pricing american interest rate options under the jump-extended vasicek model

FALL 2008 THE JOURNAL OF DERIVATIVES 29

This article shows how to price American interest rateoptions under the exponential jumps-extendedVasicek model, or the Vasicek-EJ model. We modifythe Gaussian jump-diffusion tree of Amin [1993]and apply it to the exponential jumps-based shortrate process under the Vasicek-EJ model. The treeis truncated at both ends to allow fast computationof option prices. We also consider the time-inhomo-geneous version of this model, denoted as the Vasicek-EJ++ model, that allows exact calibration to theinitially observable bond prices. We provide an ana-lytical solution to the deterministic shift term usedfor calibrating the short rate process to the initiallyobservable bond prices, and show how to generatethe jump-diffusion tree for the Vasicek-EJ++ model.Our simulations show fast convergence of Europeanoption prices obtained using the jump-diffusion treeto those obtained using the Fourier inversion method( for options on zero-coupon bonds, or caplets) andthe cumulant expansion method ( for options oncoupon bonds, or swaptions).

Backus, Foresi, and Wu [1997], Das[1998, 2002], Das and Foresi [1996],Johannes [2004], and Piazzesi [1998,2005] find that jumps caused by

market crashes, interventions by the FederalReserve, economic surprises, shocks in theforeign exchange markets, and other rareevents, play a significant role in explaining thedynamics of interest rate changes. Das [2002]finds that a mix of ARCH processes withjumps explains the behavior of short ratechanges well. The ARCH features are required

to capture the high persistence in short ratevolatility, and jump features are required toexplain the sudden large movements that leadto leptokurtosis. Das finds jumps to be morepronounced during the two-day meeting datesof the Federal Open Market Committee, andcuriously also on many Wednesdays due to theoption expiry effects on that day. In a recentarticle, Jarrow, Li, and Zhao [2007] find thata stochastic volatility extension of the LIBORmarket model can be modeled to generate asymmetric-shaped smile in the Black-impliedvolatility of caps. However, the asymmetrichockey-shaped smile observed in the caps datacan be generated only by using downwardjumps under the risk-neutral measure.

Jump models assume that the size of thejump is a random variable with some proba-bility distribution, which is independent ofthe length of the time interval over which thejump occurs. The probability of jump isassumed to be directly proportional to thelength of the time interval. Though compu-tational methods exist for pricing of Europeaninterest rate options under jump models, fewnumerical methods for pricing Americaninterest rate options—such as optionsembedded in callable and convertible bonds,mortgage prepayment options, Americanswaptions, and so forth—exist in the fixed-income literature. In Nawalkha, Beliaeva, andSoto (NBS [2007]), we show how to priceAmerican interest rate options using the expo-nential jumps-extended Vasicek model of

Pricing American InterestRate Options under the Jump-Extended Vasicek ModelNATALIA A. BELIAEVA, SANJAY K. NAWALKHA,AND GLORIA M. SOTO

NATALIA A. BELIAEVA

is an assistant professor offinance in the Sawyer Busi-ness School at Suffolk Uni-versity in Boston, [email protected]

SANJAY K. NAWALKHA

is an associate professor offinance in the IsenbergSchool of Management atthe University of Massachu-setts in Amherst, [email protected]

GLORIA M. SOTO

is a tenured professor ofapplied economics andfinance at the University ofMurcia in Murcia, [email protected]

IIJ-JOD-BELIAEVA 8/20/08 6:41 PM Page 29

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Chacko and Das [2002], denoted as the Vasicek-EJ model.The advantage of the Vasicek-EJ model is that it allowsseparate distributions for the upward jumps and downwardjumps. This can significantly reduce the probability ofnegative interest rates by allowing more flexibility in esti-mating parameters for different interest rate regimes. Forexample, when interest rates are at their lows at the bottomof a business cycle, the downward jumps may have a lowersize and a lower frequency of occurrence than upwardjumps.

To price American interest rate options under theVasicek-EJ model, we construct a jump-diffusion tree byextending the work of Amin [1993]. The jump-diffusiontree allows an arbitrarily large number of nodes at each stepto capture the jump component, while two local nodes areused to capture the diffusion component. We also demon-strate how to calibrate the jump-diffusion tree to fit an ini-tial yield curve or the initially observable zero-coupon bondprices, by allowing a time-dependent drift for the short rateprocess. In this article, we build on the earlier results pre-sented in NBS [2007], and show how to price options oncoupon bonds (or swaptions) under the Vasicek-EJ model.We first extend the cumulant-expansion method of Collin-Dufresne and Goldstein [2002] to price European optionson coupon bonds under the Vasicek-EJ model. Next, wedemonstrate the application of the jump-diffusion tree toprice American options on coupon bonds (or Americanswaptions) under this model.

Our simulations show fast convergence of Europeanoption prices obtained using the jump-diffusion tree tothose obtained using the Fourier inversion method (foroptions on zero-coupon bonds, or caplets) and the cumu-lant expansion method (for options on coupon bonds, orswaptions).

We also consider the single-plus and double-plusextensions of these models given as the Vasicek-EJ+ andVasicek-EJ++ models, based upon the new taxonomy ofterm structure models given by NBS [2007, pp. 106–112].As shown by NBS, the Vasicek-EJ+ model allows inde-pendence from the market price of diffusion risk (repre-senting the one plus), and Vasicek-EJ++ model allowsindependence from market price of diffusion risk, as wellas calibration to the initial zero-coupon bond prices (rep-resenting the two plusses). Independence of these modelsfrom the market price of diffusion risk makes these modelspartially preference-free. These models still require the

market prices of jump risk, which cannot be eliminatedusing the single-plus and double-plus extensions.

THE VASICEK-EJ MODEL

Consider the short rate process given by Vasicek[1977] with an added Poisson-jump component, givenas follows:

(1)

where α, m, and σ define the speed of mean-reversion,long-term mean, and the diffusion volatility, respectively.The variable Z(t) is the Wiener process distributed inde-pendently of N(λ). The variable N(λ) represents a Poissonprocess with an intensity λ. The variable λ gives the meannumber of jumps per unit of time. The variable J denotesthe size of the jump and is assumed to be distributed inde-pendently of both Z(t) and N(λ). The choice of jump dis-tribution has a significant effect on the pricing of bonds.For example, the jump size can follow a Gaussian, a log-normal, or an exponential distribution. This article con-siders jumps that follow exponential distributions.

The probability of the occurrence of one Poissonevent over the infinitesimal time-interval dt is given asλdt. Upon the occurrence of the Poisson event, the changein the Poisson variable, dN(λ), equals 1, otherwise dN(λ)equals 0. The probability that more than one jump occursduring the interval dt is less than 0(dt), hence as dt con-verges to an infinitesimally small number, only one jumpwith probability λdt needs to be modeled over eachinterval dt for numerical applications.

Chacko and Das [2002] extend the Vasicek modelwith two exponential jump processes, such that the jumpsize and the jump intensity of the upward jumps can bedifferent from those of the downward jumps. The doubleexponential jump model, denoted as the Vasicek-EJmodel, also allows an analytical closed-form solution tothe bond price.

The short rate process under the Vasicek-EJ modelis given as follows:

(2)

where α, m, and σ are the speed of mean-reversion,long-term mean, and the diffusion volatility, respectively.

dr m r t dt dZ t

J dN J dNu u u d d d

= − ++ −α σ

λ λ( ( )) ( )

( ) ( )

dr t m r t dt dZ t JdN( ) ( ( )) ( ) ( )= − + +α σ λ

30 PRICING AMERICAN INTEREST RATE OPTIONS UNDER THE JUMP-EXTENDED VASICEK MODEL FALL 2008


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The up-jump variable Ju and the down-jump variable Jdare exponentially distributed with positive means 1/ηuand 1/ηd, respectively. The two Poisson variables dNu(λu)and dNd(λd) are distributed independently, with intensi-ties λu and λd, respectively. The probability density of theexponential variable J is defined as follows:

(3)

The advantage of this model is that it allows sepa-rate distributions for the upward jumps and downwardjumps. This can significantly reduce the probability ofnegative interest rates caused by downward jumps byallowing more flexibility in choosing the parameters underdifferent interest rate regimes. For example, when interestrates are low at the bottom of a business cycle, the down-ward jumps in interest rates can be chosen to be lower bothin magnitude and the frequency of occurrence than theupward jumps.

Using the stochastic discount factor, the risk-neu-tral process for the short rate is given as follows:1

(4)

where

(5)

where γ0 and γ1 give the more general forms of marketprices of diffusion risk as in Duffee [2002], and γuJ and γdJgive the market prices of upward jumps and downwardjumps, respectively.

Let the short rate be given as follows:

(6)

where the state variable Y(t) follows the following risk-neutral process,

r t m Y t( ) ( )= +�

�

�

�

�

α α γα γα γ

λ λ γ

λ λ γ

= +

=−+

= −

= −

1

0

1

1

1

mm

u u uJ

d d

( )

( ddJ )

dr m r t dt dZ t

J dN J dNu u u d d

= − +

+ −

� � �

� �α σ

λ( ( )) ( )

( ) (λλd )

f J e JJ( ) ,= × ≤ ≤ ∞− ×η η for 0

(7)

Using Ito’s lemma, it is easy to confirm that the short rateprocess given in Equation (4) is consistent with the statevariable process given in the preceding equation.

Expressing the bond price as a function of Y(t) andt, using Ito’s lemma, and taking risk-neutral expectationof the bond price, the partial differential difference equa-tion (PDDE) for the bond price can be given as follows:

(8)

subject to the boundary condition P(T,T ) = 1.

Bond Price Solution

The bond price solution for the Vasicek-EJ modelis given as follows:

(9)

where τ = T – t, and

(10)

(11)

and

A BB

u( ) ( )( )τ τ τ σ

ασ τ

αλ λ= −( )⎛

⎝⎜⎞⎠⎟− − +

2

2

2 2

2 4� ��

dd

u u

u u u

e

( )

++

+⎛

⎝⎜⎞

⎠⎟−

+

τ

λ ηαη αη αη

ατ�

� � ��

11

1 1ln

��

� � ��λ η

αη αη αηατd d

d d d

e−

−⎛

⎝⎜⎞

⎠⎟+

11

1 1ln

H t T mdu m T t mt

T( , ) ( )= = − =∫ � � �τ

P t T eA B Y t H t T( , ) ( ) ( ) ( ) ( , )= − −τ τ

∂∂

−∂∂

+∂∂

+

+

P

tY t

P

Y

P

YE P Y t

J t

t

u

� �α σ( ) [ ( ( )

, ,

1

22

2

2

TT P Y t t T E P Y t

J t T P Yu t

d

) ( ( ), , )] [ ( ( )

, , ) ( (

− +

− −

� �λ

tt t T r t P t T

m Y t P t Td), , )] ( ) ( , )

( ) ( , )

�

�

λ =

= +( )

dY t Y t dt dZ t J dN

J dN

u u u

d d

( ) ( ) ( ) ( )

(

= − + +

−

� � �

�α σ λ

λdd )


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(12)

This solution is identical to that given by Chacko andDas [2002], although expressed in a slightly different form.As shown later, expressing the solution in this form allowsit to be extended to fit the initial bond prices by simplychanging the definition of the term H(t,T ).

Jump Diffusion Tree

This section explains how to build a jump-diffusiontree for the Vasicek-EJ model for pricing Americanoptions. The following approximation based on Amin[1993] is used to model the short rate tree with expo-nential jumps:

B t Te

( , ) =−⎛

⎝⎜⎞⎠⎟

−1 �

�

ατ

α(13)

It can be shown that Equation (13) deviates fromEquation (4) only in the order o(dt) and, hence, the twoequations converge in the limit as dt goes to zero. Toapproximate the jump-diffusion process in Equation (13),we use a single n-step recombining tree with M branches,where M is always an odd number. This tree is displayedin Exhibit 1. Given the initial value of the short rate is r(0),the short rate can either undergo a local change due tothe diffusion component shown through two localbranches (represented by the segmented lines) or experi-ence a jump shown through M – 2 remaining branches,

dr t

m r t dt dZ t

( )

( ( )) ( ),

=

− +� � �α σ with probabilityy

with probability

1− −� �

�λ λ

λu d

u u

dt dt

J dt

,

, ,

−−

⎧

⎨

⎪⎪

⎩

⎪⎪

⎫

⎬

⎪⎪

⎭

⎪⎪J dtd d, ,with probability �λ


E X H I B I T 1Jump-Diffusion Interest Rate Tree

+ Δ

+ Δ

+ Δ

− Δ

− Δ

− Δ

− Δ

− Δ

− Δ

− Δ


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including the central branch, where the short rate doesnot change (represented by the solid lines).

To keep the analysis general, we consider an arbi-trary node r(t), at time t = iΔt, for i = 0, 1, 2, ..., n – 1,where length of time T is divided into n equal intervalsof Δt. At time (i + 1)Δt, the M different values for theinterest rate are given as follows:

We specify M probabilities associated with the M

nodes. The short rate can either undergo a local change

with probability , or experience an up-jump

with probability or a down-jump with probability

.The two local probabilities are computed as follows:

(14)

The upward jump nodes and downward jump nodesare approximated separately with two exponential curves(see Exhibit 2).

The total risk-neutral probability of upward jumps

is Δt and the total risk-neutral probability of down-

ward jumps is Δt. The upward jump probability and

the downward jump probability are allocated to two sep-

arate exponential distributions, one drawn from zero to

infinity above the center point in Exhibit 2, and the other

drawn from zero to negative infinity below the center

point. The entire line from negative infinity to positive

infinity in Exhibit 2 consists of M points and M non-

overlapping intervals that cover both the exponential dis-

tributions for upward and downward jumps. The

probability mass over each of these M intervals is assigned

�λd

�λu

P r t tm r t

t( )( ( ))+⎡

⎣⎤⎦ = + −⎛

⎝⎜⎞⎠⎟

−

σ ασ

λ

Δ Δ12

12

1

� �

�uu dt t

P r t tm r t

Δ Δ

Δ

−( )−⎡

⎣⎤⎦ = − −

�

� �

λ

σ ασ

( )( ( ))1

212

ΔΔ

Δ Δ

t

t tu d

⎛⎝⎜

⎞⎠⎟

− −( )1 � �λ λ

�λddt

�λudt

1− −� �λ λu ddt dt

r t j t jM

( ) , , , ,± =−σ Δ 0 1

1

2…

to the point between each of these intervals using the

following three steps.Step 1:

The first step computes the probabilities of all jumpnodes except the central node, the top node, and thebottom node, as follows. The probability mass for theintervals corresponding to the upward jump nodes isassigned to the points at the center of these intervals asfollows:

(15)

where Fu(x) is the cumulative exponential distributionfor upward jumps defined as follows:

(16)

and

Fu ( )x e dJ euJ

x

uxu u u= = −− −∫η η η

0

1

P r t k t x x t

kM

u[ ( ) ] ( ) ( ) ,

, ,

+ = −⎡⎣ ⎤⎦

=

σ Δ ΔF Fu u1 2

2

�

…

λ−− 3

2


E X H I B I T 2Approximation of the Jump Distribution Using TwoExponential Curves


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The expression Fu(x1) – Fu(x2) gives the differencebetween two cumulative exponential distribution func-tions and, hence, gives the probability measured by thearea under the exponential curve over one of the regionscorresponding to the upward jumps in Exhibit 2. Multi-plication of this probability with the total risk-neutraljump probability of upward jumps Δt, gives the prob-ability of the short rate node at the point between of x1and x2 as shown in Equation (15).

Similarly, the probability mass for the intervals cor-responding to the downward jump nodes is assigned to thepoints at the center of the following intervals as follows:

(17)

where Fd(x) is the cumulative exponential distribution

for downward jumps defined as follows:

(18)

where

Step 2:Since we only consider the nodes in the range

in Exhibit 2 the probability mass outside this region shouldbe assigned to the end nodes. The probability mass assignedto the top node corresponding to the upward jump isgiven as

r tM

t r tM

t( ) , ( )+−

−−⎡

⎣⎢⎤⎦⎥

1

2

1

2σ σΔ Δ

x k t

x k t

1

2

0 5

0 5

= +

= −

( . )

( . )

σ

σ

Δ

Δ

Fd( )x e dJ edJ

x

dxd d d= = −− −∫η η η

0

1

P r t k t x x t

kM

d[ ( ) ] ( ) ( ) ,

, ,

− = −⎡⎣ ⎤⎦

=

σ λΔ ΔF Fd d1 2

2

�

…−− 3

2

�λu

x k t

x k t

1

2

0 5

0 5

= +

= −

( . )

( . )

σ

σ

Δ

Δ(19)

where

Similarly, the probability mass assigned to the bottomnode corresponding to the downward jump is given as

(20)

where

Step 3:The remaining probabilities from both exponential

distributions for upward and downward jumps are allo-cated to the central node as follows:

(21)

where

Equation (14) defines the probabilities at two nodesthat give the local change due to the diffusion process,and Equations (15), (17), (19), (20), and (21) define theprobabilities at the remaining M – 2 nodes that give thechange due to the two exponential jumps. Together, allof these probabilities sum up to one.

The number of nodes in a jump-diffusion tree is ofthe order O(N 2M). To increase the efficiency we imple-ment the following truncation methodology. The tree istruncated from above and below at (M – 1)/2. This ideais illustrated in Exhibit 3. When M is large, the probabilityof reaching the top nodes is effectively zero. Therefore,the whole top part of the tree can be truncated without

x t1 1 5= . σ Δ

P u u[ ( )] ( ) ( )r t F x t F x td= 1 1⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦� �λ λΔ Δ+ d

xM

t2

1

20 5=

−−⎛

⎝⎜⎞⎠⎟

. σ Δ

P r tM

t x td( ) ( )− −⎛⎝⎜

⎞⎠⎟= ⎡⎣ ⎤⎦

12

1 2σ λΔ Δ– Fd�

xM

t2

1

20 5=

−−⎛

⎝⎜⎞⎠⎟

. σ Δ

P r tM

t x tu( ) ( )+−⎛

⎝⎜⎞⎠⎟= −⎡⎣ ⎤⎦

1

21 2σ λΔ ΔFu

�



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loss of accuracy. The probability mass above the trunca-tion line is assigned to the node that lies on the trunca-tion line. Because of truncation, the number of nodes ina tree is reduced to O(N × M).

THE VASICEK-EJ+ AND VASICEK-EJ++MODELS

This section considers single-plus and double-plusextensions of the Vasicek-EJ model, given as the Vasicek-EJ+ model and the Vasicek-EJ++ model, based on thenew taxonomy of term structure models given by NBS[2007]. As shown by NBS, the Vasicek-EJ+ model allowsindependence from the market price of diffusion risk(representing the one plus), and the Vasicek-EJ++ modelallows not only independence from the market price ofdiffusion risk but also calibration to exogenously giveninitial zero-coupon bond prices (representing the twoplusses).2

Though all of the valuation formulas and numericalsolutions are identical under the Vasicek-EJ model andthe Vasicek-EJ+ model, the empirical estimates of therisk-neutral parameters can be very different under thesetwo models since the former model imposes restrictivelinear functional forms on the diffusion-related marketprice of risk (MPR). In contrast, the latter model is inde-pendent of the diffusion-related MPR and can implicitlyallow arbitrary nonlinear forms of diffusion-related MPR,which may even depend on multiple state variables (seeNBS [2007, Chapter 4]).

The Vasicek-EJ+ model can be further extended tofit the initially observable bond prices at time zero. Underthe extended model, denoted as the Vaseick-EJ++ model,the short rate process follows a time-inhomogeneous

process, given as a sum of a deterministic term and a statevariable, as follows:

(22)

where δ(t) is the deterministic term used for calibrationto the initial bond prices, and the risk-neutral stochasticprocess for the state variable Y(t) is given as follows:

(23)

Expressing the bond price as a function of Y(t) and t,using Ito’s lemma, and taking a risk-neutral expectationof the bond price, the PDDE for the bond price can begiven as follows:

(24)

The solution to the above PDDE is given as follows:

(25)

where τ = T – t, and

(26)

subject to A(0) = B(0) = 0, and by definition H(T, T ) = 0.

H t T u dut

T

( , ) ( )= ∫δ

P t T eA B Y t H t T( , ) ( ) ( ) ( ) ( , )= − −τ τ

∂∂

−∂∂

+∂∂

+

+

P

tY t

P

Y

P

YE P Y t

J t

t

u

� �α σ( ) [ ( ( )

, ,

1

22

2

2

TT P Y t t T E P Y t

J t T P Y

u t

d

) ( ( ), , )] [ ( ( )

, , ) ( (

− +

− −

� �λ

tt t T r t P t T

Y t t P t Td), , )] ( ) ( , )

( ) ( ) ( , )

�λδ

=

= +( )


J dNu u u

d d

( ) ( ) ( ) ( )

(

= − + +

−

� � �

�α σ λ

λdd Y), ( )0 0=

r t t Y t( ) ( ) ( )= +δ


E X H I B I T 3Jump-Diffusion Tree Truncation


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(27)

and

(28)

Note that the solutions of the bond price under theVasicek-EJ model (see Equation (9)) and the Vasicek-EJ++model (see Equation (25)) are identical except for the H(t,T )term, which becomes time-inhomogeneous under the lattermodel. To solve δ(t) and H(t,T ), under the Vasicek-EJ++model, consider the initially observable zero-coupon bondprice function given as P(0,T). By fitting this price exactlyto the Vasicek-EJ++ model, the log of the bond price eval-uated at time 0 is given as follows:

(29)

where by definition, Y(0) = 0. Differentiating the pre-ceding equation with respect to T, we get

(30)

where f(0,T) = – lnP(0,T )/ T is the initial forward ratecurve at time 0. Substituting the partial derivative of A(T )and replacing T with t, we get

(31)

where

L E eB

Bu tB J

u

u( )( )

( )( )τ τ

η ττ= ( ) − =

−+

−� 1

δ σ λ λ( ) ( , ) ( ) ( ) ( )t f t B t L t L tu u d d= + + +01

22 2 � �

∂∂

δ( ) ( , )( )

T f TA T

T= +

∂∂

0

ln ( , ) ( ) ( , )P T A T H T0 0= −

B t Te

( , ) =−⎛

⎝⎜⎞⎠⎟

−1 �

�

ατ

α

A BB

u d( ) ( )( )τ τ τ σ

ασ τ

αλ λ τ= −( ) − − +( )

+

2

2

2 2

2 4� ��

�λλ ηαη αη αη

λ η

ατu u

u u u

d d

e� � �

�

�

�

++

⎛

⎝⎜⎞

⎠⎟−

+

11

1 1ln

ααη αη αηατ

d d d

e−

−⎛

⎝⎜⎞

⎠⎟+

11

1 1ln

� ��

Using Equations (26) and (29), the function H(t,T) canbe expressed as follows:

(32)

Since P(0, t) and P(0,T ) are the initially observed bondprices, the preceding equation, together with Equations(27) and (28), gives the full closed-form solution of thebond price in Equation (25).

Jump Diffusion Tree

A slight modification of the jump diffusion treederived for the Vasicek-EJ model immediately gives thejump diffusion tree for the Vasicek-EJ++ model. Con-sider the short rate process and the state variable processshown in Equations (4) and (23), respectively, given asfollows:

(33)

(34)

The processes in the above two equations have iden-tical forms except that the risk-neutral long-term mean m~

equals 0 for the Y(t) process, and the starting value for theY(t) process is given as Y(0) = 0. Hence, the jump diffu-sion tree for the Y(t) process can be built exactly as it wasbuilt for the short rate under the Vasicek-EJ model, byequating m~ to 0 and using the starting value Y(0) = 0. Oncethe jump diffusion tree for the Y(t) process has been built,the jump diffusion tree for the short rate r(t) can beobtained by adding the deterministic term δ(t) to Y(t) ateach node of the tree. The term δ(t) can be solved usingtwo different methods. Using the first method, δ(t) issolved analytically using Equation (31). However, thissolution is valid only in the continuous-time limit, andso it allows an exact match with the initial zero-coupon


J dNu u u

d d

( ) ( ( )) ( ) ( )= − + +

−

� � �α σ λ0

(( ), ( )�λd Y 0 0=

dr m r t dt dZ t J dN J dNu u u d d= − + + −� � � � �α σ λ( ( )) ( ) ( ) (λλd )

H t T H T H t

A T A t P T P

( , ) ( , ) ( , )

( ) ( ) ln ( , ) ln

= −= − − +

0 0

0 (( , )0 t

L E eB

Bd tB J

d

d( )( )

( )( )τ τ

η ττ= ( ) − =

−� 1



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bond prices only when using a large number of steps inthe tree. If an exact match is desired for an arbitrary smallnumber of steps, then δ(t) can be obtained numericallyfor 0 < t < T by defining a pseudo bond price P*(0, t),as follows:

(35)

where P(0, t) is the initially observed bond price r andP*(0, t) is the pseudo bond price obtained by taking dis-counted risk-neutral expectation using Y-process as thepseudo short rate. The jump diffusion tree for Y-processcan be used to obtain P*(0, t) for different values of t (suchthat 0 < t < T). Given the values of P(0, t) and P*(0, t)for different values of t, δ(t) can be obtained numerically,as follows:

(36)

where Δt = T/n and the integral in H(0, t) is approxi-mated as a discrete sum by dividing t into t/Δt number ofsteps. The values of δ ( jΔt), for j = 0, 1, 2, ..., n – 1, canbe obtained iteratively by using successive values of t asΔt, 2Δt, ..., and nΔt, such that an exact match is obtainedwith the initial bond price function P(0,T ). The valuesof δ(t) can then be added to the appropriate nodes on then-step Y(t)-tree to obtain the corresponding tree for theshort rate.

VALUING OPTIONS

In this section, we value European options on zero-coupon bonds, or caplets, using the Fourier inversionmethod, and European options on coupon bonds, orEuropean swaptions, using the cumulant expansionmethod.

Valuing European Options on Zero-CouponBonds, or Caplets: The Fourier Inversion Method

Heston [1993] introduced the Fourier inversionmethod to solve the price of an option when the under-lying stock follows a stochastic volatility process. Recently,a number of researchers, including Duffie, Pan, and

H t j t t P t P tj

t t

( , ) ( ) ln ( , ) ln ( ,/

0 0 00

1

= = −=

−

∑ δ Δ ΔΔ

* ))

P t P t e P t eH t v dvt

( , ) ( , ) ( , )( , ) ( )0 0 00 0= = ∫− −* * δ

Singleton [2000], Bakshi and Madan [2000], and Chackoand Das [2002], have shown this method to be versatilein obtaining quasi-analytical formulas for a variety of moredifficult option pricing problems. For example, thismethod can be used to solve option prices when theunderlying price processes include jumps and stochasticvolatility, as well as to solve the prices of more complexderivatives like Asian options. We now demonstrate thismethod to solve for the price of an option on a zero-coupon bond under the Vasicek-EJ and Vasicek-EJ++models. Because an interest rate caplet is equivalent to aEuropean put option on a zero-coupon bond, and sincean interest rate cap is a portfolio of caplets, this methodcan be also used to price caps.

Recall that we express the bond price solution in aslightly nontraditional form as follows:

(37)

where A(τ ) and B(τ ) are time-homogenous functions ofτ = T – t. The motivation for using this form is that itallows a common framework for obtaining the formulasfor the bond price under the Vasicek-EJ and Vasicek-EJ++ models, since the solutions of A(τ ) and B(τ ) areidentical under both models and the only difference resultsfrom the H(t,T) term. In this section, we derive formulasfor pricing options under the Vasicek-EJ and Vasicek-EJ++ models and show that differences in the formulasbetween these models result only from the H(t,T ) term.

The price of a call option expiring on date S, writtenon a $1 face-value zero-coupon bond maturing at time T,can be computed as follows (see Chacko and Das [2002]):

(38)

where

(39)

(40)Π2 221

2

1t t

K

i Ktf y dy

e g

i= = +

⎡

⎣⎢

∞ −

∫ ( ) Re( )

ln

ln

πω

ω

ω ⎤⎤

⎦⎥

∞

∫ dω0

Π1 111

2

1t t

K

i Ktf y dy

e g

i= = +

⎡

⎣⎢

∞ −

∫ ( ) Re( )

ln

ln

πω

ω

ω ⎤⎤

⎦⎥

∞

∫0

dω

c t P t T KP t St t( ) ( , ) ( , )= −Π Π1 2

P t T eA B Y t H t T( , ) ( ) ( ) ( ) ( , )= − −τ τ



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The terms Π1t and Π2t can be interpreted as risk-neutralprobabilities. Re(.) function denotes the real part of theexpression contained in the brackets. The expression insidethe bracket of Re(.) contains complex numbers. All com-plex numbers can be written as a + bi, where i is the imag-inary number, and a is the real part.

The characteristic functions g1t(w) and g2t(w) aregiven in closed-form as follows:3

(41)

(42)

where H(t, S) and H(S, T) are defined by Equation (10)under the Vasicek-EJ model and by Equation (32) underthe Vasicek-EJ++ model, and A*

i(s) and B*i(s) are given as

(43)

and

(44)

where ai and bi for i = 1 and 2 are defined as

(45)

and

a A U i

b B U i1

1

1

1

= += +

( )( )

( )( )

ωω

B s B s b ei is*( ) ( )= + − �α

A s as b B s

bi i

i

i

*( )( ) ( )

( )= +

− −

+ −1

2

2 1

1

21

2

2

σα

α

α�

�

� 22 2

1

1 1

B s

eu u

u

us

( )

ln( )

⎛

⎝⎜⎜

⎞

⎠⎟⎟

++

+ −�

�

� �λ ηαη

αη α −−( )+

+−

−

�

�

�

�

� �

αα η

λ ηαη

αη α

b

b

e

i

u i

d d

d

ds

( )

ln( )

1

1 ++ −( )−

− +( )

1 �

�

� �

αα η

λ λ

b

b

s

i

d i

u d

( )

ge

t

A s B s Y t H S T i H t S

2

2 2

( )( ) ( ) ( ) ( , )( ) ( , )

ωω

=− − −* *

PP t Ss S t

( , ), –=

ge

t

A s B s Y t H S T i H t

1

11 1

( )( ) ( ) ( ) ( , )( ) ( ,

ωω

=− − + −* * SS

P t Ts S t

)

( , ), –=

(46)

where U = T – S and A(.) and B(.) are defined in Equa-tions (11) and (12), respectively.

The solutions of the probabilities Π1t and Π2t canalso be used to price a European put option by applyingthe put-call parity relation, as follows:

(47)

Note that the H(.) terms in Equations (41) and (42),under the Vasicek-EJ++ model, are given in closed-form inEquation (32). Hence, the preceding method has an ana-lytical advantage over the calibration method suggested byChacko and Das [2002], under which the time-dependentlong-term mean is determined numerically using an iter-ative method for pricing options.

Valuing European Options on Coupon Bondsor European Swaptions: The CumulantExpansion Method

Collin-Dufresne and Goldstein (CDG) [2002] pro-posed a fast and accurate cumulant expansion approxi-mation for pricing European options on coupon bonds,or European swaptions, under multifactor affine models,giving explicit solutions for the case of the two-factorCIR model and the three-factor Vasicek model. Extendingthe CDG approach, NBS [2007] showed the link betweenthe Fourier inversion-based approach for pricing caps andthe cumulant expansion-based approach for pricing swap-tions. Using this link, NBS derived explicit solutions forcaps and swaptions under the simple multifactor affinemodels4 and also under the subfamily of quadratic termstructure models with uncorrelated state variables and nointerdependencies.

However, neither CDG nor NBS derived explicitanalytical solutions for European options on coupon bonds(or European swaptions) for the case of the jump-extendedVasicek model of Chacko and Das [2002]. The previoussection showed how to price options on zero-couponbonds or caps under the jump-extended Vasicek modelusing the Fourier inversion-based approach. This sectionextends the results obtained in the previous section to

p t KP t S P t Tt t( ) ( , )( ) ( , )( )= − − −1 12 1Π Π

a A U i

b B U i2

2

==

( )( )

( )( )

ωω



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price European options on coupon bonds or Europeanswaptions using the cumulant expansion-based approach.

As shown by CDG, the price of a European calloption with expiration date S and exercise price K, writtenon a coupon bond with cash flows CFi at time Ti > S (fori = 1, 2, ..., n), is given as follows:

(48)

where is the price of thecoupon bond at the option expiration date, and

is defined as the proba-bility of the option being in the money under the forwardprobability measure associated with maturity W.5

The method consists of approximating n + 1 forwardprobabilities associated with maturity W,for W = S, T1, T2, ..., Tn, using a cumulant expansion ofthe characteristic function associated with the randomvariable Pc(S). Using the cumulant expansion, CDGshowed that the forward probabilities can be approxi-mated as follows:

(49)

where Q can be chosen for the desired level of accuracy.CDG proposed a value of Q = 7 for most affine models.The appendix gives the solutions of the coefficients κj

Q

and λj for Q = 7, as functions of the seven cumulants c1,c2, c3, ..., c7 associated with the distribution of Pc(S). Usingthe mathematical relationship between the cumulants andthe moments of any distribution, the seven cumulants canbe obtained using the first seven moments of the distri-bution of Pc(S ), as follows:

(50)

The solutions of the coefficients κjQ and λj (given in the

appendix), and the moments and cumulants given inEquation (50) are independent of the forward measure beingused. The forward-measure specific coefficients κj

Q,W and

cj

icj j j i i

i

j

= −−⎛

⎝⎜⎞⎠⎟ −

=

−

∑μ μ1

1

1

ΠtW

c jQ W

jW

j

Q

P S K( ( ) ) ,> ≈=∑κ λ

0

ΠtW

cP S K( ( ) )>

ΠtW

c tW

P S KP S K Ec

( ( ) ) [ ]( )> = >1

P S CF P S Tc in

i i( ) ( , )= ∑ =1

c t CF P t T P S K

K P t S

ii

n

i tT

ci( ) ( , ) ( )

( , )

= >( )−=∑

1

Π

ΠΠtS

cP S K( ) >( )

λjW in Equation (49) can be obtained by simply replacing

the seven moments μ1, μ2, μ3, ..., μ7, in Equation (50) withthe moments, μ1

W, μ2W, μ3

W, ..., μ7W, defined as follows:

(51)

where the expectation is taken under the forward mea-sure associated with the maturity W.

NBS [2007] provided a general solution for momentsin Equation (51), which used solutions of the ordinary dif-ferential equations (ODEs) associated with the charac-teristic functions, already obtained for valuing options onzero-coupon bonds, or caps, using the Fourier-inversionmethod. Applying this method, the general solution forthe moments under the jump-extended Vasicek modelsis given as follows:

(52)

where s = S – t, and the solutions of AG* (s) and BG

* (s) areidentical to the solutions of Ai

*(s) and Bi*(s), respectively,

obtained in Equations (43) and (44), respectively, with aiand bi replaced by the following:

(53)

and A(.) and B(.) are the solutions of the ODEs associ-ated with the PDDE of the bond price, given in Equa-tions (27) and (28), respectively. The function H(t,T ) isgiven by Equation (10) for the Vasicek-EJ model and isgiven by Equation (32) for the Vasicek-EJ++ model.

a A W S A U

b B W S B U

U

i lk

h

i lk

h

k

k

= − +

= − +

=

=

∑

∑

( ) ( )

( ) ( )

1

1

ll lk kT S= −

μhW

tW

ch

l l ll

n

l

E P S

CF CF CFh

h

= ⎡⎣ ⎤⎦

= ( )=∑

( )

...1 2

1

…221 11 ==∑∑

× − − −∑

n

l

n

G G kA s B s Y t H t Wexp ( ) ( ) ( ) ( , )* *==( )1

hlH S T

P t Wk

( , )

( , )

μhW

tW

chE P S h Q= ⎡⎣ ⎤⎦ =( ) , , , ,for 1 2 …



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Valuing American Options

The Fourier inversion method and the cumulantexpansion method presented in the previous two subsec-tions give prices for the European options on zero-couponbonds (caplets) and European options on coupon bonds(European swaptions), respectively. For pricing Americanoptions, we use the truncated-tree approach introducedearlier for the Vasicek-EJ and Vasicek-EJ++ models. Tocompute American option prices, the value of the interestrate and the probability vector are computed at each node,according to Equations (14) through (21). The interestrate tree is then used to compute the terminal optionvalues at each terminal node, and the option values at theremaining nodes are obtained by taking the maximum ofthe discounted option values and the correspondingintrinsic values.

The accuracy for pricing American options can beinvestigated by comparing the European put option pricesobtained using the truncated tree approach and the Euro-

pean put prices obtained through the Fourier inversionapproach, under the Vasicek-EJ model. Exhibit 4 makesthis comparison using the following parameter values: cur-rent value of the interest rate, r0 = 10%; risk-neutral speedof mean reversion, α~ = 0.2; risk-neutral long-run mean,m~ = 10%; and interest rate volatility, σ = 2%. The matu-rity of bonds was varied from T = 1 year to T = 10 years,the maturity of options was varied from 1 month to 5years, and the bond face value, F, is assumed to be $100.The intensities of the Poisson process, λu = 3, λd = 3, andthe jump variables, Ju and Jd, are distributed exponentiallywith the means, 1/ηu = 0.005 and 1/ηd = 0.005. Theoption prices are solved using N = 200 steps and N =1,000 steps, for which the truncation levels are at M = 150and M = 400, respectively.

Exhibit 4 shows fast convergence between Euro-pean put values computed using the truncated tree pro-cedure and the Fourier inversion method. The accuracyfalls as bond maturity increases, but improves with anincrease in the number of steps, N, used in the tree.


E X H I B I T 4European Put Option Prices under the Vasicek-EJ Model Using the Jump-Tree and Fourier Inversion Method


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Exhibit 5 compares the prices for European swaptionsobtained using the cumulant expansion method with Q = 7,and the truncated tree approach with N = 200 and M =150, and with N = 1000 and M = 400. The parameter valuesare kept as in Exhibit 4. The swap length varies from oneto five years, and the option expiration ranges from onemonth to three years. The swap strike rates are 2%, 6%, and10%. The tenor of the fixed leg is 0.5 years, and the notionalprincipal is assumed to be $100.

Exhibit 5 shows the accuracy gains in the tree valueswhen the number of steps increases and, also, in most of thecases, a convergence of the tree values to those obtained fromthe cumulant expansion approach. Also, the differencesbetween the prices obtained using the truncated tree and thecumulant expansion approach are in line with the differencesfound between the tree approach and the Fourier expansionmethod in Exhibit 4.

SUMMARY AND CONCLUSION

In this article, we modify the Gaussian jump-diffusiontree of Amin [1993] and apply it to the Vasicek-EJ, Vasicek-EJ+, and Vasicek-EJ++ models for pricing Americaninterest rate options. Our simulations show fast conver-gence of European option prices obtained using the trun-cated jump-diffusion tree, to those obtained using the Fourierinversion method (for options on zero-coupon bonds, orcaplets), and the cumulant expansion method (for optionson coupon bonds, or swaptions). As additional contribu-tions, we also provide 1) an analytical solution for the deter-ministic drift of the short rate process under theVasicek-EJ++ model, which allows this model to be exactlycalibrated to the initially observable bond prices; and 2)new analytical solutions for pricing European interest rateswaptions under the Vasicek-EJ, Vasicek-EJ+, and Vasicek-EJ++ models, using the cumulant expansion method ofCollin-Dufresne and Goldstein [2002].


E X H I B I T 5European Swaption Prices under the Vasicek-EJ Model Using the Jump-Tree and Cumulant-Expansion Method


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A P P E N D I X

The solutions of the coefficients κjQ and λj for Q = 7 as

functions of the seven cumulants, c1, c2, c3, ..., c7:

(54)

and

(55)where

λλλ

λ

0

1 2

2 2 2 1

3 2 12

222

=== + −

= − +

D

c d

c D c K c d

c K c c

( )

( )(( )= + − + −( )=

d

c D c K c c K c d

c K

λ

λ4 2

22 1

322

1

5 2

3 3( ) ( )

( −− + − +( )= + −

c c K c c d

c D c K c

14

22

12

23

6 23

2 1

4 8

15

) ( )

(λ )) ( ) ( )

( )

522

13

23

1

7 2 16

5 15+ − + −( )= − +

c K c c K c d

c K cλ 66 24 4822

14

23

12

24c K c c K c c d( ) ( )− + − +( )

κ

κ

07 4

22

23

6 32

2

17

13

4

15

6 2 3= + − +

⎛

⎝⎜⎞

⎠⎟c

c c

c c

! ! ( !)

== − + − +⎛⎝

3

3

15

5

105

7 3 43

22

5

23

24

7 3 4c

c

c

c c

c c c

! ! ! ! !⎜⎜⎞⎠⎟

= − + +⎛

⎝⎜⎞

⎠κ 2

7 4

23

24

6 32

2

6

4

45

6 2 3

c

c c

c c

! ! ( !) ⎟⎟

= − + +κ 37 3

23

5

24

25

7 3 4

3

10

5

105

7 3 4

c

c

c

c c

c c c

! ! ! ! !

⎛⎛⎝⎜

⎞⎠⎟

= − +⎛

⎝⎜⎞

⎠κ 4

7 4

24

25

6 32

24

15

6 2 3

c

c c

c c

! ! ( !) ⎟⎟

= − +⎛⎝⎜

⎞⎠⎟

=

κ

κ

57 5

25

26

7 3 4

67

5

21

7 3 4

1

c

c c

c c c

! ! ! !

cc

c c

c

c c c

26

6 32

2

77

27

7 3 4

6 2 3

1

7 3

! ( !)

! !

+⎛

⎝⎜⎞

⎠⎟

= +κ44!

⎛⎝⎜

⎞⎠⎟

(56)

N(.) gives the cumulative normal distribution function, and cj( j = 1, 2, ..., Q = 7) are the cumulants of the distribution ofPc(S).

ENDNOTES

1The derivation is similar to Das and Foresi [1996] andcan be obtained from the authors.

2The notation is borrowed from Brigo and Mercurio[2001] and NBS [2007, Chapter 5].

3The proof can be obtained from the authors uponrequest.

4NBS [2007, Chapter 9] defined simple multifactor affinemodels (i.e., simple AM(N) models) as models that have N – Mnormally distributed processes, which may be correlated witheach other but are uncorrelated with the M independently dis-tributed square root processes, for N > M.

51Pc(S) > K is the indicator function, which equals 1 ifPc(S) > K, and 0 otherwise.

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pricing american interest rate options under the jump-extended vasicek model

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