Markov Interest Rate Models - Hagan and Woodward

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Markov interest rate models

P AT R I C K S . HA G A N 1 and DIANA E. WOODWARD2

1 NumeriX, 546 Fifth Avenue, 17th Floor, New York, NY 10036.2The Bank of Tokyo-Mitsubishi, Ltd., 1251 Avenue of the Americas, New York, NY 10020.

Received October 1997. Revised March 1998. Accepted October 1999.

A general procedure for creating Markovian interest rate models is presented. The models created by thprocedure automatically t within the HJM framework and t the initial term structure exactly. Therefore theare arbitrage free. Because the models created by this procedure have only one state variable per factor, twand even three-factor models can be computed efciently, without resorting to Monte Carlo techniques. Thcomputational efciency makes calibration of the new models to market prices straightforward. Extended Hull

White, extended CIR, Black Karasinski, Jamshidians Brownian path independent models, and Flesaker anHughstons rational log normal models are one-state variable models which t naturally within this theoreticframework. The separable n-factor models of Cheyette and Li, Ritchken, and Sankarasubramanian whicrequire n(n 3)=2 state variables are degenerate members of the new class of models with n(n 3)=factors. The procedure is used to create a new class of one-factor models, the bg models. These models camatch the implied volatility smiles of swaptions and caplets, and thus enable one to eliminate smile error. Thbg models are also exactly solvable in that their transition densities can be written explicitly. For these modeaccurate but not exact formulas are presented for caplet and swaption prices, and it is indicated how theclosed form expressions can be used to efciently calibrate the models to market prices.

1. Introduction

Heath, Jarrow and Morton (1990, 1992) created a broad framework for developing arbitrage-freterm structure models. In general, HJM models are non-Markovian and require extensive MonCarlo simulation to calibrate model parameters to market prices, to value contingent claims, anto determine hedges. This computational burden can be greatly reduced by using special cases othe HJM models which are Markovian. Notable among these special cases are the separable nfactor models of Cheyette (1992) and Li et al. (1995) and Ritchken and Sankarasubramania(1995). Although the separable n factor models are Markovian, they require n(n

3)=2 sta

variables, which still imposes a stiff computational burden. Moreover, to date there has been nsystematic procedure for nding HJM models which are Markovian.

Here we use the method of the undetermined numeraire to develop a general procedure focreating Markovian term structure models. The resulting models automatically t within the HJMframework and match the initial discount curve. Thus they are arbitrage free. Extended HullWhi(1990a, b), extended CIR (Cox et al., 1985), Black-Karasinski (1991), Jamshidians (1991) Browniapath independent models, and Flesaker and Hughstons rational log normal models (1996) all naturally within our theoretical framework.

Unlike the separable models, the new models require only n state variables for an n-facto

Applied Mathematical Finance 6, 233260 (1999)

Applied Mathematical Finance ISSN 1350-486X print/ISSN 1466-4313 online # 1999 Taylor & Francis Ltdhttp://www.tandf.co.uk/journals


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model. This greatly reduces their computational complexity, allowing two- and even three-factmodels to be used without requiring Monte Carlo simulations. Although we can re-create tseparable models within our framework as degenerate models with n(n 3)=2 factors and stavariables, we can also create models which are direct analogues of the separable models and requionly n factors and state variables. These analogues are asymptotically equivalent to their separabcounterparts in the limit of small volatilities, and, at normal market volatilities, the two sets models should give essentially identical results.

We use our procedure to create a new class of one factor models, the bg models. Unlike othone-factor models, these models can match the implied volatility smiles of swaptions and capleand thus enable us to largely eliminate smile error from our books. The bg models are also exactsolvable in that their transition densities (Greens functions) can be written exactly. For these modewe present accurate but not exact formulas for caplet and swaption prices, and indicate hothese closed form expressions can be used to efciently calibrate the models to exact market price

2. Derivation

Consider a continuous trading economy on the interval [0, Tmax]. To x notation, let f0(T)todays instantaneous forward rate for date T for this economy. Then the current value of a zecoupon bond which pays $1 at maturity T is

D(0, T) e-

T0

f0(T9) dT9 (2:

and the current discount factor from time t to T is D(t, T) D(0, T)=D(0, t).To model this economy, recall that arbitrage-free term structure models have three essenti

elements. First is a set of stochastic processes which drive the evolution of interest rates. S econd a numeraire, a tradable instrument with positive value and no cash ows. Commonly the monmarket account or a specic zero coupon bond is used as the numeraire. The nal element isvaluation formula which states that, in the absence of any cash ows, the value of any tradabinstrument1 is a Martingale when expressed in units of the numeraire (Harrison and Pliska, 198Harrison and Kreps, 1979).

In a Markovian model, the entire term structure at any time t must be determined solely by tvalues X1, X2, . . . , Xn of a nite set of random state variables X1(t), X2(t), . . . , Xn(t). We assumthat these state variables evolve according to the Ito processes

dXi(t) li(t, X) dt ri(t, X) dbWi(t), Xi(0) 0, i 1, 2, . . . , n (2:2Here bW1(t), bW2(t), . . . , bWn(t) represent n Brownian motions, which can be correlated

dbWi(t) dbWj(t) rij(t) dt (2:2To obtain a Markovian model, we need to choose a numeraire whose value N depends only on th

values x1, x2, . . . , xn of the state variables : N eH( t,x). We stress that at this point, the functio

1 The term tradable instruments is used to refer to both fundamental instruments that are actively traded and all derivativ

and synthetic instrument that can be created.

234 Hagan and Woodwa


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H(t, x) is arbitrary and we do not need to know which nancial instrument is represented by thnumeraire. Simply requiring N(t, x) to be a market instrument (with no cash ows) will enable uto derive all interest rates.

To express the valuation formula, consider a tradable instrument which has the value V(T, X(Tat some date T and pays a cash ow C(t, X(t)). The value of the instrument expressed in units othe numeraire is V(t, x)=N(t, x) V(t, x) e- H( t,x). Requiring V(t, x)=N(t, x) to be a Martingale ithe absence of any cash ows then yields

V(t, x) eH( t,x)bE V(T, X(T)) e- H(T,X(T)) Tt

C(t9, X(t9)) e- H( t9,X( t9)) dt9 X(t) x( )

,

(2:3

where x X(t) is the state of the economy at time t. Equation 2.3 gives the value of thinstrument at any time t, T. Note that bE is the expected value under the probability measuredetermined by Equation 2.2.

Without loss of generality, we re-write the numeraire as

N(t, x) 1D(0, t)

eh( t,x)A( t) (2:4a

where

h(t, 0) 0, A(0) 0 (2:4bThen the valuation formula becomes

V(t, x) e h( t,x)A( t)

bE V(T, X(T))D(t, T) e- h(T,X(T))- A(T)

T

t

C(t9, X(t9))D(t, t9) e- h(t9,X( t9))- A( t9) dt9 X(t) x (2:5

The valuation formula (2.5), with the state variables X(t) determined by the stochastic proce(2.2), denes a term structure model. As we shall see, if this model is consistent with the initidiscount curve then it is arbitrage free. In particular, this model automatically satises the cruciforward rate drift restriction of HJM (1992) theory.3 We stress that apart from some mathematicregularity conditions, the drift rates li(t, x), the volatilities ri(t, x), and the function h(t, x) aarbitrary. The remaining function A(t) will be determined by requiring the valuation formula (2.5) t

be consistent with the initial discount curve.To impose consistency, dene Z(t, x; T) to be the value at t, x of a zero coupon bond which pay$1 at maturity T. From (2.5),

2 Equation 2.2 thus denes the risk neutral probability measure induced by the numeraire N(t, x), and does not represent reworld probabilities.3 This is not surprising: (2.2) and (2.5) self-consistently dene the probability measure of the risk neutral-world underspecic (if unknown) numeraire N( t, x); this then uniquely denes the risk neutral probability measure under every othnumeraire, including the money market numeraire; and HJM (1992) showed that for any self-consistent risk-neutral world, thinstantaneous forward rates must satisfy the forward drift restriction under the probability measure induced by the mon

market numeraire.

Markov interest rate models 23


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Z(t, x; T) D(t, T) eh( t,x)A( t)- A(T)S(t, x; T) (2:6where

S(t, x; T)

bEfe- h(T,X(T)) j X(t) xg (2:6

As we are taking X(0)

0, consistency with the initial discount curve requires Z(0, 0; T)

D(0, T). Since h(t, 0) A(0) 0, this requirement becomesA(T) log S(0, 0; T) logbEfe- h(T,X(T)) j X(0) 0g for all T.0 (2:

Thus consistency with the initial discount curve requires choosing A(T) so that the expectvalue of e- h(T,X(T))- A(T) is 1 for all T.0.

Note that the current discount factors D(0, T) do not appear in (2.7); choosing A(T) so that thmodel is consistent with the initial term structure is independent of the actual initial term structurOnce A(T) has been selected according to (2.7), then if the initial term structure changes, all wneed do is replace the discount factors D(t, T) with the new discount factors, and the model will bconsistent with the new initial term structure. This is useful during calibration since it separates thvolatility functions ri(t, X), li(t, X), and h(t, X) endogenous to the model from the currediscount factors D(t, T), which are exogenous.

With the economy in state X(t) x at date t, the instantaneous forward rates f(t, x; T) adened implicitly by Z(t, x; T) expf-

Tt

f(t, x; T9) dT9g. From (2.6),Tt

f(t, x; T9) dT9 T

t

f0(T9) dT9 - h(t, x) - A(t) A(T) - log S(t, x; T) (2:

Hence the forward rate curve at t, x is

f(t, x; T)

f0(T)

A9(T) - ST(t, x; T)=S(t, x; T) (2:

where we are using subscripts to denote partial derivatives. The short rate r(t, x) is dened f(t, x; t), so we have

r(t, x) f0(t) A9(t) - ST(t, x; t)=S(t, x; t) (2:10In Appendix A it is shown that this expression for the short rate is equivalent to

r(t, x) f0(t) A9(t) h t(t, x)X

i

li(t, x)hxi (t, x)

1

2Xij

rij(t)ri(t, x)rj(t, x)[hxixj (t, x)-

hxi (t, x)hxj (t, x)] (2:10

With A(t) dened by (2.7), the valuation formula (2.5) and random processes (2.2) uniquedene a term structure model which is consistent with the initial discount curve. The instantaneoforward interest rates and short rate for this model are given by (2.9) and (2.10), respectively. At thpoint it would be natural to use Martingale theory (Harrison and Pliska, 1981; Harrison and Krep1979) to show that this model is arbitrage free. Instead, in Appendix A we show that this model within the HJM framework. There it is found that using the money market as a numeraire, tprocess for the instantaneous forward rate, F(t, T) f(t, X(t); T), satises



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dF(t, T) X

ij

rij(t)aiT(t, X; T)a

j(t, X; T) dt -X

i

aiT(t, X; T) deWi(t) (2:11a

in the risk-neutral probability world, with

a i(t, x; T)

ri(t, x)

Sxi ( t, x; T)

S(t,x

; T)-

Sxi (t, x; t)

S(t,x

; t)i

1, 2, . . . , n: (2:11b

Inspection shows that the drift terms in (2.11a) satisfy the forward drift restriction of HJM(1992) theory. Consequently, any term structure model given by (2.5), (2.2), and (2.7) ts withithe HJM framework, and provided it satises innocuous regularity conditions, the model arbitrage free.4

Equation 2.5 can be used to directly value any contingent claim whose payoff V(T, X(T)) ancash ow C(t, X(t)) can be expressed as explicit functions of the state variables. Since (2.9expresses the entire forward rate curve f(t, x; T) explicitly, this includes all instruments whopayoff and cash ow depend only on the then current yield curve. This includes EuropeaBermudan, and American swaptions, caps, and most yield curve options. However, there is nguarantee that instruments whose payoff or cash ow are path dependent (e.g. instruments wipayments determined by the average short rate over the preceding period) can be evaluated withouadding extra state variables or resorting to path dependent valuation techniques.

3. One factor models

We now use the general framework for arbitrage-free Markovian models developed in Section to create a special class of one factor models. In S ection 4 we will further specialize this cla

of models to obtain the bg models.Although one-factor models are usually expressed in terms of the money market numeraire usin

the short rate r as the state variable, any such short rate model can be re-cast within thframework of Section 2 by using a zero coupon bond as the numeraire. Conversely, any one-factomodel within the framework of Section 2 can be transformed into a short rate model. Thestransformations are carried out in Appendix B.

Within the framework of Section 2, a one-factor model is dened by the stochastic process for thstate variable,

dX(t)

l(t, X) dt

r(t, X) dbW(t) X(0) 0 (3:1and the numeraire

N(t, x) 1D(0, t)

eh( t,x)A( t) (3:2

which creates the link between the state variable and nancial markets.

4 We do not restate these conditions here; we refer the reader to HJM (1992). Although these conditions are innocuomathematically, each points out a more stringent business condition. For example, the mathematical requirement mark

completeness points out that risks must be hedgeable with liquid instruments with minimal transaction costs.



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We argue that with our current knowledge about interest rates, we are not justied in using bothcomplicated function h(t, x) in the numeraire and a complicated stochastic process for X(t). Insteawe should choose either a relatively simple function h(t, x) or a relatively simple stochastic procesWe arbitrarily choose to use a simple numeraire

N(t, x)

1

D(0, t)

ek( t)xA( t) (3:

for our model. The consistency condition (2.7) then becomes

A(t) logbEfe-k( t)X( t) j X(0) 0g (3:Now consider the drift term, l(t, X) dt. Suppose the model had, say, a positive drift so that X

increased on average. Then A(t) would be negative and would effectively cancel out the averaeffect of the drift term in the numeraire. Similarly, if the drift term was negative, A(t) would positive, again canceling out most of the drift. Indeed (3.4) can be written as

bEfe-k( t)X(t)- A( t) j X(0) 0g 1 (3:Since A(t) cancels out the main effects of the drift term, any residual inuence of the drift terl(t, X) dt on the model would be quite subtle. Therefore, in the interests of simplicity, we taour model to be driftless.

So consider the special class of one factor models

dX(t) r(t, X) dbW(t) X(0) 0 (3:with the numeraire (3.3). As shown in Appendix C, this class of models includes the extendHullWhite, the extended CIR, and BlackKarasinski like models. However, the BlackKarasinski model itself cannot be included without adding a drift term to (3.6).

With the numeraire (3.3), the valuation formula becomes

V(t, x) ek( t)xA( t)bE V(T, X(T))D(t, T) e- k(T)X(T)- A(T)T

t

C(t9, X(t9))D(t, t9) e- k(t9)X( t9)- A( t9) dt9 X(t) x (3:

so the value of a zero coupon bond becomes

Z(t, x; T)

D(t, T) e- [k(T)-k( t)]x- [A(T)- A(t)]M( t,x;T) (3:

Here we have dened

M(t, x; T) logbEfe- k(T)[X(T)-x] j X(t) xg k(T)x log S(t, x, T) (3:9and the consistency condition is now

A(T) M(0, 0; T) for all T (3:9It is instructive to examine the forward rate curve. Since Z(t, x; T) expf- T

tf(t, x; T9)dT9

Equation 3.8 shows that if the economy is in state X(t) at time t, then the instantaneous forwarate for date T would be



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f(t, X(t); T) f0(T) k9(T)X A9(T) - MT(t, X; T) (3:10aThe corresponding short rate would be

r(t, X(t)) f0(t) k9(t)X A9(t) - MT(t, X; t) (3:10bFor the moment, let us neglect the last two terms in (3.10a) and (3.10b). Then the forward ra

curve is just the curve k9(T)X added to the original forward curve f0(T). Note that k9(T) is gearing factor: if the state variable X(t) is shifted by an amount D, then the short rate would shiby dr(t) k9(t)D and the forward rates f(t, X; T) would shift by df k9(T)D k9(T)dr=k9(tSince we expect shocks to affect short maturities more than longer maturities, we expect k9(T) to ba decreasing function of T.

Note that Equation 3.6 implies that X(t) is a Martingale, so for all times T. t, the expectevalue of X(T) is just X(t). Thus, if X(t) was shifted by an amount D, then for all later times T thexpected value of X(T) would also shift by D. Consequently, if a shock shifted the short rate bdr(t) k9(t)D, then for all later times T the expected value of the short rate r(T, X(T)) would shiby k9(T)dr=k9(t). As T increases, the expected value of the short rate returns to the initial forwarcurve f0(T) at a rate determined by how rapidly k9(T) decreases. So mean reversion of interest rateis directly determined by the function k(T).

With f(t, X; T) f0(T) k9(T)X . . . , the distribution of forward rates is determined by thdistribution of X(t). The width of this distribution is determined mainly by the overall magnitude or(t, x), and the shape of this distribution (i.e. the deviation from Gaussian or log normal behaviois determined mainly by the functional form of the dependence of r(t, x) on x. As noted abovmean reversion of interest rates is determined mainly by the function k(T). Since skews and smile(the changes in an options implied volatility as the strike changes) depend mainly on the shape othe distribution, and since the change in the implied volatility as the duration of the underlyin

instrument changes depends mainly on mean reversion, this separation makes it easy to calibrathese models to both at-the-money and off-market instruments of differing durations. This makethese models very useful for pricing, hedging, and understanding instruments in the presence oimplied volatility skews and smiles.

The third term A9(T) in (3.10) embodies the consistency requirement, and is much smaller thathe rst two terms. Even though (3.6) implies that X(t) has mean zero, the convex relation betweebond prices and interest rates would cause bond prices to drift, on average, as the distribution oX(t) spreads out. The term A9(T) cancels the expected value of this convexity effect. This is whthe consistency term A(T) depends on the functions k(T) and r(t, x), but not on the initial termstructure f0(T). The last term MT(t, x; T) arises because as X(t) evolves away from X(0)

0, th

expected value of the convexity effect changes.We can clarify the relation between the distribution of X(T) and the convexity terms A(T) an

M(t, x; T) by expanding (3.9) in powers of k. Dene the moments

m k(t, x; T) bEf[X(T) - x]k j X(t) xg k 1, 2, . . . (3:11and note that m1(t, x; T) 0. Expanding (3.9) yields

M(t, x; T) 12k2(T)m2(t, x; T) - 16k3(T)m3(t, x; T) 124k4(T)[m4(t, x; T) - 3m22(t, x; T)] .(3:12a



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A(T) 12k2(T)m2(0, 0; T) - 16k3(T)m3(0, 0; T) 124k4(T)[m4(0, 0; T) - 3m22(0, 0; T)] . .(3:12

Our experience calibrating one-factor models to US swaption and caplet markets shows that tmagnitude of the quadratic term 12k

2(T)m2(t, x; T) is roughly (T - t)T23 10- 4, where time

measured in years. The cubic and quartic terms, which arise from the skewness and kurtosis of th

distribution, are much smaller and have magnitudes of roughly (T-

t)2T3 3 10-8 a(T - t)2T4 3 10-9, respectively.5 We conclude that it is safe to truncate after the quadratic terexcept for unusually sensitive instruments, unusually long durations, or unusually competitimarkets.

Truncating after the quadratic terms yields the forward rates

f(t, x; T) f0(T) k9(T)x 12

@

@Tfk2(T)[m2(0, 0; T) - m2(t, x; T)]g . . . (3:1

Consequently, the expected forward rate is

bEff(t, X(t); T) j X(0) 0g f0(T) k9(T)k(T)m2(0, 0; t) . . . (3:14Note that m2(0, 0; t) is the variance of X(t), not X(T). The covariance between forward rates different maturities is

Covff(t, X(t); T1), f(t, X(t); T2) j X(0) 0g k9(T1)k9(T2)m2(0, 0; t) . . . (3:14showing that the two rates are perfectly correlated through this order.

Before continuing, let us briey address the unknown numeraire in (3.2). By constructioN(t, x) is always a valid numeraire.6 Suppose we arbitrarily set k(Tref ) 0 for some date Tref. Th(3.9) would imply that A(Tref ) 0 and M(t, x; Tref ) 0, so the value of the zero coupon bond maturity T

refwould be D(t, T

ref) ek(t)

A( t) (see (3.8)). So if we made this choice, then the unknow

numeraire N(t, x) would represent a zero coupon bond paying 1=D(0, Tref ) dollars at maturity Tr at least for times t< Tref. Although N(t, x) would remain a valid numeraire for times t. Tref,would no longer represent any simple security. In the same spirit, we could restrict the k(t) curve that k(t)!0 as t!1; then the numeraire would correspond to Flesaker and Hughstons (199absolute terminal measure .

4. The bgmodels

We now specialize to volatilities of the form r(t, x) a(t)[1 bX(t)]g, where b and g aconstant, and dene the bg models by

dX(t) a(t)[1 bX(t)]g dbW(t) X(0) 0 (4:15 This is equivalent to the change in interest rates over a period Dt having a standard deviation of sO(100 bps [Dt=yrs]1=2 ), variance of size O(10-4Dt=yrs), kurtosis of size O(1), and skewness of size O(10-2[Dt=yrs]1=2 )6 Substituting V( t, x) N( t, x) and C(t, x) 0 into the valuation formula (2.5) shows that for any Tfinal . 0, N(t, x) is t

value of the interest rate option which has no cash ow and has the payoff N(Tfinal, X(Tfinal )) at Tfinal.



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with the numeraire

N(t, x) 1D(0, t)

ek( t)xA( t) (4:1b

Without loss of generality, we normalize k(t) so that k9(0) 1.A key advantage ofbg models is that there is a closed form solution for the transition densit

p(t, x; t, x ) dx Probfx , X( t), x dx j X(t) xg (4:2The transition density p satises the backward equation

pt 12a2(t)j1 bxj2gpxx 0 t, t (4:3ap !d(x - x ) as t! t (4:3b

In terms of the new variables

s(t)

t

0

a2(t9) dt9 y(x)

j1 bxj1-g

b(1-

g)

(4:4

this becomes

ps 12

(pyy -2m- 1

ypy ) 0 s, s (4:5a

p ![b(1 - g)y ]-2m1d(y - y ) as s! s (4:5bwith m 1=(2 - 2g). In particular, s(t) is roughly the variance of X(t), which is around 10- 4 t iUS dollar markets, where t is measured in years.

Equation 4.5 can be solved by a combination of Laplace transform and Greens functio

techniques. For exponents 0,g,1=2 this yields

p(t, x; t, x ) (y= y )m ys - s

f12I-m(y y =(s - s)) 12Im(y y=(s - s))g e- ( y2y2)=2( s-s) dy

dx

for x . -1=b (4:6a

p(t, x; t, x ) sinpmp

(y= y )my

s - sKm(y y =(s - s)) e

- ( y 2y2)=2( s- s) dydx

for x , -1=b

(4:6b

Here

dydx

[(1 - g)by ]- 2m1 (4:7

and Km and I m are the modied Bessel functions.The bg models have an articial barrier at x -1=b, where the volatility j1 bxjg goes t

zero. For exponents g,1=2, the volatility does not decay fast enough as y !0 to prevent y frombecoming negative. If we wished to avoid negative values of y, we could put a reecting boundarcondition at y 0; this would eliminate (4.6b) and change the Bessel functions in (4.6a) from12(I-m Im) to I-m.

The probability of y approaching zero is extremely small, of order e-1=b2(1-g)2s. This is much to



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small to affect the pricing of realistic products. Since b and g can be chosen to reproduce tvolatility smile reasonably well, indicating that the model is reproducing the right shape of the risneutral probability distribution near the distributions centre, we argue that the benet of increasaccuracy near the distributions centre far outweighs the liability of having an articial barrier in thextreme tails of the distribution. In fact, we do not expect any model to be valid in the extreme taof the distribution where there is no relevant market experience.

For exponents 1=2 < g, 1 we have

p(t, x; t, x ) (y= y )m ys - s

Im(y y =(s - s)) e- ( y 2y2 )=2( s- s) dy

dx C(m, y

2=2(s - s))

C(m)d(x 1=

(4:

where C(m, h) is the incomplete gamma function. For these exponents the volatility declines faenough to prevent y from becoming negative, but it doesnt prevent y from reaching the origiThis causes a very small, but nite, probability of y being exactly zero. As before, thprobability is much too small to affect the pricing of realistic deals.

For both sets of exponents, the transition probability can be expanded as

p(t, x; t, x ) (y= y )m- 1=2

[2p(s - s)]1=2e- ( y - y)

2=2( s- s) dy

dx1 -

4m2 - 1

8y y(s - s) . . .

( )(4:

for small values of the variance s - s.Finally, for exponent g 1 we need to re-dene y as

y(x) b-1 log (1 bx) (4:10Then the transition density is simply

p(t, x; t, x ) e-by

[2p(s - s)]1=2e- ( y - yb( s- s)=2)

2=2(s- s) (4:10

With the transition density in hand, M(t, x; T) can be found by integration,

M(t, x; T) log

p(t, x; T, x ) e-k(T)(x - x) dx (4:11

Close inspection of (4.5) and (4.11a) reveals that M(t, x; T) is of the form

M(t, x; T) M(S, H) (4:11

where

S b2(s - s)=(1 bx)2(1-g) H k(T)(1 bx)=b (4:11This is useful since for each exponent g, the function M(S, H) can be calculated and tabulatonce, eliminating most of the computational burden of the bg model. Once these tables habeen calculated, the values Z(t, x; T) of zero coupons bonds, the forward rate curve f(t, x; Tand the short rate r(t, x) can be obtained directly from (3.8) (3.10).

Three exponents merit special attention for their simplicity: g 0, g 1=2, and g 1. Appendix C we analyse the bg model for these exponents, allowing b to depend on time:



11/28

dX(t) a(t)[1 b(t)X(t)]g dbW(t) (4:12Here we quote only the results with b constant.

When g 0, the bg model becomes the extended HullWhite model. In this case A(T) anM(t, x; T) are given by

A(t)

1

2

k2(T)s M(t, x; T)

1

2

k2(T)(s - s) (4:13

where

sT

0a2(t9) dt9 s

t0a2(t9) dt9 (4:14

Equations 3.83.10 now yield explicit expressions for the value of zero coupon bonds, thforward rate curve, and the short rate.

Similarly, A(T) and M(t, x; T) can be written explicitly when g 1=2,

A(T)

k2(T)s

2 bk(T)sM(t, x; T)

(1 bx)k2(T)(s - s)

2 bk(T)(s-s)

(4:15

where s and s are given by (4.14). This again enables Z(t, x; T), f(t, x; T) and r(t, x) to bobtained from (3.8)(3.10). Besides deriving the corresponding formulas when b b(t), iAppendix C we nd that b(t) can be chosen so that this model is exactly the CIR modeAlternatively, one can use the extra freedom of choosing b(t) to calibrate the model to a serieof off-market instruments as well as a series of at-the-money instruments, and thus reproduce thvolatility smile.

The exponent g 1 (Black Karasinski like models) is more difcult. Although we have beeunable to derive explicit expressions for A(T) and M(t, x; T), the moments m k(t, x; T) can b

written explicitly. When b is constant,m2(t, x; T) [(1 bx)=b]2[eb2( s- s) - 1] (4:16am3(t, x; T) [(1 bx)=b]3[e3b2( s- s) - 3 eb2( s- s) 2] (4:16bm4(t, x; T) [(1 bx)=b]4[e6b2( s- s) - 4 e3b2( s- s) 6 eb2( s- s) - 3] (4:16c

We can use the rst few terms in expansion (3.12) for M(t, x; T) and A(T), and substitute thesexpansions into (3.8) and (3.10) to obtain Z(t, x; T), f(t, x; T), and r(t, x). Even if only thsecond moment is used, this should be accurate enough to price all but the most sensitivinstruments.

4.1. European option prices

A vanilla receiver swaption is a European option which gives the holder the right to receive predetermined series of cash payments Ci on dates ti, i 1, 2, . . . , n, in return for essentiallypaying K on the settlement date ts. A payor swaption is a European option which gives th

7 We are assuming that the oating leg re-values to K at the settlement date.



12/28

holder the right to receive the strike K at settlement in return for making a predetermined seriof payments. Similarly, a caplet or oorlet is a European option which gives the holder the rigto exchange specic payments with the issuer.

Consider a receiver swaption with exercise date te. Since N(0, 0) 1, the value of the optiotoday is

V(0, 0) D(0, te ) p(0, 0; te, xe )Q(te, xe ) e-k( te )xe- A( te ) dxe (4:17Here

Q(te, xe ) X

i

CiZ(te, xe ; ti ) - KZ(te, xe ; ts )

( )(4:17

is the value of the payoff if the economy is in state X(te ) xe at expiry. Substituting (3.8) fthe zero coupon bonds gives

V(0, 0) p(0, 0; te, xe ) Xi

CiD(0, ti ) e- kixe- AiM(te,xe ;ti )(

- KD(0, ts ) e- ksxe- AsM( te ,xe ; ts )

dxe (4:1

where ki k(ti ), ks k(ts ), etc. Since the transition densities p(0, 0; te, xe ) are known and tfunction M(t, x; T) is pre-tabulated, these options can be valued exactly with a single integratio

Let F0 be todays forward value of the cash ow at settlement,

F0 Xi

CiD(ts, ti ) (4:1

The implied price vol of the option8 is dened as the volatility rB for which Blacks formula,

V0 D(0, ts )[F0N(d1) - KN(d2)] (4:20

d1,2 log F0=K

12r

2B te

rB

tp

e

(4:20

yields the correct price V0 given by (4.18).

In Hagan and Woodward (in preparation) singular perturbation techniques are used to obtaaccurate approximations to the option price (4.18), with the results stated in terms of implied privols. There it is found that to leading order the option price is

rB 1te

te0a2(t9) dt9

1=2

K1(1 bx0)g . . . (4:21

where x0 is dened implicitly by

8 The price vol dened here should not be confused with the more commonly quoted rate vol.



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12(F0 K)

Xi

CiD(ts, ti ) e- (ki-ks )x0 (4:21b

Here

K1 Pi CiD(ts, ti )(ki - ks ) e

-kix0

P iCiD(ts, ti ) e- kix0 (4:21cA much more accurate formula is quoted in Appendix F.

Equation 4.21a illustrates the roles played by local volatility, mean reversion, and skew ipricing. The local volatility a(t) only enters (4.21a), which shows that the implied price vol rBproportional to the mean-square-average of the local volatility between today and the expiratiodate.

Recall that k9(t) is a decreasing function, and that mean reversion refers to how rapidly k9(decreases (see (3.10) et seq). As mean reversion increases, k(ti ) - k(ts ) ki - ks decreasewhich decreases the value of K1. Thus, as mean reversion increases, the price vol rB of thswaption decreases, as expected. Increasing the mean reversion also increases the importance o

the earlier payments Ci relative to the later payments.Equations 4.19 and 4.21b imply that x0 0 when the option is struck at-the-money (K F0), an

that x0 is a decreasing function of the strike K. Thus, the price of an at-the-money swaption independent of band g, at least to within the approximations used to obtain (4.21). Ifbor g is zerothe price vol is independent of the strike K; as b and g increase from zero, the implied price vobecomes a decreasing function of the strike K.

Besides giving insight into how the model parameters affect option prices, Equation 4.21 cabe used effectively in model calibration. Calibration is the process of choosing the modparameters a(t) and k(t) to minimize or eliminate the errors between the models predicte

prices and the actual prices of a set of standard market instruments. This is typically aiterative process, with most of the computational time spent calculating derivatives of thpredicted prices with respect to variations in the model parameters. By using the approximaformula (4.21) to obtain a shrewd initial guess for the model parameters, and then using thderivatives of (4.21) in place of the actual derivatives, one can greatly speed up the calibratioprocess.

5. Conclusions.

We used HJM theory to develop a framework for creating arbitrage-free Markovian term structurmodels. This framework makes designing arbitrage-free interest rate models which match thinitial discount curve a straightforward task, freeing the designer to concentrate on imbuing thmodel with less fundamental attributes. Not only are the resulting models Markovian, but thehave no more state variables than are needed to handle the random factors. This parsimony openup new methods for evaluating and using multi-factor models: trees, explicit and implicit nidifference schemes, and Greens function techniques can be used as well as Monte Carlmethods.

All arbitrage-free Markovian term structure models we are aware of t within this framewor



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Appendices B and C explicitly demonstrate that the extended HullWhite (1990a, b), extended CI(Cox et al. 1985), Black Karasinski (1991), and Jamshidians (1991) Brownian path independemodels t within this framework, and Appendix D shows that Flesaker and Hughstons (199rational interest rate models also t within the framework. The separable n-factor models are modifcult since they require n(n 3)=2 state variables. Appendix E shows that these models can constructed as degenerate n(n

3)=2 factor models. There we also create n-factor models which a

analogues of the separable models and are asymptotically equivalent to their separable counterparin the limit of small volatilities.

In Section 4 we used this framework to create the bg models, a class of one-factor modedened by the process

dX(t) a(t)[1 bX(t)]g dbW(t) X(0) 0 (5:and the numeraire

N(t, x)

1

D(0, t)

ek( t)xA( t) (5:

These models enable us to account for local volatility, mean reversion, and skew, and allow boat-the-money and off-market instruments of varying maturities to be priced simultaneously. Theanalytic tractability is an added benet.

Describing local volatility, mean reversion, and skew is as much as can be expected from a onfactor model. However, some products are sensitive to risks which can not be described by a onfactor model. The two most common risks are sensitivity to rate decorrelation/curve exing aforward/stochastic volatilities. Decorrelation risk occurs because forward rates become progressiveless correlated as the difference between the maturity dates (and start dates) increases; th

decorrelation tends to make the forward rate curve ex. In contrast, all forward rates in a onfactor model are essentially perfectly correlated. Forward volatility risk occurs in deals whicontain an option on an option. Examples are captions (options on caps), and Bermudan aAmerican swaptions. In each case exercising the option involves either receiving or giving up option, so the exercise decision must balance the intrinsic value of the payoff against the value the option received or lost, which is determined by the volatility environment that pertains at texercise date.

The theoretical framework makes it easy to extend popular one-factor models to include therisks. For example, we can convert the bg model to a stochastic volatility model by changing tequation for the state variable to

dX(t) a(t)[1 b(t)X(t)]gY(t) dbW1(t) X(0) 0 (5:3dY(t) c(t)Y(t) dbW2(t) Y(0) 1 (5:3

and using the same numeraire as before,

N(t, x) 1D(0, t)

ek( t)xA( t) (5:3

Finally, note that the stochastic differential equations for the state variables play only a peripher



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role in our theoretical framework; one could choose to directly model the risk neutral transitiodensities p(t, x; T; X), dispensing with these equations entirely.

Acknowledgements

We gratefully acknowledge the assistance of A. Berner, Olivier Van Eyseren, our colleagues oParibas New York Swaps desk and Paribas London FIRST team. The views presented in threport do not necessarily reect the views of NumeriX, The Bank of Tokyo-Mitsubishi, or any otheir afliates.

References

Black, F. and Karasinski, P. (1991) Bond and option pricing when short rates are lognormal, FinanciAnalysts Journal, 46 5259.

Cheyette, O. (1992) Term structure dynamics and mortgage valuation, Journal of Fixed Income 2841.Cox, J.C., Ingersoll, J.E. and Ross, S.A. (1985) A theory of the term structure of interest rates, Economics 5

385 407.Flesaker, B. and Hughston, L.P. (1996) Positive interest, in Vasicek and Beyond, Lane Hughston (ed.), (Ri

Publications, London).Hagan, P.S. and Woodward, D.E. Swaption and caplet prices for one factor models, in preparation.Harrison, J.M. and Kreps, D. (1979) Martingales and arbitrage in multiperiod securities markets, Journal

Economic Stochastic Processes and Theory, 20 381408.

Harrison, J.M. and Pliska, S.R. (1981) Martingales and stochastic integrals in the theory of continuous tradinStochastic Processes and Applications, 11 215260.

Heath, D., Jarrow. R. and Morton, A. (1990) Bond pricing and the term structure of interest rates: A discretime approximation, Journal of Finance and Quantitative Analysis, 25 419440.

Heath, D., Jarrow. R. and Morton, A. (1992) Bond pricing and the term structure of interest rates: A nemethodology, Econometrica, 60 77105.

Ho, T.S.Y. and Lee, S.B. (1986) Term structure movements and pricing interest rate contingent claims, Journof Finance, 41 10111028.

Hull, J.C. (1997) Options, Futures, and Other Derivatives (Prentice-Hall, Upper Saddle River).Hull, J. and White, A. (1990a) Pricing interest rate derivative securities, Revue of Financial Studies, 3 57359Hull, J. and White, A. (1990b) Efcient procedures for valuing European and American path-depende

securities, Journal of Finance and Quantitative Analysis, 25 87100. Jamshidian. F. (1991) Forward induction and constructin of yield curve diffusion models, Journal of Fixe

Income, 6274.Li, A., Ritchken, P. and Sankarasubramanian, L. (1995) Lattice models for pricing American interest ra

claims, Journal of Finance, 50 719737.Longstaff, F.A. and Schwartz, E.S. (1992) Interest rate volatility and the term structure: a two-factor gener

equilibrium model, Journal of Finance, 47 1259 1282.Rebonato, R. (1996) Interest-Rate Option Models (Wiley, New York).Ritchken, P. and Sankarasubramanian, L. (1995) Volatility structures of forward rates and the dynamics of th

term structure, Mathematical Finance, 5 5572.



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Appendix

A Connection with HJM

Here we show that the random process followed by the forward rate

F(t; T) f(t, X(t); T) f0(T) A9(T)-

ST(t, X(t); T)=S(t, X(t); T) (A:

satises the forward rate drift restriction of HJM.The backward Kolmogorov equation for (2.6b) implies that S(t, x; T) satises the PDE

S tX

i

li(t, x)Sxi 1

2

Xij

rij(t)ri(t, x)rj(t, x)Sxixj 0 0, t, T (A:2

S(T, x; T) e- h(T,x) (A:2Applying Itos lemma to (A.1), and using (A.2) to simplify the result, then yields

dF(t; T) 12X

ij

rij(t)ri(t, X)rj(t, X)(Sxi Sxj=S2)T dt- X

i

ri(t, X)(Sxi=S )T dbWi(t) (A:Equation A.3 appears to violate the forward drift restriction of HJM (1992) theory. However, th

restriction is phrased in terms of the risk-neutral probability measure in the money marknumeraire . To switch to the money market numeraire, we apply the backwards Kolmogorov equatioto the valuation formula

V(t, x) e h( t,x)A( t)

bE V(T, X(T))D(t, T) e- h(T,X(T))- A(T)

Tt

C(t9, X(t9))D(t, t9) e- h(t9,X( t9))- A( t9) dt9 X(t) x (A:

This shows that the value of a marketable instrument expressed in units of the numeraire,

bV(t, x) V(t, x) e- H( t,x) V(t, x) e- t0 f0(T9) dT9- h( t,x)- A( t) (A:5satises the partial differential equation

bVt

Xili(t, x)

bVxi

1

2Xijrij(t)ri(t, x)rj(t, x)

bVxixj - C(t, x) e-

t

0f0(T9) dT9- h( t,x)- A( t)

(A:5

Consequently, V(t, x) satises the partial differential equation

VtX

i

~li(t, x)Vxi 1

2

Xij

rij(t)ri(t, x)rj(t, x)Vxixj - r(t, x)V -C(t, x) (A:6

where

~li(t, x) li(t, x) - ri(t, x)X

j

rij(t)rj(t, x)hxj (t, x) (A:6



17/28

and

r(t, x) f0(t) A9(t) h t(t, x)X

i

li(t, x)hxi (t, x)

1

2Xij rij(t)ri(t, x)rj(t, x)[hxixj (t, x) - hxi (t, x)hxj (t, x)]: (A:6cExpression (A.6c) for the short rate appears to be different from (2.10a). We can show that thes

two expressions are equivalent by applying Itos lemma to e- h(T,X(T)) in (2.6b), taking the expectevalue, and letting T! t. This yields

-

ST(t, x; t)

S(t, x; t) h t

Xi

li(t, x)hxi 1

2

Xij

rij(t)ri(t, x)rj(t, x)[hxixj - hxi hxj ] (A:7a

as is needed for (A.6c) to agree with (2.10a). For future reference note that

-

Sxi

(t,x

; t)

S(t, x; t) hxi (t, x) (A:7bDene the money market numeraire B(t) by

dB(t)

B(t) r(t, X(t)) dt (A:8

Inspection of (A.6a) reveals that it is the backwards Kolmogorov equation for

V(t, x) eE

B(t)

B(T)V(T, X(T))

T

t

B(t)

B(t9)C(t9, X(t9)) dt9 X(t) x

( )(A:9

where eE refers to the expected value in a probability measure for whichdXi(t) ~li(t, X(t)) dt ri(t, X(t)) deWi(t) i 1, 2, . . . , (A:10a

Here eW1(t), eW2(t), . . . , eWn(t) are Brownian motions with the same correlation structure abefore:

d eWi(t) d eWj(t) rij(t) dt (A:10bClearly, (A.9) is the valuation formula using the money market as the numeraire.

Comparing (A.10) with (2.2) shows that changing from N(t,x

) to the money market numeraire equivalent to using Girsanovs theorem to change the probability measure so that

d eWi(t) dbWi(t)Xj

rij(t)rj( t, X)hxj (t, X) dt i 1, 2, . . . , n, (A:11

are n Brownian motions with the same correlation structure as before. Substituting (A.11) intthe forward rate process (A.3) now yields

dF(t, T) X

ij

rij(t)aiT

(t, X; T)aj(t, X; T) dt -X

i

aiT(t, X; T) d

eWi(t) (A:12a



18/28

with

a i(t, x; T) ri(t, x) Sxi (t, x; T)S(t, x; T)

-

Sxi (t, x; t)

S(t, x; t)(A:12

where we have used (A.7b) to replace hxi (t, x). Inspection of the drift term in (A.12a) shows ththe forward rate process satises the HJM restriction. See Equation 18 of HJM (1992).

B Short rate models

Arbitrage free short rate models are expressed in the money market numeraire,

V(t, r) eE e- Tt R(t9) dt9V(T, R(T)) Tt

C(t9, R(t9)) e-

t9t

R( t0) dt0dt9 R(t) r

( )(B:1

and assume that the risk neutral process for the short rate R(t) is of the form

dR(t) [h(t) l(t, R)] dt s(t, R) deW(t) (B:1under this numeraire. Here the function h(t) must be selected to match the initial discount curvWith some ingenuity, h(t) can be found analytically for some models; otherwise a forwainduction scheme can be used (Jamshidian, 1991).

B.1. Recasting short rate models

We rst recast these models in the framework of Section 2 by choosing a zero coupon bond the numeraire. Equations B.1a and B.1b imply that V(t, r) satises

Vt [h(t) l(t, r)]Vr 12s2(t, r)Vrr - rV -C(t, r) (B:Let us select a maturity Tref, and dene the zero coupon bond

Z(t, r; Tref ) eEfe- Treft R( t9) dt9 j R(t) rg (B:Then Z solves the partial differential equation

Zt [h(t) l(t, r)]Zr 12s2(t, r)Zrr - rZ 0 t, Tref (B:4Z(Tref, r; Tref )

1 (B:4

Suppose we denominate the value V(t, r) of tradable instruments in units of Z(t, r; Tref ),

bV(t, r) V(t, r)Z(t, r; Tref )

(B:

Substituting (B.5) into (B.2) then yields

bVt [h(t) l(t, r)]bVr 12s2(t, r)bVrr - C(t, r)Z(t, r; Tref ) (B:6where



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l(t, r) l(t, r) s2(t, r) Zr(t, r; Tref )Z(t, r; Tref )

(B:6b

This is equivalent to

V(t, r)

Z(t, r; Tref )bE

V(T, R(T))

Z(T, R(T); Tref ) T

t

C(t9, R(t9))

Z(t9, R(t

9); Tref )

dt9 R(t)

r( ) (B:7a

where R(t) evolves according to

dR(t) [h(t) l(t, r)] dt s(t, r) dbW(t) (B:7bas can be seen by applying the backwards Kolmogorov equation to (B.7a). Note that thargument represents an extension of Jamshidians (1991) Brownian path independent models.

B.2 Recasting one-factor models as short rate models

We now consider the general class of one factor models,

dX(t) l(t, X(t)) dt r(t, X(t)) dbW(t) (B:8aunder the numeraire

N(t, x) 1D(0, t)

eh( t,x)A( t) (B:8b

To match the initial term-structure, A(t) must be chosen as

A(t)

logbEfe- h( t,X(t)) j X(0) 0g (B:8cOnce this is done, the short rate is given by

r(t, x) f0(t) A9(t) h t(t, x) l(t, x)hx(t, x) 12r2(t, x)[hxx(t, x) - h2x(t, x)] (B:9See (A.6c).

To re-write this as a short-rate model requires switching to the money market numeraire. Usin(A.11), the risk-neutral process for X(t) is

dX(t) [l(t, X) - r2(t, X)hx(t, X)] dt r(t, X) d

eW(t): (B:10

under the money market numeraire. Applying Itos lemma to (B.9) now shows that the procefor the short rate, R(t) r(t, X(t)), is

dR(t) [rt (l- r2hx )rx 12r2rxx ] dt rrx d eW(t): (B:11We see that expressing an arbitrary one-factor model (B.8a)(B.8c) as a short rate model require

inverting the relationship R r(t, x) given in (B.9) to obtain x x(t, R), and then substituting thinto (B.11) to obtain the drift and volatility terms as functions of t and R. This is a tedious (anneedless) process, but it simplies signicantly for several classes of models. For example, considethe class of models examined in Section 3. For these models l(t, x) 0 and h(t, x) k(t)x, so thcorresponding short rate models are



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dR(t) [rt - k(t)r2(t, x)rx 12r2(t, x)rxx] dt r(t, x)rx d eW(t) (B:12where

r(t, x) f0(t) A9(t) k9(t)x - 12k2(t)r2(t, x) (B:12

C Special exponents

The bg models are dened by the numeraire

N(t, x) 1D(0, t)

ek( t)xA( t) (C:1

and the process

dX(t) a(t)[1 b(t)X(t)]g db

W(t) (C:1

where A(t) M(0, 0; T). Here k(t), a(t), and b(t) are arbitrary model parameters that can used to calibrate the model to the prices of market instruments. We now consider the exponeng 0, g 1=2, and g 1 and solve for

M(t, x; T) logbEfe- k(T)[X(T)-x] j X(t) xg (C:The value Z(t, x; T) of zero coupon bonds, the instantaneous forward rates f(t, x; T), and tshort rate r(t, x) can then be read off from (3.8) and (3.10).

C.1. Extended Hull-White (g

0)

When g 0, the transition density p(t, x; t, x ) is Gaussian,

p(t, x; t, x ) e- (x -x)2=2( s- s)

[2p(s - s)]1=2(C:

where

s t

0a2(t9) dt9 s

t0a2(t9) dt9 (C:

Integrating to compute the expected value in (C.2) yieldsM(t, x; T) 12k2(T)(s - s) A(T) 12k2(T)s (C:

To show that this is identical to the extended HullWhite model, note that (3.10) yields

r(t, x) f0(t) k9(t)x k9(t)k(t)s(t) (C:Equation B.12 now shows that the equivalent short rate model is

dR(t) [h(t) - l(t)R(t)] dt s(t) d eW(t) (C:7under the money market numeraire, where



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l(t) -k0(t)=k9(t) s(t) k9(t)a(t) h(t) f90(t) l(t)f0(t) [k9(t)]2s(t) (C:7bThe extended HullWhite model is usually expressed as the short rate process (C.7a), where l(

and s(t) are arbitrary model parameters and h(t) is chosen as

h(t)

f90(t) l(t)f0(t)

t

0

s2(t9) e-2

t

t9l( t0) dt0

dt9 (C:8

to match the initial discount curve. Note that mean reversion is expressed through the parametel(t) in the extended Hull White model, while it is expressed through the gearing factor k(t) ithe equivalent bg model.

C.2. CIR-like models (g 12 )When g 12 the state variable obeys

dX(t) a(t)[1 b(t)X(t)]1=2 dbW(t) (C:9Consequently, the transition density p(t, x; t, x ) satises the forward Kolmogorov equation

p t 12a2( t)([1 b( t)x ]p)x x t. t (C:10ap !d(x - x) as t! t (C:10b

To obtain M(t, x; T) we take the two-sided Laplace transform

P(t, x; t, k)

1

-

1

e-kx p(t, x; t, x ) dx (C:11

which yields

P t 12a2( t)k2fP - b( t)Pkg t. t (C:12aP e- kx at t t (C:12b

This is a rst-order hyperbolic equation which can be solved by the method of characteristicWe obtain

log P(t, x; t, k)

-kx k2 x[B( t) - B(t)]

2k[B( t) - B(t)]

t

t

2a2(t9) dt9

(2k[B( t) - B(t9)])2( ) (C:13a

where

B(t) t

0a

2(t9)b(t9) dt9 (C:13b

Consequently, M(t, x; T) and A(T) are

M(t, x; T) k2(T) x[B(T) - B(t)]2 k(T)[B(T) - B(t)]

Tt

2a2(t9) dt9

(2 k(T)[B(T) - B(t9)])2

( )(C:14a



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A(T) k2(T)T

0

2a2(t9) dt9

(2 k(T)[B(T) - B(t9)])2 (C:14:

The values of zero coupon bonds, the forward rate curve, and the short rate can now be obtaindirectly from (3.8) and (3.10). In particular, the short rate is

r(t, x)

f0(t)

K(t)[x

C(t)] (C:15

with

K(t) k9(t) - 12k2(t)a2(t)b(t) C(t) k(t) t

0

a2(t9) dt9

(1 k(t)[B(t) - B(t9)]=2)3 (C:15

Let us cast this model as the equivalent short rate model. Using (B.12) shows that this model

dR(t) [h(t) - l(t)R(t)] dt s(t)

d(t) R(t)p

deW(t) (C:16under the money market numeraire, where

l(t) -

[K9(t)-k(t)a2(t)b(t)]=K(t) (C:16:

h(t) f90(t) l(t)f0(t) K(t)C9(t) - k(t)a2(t)K(t)[1 - b(t)C(t)] (C:16s(t) a(t)

K(t)b(t)

pd(t) K(t)=b(t) - f0(t) - K(t)C(t) (C:16

The extended CIR model is usually expressed as

dR(t) [h(t) - l(t)R(t)] dt s(t)

R(t)p

d eW(t) (C:1Here l(t) and s(t) are arbitrary model parameters, and h(t) must be chosen to t the initial terstructure. To state this requirement explicitly, recall that the value of a zero coupon bond is

Z(t, r; T) eEfe- Tt R( t9) dt9 j R(t) rg (C:1Equivalently, Z(t, r; T) is the solution of the backwards equation

Zt [h(t) - l(t)r]Zr 12s2(t)rZrr - rZ 0 t, T (C:19with

Z(T, r; T) 1 (C:19The solution of (C.19) is

Z(t, r; T) e- B(t,T)r- A( t,T) (C:20where B(t, T) satises the Ricatti equation

B t l(t)B 12s2(t)B2 - 1 t, T (C:20with the boundary condition B(T, T) 0, and

A(t, T) T

t

h(t9)B(t9, T) dt9 (C:20



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Matching the initial discount curve requires Z(0, 0; T) D(0, T) e -T

0f0(T9) dT9, which require

h(t) to be chosen so thatT0h(t9)B(t9, T) dt9

T0

f0(T9) dT9 - B(0, T)f0(0) (C:21

We see that matching the extended CIR model to the initial discount curve requires solving Ricatti equation to obtain the zero coupon bond values. This can be done efciently, but it difcult to understand the qualitative impact of yield curve shifts on pricing.

Comparing (C.16) and (C.17) shows that the extended CIR model is a special case of the bmodel with g 1=2 and with b(t) chosen so that d(t) 0. Unsurprisingly, making this choice ob(t) also requires solving a Ricatti equation.

A simpler alternative is to use the g 1=2 model without requiring d(t) 0. The g 1=2 modthen gives the zero coupon bond values directly, without solving a Ricatti equation. In addition, b(can be used to match the implied volatility smile by calibrating the model to the prices of ofmarket instruments. Although the CIR model is more aesthetically appealing, with its natural barri

occuring at R(t) 0 and its simple relation between the volatility and the short rate, we stress thneither model can be expected to be valid near R(t) 0. Since the barrier in the g 1=2 model irrelevant for pricing commonly traded US instruments, it seems better to use b(t) to match thimplied volatility smile.

C.3. BlackKarasinski-like models (g 1)When g 1 we have been unable to nd the Laplace transform of the transition density, anthus M(t, x; T). However, we can nd the moments m k(t, x; T) explicitly, and then use th

expansions

M(t, x; T) 12k2(T)m2(t, x; T) - 16k3(T)m3(t, x; T) . . . (C:22aA(T) 12k2(T)m2(0, 0; T) - 16k3(T)m3(0, 0; T) . . . (C:22b

Equations 3.8 and 3.10 then determine the zero coupon values, the forward rate curve, and thshort rate as before. Although these expressions are only approximate, using the rst termsufces to price all but the most sensitive instruments.

With the exponent g 1,

dX(T) a(T)[1 b(T)X(T)] dbW(T) (C:23and applying Itos lemma yields

d[X(T) - x]k 12k(k - 1)a2(T)[1 b(T)X]2[X - x]k- 2 dT

ka(T)[1 b(T)X][X - x]k- 1 dbW(T) (C:24Taking the expected value then yields equations for the moments m k(t, x; T):

dm kdT

12k(k - 1)a2(T)f(1 bx)2m k-2 2b(1 bx)m k-1 b2m kg (C:25



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See (3.11). Clearly at T t we have m0(t, x; t) 1 and m k(t, x; t) 0 for all k.1. Solvithese equations successively yields

m0(t, x; T) 1 m1(t, x; T) 0 (C:26

m2(t, x; T)

T

t

a2(t9)[1 b(t9)x]2exp T

t9

a2(T9)b2(T9) dT9( )dt9 (C:26m3(t, x; T) 6

Tt

t9t

a2(t9)a2(t0)b(t9)[1 b(t9)x][1 b(t0)x]2

. exp 3

Tt9

a2(T9)b2(T9) dT9 t9

t0

a2(T9)b2(T9) dT9

( )dt0 dt9 (C:26

As earlier, Equation B.12 can be used to obtain the equivalent short rate model. Even though thdistribution of X(t) is roughly lognormal, the short rate model is not the Black Karasinski modregardless of how b(t) is chosen. The Black Karasinski model requires adding a drift term to tstochastic differential equation for X(t).

D Rational interest rate models

Consider the interest rate model dened by the valuation formula

V(t, x) N(t, x)

bE

V(T, X(T))

N(T, X(T))T

t

C(t9, X(t9))

N(t9, X(t9))dt9 X(t) x

( )(D:1

with the numeraire

N(t, x) 1D(0, t) b(t) . x (D:1

and assume that the state variables X(t) evolve according to

dXi(t) ri(t, X) dbWi(t) Xi(0) 0 i 1, 2, . . . , n (D:1under this numeraire in the risk-neutral world. Noting that X(t) is a Martingale, we nd that tzero coupon bond prices are

Z(t, x; T) bE D(0, T) b(T) . X(T)D(0, t) b(t) . x X(t) x( ) D(0, T) b(T) .xD(0, t) b(t) . x (D:

Consequently, the instantaneous forward rate curve is

f(t, x; T) - DT(0, T) b9(T).x

D(0, T) b(T) . x (D:

Flesaker and Hughstons (1996) rational lognormal model is the special cari(t, x) ai(t)(1 xi ). These models have the advantage that all forward rates f(t, x; T) aguaranteed to remain positive provided that b9i(T) < 0 for each i and DT(0, T),P

b9i(T) for all



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Moreover, the one-factor version of this model has closed-form prices for swaptions and caple(Flesaker and Hughston, 1996).

E Separable models

The general n-factor HJM model for the forward rate is

dF(t, x(t); T) X

ij

rij(t)aiT(t, x(t); T)a

j(t, x(t); T) dt -X

i

aiT(t, x(t); T) dfWi(t) (E:1in the risk-neutral world under the money market numeraire:

V(t)

eE e

-

Tt

R( t0,x0) dt0V(T, x(T))

Tt

C(t9, x(t9)) e-

t9t

R( t0,x0) dt0dt9Ft

( )(E:2

Here x(t) indicates that the volatilities aiT, payoffs, and cash ows can depend on all measurab

events that can be resolved by time t.The separable models (Li et al., 1995; Ritchken and Sankarasubramanian, 1995; Cheyette, 1992

are derived by presuming that the volatilities have the form

a iT(t, x(t); T) k9i(T)ri(t, x(t)) i 1, 2, . . . , n (E:3aso that

a i(t, x(t); T) [ki(T) - ki(t)]ri(t, x(t)) i 1, 2 . . . , n (E:3bThen the stochastic process for the forward rates becomes

dF(t, x(t); T) Xij

k9i(T)[kj(T) - kj(t)]rij(t)ri(t, x(t))rj(t, x(t)) dt

-

Xi

k9i(T)ri(t, x(t)) deWi(t) (E:4Integrating over t now yields

F(t, x; T) f0(T)

Xik9i(T)Xi(t, x)

Xijk9i(T)kj(T)Yij(t, x) (E:5

where

Xi(t, x) - t

0ri(t9, x9) d eWi(t9) - X

j

t0kj(t9)rij(t9)ri(t9, x9)rj(t9, x9) dt9 i 1, . . . ,

(E:6a

Yij(t, x) t

0rij(t9)ri(t9, x9)rj(t9, x9) dt9 i, j 1, . . . , n (E:6b

One now assumes that the coefcients ri(t, x(t)) depend on the path taken by the Brownia



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motion only through the value of the state variables X(t), Y(t) at time t. This is realistic since (E.expresses the entire term structure in terms of these variables. With this assumption, (E.6) becom

dXi(t) -ri(t, X, Y) d eWi(t)Xj

kj(t)rij(t)rj(t, X, Y) dt

( )Xi(0) 0 i 1, . . . ,

(E:7dYij(t) rij(t)ri(t, X, Y)rj(t, X, Y) dt Yij(0) 0 i, j 1, . . . , n (E:7

and the instantaneous forward rate curve is

f(t, x, y; T) f0(T)X

i

k9i(T)xi X

ij

k9i(T)kj(T)yij (E:

where xi Xi(t) and yij Yij(t) are the values of the state variables at time t.Under the money market numeraire, the general separable model is dened by the forward ra

curve (E.8) and the stochastic processes (E.7). We can put this model into the context of Section

by switching to the numeraire

N(t, x, y) 1D(0, t)

expX

i

ki(t)xi 12X

ij

ki(t)kj(t)yij

( )(E:

Under this numeraire the value of a tradable instrument is

V(t, x, y)

N(t, x, y)

bE

V(T, X(T), Y(T))

N(T, X(T), Y(T))

T

t

C(t9, X(t9), Y(t9))

N(t9, X(t9), Y(t9))dt9 X(t) x, Y(t) y

( )(E:1

Reversing the steps in Appendix A, we nd that the state variables evolve according to

dXi(t) -ri(t, X, Y) dbWi(t) Xi(0) 0 i 1, . . . , n (E:11dYij(t) rij(t)ri(t, X, Y)rj(t, X, Y) dt Yij(0) 0 i, j 1, . . . , n (E:11

under the new numeraire. Finally, integrating the forward rate curve (E.8) with respect to tmaturity T yields the zero coupon bond prices:

Z(t, x, y; T) D(t, T)exp -Xi

[ki(T) - ki(t)]xi -1

2Xij[ki(T)kj(T) - ki(t)kj(t)]yij( )

(E:1

Note that Z(0, 0, 0; T) D(0, T), so the separable models are automatically consistent with tinitial term structure.

Since Yij(t) Yji(t), the n-factor separable model generally requires n(n 3)=2 state variableAs the random processes Yij(t) are not being driven directly by Brownian motions, we can only viethe separable models within the framework of Section 2 by considering (E.11) as a highdegenerate n(n 3)=2 factor model.

The zero coupon bond prices in (E.12) show that the extra state variables yij are needed



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account for convexity effects in an arbitrage-free manner. Since convexity effects are quite smaland since (E.11b) shows that the Yij(t) are only driven indirectly by the stochastic processes dcWi(tthe distributions of Yij(t) are highly peaked about their means, which are near zero. Specically, athe volatilities ri decrease, the means of the variables Yij(t) scale like r

2 t and the variances oYij(t) scale like r

6t3. This suggests that the convexity effects can be accounted for by replacing thdistributions of Yij(t) by d-functions, selecting the position of the d-functions to make the theorarbitrage free. Carrying this idea through leads to the arbitrage-free n-state variable model deneby the valuation formula

V(t, x) N(t, x)bE V(T, X(T))N(T, X(T))

T

t

C(t9, X(t9))

N(t9, X(t9))dt9 X(t) x

( )the numeraire

N(t, x) 1D(0, t)

ek( t).xA( t)

and the risk neutral processes

dXi(t) -ri(t, X) dbWi(t) Xi(0) 0 i 1, . . . , nunder this numeraire. Here A(T) is dened by

A(T) logbEfe- k(T).X(T)jX(0) 0gThis model is the multi-factor generalization of the one-factor models in Section 3. Following th

analysis there shows that the zero coupon bond prices are

Z(t, x; T)

D(t, T)e- [k(T)- k( t)]

.x- A(T)A( t)M( t,x;T)

where

M(t, x; T) logbEfe-k(T).[X(T)-x] j X(t) xgConsequently the instantaneous forward rates are

f(t, x; T) f0(T) k9(T) . x A9(T) - MT(t, x; T)Typically k9i(T)ri is around 10

- 2 in US dollar markets. At these volatilities the n-state variabmodel (E.13) is virtually identical to the analogous separable model in (E.9)(E.11). Although thn-state variable model has the disadvantage of requiring computation of M(t, x; T) to obtain th

zero coupon bond prices, this is usually outweighed by having many fewer state variables than thseparable models.

F Approximate prices of European options

As in Section 4.1, let te be the expiration date of a European option to receive the cash owCi on dates ti, i 1, 2, . . . , n in return for paying the strike K on the settlement date ts. Alsdene



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F0 X

i

CiD(ts, ti ) se

te0a2(t9) dt9 (F:

as before. To express the option prices succinctly, dene t0 by the relation

t0

0

a2(t9) dt9

se=2 (F:

dene x by the implicit relation

12(F0 K)

Xi

CiD(ts, ti ) e- [ki-ks ]x - AiAsM( t0,x ;ti)- M( t0,x ;ts ) (F:

and dene Kk by

Kk P

i CiD(ts, ti )[ki - ks - Mx(t0, x ; ti ) Mx(t0, x ; ts )]k e-kix - A iM( t0,x ; ti )

Pi CiD(ts, ti ) e-kix - AiM( t0,x ;ti )

(F:

Then the implied price vol of the option is (Hagan and Woodward, in preparation)

rB K1(se= te )1=2(1 bx )g 1 124(1 bx )2gse[K21 -g(2 - g)b2

(1 bx )2 2K3

K1- 3

K22

K21

]

(

(F0 - K)2

6K21(F0 K)2[2K21 -

g(1 g)b2(1 bx )2

3gbK2(1 bx )K1

K3K1

- 3K

22

K21

] . . . (F:

This expression yields prices that are usually accurate to within a tenth of a basis point, althougthe error can be several times larger in unusual situations.


Markov Interest Rate Models - Hagan and Woodward

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