Stochastic Duration and Fast Coupon Bond Option Pricing in ... · Stochastic Duration and Fast...

Review of Derivatives Research, 3, 157–181 (1999)c© 2000 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands.

Stochastic Duration and Fast Coupon Bond OptionPricing in Multi-Factor Models

CLAUS MUNK [email protected] of Management, Odense University, DK-5230 Odense M, Denmark

Abstract. Generalizing Cox, Ingersoll, and Ross (1979), this paper defines the stochastic duration of a bond ina general multi-factor diffusion model as the time to maturity of the zero-coupon bond with the same relativevolatility as the bond. Important general properties of the stochastic duration measure are derived analytically,and the stochastic duration is studied in detail in various well-known models. It is also demonstrated by analyticalarguments and numerical examples that the price of a European option on a coupon bond (and, hence, of a Europeanswaption) can be approximated very accurately by a multiple of the price of a European option on a zero-couponbond with a time to maturity equal to the stochastic duration of the coupon bond.

Keywords: the term structure of interest rates, stochastic duration, multi-factor models, coupon bond optionpricing, swaption pricing.

JEL classification: E43, G13.

To quantify the interest rate risk of portfolios of bonds and other term structure derivativesvarious duration concepts have been suggested in the literature with the socalled Macaulayduration and Fisher-Weil duration as the two prevalent measures. However, as discussedby Ingersoll, Skelton, and Weil (1978) and Cox, Ingersoll, and Ross (1979), these measuresare not consistent with any reasonable, arbitrage-free dynamic model of the term structureof interest rates. For a general dynamic model with the short-term interest rate as the onlystate variable, Cox, Ingersoll, and Ross (1979) define a stochastic duration concept whichthey argue is a superior measure of basis risk, i.e. the relative change in the price of a bonddue to an unexpected change in the short rate.

In this paper, I generalize the concept to multi-factor diffusion models by defining thestochastic duration of a coupon bond as the time to maturity of the zero-coupon bond havingthe same relative volatility as the coupon bond. I derive some important properties of thestochastic duration measure analytically. For example, I show that under conditions satisfiedby most popular term structure models the familiar Fisher-Weil duration overestimates theinterest rate risk of coupon bonds. The stochastic duration measure is studied in detail in theone-factor models of Vasicek (1977), Cox, Ingersoll, and Ross (1985), and Hull and White(1990), the two-factor model of Longstaff and Schwartz (1992), and in various models ofthe HJM-class introduced by Heath, Jarrow, and Morton (1992).

Stochastic duration has an interesting application in option pricing. Wei (1997) shows bynumerical examples that in the Vasicek and CIR one-factor models the price of a Europeancoupon bond option is very closely approximated by a multiple of the price of a Europeanoption on the zero-coupon bond having a time to maturity equal to the stochastic duration

158 CLAUS MUNK

of the coupon bond. Closed-form solutions for the prices of European coupon bond optionsdo exist in these models, cf. Jamshidian (1989) and Longstaff (1993), but the approximateprice is faster to compute. Since a European swaption is equivalent to a European couponbond option, fast computation of such prices is of great importance to the financial industry.1

I show by numerical examples and an analytical argument that the approximation usingmy generalization of stochastic duration is also very precise in multi-factor models. Theapproximation is intuitively appealing since the zero-coupon bond involved is the one withthe same relative volatility as the coupon bond. In many multi-factor models there isa closed-form expression for the price of European zero-coupon bond options, but nonefor European coupon bond options, so the alternative is pricing by numerical methods.The use of multi-factor models is of great importance for pricing coupon bond optionsand swaptions since these assets depend on the variances and the correlations of variousforward rates and it is well-known that one-factor models are unable to match the observednon-perfect correlation structure of forward rates.

I continue in Section 1 with the general definition of stochastic duration in any diffusion-type continuous-time model of the term structure of interest rates. I also state and prove someimportant properties of the stochastic duration measure. In Section 2, I study the stochasticduration measure in detail in various popular term-structure models. The coupon bondoption pricing approximation is examined in Section 3. Finally, Section 4 summarizes thepaper and discusses the applicability of stochastic duration for interest rate risk management.

1. Stochastic Duration: Definition and Properties

1.1. The Stochastic Duration Measure

Consider a model of the term structure of interest rates where the evolution of the bondprices is affected by changes inK independent standard Brownian motionsW1, . . . ,WK .To be precise, assume that, for anyT > 0, the price of the zero-coupon bond maturing attime T evolves according to the equation

d P(t, T) = P(t, T)

[µ(t, T, Ät )dt +

K∑k=1

νk(t, T, Ät )dWk(t)

], (1)

for some drift functionµ and some functionsν1, . . . , νK with νk(T, T, ÄT ) = 0 for allk andall T .2 I will refer to the functionsνk as the factor sensitivities of the model. TheÄt term inµand theνk’s represents all those values of interest rates at and before timet , which are relevantfor the evolution of bond prices immediately after timet . The formulation (1) thereforeencompasses both standard Markov term structure models and non-Markov models of, e.g.,the HJM-class.

Consider a coupon bond (or, more generally, any asset with a stream of predeterminedpayments) payingxi units of accounts at timeti , i = 1,2, . . . , N, wheret1 < . . . < tn.Ruling out obvious arbitrage, the price of this bond at any timet < t1 is given byB(t) =

MULTI-FACTOR MODELS 159

∑ni=1 xi P(t, ti ). A simple application of Itˆo’s Lemma yields that

d B(t) =n∑

i=1

xi P(t, ti )µ(t, ti , Ät )dt +n∑

i=1

xi P(t, ti )K∑

k=1

νk(t, ti , Ät )dWk(t)

= B(t)

[n∑

i=1

w(t, ti )µ(t, ti , Ät )dt+K∑

k=1

(n∑

i=1

w(t, ti )νk(t, ti , Ät )

)dWk(t)

],

where I have introduced the weights

w(t, ti ) = xi P(t, ti )∑ni=1 xi P(t, ti )

,

which are non-negative and sum to one.3

I define the stochastic durationD(t) of the coupon bond as the time to maturity of thezero-coupon bond having the same relative volatility as the coupon bond, i.e. the sameinstantaneous variance of relative price changes. More formally,D(t) is given by

K∑k=1

νk(t, t + D(t),Ät )2 =

K∑k=1

(n∑

i=1


)2

. (2)

In the next subsection, I shall state conditions under which the stochastic duration measureis well-defined and derive some important general properties of the measure. In Section 2.1it is shown that the above definition of stochastic duration indeed is a generalization of thatin Cox, Ingersoll, and Ross (1979).

Among the many proposedone-dimensional risk measures, the stochastic duration, as Ihave defined it above, has some notable qualities. Firstly, it is consistent with an arbitrage-free dynamic model of the term structure and measures the price sensitivity with respectto anychange of the term structure within the model.4 Secondly, it is directly linked to thevolatility of relative prices, a familiar concept in the option pricing industry. Thirdly, it ismeasured in units of time as traditional duration measures. Fourthly, it measures only theunexpected changes in prices. Fifthly, it is easily computable for standard term structuremodels as will be shown in the following section. Finally, the stochastic duration is the keynumber to implement the accurate coupon bond option pricing approximation techniquediscussed in Section 3. The applicability of stochastic duration for hedging purposes isdiscussed in Section 4.

1.2. General Properties of Stochastic Duration

A priori, it is not certain that there is a solutionD(t) to (2) and, if so, whether it is unique.Before I address this important question, note the following result, which can easily beproven by an application of the chain rule.

Lemma 1 If the factor sensitivitiesνk(t, T, Ät ) are positive and increasing or negativeand decreasing in T , then

∑Kk=1 νk(t, T, Ät )

2 is increasing in T . If the factor sensi-tivities νk(t, T, Ät ) are positive and decreasing or negative and increasing in T , then∑K

k=1 νk(t, T, Ät )2 is decreasing in T .

160 CLAUS MUNK

It is now rather simple to show that when the factor sensitivitiesνk are either uniformlyincreasing or uniformly decreasing, a unique solution to (2) exists. Furthermore, the solutionlies between the time to the next payment of the bond and the time to maturity of thebond. Indeed, the stochastic duration can, as both Macaulay and Fisher-Weil duration, beinterpreted as a (very complex) weighted average of the time distances to the payment dates.

Theorem 1 If the factor sensitivitiesνk(t, T, Ät ) are all either increasing or decreasingin T , a unique solution D(t) to (2) exists and t1− t ≤ D(t) ≤ tn − t .

Proof: It is clear that∑n

i=1w(t, ti )νk(t, ti , Ät ) is betweenνk(t, t1, Ät ) andνk(t, tn, Ät ),and hence that

∑Kk=1(

∑ni=1w(t, ti )νk(t, ti , Ät ))

2 =∑Kk=1 νk(t, t + D(t),Ät )

2 is between∑Kk=1 νk(t, t1, Ät )

2 and∑K

k=1 νk(t, tn, Ät )2. From this observation and Lemma 1, the claim

immediately follows.

Recall that the Fisher-Weil duration measure is defined by

DFW(t) =∑n

i=1 xi P(t, ti )(ti − t)∑ni=1 xi P(t, ti )

=n∑

i=1

w(t, ti )(ti − t) =n∑

i=1

w(t, ti )ti − t.

The next theorem shows that for models with linear factor sensitivities, i.e.

νk(t, T, Ät ) = (T − t)bk(t, Ät ) (3)

for some maturity-independent functionbk, the Fisher-Weil duration and the stochasticduration are identical. As will be discussed in Section 2, the condition of the theorem issatisfied, e.g., for a one-factor model with a deterministic short rate drift and volatility andin any HJM model with maturity-independent forward-rate volatilities.

Theorem 2 If the factor sensitivitiesνk(t, T, Ät ) are of the form (3), then the stochasticduration and the Fisher-Weil duration are identical for all bonds.

Proof: Linear factor sensitivities imply that

n∑i=1

w(t, ti )νk(t, ti , Ät ) = νk

(t,

n∑i=1

w(t, ti )ti , Ät

), (4)

and hence that

K∑k=1

(n∑

i=1


)2

=K∑

k=1

νk

(t,

n∑i=1

w(t, ti )ti , Ät

)2

. (5)

The left-hand side of this expression equals∑K

k=1 νk(t, t+D(t),Ät )2, while the right-hand

side equals∑K

k=1 νk(t, t + DFW(t),Ät )2. Therefore,D(t) = DFW(t).

For a one-factor model where the factor sensitivity has the same sign for all maturities, it isclear that (5) implies (4), and therefore the converse to Theorem 2 holds: If the stochastic


duration and Fisher-Weil duration are identical for all bonds, the factor sensitivity must belinear. For multi-factor models, (5) will only imply (4) under stronger conditions on thefactor sensitivity.

Let us study the implications of linear factor sensitivity for yield curve movements.Equation (1) and Itˆo’s Lemma imply that the (continuously compounded) yieldy(t, T) =−(ln P(t, T))/(T − t) on the zero-coupon bond maturing at timeT has dynamics

dy(t, T) =[

y(t, T)

T − t− µ(t, T, Ät )

T − t+ 1

2(T − t)

K∑k=1

νk(t, T, Ät )2

]dt

− 1

T − t

K∑k=1

νk(t, T, Ät )dWk(t).

With factor sensitivities of the form (3), this simplifies to

dy(t, T) =[

y(t, T)

T − t− µ(t, T, Ät )

T − t+ 1

2(T − t)K∑

k=1

bk(t, Ät )2

]dt

−K∑

k=1

bk(t, Ät )dWk(t).

Hence, the unexpected changes in the zero-coupon yields are maturity-independent, butsince the expected changes can, in general, be maturity-dependent, the yield curve move-ments need not be parallel (cf. Example 1). Therefore, the Fisher-Weil duration can be arelevant measure of interest rate risk even when the yield curve does not move in parallelshifts. At first sight, this may seem to contradict the traditional perception that Fisher-Weilduration should only be used for parallel yield curve shifts. But, as shown by Ingersoll,Skelton, and Weil (1978), the precise relation is that the percentage change in the bondprice is proportional to the Fisher-Weil duration of the bond, if and only if the yield curvecan only move by parallel shifts. The percentage price change has both an expected part(stemming from the drift) and an unexpected part (stemming from the volatilities), whereasthe risk measures I focus on in this paper only quantify the unexpected part.

The next theorem provides conditions on the factor sensitivities under which the stochasticduration is smaller or greater, respectively, than the Fisher-Weil duration.

Theorem 3 Suppose all the factor sensitivitiesνk(t, T, Ät ) have the same sign. If theνk’sare all convexly decreasing or concavely increasing in T , the stochastic duration D(t) willbe smaller than the Fisher-Weil duration DFW(t). If theνk’s are all concavely decreasing orconvexly increasing in T , the stochastic duration D(t) will be greater than the Fisher-Weilduration DFW(t).

Proof: Let us consider the case, where theνk’s are negative and convexly decreasing inT . By Jensen’s Inequality,

n∑i=1

w(t, ti )νk(t, ti , Ät ) ≥ νk

(t,

n∑i=1

w(t, ti )ti , Ät

),

162 CLAUS MUNK

and hence

K∑k=1

(n∑

i=1


)2

≤K∑

k=1

νk

(t,

n∑i=1

w(t, ti )ti , Ät

)2

.

This equation is equivalent to

K∑k=1

νk(t, t + D(t),Ät )2 ≤

K∑k=1

νk(t, t + DFW(t),Ät )2.

Considering Lemma 1, the claim follows. The other cases are handled similarly.

As discussed in Section 2, traditional formulations of many popular models display negative,convexly decreasing factor sensitivities. Hence, the Fisher-Weil duration overestimates thebond price sensitivity to term structure changes in those models.

Next, consider the stochastic duration of a portfolio of bonds. LetM be the number ofdifferent bonds in the portfolio, lett1, . . . , tn be the payment dates of the portfolio, and letxmi be the total payment from the holdings of bondm= 1, . . . ,M at dateti , i = 1, . . . ,n.The value of the holdings of bondm is Bm(t) =

∑i xmi P(t, ti ), and the total value of the

portfolio is

V(t) =M∑

m=1

Bm(t) =M∑

m=1

n∑i=1

xmi P(t, ti ) =n∑

i=1

(M∑

m=1

xmi

)P(t, ti ).

The value of the holdings of bondm constitutes the fractionzm(t) = Bm(t)/V(t) of thetotal value of the portfolio. LetDm(t) denote the stochastic duration of bondm.

Theorem 4 In a one-factor model where the factor sensitivity is either convexly decreasingor concavely increasing (in T ), the stochastic duration DV (t) of a portfolio of bonds issmaller than the value-weighted average of the stochastic durations of the bonds in theportfolio, i.e.

DV (t) ≤M∑

m=1

zm(t)Dm(t). (6)

The converse relation holds if the factor sensitivity is either concavely decreasing or convexlyincreasing.

Proof: The stochastic durationDV (t) of the portfolio is given by

ν(t, t + DV (t),Ät ) =n∑

i=1

∑Mm=1 xmi P(t, ti )

V(t)ν(t, ti , Ät )

=M∑

m=1

Bm(t)

V(t)

n∑i=1

xmi P(t, ti )

Bm(t)ν(t, ti , Ät )

=M∑

m=1

zm(t)ν(t, t + Dm(t),Ät ).


If τ 7→ ν(t, t + τ,Ät ) is convex, then

ν(t, t + DV (t),Ät ) ≥ ν(

t, t +M∑

m=1

zm(t)Dm(t),Ät

).

If, furthermore,τ 7→ ν(t, t + τ,Ät ) is decreasing, Equation (6) follows. The other casesare handled similarly.

In the next section it is shown that the factor sensitivity in the Vasicek model, the Hull-Whitemodel, and (for reasonable parameter values) in the CIR model is convexly decreasing inT . In those models, therefore, the stochastic duration of a portfolio of bonds is smallerthan the value-weighted average of the stochastic durations of the bonds in the portfolio.This isnot reflecting some kind of diversification effect, since the prices of all bonds areinstantaneously perfectly correlated in one-factor models, and, hence, the volatility of aportfolio equals the value-weighted average of the volatilities of the bonds in the portfolio.It is the mapping of the volatility to time through the non-linear functionτ 7→ ν(t, t+τ,Ät ),which causes the non-linearity of the stochastic duration.

A diversification effectis present for multi-factor models, so the instantaneous varianceof a portfolio of bonds,

K∑k=1

(n∑

i=1

xmi P(t, ti )

V(t)νk(t, ti , Ät )

)2

=K∑

k=1

(M∑

m=1

zm(t)n∑

i=1

xmi P(t, ti )

Bm(t)νk(t, ti , Ät )

)2

,

is smaller than the value-weighted average of the instantaneous variances of the bonds inthe portfolio,

M∑m=1

zm(t)K∑

k=1

(n∑

i=1

xmi P(t, ti )

Bm(t)νk(t, ti , Ät )

)2

.

This follows immediately from Jensen’s Inequality. By definition of the stochastic durationsDV (t) andDm(t), the inequality

K∑k=1

νk(t, t + DV (t),Ät )2 ≤

M∑m=1

zm(t)K∑

k=1

νk(t, t + Dm(t),Ät )2

follows. If the functionτ 7→∑Kk=1 νk(t, t+τ,Ät )

2 is concave, I get by Jensen’s Inequalitythat

K∑k=1

νk(t, t + DV (t),Ät )2 ≤

K∑k=1

νk

(t, t +

M∑m=1

zm(t)Dm(t),Ät

)2

.

If, furthermore, the functionτ 7→ ∑Kk=1 νk(t, t + τ,Ät )

2 is increasing the inequality (6)follows. While for most models the functionτ 7→∑K

k=1 νk(t, t + τ,Ät )2 is increasing, it

164 CLAUS MUNK

is generally not concave (nor convex) for all positiveτ . Therefore, a relation like (6) is notgenerally valid in multi-factor models.

Another possible generalization of stochastic duration to multi-factor settings is to definea duration measure for each factor analogously to the stochastic duration in a one-factormodel. Define thek’th factor durationasD(k)(t), where

νk(t, t + D(k)(t),Ät ) =n∑

i=1

w(t, ti )νk(t, ti , Ät ), (7)

i.e. D(k)(t) is the maturity of the zero-coupon bond having the same relative price sensitivityto changes in thek’th Wiener process as the coupon bond. Obviously, each of the factorduration measuresD(k)(t) satisfies the results of Theorem 1-4. The next result shows thatthe stochastic duration is a complex average of the factor durations. See also Section 4 forfurther discussion of factor durations.

Theorem 5 If the factor sensitivitiesνk are all either increasing or decreasing and ofthe same sign, then the stochastic duration is a complex average of the factor durationsD(1)(t), . . . , D(K )(t). In particular,

mink=1,...,K

D(k)(t) ≤ D(t) ≤ maxk=1,...,K

D(k)(t).

Proof: The claim follows from Lemma 1 and the fact that the stochastic durationD(t)satisfies

K∑k=1

νk(t, t + D(t),Ät )2 =

K∑k=1

(n∑

i=1


)2

=K∑

k=1

νk(t, t + D(k)(t),Ät )2.

2. Stochastic Duration in Well-Known Term Structure Models

2.1. One-Factor Short Rate Models

In traditional one-factor models the movements of the entire term structure are governedby the (continuously compounded) short-term interest rate,r (·), which evolves accordingto a stochastic differential equationdr(t) = m(t, r (t))dt + σ(t, r (t))dW(t). The marketprice ofr -risk is a functionλ(t, r ), such that the risk-adjusted short rate drift ism(t, r ) =m(t, r ) − λ(t, r )σ (t, r ). The price of a zero-coupon bond is a functionP(t, T, r ) of theshort rate and, by Itˆo’s Lemma, the relative volatility (and factor sensitivity) is

ν(t, T, r ) =∂P∂r (t, T, r )

P(t, T, r )σ (t, r ).


The stochastic duration of a coupon bond is then given by the numberD(t), which solvesthe equation

∂P∂r (t, t + D(t), r )

P(t, t + D(t), r )=∑n

i=1 xi∂P∂r (t, ti , r )

B(t).

In the class of affine one-factor models of the term structure, which includes both theVasicek (1977) model, the Hull and White (1990, 1995) model, and the Cox, Ingersoll, andRoss (1985) square-root model, the zero-coupon bond price is of the formP(t, T, r ) =exp{a(t, T)+ b∗(t, T)r } for some continuously differentiable functionsa andb∗, cf. Duffie(1996, Chapter 7). Hence, the stochastic duration is the solution to

b∗(t, t + D(t)) =n∑

i=1

w(t, ti , r )b∗(t, ti ).

If the functionb(τ ) = b∗(t, t + τ) is well-defined and invertible, then

D(t) = b−1

(n∑

i=1

w(t, ti , r )b(ti − t)

).

Example 1. In the continuous-time version of the Ho and Lee (1986) model, wherem(t, r ) = m(t) andσ(t, r ) = σ(t), the bond price is of the exponential-affine form withb∗(t, T) = −(T − t). According to Theorem 2,D(t) = DFW(t). The unexpected changesin all zero yields are identical. Only for constant coefficients are the expected, and hencethe total, changes identical, and in that case the only possible shifts in the term structure areparallel shifts.

Example 2. In the Hull-White (extended-Vasicek) model, wherem(t, r ) = κ[θ (t) − r ]andσ(t, r ) = σ for positive constantsκ andσ , the bond price volatility isν(t, T, Ät ) =σb(T − t), whereb(τ ) = −[1 − e−κτ ]/κ. Sinceb′(τ ) < 0 andb′′(τ ) > 0, Theorem 3implies thatD(t) ≤ DFW(t). In fact,b−1(y) = −[ln(1+ κy)]/κ, and therefore

D(t) = −1

κln

(1+ κ

n∑i=1

w(t, ti , r )b(ti − t)

)

= −1

κln

(n∑

i=1

w(t, ti , r )e−κ[ti−t ]

). (8)

The stochastic duration for the original Vasicek model (θ (t) constant) is also given by (8),since theb(·) function is the same in the Vasicek model as in the Hull-White model.

Example 3. In the CIR square-root model, wherem(t, r ) = κθ − (κ + λ)r andσ(t, r ) =σ√

r for positive constantsκ, θ andσ , theb(·) function is given by

b(τ ) = − 2(eγ τ − 1)

(β + γ ) (eγ τ − 1)+ 2γ,

166 CLAUS MUNK

whereβ = κ + λ, γ =√β2+ 2σ 2, andλ is the market price of risk parameter. A

straightforward, but lengthy, computation shows thatb′(τ ) < 0, and thatb′′(τ ) is positivefor all τ > 0 if β ≥ 0, but if β < 0, b′′(τ ) is negative forτ ∈ (0, τ ∗) and positive forτ ∈ (τ ∗,∞), whereτ ∗ = γ−1 ln ([γ − β]/[γ + β]). If β > 0, it follows from Theorem 3that D(t) ≤ DFW(t).5 Since

b−1(y) = 1

γln

(1− 2γ y

2+ (β + γ )y),

it follows that

D(t) = 1

γln

(1− 2γ

[β + γ + 2∑n

i=1w(t, ti , r )b(ti − t)

]−1).

2.2. The Longstaff-Schwartz Model

Longstaff and Schwartz (1992) consider a two-factor CIR-type model with the short rater and the instantaneous varianceV of the short rate as the two term structure generatingvariables. Let us write

dr(t) = mr (r (t),V(t))dt + α√βr (t)− V(t)

α(β − α) dW1(t)

+ β√

V(t)− αr (t)

β(β − α) dW2(t), (9)

dV(t) = mV (r (t),V(t))dt + α2

√βr (t)− V(t)

α(β − α) dW1(t)

+ β2

√V(t)− αr (t)

β(β − α) dW2(t), (10)

whereW1 andW2 are independent Brownian motions, and the drift terms are

mr (r,V) = αγ + βη − βδ − αξβ − α r − ξ − δ

β − αV,

mV (r,V) = α2γ + β2η − αβ(δ − ξ)β − α r − βξ − αδ

β − α V.

The parametersα, β, γ, δ, η, andξ are all positive.The zero-coupon bond price is of the form

P(t, T, r,V) = a(T − t)2γb(T − t)2η exp{κ[T − t ] + c(T − t)r + d(T − t)V} ,


where

a(τ ) = 2ϕ

(δ + ϕ)(eϕτ − 1)+ 2ϕ>0, c(τ )= αϕ(e

ψτ − 1)b(τ )− βψ(eϕτ − 1)a(τ )

ϕψ(β − α) ,

b(τ ) = 2ψ

(ε + ψ)(eψτ − 1)+ 2ψ>0, d(τ )= ψ(e

ϕτ − 1)a(τ )− ϕ(eψτ − 1)b(τ )

ϕψ(β − α) ,

andε = ξ +λ, ϕ = √2α + δ2,ψ =√

2β + ε2, andκ = γ [δ+ϕ]+η[ε+ψ ]. The marketprice of risk parameterλ (and henceε) can be negative.6 By Ito’s Lemma, the dynamics ofthe zero-coupon bond priceP(t, T) = P(t, T, r (t),V(t)) is given by

d P(t, T) = P(t, T) [µ(t, T, r (t),V(t))dt + ν1(t, T, r (t),V(t))dW1(t)

+ ν2(t, T, r (t),V(t))dW2(t)] ,

where the factor sensitivities can be written as

ν1(t, T, r,V) = −αϕ

√βr − V

α(β − α)(eϕ[T−t ] − 1)a(T − t), (11)

ν2(t, T, r,V) = − βψ

√V − αr

β(β − α)(eψ [T−t ] − 1)b(T − t). (12)

Obviously,ν1 andν2 are negative for allT . By straightforward, but lengthy, computationsit can be shown thatν1 and ν2 are decreasing inT , and ν1 is convex for allT > t .Furthermore,ν2 is convex for allT > t , if ε > 0, whereas ifε < 0, ν2 is concave forT ∈ (t, t + τ ∗) and convex forT ∈ (t + τ ∗,∞), whereτ ∗ = ψ−1 ln ([ψ − ε]/[ψ + ε]).In the caseε > 0, it follows from Theorem 3 thatD(t) ≤ DFW(t) for all bonds.

The instantaneous variance of relative changes in the zero-coupon bond price is

ν1(t, t + τ, r,V)2+ ν2(t, t + τ, r,V)2 = α(βr − V)

ϕ2(β − α) (eϕτ − 1)2a(τ )2

+ β(V − αr )

ψ2(β − α) (eψτ − 1)2b(τ )2.

Similarly, the instantaneous variance of relative changes in the price of a coupon bond is(n∑

i=1

w(t, ti )ν1(t, ti , r,V)

)2

+(

n∑i=1

w(t, ti )ν2(t, ti , r,V)

)2

= α(βr − V)

ϕ2(β − α) K 2a

+ β(V − αr )

ψ2(β − α) K 2b,

where

Ka=n∑

i=1

w(t, ti )(eϕ[ti−t ]−1)a(ti−t), Kb=

n∑i=1

w(t, ti )(eψ [ti−t ]−1)b(ti−t).

168 CLAUS MUNK

Therefore, the stochastic duration of the coupon bond is given by the solutionD(t) to theequation

α(βr − V)

ϕ2

((eϕD(t) − 1)2a(D(t))2− K 2

a

)+ β(V − αr )

ψ2

((eψD(t) − 1)2b(D(t))2− K 2

b

) = 0. (13)

The equation has to be solved numerically.The factor durationsD(1)(t) andD(2)(t) are given by Equation (7) which using (11)–(12)

yields

D(1)(t) = 1

ϕln

(2ϕ

2ϕKa− (δ + ϕ) + 1

), D(2)(t) = 1

ψln

(2ψ

2ψKb− (ε + ψ) + 1

).

For large values ofψ , the term(eψτ −1)b(τ ) approaches its asymptotic value (forτ →∞)2ψ/(ε + ψ) very fast.7 In that case,Kb will (except for very short bonds) be very close to2ψ/(ε+ψ), and the last term of the sum in (13) will be very close to zero (except for verysmall values ofD(t)). Therefore, the stochastic durationD(t) will be very close to the firstfactor durationD(1)(t), which is given above in closed-form.

2.3. The Heath-Jarrow-Morton Models

In the Heath, Jarrow, and Morton (1992) class of models, the prevailing term structure istaken as given and it is assumed that, for eachT , theT-maturity instantaneous forward ratef (t, T) evolves according to

d f (t, T) = m(t, T, Ät )dt +K∑

k=1

σk(t, T, Ät )dWk(t), 0≤ t ≤ T, (14)

where W1, . . . , WK are independent Wiener processes under the equivalent martingalemeasure, andm is given by the drift restriction

m(t, T, Ät ) =K∑

k=1

σk(t, T, Ät )

∫ T

tσk(t,u, Ät )du. (15)

From the relationP(t, T) = exp{− ∫ T

t f (t,u)du}

and (14), it follows that

P(t, T) = exp

{−∫ T

tf (0,u)du−

∫ T

t

[∫ t

0m(s,u, Äs)ds

]du

−K∑

k=1

∫ T

t

[∫ t

0σk(s,u, Äs)dWk(s)

]du

}(16)


and that the factor sensitivitiesνk are given by

νk(t, T, Ät ) = −∫ T

tσk(t,u, Ät )du, k = 1,2, . . . , K . (17)

Therefore, the stochastic durationD(t) of the coupon bond is the solution to the equation

K∑k=1

[∫ t+D(t)

tσk(t,u, Ät )du

]2

=K∑

k=1

(n∑

i=1

w(t, ti )∫ ti

tσk(t,u, Ät )du

)2

,

or, equivalently,

K∑k=1

[∫ D(t)

0σk(t, t + u, Ät )du

]2

=K∑

k=1

(n∑

i=1

w(t, ti )∫ ti−t

0σk(t, t + u, Ät )du

)2

. (18)

For one-factor models, this can be simplified to∫ D(t)

0σ1(t, t + u, Ät )du=

n∑i=1

w(t, ti )∫ ti−t

0σ1(t, t + u, Ät )du.

If the forward rate volatilities are all uniformly of the same sign, it follows from Theorem 1that there is a unique solutionD(t) to (18). If the forward rate volatilities are all maturity-independent, i.e.σk(t, T, Ät ) = σk(t, Ät ), the bond price volatilitiesνk(t, T, Ät ) are alllinear inT . It then follows from Theorem 2 that the stochastic duration equals the Fisher-Weil duration. This is, e.g., the case in the one-factor model, whereσ(t, T, Ät ) = σ , whichis equivalent to the Ho-Lee model of Example 1. If the forward rate volatilitiesσk(t, T, Ät )

are all positive and decreasing inT , Equation (17) implies that all the bond price volatilitiesνk(t, T, Ät ) are negative, decreasing, and convex inT . Hence, it follows from Theorem 3that the stochastic duration is smaller than the Fisher-Weil duration. This is, e.g., the case inthe one-factor model whereσ(t, T, Ät ) = σe−κ[T−t ] , which is equivalent to the Hull-Whitemodel of Example 2.

Example 4. Heath, Jarrow, and Morton (1992) and Brenner and Jarrow (1993) study thetwo-factor model withσ1(t, T, Ät ) = σ1 andσ2(t, T, Ät ) = σ2e−κ[T−t ] , whereσ1, σ2, andκ are positive constants. Again,D(t) ≤ DFW(t). From (18) the stochastic duration is thesolutionD(t) to the equation

σ 21 D(t)2+ σ

22

κ2

(1− e−κD(t)

)2 = σ 21

(n∑

i=1

w(t, ti )(ti − t)

)2

+ σ22

κ2

(n∑

i=1

w(t, ti )(1− e−κ[ti−t ]

))2

.

The equation has to be solved by numerical methods.

170 CLAUS MUNK

Example 5. Ritchken and Sankarasubramanian (1995) study a one-factor model with theforward rate volatility given by

σ(t, T, Ät ) = σ(t, t, Ät )e−∫ T

tκ(x)dx

,

whereκ is some deterministic function. Note thatσ(t, t, Ät ) is the volatility of the shortrate, and that this volatility is allowed to depend on both present and past interest rates.Ritchken and Sankarasubramanian show that the entire past-dependency can be capturedby a single variable

ϕ(t) =∫ t

0σ(u, t, Äu)

2 du=∫ t

0σ(u,u, Äu)

2e−2∫ t

uκ(x)dx du,

the accumulated forward rate variance. If, furthermore, the short rate volatilityσ(t, t, Ät )

can be written as a function of (at most)t , r (t), andϕ(t), the term structure can be representedby the two-state Markov process(r, ϕ).

In this setting, the factor sensitivity becomes

ν(t, T, Ät ) = −∫ T

tσ(t, y, Ät )dy= −σ(t, t, Ät )

∫ T

te−∫ y

tκ(x)dx dy.

If σ(t, t, Ät ) is positive, the factor sensitivity is negative and decreasing inT . If, fur-thermore,κ(·) is positive, the factor sensitivity is convex, and Theorem 3 implies thatD(t) ≤ DFW(t). The stochastic durationD(t) is the solution of the equation∫ D(t)

0e−∫ y+t

tκ(x)dx dy=

n∑i=1

w(t, ti )∫ ti

te−∫ y+t

tκ(x)dx dy.

In particular, whenκ(x) is identically equal to some constantκ, the solution is

D(t) = −1

κln

(n∑

i=1

w(t, ti )e−κ[ti−t ]

),

which is identical to the stochastic duration in the much simpler Hull-White model, cf. (8).

3. Stochastic Duration Based Approximate Coupon Bond Option Pricing

3.1. The General Idea of the Approximation

In a one-factor framework, Wei (1997) suggests that the price of a European call option ona coupon bond can be approximated by a multiple of the price of a European call option ona zero-coupon bond with a time to maturity equal to the stochastic duration of the couponbond. LetC(t; t∗, T, X) be the timet price of a European call option maturing at timet∗,written on a zero-coupon bond paying one dollar at timeT , and having an exercise priceof X. Also, letCB(t; t∗, X) be the timet price of a European call option maturing att∗,


written on a coupon bond and having an exercise priceX. The payoff, and hence theprice of the coupon bond option, does not depend on the payments of the coupon bondbefore the exercise date of the option. Assume that the coupon bond has paymentsxi

at time ti > t∗, i = 1, . . . ,n. Let B(t) denote the timet value of these payments, i.e.B(t) =∑n

i=1 xi P(t, ti ). Wei’s approximation can then be stated as

CB(t; t∗, X) ≈ CappB (t; t∗, X) ≡ ζC(t; t∗, t + D(t), X/ζ ), (19)

whereD(t) is the stochastic duration of the bond payments after expiration of the option,andζ = B(t)/P(t, t + D(t)).

Wei does not provide any analytical justification for his suggested approximation, but heshows by numerical examples that the approximation is very precise in the one-factor modelsof Vasicek (1977) and Cox, Ingersoll, and Ross (1985). Also, he claims that forK -factormodels a coupon bond must be approximated by a portfolio ofK zero-coupon bonds. I willdemonstrate that, with my general definition of stochastic duration, the approximation (19)is also very efficient for multi-factor models. Only one zero-coupon bond is involved in theapproximation.

First, I will study the approximation analytically. Under theT-forward measureQT , allprice processes relative toP(·, T) are martingales. Hence, the error in the approximationis

CB(t; t∗, X)− B(t)

P(t, T)C(t; t∗, T, P(t, T)X/B(t)

)= P(t, T)EQT

t

[max(B(t∗)− X,0)

P(t∗, T)

]− B(t)

P(t, T)P(t, T)EQT

t

[max(P(t∗, T)− X P(t, T)/B(t),0)

P(t∗, T)

]= P(t, T)EQT

t

[max

(B(t∗)

P(t∗, T)− X

P(t∗, T),0

)−max

(B(t)

P(t, T)− X

P(t∗, T),0

)].

Note that

EQTt

[B(t∗)

P(t∗, T)

]= B(t)

P(t, T)(20)

and

EQTt

[X

P(t∗, T)

]= X P(t, t∗)

P(t, T). (21)

For deep-in-the-money calls, both maximum terms yield the first argument with a probabilityclose to one, so (20) implies that the absolute pricing errors should be close to zero. Fordeep-out-of-the-money calls, both maximum terms are zero with a probability close to one,so again the absolute pricing errors (as well as the prices themselves) should be negligible.The errors arise because of the intermediary realizations where one, but not both, of themaximum terms is non-zero. This is the case whenB(t∗) and P(t∗, T) are such that

172 CLAUS MUNK

X/P(t∗, T) ends up betweenB(t∗)/P(t∗, T) andB(t)/P(t, T). As (20) and (21) suggest,this is important for (forward) near-the-money calls, for which the absolute pricing errorstherefore should be largest.

Note that the considerations above hold for all values ofT . To reduce the probability ofending up in the intermediary case, the maturityT can be chosen such thatB(t∗)/B(t) tendsto stay nearP(t∗, T)/P(t, T). One could consider choosingT as to minimize the varianceVarQT

t [B(t∗)/B(t)− P(t∗, T)/P(t, T)], but because of the complexity of the expressionsfor P(t∗, T) and, especially,B(t∗) this seems impracticable. Alternatively, one can chooseT such that the relative changes inB(t) andP(t, T) over the next infinitesimal interval are,in some sense, close. This is achieved by takingT = t+D(t), whereD(t) is the stochasticduration of the payments of the coupon bond after expiration of the option.

Another promising candidate forT is

T∗ = argminT VarQTt

[d B(t)

B(t)− d P(t, T)

P(t, T)

]and aminimum variance durationmeasure of the coupon bondB(t) can be defined asDmv(t) = T∗ − t . Letting σB denote the standard deviation ofd B(t)/B(t), σT the stan-dard deviation ofd P(t, T)/P(t, T), andρB,T the correlation between these relative pricechanges,T∗ satisfies the equation

∂σT

∂T

[σT − σBρB,T

]− σBσT∂ρB,T

∂T= 0,

assuming sufficient differentiability. For one-factor models,ρB,T = 1, and ifσT is mono-tone the equation reduces toσT = σB. Since this is the equation defining stochastic duration,it follows that Dmv(t) = D(t) for all bonds in one-factor models. In multi-factor modelsthe two measures are generally not equal, but numerical experiments indicate that for mostparameter combinations (including the most realistic combinations), they are very close toeach other. This is becauseρB,T typically is close to one and is rather insensitive toT . Inthe numerical two-factor examples presented below, there is therefore no clear differencein the quality of the approximation (19) using stochastic duration and that using minimumvariance duration. For parameter combinations where the two duration measures differsignificantly, the approximation using my definition of stochastic duration remains veryprecise, whereas the approximation using the minimum variance duration becomes lessaccurate.

Next, I will discuss the usefulness of the approximation. From Jamshidian (1989) andLongstaff (1993) it is known that in the Vasicek, the Hull-White, and the CIR model, the ex-act price of the European coupon bond call option is given in closed-form asCB(t; t∗, X) =∑n

i=1 xi C(t; t∗, ti , Xi ), whereXi = P(t∗, ti , r ∗), andr ∗ is the solution to the equation∑ni=1 xi P(t∗, ti , r ∗) = X. As noted by Hull and White (1993), the same analysis holds for

one-factor HJM models with a constant or exponentially decaying forward rate volatility.Hence, to compute the exact price, one equation in one unknown must be solved (numer-ically) andn zero-coupon bond call option prices must be computed. The approximateprice follows from a single zero-coupon bond option price and is, therefore, much fastercomputed than the exact price.


In many of the proposed multi-factor term structure models there is a closed-form ex-pression for the price for a European zero-coupon bond option, but not for the price of aEuropean coupon bond option. In these models, an approximation like (19), if sufficientlyaccurate, is even more valuable than in the one-factor models, since the alternative is tocompute the price by numerical techniques, such as finite difference or Monte Carlo meth-ods. Below, I shall examine the performance of the approximation in the two-factor modelof Longstaff and Schwartz (1992), discussed in Section 2.2, and in the two-factor GaussianHJM-model of Example 4.

3.2. The Longstaff-Schwartz Model

In the Longstaff and Schwartz model the price of a European call option on a zero-couponbond is given by an expression of the form

C(t, r,V; t∗, T, X) = P(t, T, r,V)9(θ1, θ2;4γ,4η, ω1, ω2)

− X P(t, t∗, r,V)9(θ3, θ4;4γ,4η, ω3, ω4), (22)

where9 is the cumulative distribution function of a bivariate non-centralχ2-distributedrandom variable. There is no known closed-form expression for the price of a Europeancall on a coupon bond in this model.

To evaluate the precision of the approximation (19), I compareCappB (t)with a priceCB(t)

computed by using Monte Carlo simulations.8 I have computed the approximate price bysimulation using the same sample paths as in the computation ofCB(t). I shall comparethe simulation pricesCapp

B (t) andCB(t). Of course, I could have compared the simulatedprice CB(t) with the approximate priceCapp

B (t) based on an evaluation of Equation (22),but the latter involves both a numerical integration and the computation of modified Besselfunctions.9 More importantly, a comparison of the two simulated prices (using the samesample paths) filters out potential biases stemming from the use of the simulated couponbond option priceCB(t), rather than the true, but unknown, coupon bond option priceCB(t).

I have implemented the Longstaff-Schwartz model with the parameter valuesα = 0.01,β = 0.08, γ = 0.1, δ = 0.33, η = 16, ξ = 14, andλ = 0.10 I consider 2-month and6-month call options written on a 2-year bond and a 10-year bond, respectively. The bondspay an annual coupon of 8%. The initial short rate level and short rate volatility are takento ber = 0.08 andV = 0.002, respectively. Table 1 shows results for the options on the2-year bond, for a range of exercise prices around theforward at-the-moneyvalue ofX, i.e.B(t)/P(t, t∗). Similar results for the options on the 10-year bond are given in Table 2. Theabsolute deviation reported isCapp

B (t)− CB(t) and the percentage deviation is the absolutedeviation divided by the simulated “correct” priceCB(t). The tables also provide the MonteCarlo standard deviation of the simulated difference between the “correct” price and theapproximate price.

All the displayed approximate option prices are accurate to the penny, and the percentagedeviations are also very small for the option and bond maturities studied. For all optionsconsidered the absolute deviation is much smaller than the Monte Carlo standard deviationindicating that the pricing errors are insignificant. As expected, the approximation seems

174 CLAUS MUNK

Table 1.Prices of 2-month and 6-month European call options on a 2-year bond in the Longstaff-Schwartz model.The bond has an annual coupon of 8%, a current price of 89.3400, a 2-month forward price of 91.2042, a 6-monthforward price of 95.7687, and a stochastic duration of 1.9086.X denotes the exercise price of the option,Capp

B (t)is the simulated approximate price, “abs. dev.” and “pct. dev.” the absolute and percentage difference betweenthe simulated approximate price and the simulated correct price, and “std. dev.” is the standard deviation of theMonte Carlo simulated differences between the approximate and the correct price.

2-month options 6-month options

X CappB (t) abs. dev. pct. dev. std. dev. X Capp

B (t) abs. dev. pct. dev. std. dev.

86 5.08407 0.1·10−5 0.000% 1.8·10−4 91 4.45364 2.0·10−5 0.000% 4.8·10−4

87 4.10368 0.2·10−5 0.000% 1.7·10−4 92 3.53250 4.9·10−5 0.001% 4.0·10−4

88 3.12553 0.7·10−5 0.000% 1.5·10−4 93 2.63434 8.2·10−5 0.003% 3.4·10−4

89 2.16242 2.1·10−5 0.001% 1.2·10−4 94 1.79287 9.6·10−5 0.005% 3.2·10−4

90 1.26608 3.2·10−5 0.003% 1.0·10−4 95 1.06678 5.5·10−5 0.005% 3.1·10−4

91 0.56030 0.7·10−5 0.001% 0.9·10−4 96 0.52036 −3.3 · 10−5 −0.006% 2.4·10−4

92 0.15992 −2.7 · 10−5 −0.017% 0.7·10−4 97 0.19074 −9.6 · 10−5 −0.050% 2.0·10−4

93 0.02442 −2.0 · 10−5 −0.083% 0.7·10−4 98 0.04576 −7.7 · 10−5 −0.168% 2.0·10−4

94 0.00163 −0.4 · 10−5 −0.253% 0.4·10−4 99 0.00576 −2.7 · 10−5 −0.474% 1.5·10−4

95 0.00001 −0.0 · 10−5 −1.545% 0.1·10−4 100 0.00021 −0.2 · 10−5 −1.051% 0.5·10−4

to work better the shorter the maturity of the option and the shorter the maturity of theunderlying bond, but it is very precise even for the 6-month calls on the 10-year bond.The accuracy of the approximation is rather insensitive to the initial values ofr and Vand the parameter values. The approximation tends to slightly overestimate in-the-moneycalls and slightly underestimate out-of-the-money calls. Wei (1997) reports similar resultsfor the one-factor models of Vasicek and Cox-Ingersoll-Ross. It can also be seen that, in

Table 2. Prices of 2-month and 6-month European call options on a 10-year bond in the Longstaff-Schwartzmodel. The bond has an annual coupon of 8%, a current price of 76.9324, a 2-month forward price of 78.5377, a6-month forward price of 82.4682, and a stochastic duration of 4.8630.X denotes the exercise price of the option,Capp

B (t) is the simulated approximate price, “abs. dev.” and “pct. dev.” the absolute and percentage differencebetween the simulated approximate price and the simulated correct price, and “std. dev.” is the standard deviationof the Monte Carlo simulated differences between the approximate and the correct price.

2-month options 6-month options

X CappB (t) abs. dev. pct. dev. std. dev. X Capp


74 4.42874 1.2·10−4 0.003% 1.9·10−3 78 4.27344 1.1·10−3 0.027% 4.4·10−3

75 3.46569 2.5·10−4 0.007% 1.6·10−3 79 3.42836 1.3·10−3 0.037% 4.3·10−3

76 2.53643 3.9·10−4 0.015% 1.4·10−3 80 2.64289 1.2·10−3 0.045% 4.3·10−3

77 1.69005 4.0·10−4 0.024% 1.4·10−3 81 1.93654 0.8·10−3 0.042% 4.2·10−3

78 0.98799 1.8·10−4 0.018% 1.3·10−3 82 1.33393 0.2·10−3 0.015% 3.8·10−3

79 0.48542 −1.7 · 10−4 −0.036% 1.0·10−3 83 0.85064 −0.5 · 10−3 −0.063% 3.1·10−3

80 0.19080 −3.9 · 10−4 −0.202% 0.9·10−3 84 0.49430 −1.1 · 10−3 −0.220% 2.8·10−3

81 0.05666 −3.2 · 10−4 −0.570% 0.9·10−3 85 0.25641 −1.3 · 10−3 −0.508% 2.6·10−3

82 0.01267 −1.6 · 10−4 −1.263% 0.8·10−3 86 0.11491 −1.2 · 10−3 −1.001% 2.7·10−3

83 0.00185 −0.5 · 10−4 −2.424% 0.5·10−3 87 0.04372 −0.8 · 10−3 −1.786% 2.6·10−3


Figure 1. The accuracy of the pricing approximation for 2-month calls on 2-year bonds in the Longstaff-Schwartzmodel. The graphs show the dependency of the simulated absolute pricing errorCapp

B (t)− CB(t) on the exerciseprice for different maturities of the approximating zero-coupon bond. The stochastic duration of the underlying2-year bond isD(t) = 1.9086, and the 2-month forward price of the bond is 91.2042.

accordance with the analysis in the preceding subsection, the absolute deviations are smallestfor deep-out-of- and deep-in-the-money options and somewhat higher for near-the-moneyoptions.

Figure 1 shows the dependency of the absolute deviation of the approximation on theexercise price for different values of the maturity of the zero-coupon bond involved. Thegraph is for 2-month calls on the 2-year bond, but a similar picture can be drawn for the otheroptions studied above. For deep-out-of- and deep-in-the-money the approximation is veryaccurate for all zero-coupon bond maturities, but for near-the-money options the accuracyis highly dependent on the maturity. For a maturity equal to the stochastic duration of theunderlying bond, in this caseD(t) = 1.9086, the absolute pricing error is very close tozero as Table 1 has already indicated. Again these results are consistent with the analyticaldiscussion in Section 3.1.

Finally, note that, with the stated parameter values, the stochastic durationD(t) of thebonds studied above is almost indistinguishable from the first factor durationD(1)(t) forwhich an explicit expression is available as discussed in the final paragraph of Section 2.2.This reduces the computation time of the approximate coupon bond option priceCapp

B (t).

3.3. A Two-Factor HJM Model

Consider the two-factor HJM model with forward rate volatilitiesσ1(t, T, Ät ) = σ1 andσ2(t, T, Ät ) = σ2e−κ(T−t), which was studied in Example 4. For this model specification,

176 CLAUS MUNK

Heath, Jarrow, and Morton (1992) found that the price of a European call option on azero-coupon bond is given by

C(t; t∗, T, X) = P(t, T)N(h)− X P(t, t∗)N(h− σP), (23)

where

h =ln(

P(t,T)X P(t,t∗)

)σP

+ 12σP,

and

σ 2P = σ 2

1 (T − t∗)2(t∗ − t)+ σ 22

2κ3

(1− e−κ[T−t∗])2 (1− e−2κ[t∗−t ]

).

Again, there is no known closed-form expression for the price of a European call on acoupon bond.

To evaluate the precision of the approximate option priceCappB (t), I compare with a

price CB(t) computed by using Monte Carlo simulations. As in the previous section, Ialso compare the simulated “true” priceCB(t) with a simulated approximate priceCapp

B (t),computed by using the same sample paths, to filter out possible simulation biases. Detailson the simulation approach are given in Appendix A.

For the initial term structure, I take the term structure generated by the CIR square-root model of Example 3 with a long-term levelθ = 0.085, a mean-reversion parameterκ = 0.25, a short rate volatility parameter ofσ = 0.05, and a zero market price of riskparameter.11 The short rate is initialized at 8%. The forward rate volatility parameters areσ1 = σ2 = 0.02, andκ = 0.5.

Table 3 provides results for 4-month call options on a 5-year bond with an annual couponof 8%. The approximate price deviates very little from the true price, especially whensimulated prices are compared. Again, the absolute difference between the simulated pricesis much less than the standard deviation of the simulated differences. I get very similarresults for other options and bonds and for other parameter values. Figure 2 displays theeffect of different choices of the maturity of the zero-coupon bond involved on the accuracyof the approximation. Again, it is clear that the approximation works well for deep-out-of-and deep-in-the-money no matter what maturity is chosen, but for near-the-money optionsonly a maturity close to the stochastic duration of the underlying bond can achieve a highprecision.

4. Concluding Remarks

Stochastic duration measures the sensitivity of the price of a term structure derivative, suchas a bond or a portfolio of bonds, with respect to any change in the term structure consistentwith the underlying dynamic model. I have proved some important general propertiesof stochastic duration, and studied the concept in detail in some popular term structuremodels, both one-factor and multi-factor models. I have also demonstrated that the price


Figure 2. The accuracy of the pricing approximation for 4-month calls on 5-year bonds in the two-factor HJMmodel. The graphs show the dependency of the simulated absolute pricing errorCapp

B (t)− CB(t) on the exerciseprice for different maturities of the approximating zero-coupon bond. The stochastic duration of the underlying5-year bond isD(t) = 4.2748, and the 4-month forward price of the bond is 100.6334.

Table 3. Prices of 4-month European call options on a 5-year bond in the two-factor HJM model. Thebond has an annual coupon of 8%, a current price of 97.9788, a forward price of 100.6334, and a stochasticduration of 4.2748.X denotes the exercise price of the option,CB(t) the simulated correct price,Capp

B (t) the

approximate price computed using the closed-form expression (23), andCappB (t) the simulated approximate

price. The columns “abs. dev.” and “pct. dev.” show the absolute and percentage difference between theapproximate price and the simulated correct price, and “std. dev.” is the standard deviation of the MonteCarlo simulated differences between the approximate and the correct price.

Closed-form approx. price Simulated approx. price

X CB(t) CappB (t) abs. dev. pct. dev. Capp


95 5.74963 5.76101 1.1·10−2 0.135% 5.75247 2.8·10−3 0.049% 2.9·10−2

96 4.91829 4.93245 1.4·10−2 0.288% 4.92143 3.1·10−3 0.064% 3.3·10−2

97 4.14060 4.15706 1.6·10−2 0.398% 4.14384 3.2·10−3 0.078% 3.7·10−2

98 3.42518 3.44428 1.9·10−2 0.558% 3.42812 2.9·10−3 0.086% 4.1·10−2

99 2.78194 2.80201 2.0·10−2 0.722% 2.78431 2.4·10−3 0.085% 4.5·10−2

100 2.21521 2.23573 2.1·10−2 0.926% 2.21682 1.6·10−3 0.072% 4.9·10−2

101 1.72836 1.74791 2.0·10−2 1.131% 1.72902 0.7·10−3 0.038% 4.9·10−2

102 1.32045 1.33784 1.7·10−2 1.317% 1.32018 −0.3 · 10−3 −0.020% 4.5·10−2

103 0.98671 1.00178 1.5·10−2 1.527% 0.98563 −1.1 · 10−3 −0.110% 4.1·10−2

104 0.72196 0.73344 1.1·10−2 1.590% 0.72028 −1.7 · 10−3 −0.233% 3.7·10−2

105 0.51593 0.52481 0.9·10−2 1.721% 0.51366 −2.3 · 10−3 −0.440% 3.2·10−2

106 0.36116 0.36689 0.6·10−2 1.587% 0.35858 −2.6 · 10−3 −0.713% 2.9·10−2

178 CLAUS MUNK

of a European option on a coupon bond is very accurately approximated by a multiple ofthe price of a European option on a zero-coupon bond with a time to maturity equal to thestochastic duration of the coupon bond. While this was a well-known property of someone-factor models, I have extended the idea to multi-factor models, and I have providedanalytical arguments explaining the success of the approximation.

Typically, duration measures are used for interest rate risk management. It is clear that toobtain a perfect hedge of the model consistent interest rate shocks of a given position, allthe factor durations defined in (7) must be matched. Whether or not the simpler strategyof matching stochastic durations will provide a reasonable hedge performance is an openquestion. The success of the option pricing approximation outlined in Section 3 indicatesthat stochastic duration matching may produce reasonable hedging results.12

Practitioners often apply key rate duration numbers, as introduced by Ho (1992), repre-senting price sensitivities with respect to selected points on the yield curve. However, thekey rate duration measures are not derived within an arbitrage-free dynamic model of theevolution of the term structure and it is unclear whether they are consistent with any suchmodel. Therefore, key rate durations may be subject to the same criticism as the traditionalMacaulay and Fisher-Weil duration measures.

Acknowledgments

I am grateful to Kristian R. Miltersen for fruitful discussions and suggestions that haveimproved the paper significantly. I also thank two anonymous referees, Bent Jesper Chris-tensen, Peter Honor´e, Peter Løchte Jørgensen, Jan Nygaard Nielsen, and participants atpresentations atthe 4th Nordic Symposium on Contingent Claims Analysis in Financeand Insurancein Copenhagen, May 1998, and atDanske Bank Symposium on Securitieswith Embedded Optionsin Middelfart, Denmark, December 1998, for comments and sug-gestions. Financial support from the Danish Research Councils for Natural and SocialSciences and the School of Business and Economics at Odense University is gratefullyacknowledged.

Appendix A. Simulation of the Two-Factor HJM Model

Based on the relations (14), (15), and (16), it can be shown that

P(t∗, T) = P(t, T)

P(t, t∗)exp

{−σ1(T − t∗)

∫ t∗

tdW1(u)− 1

2σ21 (T − t∗)(T − t)(t∗ − t)

+ σ2

κ

(e−κT − e−κt∗) ∫ t∗

teκu dW2(u)

− σ 22

κ3

[1

4

(e−2κT − e−2κt∗) (e2κt∗ − e2κt

)− (

e−κT − e−κt∗) (eκt∗ − eκt)]}

,


and that∫ t∗

tr (y)dy = − ln P(t, t∗)+ 1

6σ 2

1 (t∗ − t)3+ σ1

∫ t∗

t(t∗ − u)dW1(u)

+ 2σ 22

κ3

[κ(t∗ − t)− 4

(1− e−κ(t

∗−t)/2)+ (1− e−κ(t

∗−t))]

+ 2σ2

κ

∫ t∗

t

(1− e−κ(t

∗−u)/2)

dW2(u).

Therefore, to simulate∫ t∗

t r (y)dy and B(t∗), I only have to simulate the four randomvariables

Y1 =∫ t∗

tdW1(u), Y2 =

∫ t∗

t(t∗ − u)dW1(u),

Y3 =∫ t∗

teκu dW2(u), Y4 =

∫ t∗

t

(1− e−κ(t

∗−u)/2)

dW2(u),

whereW1 and W2 are independent standard Brownian motions. Applying well-knownproperties of Itˆo-integrals, I get that theYi ’s are normally distributed with mean zero and

621 ≡ Vart (Y1) = t∗ − t,

622 ≡ Vart (Y2) = 1

3(t∗ − t)3,

623 ≡ Vart (Y3) = 1

2κ

(e2κt∗ − e2κt

),

624 ≡ Vart (Y4) = t∗ − t − 4

κ

(1− eκ(t

∗−t)/2)+ 1

κ

(1− e−κ(t

∗−t)),

612 ≡ Covt (Y1,Y2) = 12(t∗ − t)2,

634 ≡ Covt (Y3,Y4) = 1

κ

(eκt∗ − eκt

)− 2

3κeκt∗ (1− e−3κ(t∗−t)/2

),

and the remaining covariances are all zero. Hence, I can simulateYi as

Y1 = 61ε1, Y2 = (612/61)ε1+√62

2 − (612/61)2ε2,

Y3 = 63ε3, Y4 = (634/63)ε3+√62

4 − (634/63)2ε4,

whereε1, . . . , ε4 are four independent draws from a standard normal distribution. Theseare generated as

ε1 =√−2 lnU1 sin(2πU2), ε2 =

√−2 lnU1 cos(2πU2)

ε3 =√−2 lnU3 sin(2πU4), ε4 =

√−2 lnU3 cos(2πU4),

180 CLAUS MUNK

whereU1, . . . ,U4 are independent draws from a uniform(0,1) distribution. I use therandom number generator “ran1” described in Press, Teukolsky, Vetterling, and Flannery(1992, p. 280) and the antithetic variables technique. The results shown in the text are basedon 10000 (pairs of) samples of(U1, . . . ,U4).

Notes

1. Recall that a European option to enter a pay-float, receive fixed rater swap is equivalent to a European calloption on a bond with a coupon ofr and a face value equal to the notional principal of the swap, which alsois the exercise price of the coupon bond call.

2. This is to ensure that the bond price end up at par. Furthermore, the size of the factor sensitivities is expected tobe increasing in the time to maturity. Therefore, most bond price models display either positive and increasing(in T) or negative and decreasing factor sensitivities. Traditional formulations of standard interest rate basedmodels result in negative and decreasing factor sensitivities, as will become clear in the following sections.Note that the distributional properties of future bond prices are not affected by the sign of the factor sensitivitiesνk.

3. Note that the zero-coupon bond pricesP(t, ti ) and, hence, the weightsw(t, ti ) will, in general, depend onÄt ,but I will often suppress this dependency.

4. Au and Thurston (1995) define a related duration measure in one-factor HJM models as the relative pricechange of the coupon bond due to unexpected changes in the short rate. However, it is only in very specialmodels of the HJM class that the movements of the entire term structure are captured by changes in the shortrate.

5. In the Gibbons and Ramaswamy (1993) test of the CIR model, the estimated value ofβ is positive.

6. The parameterε corresponds to the parameterν in Longstaff and Schwartz’ notation. With their parameterestimates, the termb(τ )2η is a very small number, whereas the term exp{κτ } is a very large number. Rewritingthe zero-coupon bond price as

P(t, T, r,V) = a(T − t)2γ b(T − t)2η exp{γ [δ + ϕ][T − t ] + c(T − t)r + d(T − t)V} ,

where

b(τ ) = b(τ )e(ε+ψ)τ/2 = 2ψe(ε−ψ)τ/2

(ε + ψ)(1− e−ψτ )+ 2ψe−ψτ,

the multiplication of a very small and a very large number is avoided.

7. Longstaff and Schwartz estimated the value ofψ as approximately 14.4.

8. I apply a simple Euler approximation scheme to (9)–(10) with 100 time steps per year of the time to maturityof the options, enhanced by the antithetic variable technique. The prices reported are based on 10000 (pairsof) sample paths. I use the random number generator “ran1” described in Press, Teukolsky, Vetterling, andFlannery (1992, p. 280).

9. Once the routine for evaluating the bivariate non-central chi-square cumulative distribution function9 hasbeen successfully implemented, the computation using (22) is considerably faster than using Monte Carlosimulation, as documented by Chen and Scott (1992).

10. These parameter values are very close to the estimated values of Longstaff and Schwartz. They estimateα tobe approximately−0.04, but sinceα must be positive, I have takenα to be 0.01.

11. These parameter values are identical to those assumed by Wei (1997), cf. his Table II.

12. The hedging performance may be improved by also matching a convexity measuring the interest rate risk ofthe price volatility. While it is simple to define a good convexity measure for short rate models, cf. Wei (1997),it is not clear how to do it in general, possibly non-Markov, multi-factor models. Applying such a convexitymeasure in the option pricing approximation should improve the precision to some extent, but it will alsoeliminate the computational simplicity of the approximate price. Since the approximation based on durationalone is already very precise, I have not investigated this extension further.


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