 and variance e−2κt∫ t

0e2κsσ2

sds. In summary, we have

rt = θt +Xt, dXt = −κXtdt+ σtdWt, X0 = 0, EQ[Xt] = 0, EQ[X2t ] = e−2κt

∫ t

0

e2κsσ2sds (1)

1.2 Pricing formula of zero coupon bond1.2.1 Formula

Denote by P (t, T ) the time-t price of a zero-coupon bond with maturity T , we have (note P (t, T ) is afunction of the state variable Xt)P (t, T ) = P (t, T ;Xt) =

P (0,T )P (0,t) exp

−H(T − t)

[Xt + νh(t) + 1

2ν(t)H(T − t)]

P (0, t) = exp−∫ t

0θsds+ νHt

(2)

where

h(t) = e−κt

H(t) =∫ t

0h(s)ds

ν(t) = e−2κt∫ t

0e2κsσ2

sds

νh(t) = h ∗ v(t) =∫ t

0e−κ(t−s)ν(s)ds

νH(t) = H ∗ ν(t) =∫ t

0H(t− s)ν(s)ds.

We also note that ddtν

H(t) = νh(t).

1.2.2 Derivation

For a derivation of formula (2), define αt = σteκt. It’s easy to see Xt = h(t)

∫ t

0αsdWs. For the convenience

of later computation, we note for u > t,

Xu = e−κu

∫ t

0

αsdWs + e−κu

∫ u

t

αsdWs = h(u− t)Xt + h(u)

∫ u

t

αsdWs.

Therefore, risk neutral pricing formula yields

P (t, T ;Xt = x) = EQ[e−

∫ Tt

rudu∣∣∣Ft, Xt = x

]= e−

∫ Tt

θudu−H(T−t)xEQ[e−

∫ Tt

h(u)(∫ ut

αsdWs)du]

Define η =∫ T

th(u)(

∫ u

tαsdWs)du. Then η is a Gaussian random variable with 0 mean and

η2 =

[H(u)

∫ u

t

αsdWs|u=Tu=t −

∫ T

t

H(u)αudWu

]2

= H2(T )

(∫ T

t

αsdWs

)2

− 2H(T )

∫ T

t

αsdWs

∫ T

t

H(u)αudWu +

(∫ T

t

H(u)αudWu

)2

.

2

Therefore the variance of η is equal to

EQ[η2] = H2(T )

∫ T

t

α2sds− 2H(T )

∫ T

t

H(s)α2sds+

∫ T

t

H2(s)α2sds

and

P (t, T ;Xt = x) = e−∫ Tt

θudu−H(T−t)xEQ[e−η] = e−∫ Tt

θudu−H(T−t)x exp1

2V ar(η)

= exp

−∫ T

t

θudu−H(T − t)x+1

2

[H2(T )

∫ T

t

α2sds+

∫ T

t

H2(s)α2sds

]−H(T )

∫ T

t

H(s)α2sds

.

As particular cases, we haveP (0, T ) = exp−∫ T

0θudu+ 1

2

[H2(T )

∫ T

0α2sds+

∫ T

0H2(s)α2

sds]−H(T )

∫ T

0H(s)α2

sds

P (0, t) = exp−∫ t

0θudu+ 1

2

[H2(t)

∫ t

0α2sds+

∫ t

0H2(s)α2

sds]−H(t)

∫ t

0H(s)α2

sds.

Therefore, we have

P (0, T )

P (0, t)P (t, T ;Xt = x)= exp

H(T − t)x+

1

2

[H2(T )

∫ t

0

α2sds+

∫ t

0

H2(s)α2sds

]−H(T )

∫ t

0

H(s)α2sds

· exp

−1

2

[H2(t)

∫ t

0

α2sds+

∫ t

0

H2(s)α2sds

]+H(t)

∫ t

0

H(s)α2sds

= exp

H(T − t)x+

1

2[H2(T )−H2(t)]

∫ t

0

α2sds− [H(T )−H(t)]

∫ t

0

H(s)α2sds

.

That is,

P (t, T ;Xt = x) =P (0, T )

P (0, t)exp

−H(T − t)x− 1

2[H2(T )−H2(t)]

∫ t

0

α2sds+ [H(T )−H(t)]

∫ t

0

H(s)α2sds

Note [H(T )−H(t)]eκt = e−κt−e−κT

κ eκt = H(T − t), we get

P (t, T ;Xt = x)

=P (0, T )

P (0, t)exp

−H(T − t)

[x+

H(T ) +H(t)

2e−κt

∫ t

0

α2sds− e−κt

∫ t

0

H(s)α2sds

]=

P (0, T )

P (0, t)exp

−H(T − t)

[x+H(t)e−κt

∫ t

0

α2sds+

H(T )−H(t)

2e−κt

∫ t

0

α2sds− e−κt

∫ t

0

H(s)α2sds

]=

P (0, T )

P (0, t)exp

−H(T − t)

[x+H(t)e−κt

∫ t

0

α2sds+

1

2H(T − t)ν(t)− e−κt

∫ t

0

H(s)α2sds

].

Note

H(t)e−κt

∫ t

0

α2sds− e−κt

∫ t

0

H(s)α2sds = h(t)

[H(t)

∫ t

0

α2sds− H(s)

∫ s

0

α2udu

∣∣∣∣t0

+

∫ t

0

h(s)

(∫ s

0

α2udu

)ds

]

= h(t)

∫ t

0

eκsν(s)ds = νh(t),

We have obtained

P (t, T ;Xt = x) =P (0, T )

P (0, t)exp

−H(T − t)

[x+ νh(t) +

1

2ν(t)H(T − t)

],

which gives formula (2).

3

1.3 Joint density of (∫ t

0Xsds,Xt) and value of EQ

[e−

∫ t0 Xsds

∣∣∣Xt

]under risk-neutral

measureTo price a European contingent claim with payoff f(XT ) at terminal time T , we typically need to evaluate

V0 = EQ[e−

∫ T0

rsdsf(XT )]= e−

∫ T0

θsdsEQ[e−

∫ T0

Xsdsf(XT )]= e−

∫ T0

θsdsEQ[EQ

[e−

∫ T0

Xsds∣∣∣XT

]f(XT )

].

This demands the knowledge of the joint density of (∫ t

0Xsds,Xt) or the value of EQ

[e−

∫ t0Xsds

∣∣∣Xt

]. We

derive these two quantities in this section.It’s easy to see Zt :=

∫ t

0Xsds and Xt are jointly Gaussian. In order to know their joint density, it’s

sufficient to know their respective mean and variance, as well as their covariance. In this regard, we noteE[Xt] = E[Zt] = 0. Define

vX(t) =√E[X2

t ], vZ(t) =√E[Z2

t ], ρXZ(t) =E[XtZt]

vX(t)vZ(t), cXZ(t) = E[XtZt].

Then v2X(t) = E[X2t ] = e−2κt

∫ t

0e2κsσ2

sds = ν(t), and by integration-by-parts formula, we have

XtZt =

∫ t

0

ZsdXs +

∫ t

0

X2sds = −κ

∫ t

0

XsZsds+

∫ t

0

X2sds+ mart. part.

Taking expectation on both sides gives

cXZ(t) = −κ

∫ t

0

cXZ(s)ds+

∫ t

0

ν(s)ds.

Solving this integral equation gives

cXZ(t) = e−κt

∫ t

0

eκsν(s)ds = νh(t).

Finally, note ddtE

[Z2t

]= 2E[ZtXt] = 2cXZ(t) = 2νh(t), we have v2Z(t) = E[Z2

t ] = 2∫ t

0νh(s)ds = 2νH(t). In

summary, we have v2X(t) = E[X2

t ] = ν(t)

v2Z(t) = E[Z2t ] = 2νH(t)

cXZ(t) = E[XtZt] = νh(t)

ρ2XZ(t) =(E[XtZt])

2

v2X(t)v2

Z(t)= (νh(t))2

2ν(t)νH(t)

Therefore, the covariance matrix Σt of the pair (Zt, Xt) is

Σt =

(E[Z2

t ] E[XtZt]E[XtZt] E[X2

t ]

)=

(2νH(t) νh(t)νh(t) ν(t)

)and its inverse is

Σ−1t =

1

1− ρ2XZ(t)

(1

v2Z(t)

− ρXZ(t)vX(t)vZ(t)

− ρXZ(t)vX(t)vZ(t)

1v2X(t)

)The joint density function of the pair (Xt, Zt) is therefore

g(x, z; t) =|Σt|−

12

2πe−

12 (x,z)Σ

−1t (x,z)′

=1

2πvX(t)vZ(t)√1− ρ2XZ(t)

exp− 1

2(1− ρ2XZ(t)

) [ x2

v2X(t)− 2ρXZ(t)

xz

vX(t)vZ(t)+

z2

v2Z(t)

](3)

4

To compute the other quantity, note when conditioning on Xt = x,

Zt ∼ N

(cXZ(t)

v2X(t)x, v2Z(t)−

c2XZ(t)

v2X(t)

)= N

(xνh(t)

ν(t), 2νH(t)− (νh(t))2

ν(t)

).

So

EQ[e−Zt

∣∣Xt

]= exp

−Xt

νh(t)

ν(t)+ νH(t)− (νh(t))2

2ν(t)

(4)

1.4 Pricing formula of European contingent claimTo price a European contingent claim with payoff f(XT ) at terminal time T , we note by formula (4)

V0 = EQ[e−

∫ T0

rsdsf(XT )]= e−

∫ T0

θsdsEQ[e−

∫ T0

Xsdsf(XT )]= e−

∫ T0

θsdsEQ[EQ

[e−

∫ T0

Xsds∣∣∣XT

]f(XT )

]= exp

−∫ T

0

θsds+ νH(T )− (νh(T ))2

2ν(T )

EQ

[f(XT ) exp

−XT

νh(T )

ν(T )

]

= P (0, T ) exp− (νh(T ))2

2ν(T )

EQ

[f(XT ) exp

−XT

νh(T )

ν(T )

](5)

1.5 Dynamics of short rate rt and state variable Xt under forward measureDenote by QT the T -forward measure. The Radon-Nikodym derivative D of T -forward measure QT

w.r.t. the risk-neutral measure Q is defined as

Dt :=dQT |Ft

dQ|Ft

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

e∫ t0rudu

.

Therefore, by formula (2)

d lnDt = d lnP (t, T )− d

∫ t

0

rudu

= d

(lnP (0, T )− lnP (0, t)−H(T − t)

[Xt + νh(t) +

1

2ν(t)H(T − t)

])− rtdt

= (· · · )dt−H(T − t)σtdWt.

Since Dt is a martingale under risk-neutral measure Q, we conclude dDt/Dt does not have drift term underQ. Therefore,

dDt = −DtH(T − t)σtdWt

Define Lt = −∫ t

0H(T − u)σudWu. Then Dt = E(Lt) := exp

Lt − 1

2 ⟨L⟩t

. By Girsanov’s Theorem,

WTt := Wt − ⟨W,L⟩t = Wt +

∫ t

0

H(T − u)σudu

is a Brownian motion under QT . Therefore, under the T -forward measure QT , the dynamics of state variableX becomes (see Brigo and Mercurio [2], Lemma 4.2)

Xt = e−κ(t−s)Xs +

∫ t

s

e−κ(t−u)σudWTu −

∫ t

s

e−κ(t−u)H(T − u)σ2udu, X0 = 0.

5

Note ν(t)′ = −2κν(t) + σ2t and h(t) + κH(t) = 1, we have∫ t

0

e−κ(t−u)H(T − u)σ2udu =

∫ t

0

h(t− u)H(T − u)[ν(u)′ + 2κν(u)

]du

= H(T − t)ν(t)−∫ t

0

ν(u)du

[h(t− u)H(T − u)

]+ 2κ

∫ t

0

h(t− u)H(T − u)ν(u)du

= H(T − t)ν(t)−∫ t

0

ν(u)[κh(t− u)H(T − u)− h(t− u)h(T − u)

]du+ 2κ

∫ t

0

h(t− u)H(T − u)ν(u)du

= H(T − t)ν(t) +

∫ t

0

ν(u)[κh(t− u)H(T − u) + h(t− u)h(T − u)

]du

= H(T − t)ν(t) +

∫ t

0

ν(u)h(t− u)du

= H(T − t)ν(t) + νh(t).

Therefore

Xt = h(t− s)Xs +

∫ t

s

h(t− u)σudWTu −

[H(T − t)ν(t) + νh(t)

]− h(t− s)

[H(T − s)ν(s) + νh(s)

].

Conditioning on Fs, Xt (t > s) is Gaussian with mean h(t− s)Xs−[H(T − t)ν(t)+ νh(t)

]+h(t− s)

[H(T −

s)ν(s) + νh(s)]

and variance∫ t

se−2κ(t−u)σ2

udu = ν(t)− h(t− s; 2κ)ν(s).In summary, under the T -forward measure QT , the model dynamics is

rt = θt +Xt

Xt = h(t− s)Xs +∫ t

sh(t− u)σudW

Tu −

[H(T − t)ν(t) + νh(t)

]− h(t− s)

[H(T − s)ν(s) + νh(s)

]ET [Xt|Xs] = h(t− s)Xs −

[H(T − t)ν(t) + νh(t)

]+ h(t− s)

[H(T − s)ν(s) + νh(s)

]V arT [Xt|Xs] = ν(t)− h(t− s; 2κ)ν(s).

(6)

where WT is a standard Brownian motion under forward measure QT , and Xt is Guassian conditioning onXs (0 ≤ s < t).

1.6 Pricing formula of caplet1.6.1 Formula

Denote by F (t;Ts, Te) the time-t forward Libor rate with expiry Ts and maturity Te (t ≤ Ts < Te):

F (t; , Ts, Te) =1

τ

P (t, Ts)− P (t, Te)

P (t, Te)

where τ is the year fraction of [Ts, Te]. Let tf ≤ Ts be the rate fixing time, then the time-t price of the capletis

Vt =

P (t, Te)τBl

(1+τK

τ , 1τP (t,Ts)P (t,Te)

, σB, 1)

if t < tf

P (t, Te)τ(F (tf ;Ts, Te)−K)+ if t ≥ tf(7)

whereσB = H(Te − Ts)

√h(Ts − tf ; 2κ)ν(tf )− h(Ts − t; 2κ)ν(t)

6

and

Bl(K,F, v, w) = FwΦ(wd1(K,F, v))−KwΦ(wd2(K,F, v))

d1(K,F, v) =ln(F/K) + v2/2

v

d2(K,F, v) =ln(F/K)− v2/2

v.

In particular,

V0 = P (0, Te)τBl(K +

1

τ, F (0;Ts, Te) +

1

τ,[H(Te − tf )−H(Ts − tf )

]√ν(tf ), 1

).

1.6.2 Derivation

To see a derivation of formula (7), we define

St = St(Ts, Te) :=P (t, Ts)

P (t, Te), t < Ts < Te

so that we can take advantage of Black-Scholes formula. Indeed, we note under the Te-forward measure QTe ,S is necessarily a martingale. Using pricing formula of zero coupon bond, we have

d lnSt = d

(−[H(Ts − t)−H(Te − t)

](Xt + νh(t))− 1

2ν(t)

[H2(Ts − t)−H2(Te − t)

])= (· · · )dt−

[H(Ts − t)−H(Te − t)

]dXt

= (· · · )dt+H(Te − Ts)h(Ts − t)σtdWTet

Therefore, we must have

dSt = StH(Te − Ts)h(Ts − t)σtdWTt , S0 =

P (0, Ts)

P (0, Te)

and the forward Libor rate can be written in the form of

F (t;Ts, Te) =1

τ(St(Ts, Te)− 1),

where τ is the year fraction of [Ts, Te].Denote by tf (tf < Ts < Te) the rate fixing time of the Libor for [Ts, Te]. Then the price of the caplet at

time t is (assuming unit notional)

Vt = τ · P (t, Te)ETet

[(F (tf ;Ts, Te)−K)+

]= P (t, Te)E

Tet

[(Stf − (1 + τK))+

].

Using Black-Scholes formula Bl(K,F, v, w)

Bl(K,F, v, w) = FwΦ(wd1(K,F, v))−KwΦ(wd2(K,F, v))

d1(K,F, v) =ln(F/K) + v2/2

v

d2(K,F, v) =ln(F/K)− v2/2

v,

we can write

Vt =

P (t, Te)Bl

(1 + τK, P (t,Ts)

P (t,Te), σB , 1

)if t < tf

P (t, Te)τ(F (tf ;Ts, Te)−K)+ if t ≥ tf

7

whereσB = H(Te − Ts)

√h(Ts − tf ; 2κ)ν(tf )− h(Ts − t; 2κ)ν(t).

For sake of model calibration to caplets, we can write the above formula equivalently as

Vt =

P (t, Te)τBl

(1+τK

τ , 1τP (t,Ts)P (t,Te)

, σB , 1)

if t < tf

P (t, Te)τ(F (tf ;Ts, Te)−K)+ if t ≥ tf

1.7 Calibration to caps marketWe assume there is a tenure structure 0 = T0 < · · · , Tn and the Libor for [Ti−1, Ti] is set at a time

ti ≤ Ti−1. We take as given the Black volatilities of caplets for various strikes. Suppose the i-th caplet hasstrike Ki and denote by τi the year fraction of [Ti−1, Ti]. The market price for the i-th caplet is

V mkti = P (0, Ti)τiBl(Ki, F (0;Ti−1, Ti), σ

mkti

√ti, 1)

where σmkti is market quoted Black volatility. Recall the model price of the i-th caplet is given by

V model = P (0, Ti)τiBl(Ki +

1

τi, F (0;Ti−1, Ti) +

1

τi,[H(Ti − ti)−H(Ti−1 − ti)

]√ν(ti), 1

).

If we assume the model parameter κ is exogenously given, we can solve the following equation for ν(ti)(i = 1, · · · , n):

Bl(Ki, F (0;Ti−1, Ti), σmkti

√ti, 1) = Bl

(Ki +

1

τi, F (0;Ti−1, Ti) +

1

τi,[H(Ti − ti)−H(Ti−1 − ti)

]√ν(ti), 1

)where H(Ti − ti)−H(Ti−1 − ti) can be further written as h(Ti−1 − ti)H(Ti − Ti−1).

If we assume the model’s volatility process (σt)t≥0 is piecewise constant (t0 = 0)

σt =n∑

i=1

σi−11ti−1≤t<ti + σn−11tn≤t,

we have the following system of equations to solve for each σi−1 (i = 1, · · · , n):

σ2i−1 =

e2κtiν(ti)− e2κti−1ν(ti−1)∫ titi−1

e2κudu=

ν(ti)− e−2κ(ti−ti−1)ν(ti−1)∫ ti−ti−1

0e−2κudu

=ν(ti)− h(2κ; ti − ti−1)ν(ti−1)

H(2κ; ti − ti−1)

Once σ0, · · · , σn−1 have been determined, we can easily implement ν(t), νh(t) and νH(t) for general t ≥ 0.

1.8 Pricing formulas of Asian swaps/caps/floorsTo simplify notations, we focus on the pricing of a single payment for the coupon period [Ts, Te]. We

assume t1, · · · , tn are the rate fixing times for the Libor rates being averaged, and we assume [ai, bi] (i =1, · · · , n) is the corresponding expiry-maturity pairs. Denote by wi the weight for the i-th Libor rate.Assuming unit notional, we need to value the time-t price of a payment made at time T (t < Te ≤ T ):

τ(Ts, Te)∑n

i=1 wiF (ti; ai, bi) for floaterτ(Ts, Te)

(∑ni=1 wiF (ti; ai, bi)−K

)+ for capletτ(Ts, Te)

(K −

∑ni=1 wiF (ti; ai, bi)

)+ for flooret

where τ(Ts, Te) is the year fraction of [Ts, Te] and F (ti; ai, bi) is the forward Libor rate for coupon period[ai, bi]:

F (ti; ai, bi) =1

τi

P (ti, ai)− P (ti, bi)

P (ti, bi).

Here τi is the year fraction of [ai, bi]. Note in general, we have ti ≤ ai < Te, but we could have both bi ≤ Te

and bi > Te.

8

1.8.1 Price of Asian floater

The time-t price of the floating leg for the coupon period [Ts, Te] is

Vt = P (t, T )τ(Ts, Te)

(n∑

i=1

wiF (ti; ai, bi)1ti≤t +n∑

i=1

wi

τi

[Aie

−BiXt − 1]1t<ti

)

where

Ai =P (0,ai)P (0,bi)

exp−1

2ν(ti)[H2(ai − ti)−H2(bi − ti)

]− λiαi +

12λ

2iβi

Bi = λih(ti − t)

λi = H(ai − ti)−H(bi − ti)

αi = −[H(T − ti)ν(ti) + νh(ti)

]+ h(ti − t)

[H(T − t)ν(t) + νh(t)

]βi = ν(ti)− h(ti − t; 2κ)ν(t)

If li is the leverage and si is the spread for the i-th Libor, the price formula is revised to

Vt = P (t, T )τ(Ts, Te)

(n∑

i=1

wi

(liF (0; ai, bi) + si

)1ti≤t +

n∑i=1

wiliτi

[Aie

−BiXt −(1− siτi

li

)]1t<ti

)(8)

In particular,

V0 = P (0, T )τ(Ts, Te)n∑

i=1

wiliτi

[Ai −

(1− siτi

li

)]with

Ai =P (0,ai)P (0,bi)

exp−1

2ν(ti)[H2(ai − ti)−H2(bi − ti)]− λiαi +

12λ

2i ν(ti)

λi = H(ai − ti)−H(bi − ti)

αi = −[H(T − ti)ν(ti) + νh(ti)]

To see a derivation of the above formulas, we note the time-t price of the floater is given by

Vt = P (t, T )τ(Ts, Te)

n∑i=1

wiETt [F (ti; ai, bi)] = P (t, T )τ(Ts, Te)

n∑i=1

wi

τi

(ET

t

[P (ti, ai)

P (ti, bi)

]− 1

).

Recall

P (ti, ai)

P (ti, bi)=

P (0, ai)

P (0, bi)exp

− [H(ai − ti)−H(bi − ti)]Xti −

1

2ν(ti)

[H2(ai − ti)−H2(bi − ti)

]If t ≥ ti, ET

t [F (ti; ai, bi)] = F (ti; ai, bi). So without loss of generality, we assume in the below t < ti.Then for each ti ∈ (t, T ], the dynamics of Xti under the T -forward measure QT is

Xti = h(ti − t)Xt +

∫ ti

t

h(ti − u)σudWTu −

[H(T − ti)ν(ti) + νh(ti)

]− h(ti − t)

[H(T − t)ν(t) + νh(t)

].

Conditioning on Ft, Xti is Gaussian with mean h(ti−t)Xt−[H(T − ti)ν(ti) + νh(ti)

]+h(ti−t)

[H(T − t)ν(t) + νh(t)

]and variance ν(ti)− h(ti − t; 2κ)ν(t). Therefore

ETt

[e−λXti

]= exp

−λh(ti − t)Xt + λ[H(T − ti)ν(ti) + νh(ti)]− λh(ti − t)[H(T − t)ν(t) + νh(t)]

+1

2λ2[ν(ti)− h(ti − t; 2κ)ν(t)]

.

9

This gives us the following formula

ETt

[P (ti, ai)

P (ti, bi)

]=

P (0, ai)

P (0, bi)exp

−1

2ν(ti)

[H2(ai − ti)−H2(bi − ti)

]· exp

−λih(ti − t)Xt − λiαi +

1

2λ2iβi

where

λi = H(ai − ti)−H(bi − ti)

αi = −[H(T − ti)ν(ti) + νh(ti)

]+ h(ti − t)

[H(T − t)ν(t) + νh(t)

]βi = ν(ti)− h(ti − t; 2κ)ν(t)

Combined, we have

Vt = P (t, T )τ(Ts, Te)

(n∑

i=1

wiF (ti; ai, bi)1ti≤t +n∑

i=1

wi

τi

[Aie

−BiXt − 1]1t<ti

)

where

Ai =P (0,ai)P (0,bi)

exp−1

2ν(ti)[H2(ai − ti)−H2(bi − ti)

]− λiαi +

12λ

2iβi

Bi = λih(ti − t)

λi = H(ai − ti)−H(bi − ti)

αi = −[H(T − ti)ν(ti) + νh(ti)

]+ h(ti − t)

[H(T − t)ν(t) + νh(t)

]βi = ν(ti)− h(ti − t; 2κ)ν(t)

In particular, we have

V0 = P (0, T )τ(Ts, Te)n∑

i=1

wi

τi

[P (0, ai)

P (0, bi)exp

−1

2ν(ti)

[H2(ai − ti)−H2(bi − ti)

]+λi

[H(T − ti)ν(ti) + νh(ti)

]+

1

2λ2i ν(ti)

− 1

]Sometimes, we will have leverage and spread for Libor rates. Denote by li the leverage and si the spread

for the i-th Libor, the time-t price of the floater is given by

Vt = P (t, T )τ(Ts, Te)

n∑i=1

wiETt

[liF (ti; ai, bi) + si

]So the price is revised to

Vt = P (t, T )τ(Ts, Te)

(n∑

i=1

wi

(liF (0; ai, bi) + si

)1ti≤t +

n∑i=1

wiliτi

[Aie

−BiXt −(1− siτi

li

)]1t<ti

)

1.8.2 Price of Asian caplet/flooret

We let w take either the value 1 or the value −1, with 1 for caplet and −1 for flooret. Then the time-tvalue of the contract is

Vt = P (t, T )τ(Ts, Te)ETt

[(w(eξ −K ′)

)+]= P (t, T )τ(Ts, Te)Bl(K ′, eµ+

12 σ

2

, σ, w) (9)

10

where µ and σ2 are obtained as below (i0 is the first i such that ti > t)

σ2 = ln VM2

µ = lnM − 12 σ

2

M =∑n

i=i0w′

ieµi+

12Σii

V =∑n

i,j=i0w′

iw′je

µi+µj+12 (Σii+Σjj)+Σij

K ′ =∑n

i=i0wi

τi+K +

∑i0−1i=1 wiF (ti; ai, bi)

w′i =

wi

τi

P (0,ai)P (0,bi)

exp− 1

2ν(ti)[H2(ai − ti)−H2(bi − ti)

]−[H(ai − ti)−H(bi − ti)

]νh(ti)

µi = −

[H(ai − ti)−H(bi − ti)

] (h(ti − t)Xt − [H(T − ti)ν(ti) + νh(ti)] + h(ti − t)[H(T − t)ν(t) + νh(t)]

)Σij(t) =

[H(ai − ti)−H(bi − ti)

][H(aj − tj)−H(bj − tj)

]h(ti + tj − 2ti ∧ tj)

[ν(ti ∧ tj)− h(ti ∧ tj − t; 2κ)ν(t)

]In particular, we have

V0 = P (0, T )τ(Ts, Te)ET0 [(w(e

ξ −K ′))+] = P (0, T )τ(Ts, Te)Bl(K ′, eµ+12 σ

2

, σ, w)

where

σ2 = ln VM2

µ = lnM − 12 σ

2

M =∑n

i=1 w′ie

µi+12Σii

V =∑n

i,j=1 w′iw

′je

µi+µj+12 (

∑ii +

∑jj)+

∑ij

K ′ =∑n

i=1wi

τi+K

w′i =

wi

τi

P (0,ai)P (0,bi)

exp−1

2ν(ti)[H2(ai − ti)−H2(bi − ti)

]− [H(ai − ti)−H(bi − ti)] ν

h(ti)

µi = [H(ai − ti)−H(bi − ti)][H(T − ti)ν(ti) + νh(ti)]

Σij =[H(ai − ti)−H(bi − ti)

][H(aj − tj)−H(bj − tj)

]h(ti + tj − 2ti ∧ tj)ν(ti ∧ tj)

To derive formula (9), we let w take either the value 1 or the value −1. We need to value

the time-t price of τ(Ts, Te)

(w

n∑i=1

wiF (ti; ai, bi)− wK

)+

which is paid at T (t < Te ≤ T ).

We first compute covariance matrix of Xti ’s under QT , conditioning on Ft. Given the dynamics of Xti

under QT

Xti = h(ti − t)Xt +

∫ ti

t

h(ti − u)σudWTu −

[H(T − ti)ν(ti) + νh(ti)

]− h(ti − t)

[H(T − t)ν(t) + νh(t)

],

this is easily obtained as

CovTt [XtiXtj ] = ETt

[∫ ti

t

h(ti − u)σudWTu

∫ tj

t

h(tj − v)σvdWTv

]= e−κ(ti+tj)

∫ ti∧tj

t

e2κuσ2udu

= h(ti + tj − 2ti ∧ tj) [ν(ti ∧ tj)− h(ti ∧ tj − t; 2κ)ν(t)] .

Denote by i0 the first i such that ti > t. Then

n∑i=1

wiF (ti; ai, bi)−K =

n∑i=i0

wiF (ti; ai, bi)−

(K +

i0−1∑i=1

wiF (ti; ai, bi)

)

=n∑

i=i0

wi

τi

P (ti, ai)

P (ti, bi)−

(n∑

i=i0

wi

τi+K +

i0−1∑i=1

wiF (ti; ai, bi)

)

11

Define K ′ =∑n

i=i0wi

τi+K +

∑i0−1i=1 wiF (ti; ai, bi) and for i ≥ i0, define

w′i =

wi

τi

P (0, ai)

P (0, bi)exp

−1

2ν(ti)

[H2(ai − ti)−H2(bi − ti)

]−[H(ai − ti)−H(bi − ti)

]νh(ti)

,

Zi = −[H(ai − ti)−H(bi − ti)

]Xti .

We then haven∑

i=1

wiF (ti; ai, bi)−K =n∑

i=i0

w′ie

Zi −K ′.

Conditioning on Ft, Zi is Gaussian with mean

µi = −[H(ai − ti)−H(bi − ti)

] (h(ti − t)Xt − [H(T − ti)ν(ti) + νh(ti)] + h(ti − t)[H(T − t)ν(t) + νh(t)]

)and covariance

Σij(t) =[H(ai− ti)−H(bi− ti)

][H(aj − tj)−H(bj − tj)

]h(ti+ tj −2ti∧ tj)

[ν(ti∧ tj)−h(ti∧ tj − t; 2κ)ν(t)

].

By the computation of Appendix B,n∑

i=1

wiF (ti; ai, bi)−K =n∑

i=i0

w′ie

Zi −K ′ ≈ eξ −K ′

where ξ ∼ N(µ, σ2) with σ2 = ln V

M2

µ = lnM − 12 σ

2

M =∑n

i=i0w′

ieµi+

12Σii

V =∑n

i,j=i0w′

iw′je

µi+µj+12 (Σii+Σjj)+Σij

Then the time-t value of the contract is

Vt = P (t, T )τ(Ts, Te)ETt

[(w(eξ −K ′)

)+]= P (t, T )τ(Ts, Te)Bl(K ′, eµ+

12 σ

2

, σ, w).

1.9 Pricing cross-currency European contingent claims under HW1FSuppose we have two currencies. Traded on the market are foreign money market Mf and a family of

foreign zero-coupon bonds Pf with all maturities, as well as domestic money market Md and a family ofdomestic zero-coupon bonds Pd with all maturities. Foreign assets are denominated in foreign currency anddomestic assets are denominated in domestic currency. We denote by Y the spot exchange rate that givesunits of domestic currency per unit of foreign currency.

Denominating everything in domestic currency, we have the tradable assets(Md,Pd, Y Mf , Y Pf

).

By the arbitrage theory, we can find a probability measure Qd such that(

Pd

Md ,YMf

Md , Y Pf

Md

)are Qd-martingales.

We assume under Qd, the domestic short rate follows one-factor Hull-White model

rdt = θdt +Xdt , dX

dt = −κdXd

t dt+ σdt dW

dt , X

d0 = 0

and the spot exchange rate Y satisfies the SDE

dYt = Yt(µetdt+ σe

t dWet ),

where σet is a deterministic function of time t and W e is a standard Brownian motion with dW e

t dWdt = ρdedt

(ρde is a constant). To avoid arbitrage, it’s necessary that the prices of domestic zero-coupon bonds are

12

given by P d(t, T ) = Ed

[exp

−∫ T

trdsds

∣∣∣Ft

](Ed denotes the expectation under Qd) and the exchange

rate satisfiesdYt = Yt

[(rdt − rft )dt+ σe

t dWet

]To take advantage of the single currency pricing infrastructure in each currency, we now denominate

everything in foreign currency: (Md

Y,Pd

Y,Mf , P f

).

We assume there is a probability measure Qf under which(

Md

YMf ,Pd

YMf ,Pf

Mf

)are martingales. We assume

under Qf the foreign short rate also follows one-factor Hull-White model

rft = θft +Xft , dX

ft = −κfXf

t dt+ σft dW

ft , X

f0 = 0

where W f is a standard Brownian motion under Qf , with dW ft dW

dt = ρdfdt and dW f

t dWet = ρefdt (ρdf and

ρef are constants).1Motivated by risk neutral pricing under domestic money market account measure, we need to know what

the behavior of rft is under Qd?

1.9.1 Dynamics of foreign rate under domestic money market account measure

We make the following key observation: Qf is such that the domestic currency denominated assets

(Md,Pd, Y Mf , Y Pf )

discounted by the numéraire YMf are Qf -martingales. Therefore, Qf is the martingale measure associatedwith the numéraire YMf .

Then we can “read out” the dynamics of rft directly. Indeed, the Radon-Nikodym derivative of Qd withrespect to Qf is

dQd|Ft

dQf |Ft

=1

Ytexp

∫ t

0

(rds − rfs )ds

= exp

−∫ t

0

σesdW

es +

1

2

∫ t

0

(σes)

2ds

= exp

−∫ t

0

σes(dW

es − σe

sds)−1

2

∫ t

0

(σes)

2ds

= E

(−∫ t

0

σesd(W

es −

∫ s

0

σeudu)

),

where E stands for the Dolèans-Dade exponential and W et −

∫ t

0σesds can be verified to be a Brownian motion

under Qf . So by Girsanov’s Theorem,

W ft − ⟨W f ,−

∫ ·

0

σesd(W

es −

∫ s

0

σeudu)⟩t = W f

t + ρef

∫ t

0

σesds := W f

t

is a Qd-BM. Then the dynamics of foreign rate rft under domestic money market account measure Qd

becomesrft = θft + Xf

t , dXft = −κf Xf

t dt+ σft dW

ft , X

f0 = 0

where θft = θft − ρefe−κf t

∫ t

0eκ

fsσfs σ

esds.

1Since the quadratic variation of W d and W f can be calculated as the a.s. limit of∑

i(Wdti−W d

ti−1)(W f

ti−W f

ti−1) and since

Qd and Qf are equivalent, ⟨W d,W f ⟩ is uniquely defined regardless the measure under consideration. Therefore, the notion ofinstantaneous correlation ρdf is well-defined. Similar argument holds for ρef .

13

1.9.2 Pricing of European contingent claims

We summarize the dynamics of domestic rate, foreign rate, and exchange rate under the domestic moneymarket account measure Qd as follows (W d, W f , and W e are all standard BM under Qd):

rdt = θdt +Xd

t , dXdt = −κdXd

t dt+ σdt dW

dt , X

d0 = 0

rft = θft + Xft , dX

ft = −κf Xf

t dt+ σft dW

ft , X

f0 = 0, θft = θft − ρefe

−κf t∫ t

0eκ

fsσfs σ

esds

Yt = Y0eθet+Xe

t , θet =∫ t

0(θds − θfs )ds− 1

2

∫ t

0(σe

s)2ds, Xe

t =∫ t

0(Xd

s − Xfs )ds+

∫ t

0σesdW

es

dW dt dW

et = ρdedt, dW

dt dW

ft = ρdfdt, dW

et dW

ft = ρefdt

(10)

Suppose we have a European contingent claim ξ having payoff ξ = f(g1(XdT ), g2(X

fT )YT ) at time T ,

where f , g1 and g2 are all deterministic functions. Then risk neutral pricing gives the claims’s price at time0 as (Ed stands for the expectation under domestic money market account measure Qd)

V0 = Ed[e−

∫ T0

rdt dtf(g1(XdT ), g2(X

fT )YT )

]= e−

∫ T0

θdt dtEd

[e−

∫ T0

Xdt dtf

(g1(X

dT ), g2(X

fT + θfT − θfT )Y0e

θeT+Xe

T

)]= P d(0, T )e−νH

d (T )Ed

[e−

∫ T0

Xdt dtf

(g1(X

dT ), g2(X

fT − ρefe

−κf t

∫ t

0

eκfsσf

s σesds)Y0e

XeT

·Pf (0, T )

P d(0, T )e−νH

f (T )+νHd (T )+ρef

∫ T0

e−κf t∫ t0eκ

f sσfs σ

esds− 1

2

∫ T0

(σet )

2dt)]

= A(T )Ed[e−

∫ T0

Xdt dtf

(g1(X

dT ), g2(X

fT −B(T )) · C(T )Y0e

XeT)]

,

where A(T ) = P d(0, T )e−νH

d (T )

B(T ) = ρefe−κfT

∫ T

0eκ

fsσfs σ

esds

C(T ) = P f (0,T )Pd(0,T )

e−νHf (T )+νH

d (T )+ρef

∫ T0

e−κf t∫ t0eκ

f sσfs σ

esds− 1

2

∫ T0

(σet )

2dt

So the price formula hinges on the joint density function of the centered Gaussian random vector(ZT , X

dT , X

fT , X

eT ) (ZT =

∫ T

0Xd

t dt). With a “conditioning” trick, we can reduce the problem’s dimension by1 (see Section 1.9.3) and obtain the time-0 price of contingent claim as

V0 = A(T )e12σ

2(T )Ed[e−µ(Xd

T ,XeT ,Xf

T ,T )f(g1(X

dT ), g2(X

fT −B(T )) · C(T )Y0e

XeT

)](11)

where

µ(xd, xe, xf , t) = (czd(t), cze(t), czf (t))

v2d(t) cde(t) cdf (t)c2ed(t) v2e(t) cef (t)cfd(t) cfe(t) v2f (t)

−1xd

xe

xf

and

σ2(t) = v2z(t)− (czd(t), cze(t), czf (t))

v2d(t) cde(t) cdf (t)c2ed(t) v2e(t) cef (t)cfd(t) cfe(t) v2f (t)

−1cdz(t)cez(t)cfz(t)

with the covariance matrix given in Appendix 1.9.3:

Σt = E

Zt

Xdt

Xet

Xft

(Zt, Xdt , X

et , X

ft )

=

v2z(t) czd(t) cze(t) czf (t)cdz(t) v2d(t) cde(t) cdf (t)cez(t) c2ed(t) v2e(t) cef (t)cfz(t) cfd(t) cfe(t) v2f (t)

14

1.9.3 Joint density of (∫ t

0Xd

s ds,Xdt , X

et , X

ft ) and value of Ed

[e−

∫ t0Xd

s ds∣∣∣Xd

t , Xet , X

ft

]We define Zt =

∫ t

0Xd

s ds. Recall

Xdt = e−κdt

∫ t

0

eκdsσd

sdWds , X

et =

∫ t

0

(Xds − Xf

s )ds+

∫ t

0

σesdW

es , X

ft = e−κf t

∫ t

0

eκfsσf

s dWfs ,

where W d, W f and W e are all standard Brownian motion under Qd with the following instantaneouscorrelation:

dW dt dW

ft = ρdfdt, dW

dt dW

et = ρdedt, dW

ft dW

et = ρfedt.

Note (Zt, Xdt , X

ft , X

et ) are jointly Gaussian, so it’s sufficient to calculate the covariance matrix. We can

easily deduce the following formulas:

v2z(t) = Ed[(Zt)2] = 2νHd (t)

v2d(t) = Ed[(Xdt )

2] = νd(t)

v2e(t) = 2cze(t)− v2z(t) + 2νHf (t) +∫ t

0(σe

s)2ds− 2ρef

∫ t

0e−κf ξ

∫ ξ

0eκ

fsσesσ

fs dsdξ

v2f (t) = Ed[(Xft )

2] = νf (t)

czd(t) = Ed[Xdt Zt] = νhd (t)

cze(t) = ρde∫ t

0e−κds

∫ s

0eκ

duσduσ

eududs− ρdf

∫ t

0

∫ s

0e−κd(s−ξ)

(e−(κd+κf )ξ

∫ ξ

0e(κ

d+κf )uσduσ

fudu

)dξds

−ρdf∫ t

0

∫ s

0e−κf (s−ξ)

(e−(κd+κf )ξ

∫ ξ

0e(κ

d+κf )uσduσ

fudu

)dξds+ v2z(t)

czf (t) = ρdf∫ t

0e−κf (t−s)

(e−(κd+κf )s

∫ s

0e(κ

d+κf )uσduσ

fudu

)ds

cde(t) = νhd (t)− ρdf∫ t

0e−κd(t−s)

(e−(κd+κf )s

∫ s

0e(κ

d+κf )uσduσ

fudu

)ds+ ρdee

−κdt∫ t

0eκ

dsσdsσ

esds

cdf (t) = Ed[Xdt X

ft ] = ρdfe

−(κd+κf )t∫ t

0e(κ

d+κf )sσdsσ

fs ds

cef (t) = ρdf∫ t

0e−κf (t−s)

(e−(κd+κf )s

∫ s

0e(κ

d+κf )uσduσ

fudu

)ds− νhf (t) + ρefe

−κf t∫ t

0eκ

fsσesσ

fs ds

(12)

Denote the covariance matrix by

Σt =

v2z(t) czd(t) cze(t) czf (t)cdz(t) v2d(t) cde(t) cdf (t)cez(t) c2ed(t) v2e(t) cef (t)cfz(t) cfd(t) cfe(t) v2f (t)

.

The joint density function of the pair (Xdt , X

et , X

ft , Zt) is therefore

g(xz, xd, xe, xf ; t) =|Σt|−

12

4π2e−

12 (xz,xd,xe,xf )Σ

−1t (xz,xd,xe,xf )

′.

Now we want to calculate the conditional expectation

Ed[e−Zt

∣∣Xdt = xd, X

et = xe, X

ft = xf

].

We note conditioning on (Xdt , X

et , X

ft ), Zt ∼ N(µ, σ2), with (see Anderson [1])

µ(xd, xe, xf , t) = (vzd(t), vze(t), vzf (t))

v2d(t) vde(t) vdf (t)v2ed(t) v2e(t) vef (t)vfd(t) vfe(t) v2f (t)

−1xd

xe

xf

and

σ2(t) = v2z(t)− (vzd(t), vze(t), vzf (t))

v2d(t) vde(t) vdf (t)v2ed(t) v2e(t) vef (t)vfd(t) vfe(t) v2f (t)

−1vdz(t)vez(t)vfz(t)

15

ThereforeEd[e−Zt

∣∣Xdt = xd, X

et = xe, X

ft = xf

]= exp

−µ(xd, xe, xf , t) +

1

2σ2(t)

.

To verify the above formulas, we note

Variance of Xdt : We note v2d(t) = Ed[(Xd

t )2] = e−2κdt

∫ t

0e2κ

ds(σds )

2ds = νd(t).

Variance of Xft : We note v2f (t) = Ed[(Xf

t )2] = e−2κf t

∫ t

0e2κ

fs(σfs )

2ds = νf (t).

Covariance of Xdt and Zt: We note

czd(t) = Ed[Xdt Zt] = Ed[

∫ t

0

(Xds )

2ds] + Ed[

∫ t

0

Zs(−κdXds ds+ σd

sdWds )]

=

∫ t

0

v2d(s)ds− κd

∫ t

0

czd(s)ds.

Solving the consequent differential equation, ddtczd(t) = −κdczd(t) + νd(t), we get czd(t) = νhd (t).

Variance of Zt: We note v2z(t) = Ed[(Zt)2] = 2

∫ t

0Ed[ZsX

ds ]ds = 2νHd (t).

Covariance of Xdt and Xf

t : We note

cdf (t) = Ed[Xdt X

ft ] = e−(κd+κf )tEd[

∫ t

0

eκdsσd

sdWds

∫ t

0

eκfsσf

s dWfs ]

= ρdfe−(κd+κf )t

∫ t

0

e(κd+κf )sσd

sσfs ds.

Covariance of Xdt and Xe

t : We note

cde(t) = Ed[Xdt X

et ] = Ed

[∫ t

0

Xds (X

ds − Xf

s )ds+

∫ t

0

Xes (−κdXd

s ds)

]+ ρde

∫ t

0

σdsσ

esds

So cde(t) satisfies the differential equation

d

dtcde(t) = v2d(t)− cdf (t)− κdcde(t) + ρdeσ

dt σ

et .

Therefore

cde(t)

= e−κdt

∫ t

0

eκds[v2d(s)− cdf (s) + ρdeσ

dsσ

es ]ds

= νhd (t)− e−κdt

∫ t

0

ρdfe−κfs

∫ s

0

e(κd+κf )sσd

uσfududs+ ρdee

−κdt

∫ t

0

eκdsσd

sσesds

= νhd (t)− ρdf

∫ t

0

e−κd(t−s)

(e−(κd+κf )s

∫ s

0

e(κd+κf )uσd

uσfudu

)ds+ ρdee

−κdt

∫ t

0

eκdsσd

sσesds.

Covariance of Zt and Xft : We note

dczf (t) = Ed[d(ZtXft )] = Ed[Xf

t Xdt dt+ Zt(−κf Xf

t )dt] = cdf (t)dt− κfczf (t)dt.

So

czf (t) = e−κf t

∫ t

0

eκfscdf (s)ds = e−κf t

∫ t

0

ρdfe−κds

∫ s

0

e(κd+κf )uσd

uσfududs

= ρdf

∫ t

0

e−κf (t−s)

(e−(κd+κf )s

∫ s

0

e(κd+κf )uσd

uσfudu

)ds.

16

Covariance of Xet and Xf

t : We note

dcef (t) = Ed[d(Xet X

ft )] = Ed[Xf

t (Xdt − Xf

t )dt+Xet (−κf Xf

t dt) + σet dW

et σ

ft dW

ft ]

= cdf (t)dt− v2f (t)dt− κfcef (t)dt+ ρefσetσ

ft dt.

Thereforecef (t)

= e−κf t

∫ t

0

eκfs[cdf (s)− v2f (s) + ρefσ

esσ

fs ]ds

= e−κf t

∫ t

0

(ρdfe

−κds

∫ s

0

e(κd+κf )uσd

uσfudu− e−κfs

∫ s

0

e2κfu(σf

u)2du

)ds+ ρefe

−κf t

∫ t

0

eκfsσe

sσfs ds

= ρdf

∫ t

0

e−κf (t−s)

(e−(κd+κf )s

∫ s

0

e(κd+κf )uσd

uσfudu

)− νhf (t) + ρefe

−κf t

∫ t

0

eκfsσe

sσfs ds.

Covariance of Zt and Xet : We note

dcze(t) = Ed[d(ZtXet )] = Ed[Xd

t Xet + Zt(X

dt − Xf

t )]dt = (cde(t) + czd(t)− czf (t))dt.

Thereforecze(t)

=

∫ t

0

[cde(s) + czd(s)− czf (s)] ds

=

∫ t

0

[2νhd (s) + ρdee

−κds

∫ s

0

eκduσd

uσeudu− ρdf

∫ s

0

e−κd(s−ξ)

(e−(κd+κf )ξ

∫ ξ

0

e(κd+κf )uσd

uσfudu

)dξ

−ρdf

∫ s

0

e−κf (s−ξ)

(e−(κd+κf )ξ

∫ ξ

0

e(κd+κf )uσd

uσfudu

)dξ

]ds

= 2νHd (t) + ρde

∫ t

0

e−κds

∫ s

0

eκduσd

uσeududs

−ρdf

∫ t

0

∫ s

0

(e−κd(s−ξ) + e−κf (s−ξ)

)(e−(κd+κf )ξ

∫ ξ

0

e(κd+κf )uσd

uσfudu

)dξds

Variance of Xet : We noted

dtv2e(t) = 2Ed[Xe

t dXet ] + Ed[(dXe

t )2] = 2Ed[Xe

t (Xdt − Xf

t )dt] + Ed[(σet )

2dt]

= (2cde(t)− 2cef (t) + (σet )

2)dt.

Thereforev2e(t)

=

∫ t

0

[2cde(s)− 2cef (s) + (σes)

2]ds

= 2νHd (t) + 2νHf (t) +

∫ t

0

(σes)

2ds− 2ρdf

∫ t

0

∫ ξ

0

e−κd(ξ−s)

(e−(κd+κf )s

∫ s

0

e(κd+κf )uσd

uσfudu

)dsdξ

−2ρdf

∫ t

0

∫ ξ

0

e−κf (ξ−s)

(e−(κd+κf )s

∫ s

0

e(κd+κf )uσd

uσfudu

)dsdξ + 2ρde

∫ t

0

e−κdξ

∫ ξ

0

eκdsσd

sσesdsdξ

−2ρef

∫ t

0

e−κfξ

∫ ξ

0

eκfsσe

sσfs dsdξ

= 2cze(t)− v2z(t) + 2νHf (t) +

∫ t

0

(σes)

2ds− 2ρef

∫ t

0

e−κfξ

∫ ξ

0

eκfsσe

sσfs dsdξ

17

1.9.4 Pricing of cross-currency swaption

Suppose two parties are going to exchange cash flows at times 0 < T1 < T2 < · · · < TN : the cash flow forforeign investor is (Cf

i )Ni=1 which is denominated in foreign currency, and the cash flow for domestic investor

is (Cdi )

Ni=1 which is denominated in domestic currency. For generality, we assume (Cf

i )Ni=1 and (Cd

i )Ni=1 may

depend on state variable (e.g. Cfi and Cd

i are LIBOR rates). Suppose the option is European with maturityt < T1. Then the value of the contingent claim at option maturity t, denominated in domestic currency andfrom the standpoint of domestic investor, is

maxYt

N∑i=1

Cfi P

f (t, Ti;Xft )−

N∑i=1

Cdi P

d(t, Ti;Xdt ), 0

.

Therefore, we have the time-0 value of the option as

V0 = E

e− ∫ t0rdsds

(Yt

N∑i=1

Cfi (X

ft )P

f (t, Ti;Xft )−

N∑i=1

Cdi (X

dt )P

d(t, Ti;Xdt )

)+

= A(t)e12σ

2(t)Ed[e−µ(Xd

t ,Xet ,X

ft ,t)f

(g1(X

dt ), g2(X

ft −B(t)) · C(t)Y0e

Xet

)](13)

where A(t), B(t), and C(t) are as given in formula (11) andg1(x) =

∑Ni=1 C

di (x)P

d(t, Ti;x)

g2(x) =∑N

i=1 Cfi (x)P

f (t, Ti;x)

f(x, y) = maxy − x, 0

Formula (13) can be evaluated by the results in Section 1.9.3.Since the calculation of µ(xd, xe, xf , t) involves matrix inversion, sometimes it might be numerically

more efficient to work without the “conditioning”. More precisely, we evaluate the price via 4-dimensionalintegration instead of 3-dimensional integration:

V0 = A(t)Ed

e− ∫ t0Xd

s ds

(C(t)Y0e

Xet

N∑i=1

Cfi (X

ft −B(t))P f (t, Ti; X

ft −B(t))−

N∑i=1

Cdi (X

dt )P

d(t, Ti;Xdt )

)+

1.10 Implementation1.10.1 Implementation of ν

We first recall the definition of h(t;κ) and H(t;κ):

h(t;κ) = e−κt, H(t;κ) =

∫ t

0

h(s;κ)ds =

t κ = 01−e−κt

κ κ = 0

Suppose we have a sequence of time points 0 = t0 < t1 < · · · < tn−1 < tn := ∞ and a sequence ofconstants σ0, σ1, · · · , σn−1, such that the one-factor Hull-White model under consideration has constantvolatility σi−1 on the interval [ti−1, ti) (i = 1, 2, · · · , n). Then ν(t;κ) = e−2κt

∫ t

0e2κsσ2

sds can be computedby the following algorithm.

First, we compute ν(t0;κ), ν(t1;κ), · · · , ν(tn−1;κ) recursively as follows: ν(t0;κ) = ν(0;κ) = 0. Fori = 1, · · · , n,

ν(ti;κ) = e−2κti

(∫ ti−1

0

e2κsσ2sds+

∫ ti

ti−1

e2κsσ2sds

)

= e−2κ(ti−ti−1)ν(ti−1;κ) + e−2κti · σ2i−1

∫ ti

ti−1

e2κsds

= h(ti − ti−1; 2κ)ν(ti−1;κ) + σ2i−1H(ti − ti−1; 2κ).

18

If the values of ν(t0;κ), ν(t1;κ), · · · , ν(tn−1;κ) are already obtained from calibration, we can save the resultsand omit this step of computation. Second, for general t ≥ 0. Suppose t ∈ [ti−1, ti), we have

ν(t;κ) = h(t− ti−1; 2κ)ν(ti−1;κ) +H(t− ti−1; 2κ)σ2i−1

1.10.2 Implementation of νh

We recall thatνh(t;κ) =

∫ t

0

e−κ(t−s)ν(s;κ)ds.

We compute νh(t;κ) recursively as follows: νh(t0;κ) = νh(0;κ) = 0. For t ∈ [ti−1, ti) (i = 1, · · · , n), we have

νh(t;κ) =

∫ t

0

e−κ(t−s)ν(s;κ)ds = e−κt

[∫ ti−1

0

eκsν(s;κ)ds+

∫ t

ti−1

eκsν(s;κ)ds

]

= e−κ(t−ti−1)νh(ti−1;κ) + e−κt

∫ t

ti−1

eκsν(s;κ)ds

Therefore, for t ∈ [ti−1, ti),

νh(t;κ) = h(t− ti−1;κ)νh(ti−1;κ) + ν(ti−1;κ)h(t− ti−1;κ)H(t− ti−1;κ) +

1

2σ2i−1H

2(t− ti−1;κ)

1.10.3 Implementation of νH

We consider a more general implementation of νH that evaluates

νH(t;κ, κ′) =

∫ t

0

H(t− s, κ′)ν(s;κ)ds,

where ν(t;κ) = e−2κt∫ t

0e2κsσ2

sds and H(t, κ′) =∫ t

0e−κ′sds. When κ′ = 0, we have

νH(t;κ, κ′) =

∫ t

0

1− e−κ′(t−s)

κ′ ν(s;κ)ds =1

κ′

[νh(t;κ, 0)− νh(t;κ, κ′)

].

When κ′ = 0, we have

νH(t;κ, 0) =

∫ t

0

(t− s)ν(s;κ)ds = t

∫ t

0

ν(s)ds−∫ t

0

sν(s)ds.

We compute ν0(t) :=∫ t

0ν(s)ds and ν1(t) :=

∫ t

0sν(s)ds recursively as follows: ν0(t0) = ν1(t0) = 0. For

t ∈ [ti−1, ti), we have

ν0(t) = ν0(ti−1) +

∫ t

ti−1

ν(s)ds

=

ν0(ti−1) +∫ t

ti−1[ν(ti−1) + σ2

i−1(s− ti−1)]ds κ = 0

ν0(ti−1) +∫ t

ti−1

[e−2κ(s−ti−1)ν(ti−1) + σ2

i−11−e−2κ(s−ti−1)

2κ

]ds κ = 0

=

ν0(ti−1) + ν(ti−1)(t− ti−1) +σ2i−1

2 (t− ti−1)2 κ = 0

ν0(ti−1) +[ν(ti−1)−

σ2i−1

2κ

]1−e−2κ(t−ti−1)

2κ +σ2i−1

2κ (t− ti−1) κ = 0

19

and

ν1(t) = ν1(ti−1) +

∫ t

ti−1

sν(s)ds

=

ν1(ti−1) +∫ t

ti−1[sν(ti−1) + σ2

i−1s(s− ti−1)]ds κ = 0

ν1(ti−1) +∫ t

ti−1s[e−2κ(s−ti−1)ν(ti−1) + σ2

i−11−e−2κ(s−ti−1)

2κ

]ds κ = 0

=

ν1(ti−1) +

ν(ti−1)2 (t2 − t2i−1) + σ2

i−1

[t3−t3i−1

3 − ti−1

2 (t2 − t2i−1)]

κ = 0

ν1(ti−1) +σ2i−1

4κ (t2 − t2i−1) +ν(ti−1)−

σ2i−12κ

2κ

[ti−1 − te−2κ(t−ti−1) + 1−e−2κ(t−ti−1)

2κ

]κ = 0

Once we have computed ν0(t) and ν1(t), we can accordingly compute νH as νH = tν0(t)− ν1(t).

2 Two-factor Hull-White modelIn the two-factor Hull-White short rate model, it is assumed that the short rate r follows the following

dynamics under risk-neutral measure with money market account as the numeraire:

r(t) = X1(t) +X2(t) + θ(t), r(0) = r0,

where the processes X1(t) and X2(t) satisfydX1(t) = −κ1X1(t)dt+ σ1(t)dW1(t), X1(0) = 0

dX2(t) = −κ2X2(t)dt+ σ2(t)dW2(t), X2(0) = 0

where (W1,W2) is a two-dimensional Brownian motion with instantaneous correlation ρ:

dW1(t)dW2(t) = ρdt,

r0, κ1, κ2 are positive constants, σ1(t) and σ2(t) are positive deterministic functions, and ρ ∈ [−1, 1]. Thefunction θ(t) is a deterministic function that will be used to fit the initial yield curve.

2.1 Dynamics of x1(t) and x2(t) under risk-neutral measureWe have

x1(t) = e−κ1(t−s)x1(s) +∫ t

se−κ1(t−u)σ1(u)dW1(u), x1(0) = 0

x2(t) = e−κ2(t−s)x2(s) +∫ t

se−κ2(t−u)σ2(u)dW2(u), x2(0) = 0

(14)

Conditioning on Fs, the random vector (x1(t), x2(t))′ is Gaussian with mean

µ(s, t) = E[(x1(t), x2(t))′ | Fs] = (µ1(s, t), µ2(s, t))

′ = (h1(t− s)x1(s), h2(t− s)x2(s))′

and covariance matrix

Σ(s, t) =

( ∫ t

se−2κ1(t−u)σ2

1(u)du∫ t

se−(κ1+κ2)(t−u)σ1(u)σ2(u)ρdu∫ t

se−(κ1+κ2)(t−u)σ1(u)σ2(u)ρdu

∫ t

se−2κ2(t−u)σ2

2(u)du

)

=

(ν11(s, t) ν12(s, t)ν21(s, t) ν22(s, t)

)

20

2.2 Dynamics of x1(t) and x2(t) under T -forward measureDenote by QT the T -forward measure. Note the Radon-Nikodym derivative of T -forward measure QT

w.r.t. the risk-neutral measure Q is

ζt = EQt

[dQT

dQ

]=

P (t, T )/P (0, T )

e∫ t0r(u)du

.

Therefore

d ln ζt = d lnP (t, T )− d

∫ t

0

r(u)du = d

[−∫ T

t

φ(u)du−2∑

i=1

Hi(T − t)xi(t) +1

2V (t, T )

]− r(t)dt

= φ(t)dt+2∑

i=1

hi(T − t)xi(t)dt−2∑

i=1

Hi(T − t)dxi(t) +1

2

∂

∂tV (t, T )− r(t)dt

=

[−

2∑i=1

xi(t) +2∑

i=1

hi(T − t)xi(t) +1

2

∂

∂tVt(t, T ) +

2∑i=1

Hi(T − t)κixi(t)

]dt−

2∑i=1

Hi(T − t)σi(t)dWi(t)

By Ito’s formula, d ln ζt = dζt/ζt − 12 (dζt)

2/ζ2t . So

(dζt)2

ζ2t= (d ln ζt)

2 =

2∑i=1

Hi(T − t)2σ2i (t)dt+ 2ρH1(T − t)H2(T − t)σ1(t)σ2(t)dt = − ∂

∂tV (t, T ).

This implies

dζtζt

= d ln ζt +1

2

(dζt)2

ζ2t

=

[−

2∑i=1

xi(t) +

2∑i=1

hi(T − t)xi(t) +

2∑i=1

Hi(T − t)κixi(t)

]dt−

2∑i=1

Hi(T − t)σi(t)dWi(t)

=

2∑i=1

[−1 + hi(T − t) +Hi(T − t)κi]xi(t)dt−2∑

i=1

Hi(T − t)σi(t)dWi(t)

= −2∑

i=1

Hi(T − t)σi(t)dWi(t),

which is as expected. Define Lt = −∑2

i=1

∫ t

0Hi(T − u)σi(u)dWi(u). Then ζt = E(Lt) := exp

Lt − 1

2 ⟨L⟩t

.By Girsanov’s Theorem,

WT1 (t) = W1(t)− ⟨W1, L⟩t = W1(t) +

∫ t

0

H1(T − u)σ1(u)du+ ρ

∫ t

0

H2(T − u)σ2(u)du

andWT

2 (t) = W2(t)− ⟨W2, L⟩t = W2(t) + ρ

∫ t

0

H1(T − u)σ1(u)du+

∫ t

0

H2(T − u)σ2(u)du

are Brownian motion under QT .Therefore, under the T -forward measure QT , we have

x1(t) = e−κ1(t−s)x1(s)−MT1 (s, t) +

∫ t

se−κ1(t−u)σ1(u)dW

T1 (u), x1(0) = 0

x2(t) = e−κ2(t−s)x2(s)−MT2 (s, t) +

∫ t

se−κ2(t−u)σ2(u)dW

T2 (u), x2(0) = 0

(15)

where (WT1 ,WT

2 ) is a two-dimensional Brownian motion under QT with instantaneous correlation ρ andMT

1 (s, t) =∫ t

se−κ1(t−u)σ1(u)[H1(T − u)σ1(u) + ρH2(T − u)σ2(u)]du

MT2 (s, t) =

∫ t

se−κ2(t−u)σ2(u)[ρH1(T − u)σ1(u) +H2(T − u)σ2(u)]du

21

Conditioning on Fs, the random vector (x1(t), x2(t))′ is Gaussian with mean

µT (s, t) = ETs [(x1(t), x2(t))

′] = (h1(t− s)x1(s)−MT1 (s, t), h2(t− s)x2(t)−MT

2 (s, t))′

and covariance matrix is the same as the covariance matrix under risk-neutral measure Q.

2.3 Dynamics of∫ T

tx1(u)du and

∫ T

tx2(u)du under risk-neutral measure

For convenience of computation, we drop the subscript. Then we have∫ T

t

x(u)du = x(T )T − x(t)t−∫ T

t

udx(u) =

∫ T

t

(T − u)dx(u) + (T − t)x(t)

=

∫ T

t

(T − u)[−κx(u)du+ σ(u)dW (u)] + (T − t)x(t)

= −κ

∫ T

t

(T − u)x(u)du+

∫ T

t

(T − u)σ(u)dW (u) + (T − t)x(t)

By equation (14), for κ = 0, we have

−κ

∫ T

t

(T − u)x(u)du = −κ

∫ T

t

(T − u)

[e−κ(u−t)x(t) +

∫ u

t

e−κ(u−s)σ(s)dW (s)

]du

= −κx(t)

∫ T

t

(T − u)e−κ(u−t)du− κ

∫ T

t

(T − u)

∫ u

t

e−κ(u−s)σ(s)dW (s)du

= x(t)

∫ T

t

(T − u)de−κ(u−t) − κ

∫ T

t

∫ u

t

eκsσ(s)dW (s)du

(∫ u

t

(T − v)e−κvdv

)Therefore for κ = 0

−κ

∫ T

t

(T − u)x(u)du

= −x(t)(T − t) + x(t)H(T − t)

−κ

[∫ T

t

eκsσ(s)dW (s)

∫ T

t

(T − v)e−κvdv −∫ T

t

(∫ u

t

(T − v)e−κvdv

)eκuσ(u)dW (u)

],

where the last term can be further simplified to (note∫ T

u(T − v)e−κvdv = (T−u)e−κu

κ −∫ Tu

e−κvdv

κ )

−κ

∫ T

t

eκuσ(u)

(∫ T

u

(T − v)e−κvdv

)dW (u) = −

∫ T

t

eκuσ(u)

((T − u)e−κu −

∫ T

u

e−κvdv

)dW (u)

= −∫ T

t

σ(u)

[(T − u)−

∫ T

u

e−κ(v−u)dv

]dW (u)

22

We can verify that when κ = 0, the above equality also holds. Therefore∫ T

t

x(u)du = −κ

∫ T

t

(T − u)x(u)du+

∫ T

t

(T − u)σ(u)dW (u) + (T − t)x(t)

= −x(t)(T − t) + x(t)H(T − t)−∫ T

t

σ(u)

[(T − u)−

∫ T

u

e−κ(v−u)dv

]dW (u)

+

∫ T

t

(T − u)σ(u)dW (u) + (T − t)x(t)

= x(t)H(T − t) +

∫ T

t

σ(u)

(∫ T

u

e−κ(v−u)dv

)dW (u)

= x(t)H(T − t) +

∫ T

t

σ(u)H(T − u)dW (u)

That is, we have ∫ T

tx1(u)du = x1(t)H1(T − t) +

∫ T

tσ1(u)H1(T − u)dW1(u)∫ T

tx2(u)du = x2(t)H2(T − t) +

∫ T

tσ2(u)H2(T − u)dW2(u)

(16)

Then conditioning on Ft, (∫ T

tx1(u)du,

∫ T

tx2(u)du)

′ is Gaussian with mean

µ(t, T )integral = E

[(∫ T

t

x1(u)du,

∫ T

t

x2(u)du

)′∣∣∣∣∣Ft

]= (H1(T − t)x1(t),H2(T − t)x2(t))

′

and covariance matrix

Σ(t, T )integral =

( ∫ T

tσ21(u)H

21 (T − u)du

∫ T

tσ1(u)σ2(u)H1(T − t)H2(T − t)ρdu∫ T

tσ1(u)σ2(u)H1(T − t)H2(T − t)ρdu

∫ T

tσ22(u)H

22 (T − u)du

)

2.4 Derivation of zero-coupon bond priceBy formula (??) and formula (16) The price at time t of a zero-coupon bond maturing at time T and

with unit face value is

P (t, T ) = EQ[e−

∫ Tt

r(u)du∣∣∣Ft

]= e−

∫ Tt

φ(u)duEQ[e−

∫ Tt

(x1(u)+x2(u))du∣∣∣Ft

]= exp

−∫ T

t

φ(u)du−H1(T − t)x1(t)−H2(T − t)x2(t) +1

2V (t, T )

and

V (t, T ) =

∫ T

t

σ21(u)H

21 (T − u)du+

∫ T

t

σ22(u)H

22 (T − u)du+ 2ρ

∫ T

t

σ1(u)σ2(u)H1(T − u)H2(T − u)du.

A Summary of Girsanov’s Theorem for continuous semimartin-gale

For sake of convenience, we record here a version of Girsanov’s Theorem as presented in Revuz and Yor[3]. We will freely use jargons in the theory of continuous semimartingales.

Suppose (Ω, (Ft)t≥0, P ) is a filtered probability space that satisfies the usual hypotheses. Q is anotherprobability measure such that Q|Ft is absolutely continuous with respect to P |Ft . We call Dt the Radon-Nikodym derivative of Q with respect to P on Ft. These random variables (Dt)t≥0 form a (Ft, P )-martingaleand can be chosen in such a way that it has cadlag path a.s..

23

Theorem A.1 (Girsanov’s Theorem). If D is continuous, every continuous (Ft, P )-semimartingale is acontinuous (Ft, Q)-semimartingale. More precisely, if M is a continuous (Ft, P )-local martingale, then

M = M −D−1.⟨M,D⟩

is a continuous (Ft, Q)-local martingale. Moreover, if N is another continuous P -local martingale,

⟨M, N⟩ = ⟨M,N⟩ = ⟨M,N⟩.

To apply Girsanov’s Theorem more conveniently, we often use the following results.Proposition A.1. If D is a strictly positive continuous local martingale, there exists a unique continuouslocal martingale L such that

Dt = expLt −

1

2⟨L,L⟩t

= E(L)t;

L is given by the formula

Lt = logD0 +

∫ t

0

D−1s dDs.

Theorem A.2. If Q = E(L) · P and M is a continuous P -local martingale, then

M = M −D−1 · ⟨M,D⟩ = M − ⟨M,L⟩

is a continuous Q-local martingale. Moreover, P = E(−L)−1 ·Q = E(−L) ·Q.

B Lognormal approximation of sum of lognormalsWe consider the following problem: suppose (Z1, · · · , Zn) is an n-dimensional Gaussian random vector

with mean µ and covariance matrix Σ. Find the lognormal approximation of∑n

i=1 wieZi where w1, · · · , wn

are constants.The typical method is moment-matching. Recall for a Gaussian random vector X ∼ N(µ,Σ), its charac-

teristic function isϕ(t) = E[eit

′X ] = eit′µ− 1

2 t′Σt,

which implies E[et′X ] = et

′µ+ 12 t

′Σt. Therefore the first moment is

M := E

[n∑

i=1

wieZi

]=

n∑i=1

wieµi+

12Σii .

The second moment is

V := E

( n∑i=1

wieZi

)2 =

n∑i,j=1

wiwjE[eZi+Zj ] =n∑

i,j=1

wiwjeµi+µj+

12 (Σii+Σjj)+Σij ,

since Zi + Zj ∼ N(µi + µj ,Σii +Σjj + 2Σij).Suppose we want to find a normal random variable ξ ∼ N(µ, σ2) such that

E[eξ] = E[∑n

i=1 wieZi ]

E[e2ξ] = E[(∑n

i=1 wieZi)2]

Then we need to solve the equation eµ+

12 σ

2

= M

e2µ+2σ2

= V

which gives σ2 = ln V

M2

µ = lnM − 12 σ

2

24

C Characteristic function of a Gaussian random vectorRecall for a Gaussian random vector X ∼ N(µ,Σ), its characteristic function is (see, for example,

Anderson [1] pp.43, Theorem 2.6.1)

ϕ(t) = E[eit′X ] = eit

′µ− 12 t

′Σt.

References[1] T. W. Anderson. An introduction to multivariate statistical analysis, 3rd edition. Wiley-Interscience,

2003. 15, 25

[2] Damiano Brigo and Fabio Mercurio. Interest rate models - Theory and practice. Springer, 2007. 5

[3] D. Revuz and M. Yor. Continuous martingales and Brownian motion, 3rd Edition. Springer, 2004. 23

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