markov interest rate models

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Markov interest rate modelsPatrick S. Hagan a & Diana E. Woodward ba NumeriX, 546 Fifth Avenue, 17th Floor, New York, NY10036b The Bank of Tokyo-Mitsubishi, Ltd., 1251 Avenue of theAmericas, New York, NY 10020Published online: 14 Oct 2010.

To cite this article: Patrick S. Hagan & Diana E. Woodward (1999) Markov interest ratemodels, Applied Mathematical Finance, 6:4, 233-260, DOI: 10.1080/13504869950079275

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Markov interest rate modelsPATRI CK S . HAGA N1 and DIANA E. WOODWARD2

1NumeriX, 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 thisprocedure automatically �t within the HJM framework and �t the initial term structure exactly. Therefore theyare arbitrage free. Because the models created by this procedure have only one state variable per factor, two-and even three-factor models can be computed ef�ciently, without resorting to Monte Carlo techniques. Thiscomputational ef�ciency makes calibration of the new models to market prices straightforward. Extended Hull–White, extended CIR, Black–Karasinski, Jamshidian’s Brownian path independent models, and Flesaker andHughston’s rational log normal models are one-state variable models which �t naturally within this theoreticalframework. The ‘separable’ n-factor models of Cheyette and Li, Ritchken, and Sankarasubramanian – whichrequire n(n ‡ 3)=2 state variables – are degenerate members of the new class of models with n(n ‡ 3)=2factors. The procedure is used to create a new class of one-factor models, the ‘ b – g models.’ These models canmatch the implied volatility smiles of swaptions and caplets, and thus enable one to eliminate smile error. Theb –g models are also exactly solvable in that their transition densities can be written explicitly. For these modelsaccurate – but not exact – formulas are presented for caplet and swaption prices, and it is indicated how theseclosed form expressions can be used to ef�ciently calibrate the models to market prices.

1. Introduction

Heath, Jarrow and Morton (1990, 1992) created a broad framework for developing arbitrage-freeterm structure models. In general, HJM models are non-Markovian and require extensive MonteCarlo simulation to calibrate model parameters to market prices, to value contingent claims, andto determine hedges. This computational burden can be greatly reduced by using special cases ofthe HJM models which are Markovian. Notable among these special cases are the ‘separable’ n-factor models of Cheyette (1992) and Li et al. (1995) and Ritchken and Sankarasubramanian(1995). Although the ‘separable’ n factor models are Markovian, they require n(n ‡ 3)=2 statevariables, which still imposes a stiff computational burden. Moreover, to date there has been nosystematic procedure for �nding HJM models which are Markovian.

Here we use the method of the undetermined numeraire to develop a general procedure forcreating Markovian term structure models. The resulting models automatically �t within the HJMframework and match the initial discount curve. Thus they are arbitrage free. Extended Hull–White(1990a, b), extended CIR (Cox et al., 1985), Black-Karasinski (1991), Jamshidian’s (1991) Brownianpath independent models, and Flesaker and Hughston’s rational log normal models (1996) all �tnaturally within our theoretical framework.

Unlike the ‘separable’ models, the new models require only n state variables for an n-factor

Applied Mathematical Finance 6, 233–260 (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-factormodels to be used without requiring Monte Carlo simulations. Although we can re-create theseparable models within our framework as degenerate models with n(n ‡ 3)=2 factors and statevariables, we can also create models which are direct analogues of the separable models and requireonly n factors and state variables. These analogues are asymptotically equivalent to their separablecounterparts in the limit of small volatilities, and, at normal market volatilities, the two sets ofmodels should give essentially identical results.

We use our procedure to create a new class of one factor models, the ‘ b – g models.’ Unlike otherone-factor models, these models can match the implied volatility smiles of swaptions and caplets,and thus enable us to largely eliminate smile error from our books. The b – g models are also exactlysolvable in that their transition densities (Green’s functions) can be written exactly. For these modelswe present accurate – but not exact – formulas for caplet and swaption prices, and indicate howthese closed form expressions can be used to ef�ciently calibrate the models to exact market prices.

2. Derivation

Consider a continuous trading economy on the interval [0, Tmax]. To �x notation, let f0(T ) betoday’s instantaneous forward rate for date T for this economy. Then the current value of a zerocoupon bond which pays $1 at maturity T is

D(0, T ) e -„ T

0f0(T 9 ) dT 9 (2:1)

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 essential

elements. First is a set of stochastic processes which drive the evolution of interest rates. Second isa numeraire, a tradable instrument with positive value and no cash �ows. Commonly the moneymarket account or a speci�c zero coupon bond is used as the numeraire. The �nal element is avaluation formula which states that, in the absence of any cash �ows, the value of any tradableinstrument1 is a Martingale when expressed in units of the numeraire (Harrison and Pliska, 1981;Harrison and Kreps, 1979).

In a Markovian model, the entire term structure at any time t must be determined solely by thevalues X 1, X 2, . . . , X n of a �nite set of random state variables X1(t), X2(t), . . . , X n(t). We assumethat these state variables evolve according to the Ito processes

dX i(t) ˆ l i(t, X )dt ‡ r i(t, X )d bW i(t), X i(0) ˆ 0, i ˆ 1, 2, . . . , n (2:2a)

Here bW1(t), bW2(t), . . . , bW n(t) represent n Brownian motions, which can be correlated

d bW i(t)d bW j(t) ˆ rij(t)dt (2:2b)

To obtain a Markovian model, we need to choose a numeraire whose value N depends only on thevalues x1, x2, . . . , xn of the state variables: N e H( t,x). We stress that at this point, the function

1 The term ‘tradable instruments’ is used to refer to both fundamental instruments that are actively traded and all derivativesand synthetic instrument that can be created.

234 Hagan and Woodward

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

To express the valuation formula, consider a tradable instrument which has the value V (T , X (T ))at some date T and pays a cash �ow C(t, X (t)). The value of the instrument expressed in units ofthe 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 inthe absence of any cash �ows then yields

V (t, x) ˆ eH ( t,x)bE V (T , X (T )) e - H (T ,X(T )) ‡…T

tC(t 9 , X (t 9 )) e - H( t 9 ,X( t 9 )) dt 9 X (t) ˆ x

( )

,

(2:3)

where x ˆ X (t) is the state of the economy at time t. Equation 2.3 gives the value of theinstrument at any time t , T . Note that bE is the expected value under the probability measure2

determined by Equation 2.2.Without loss of generality, we re-write the numeraire as

N (t, x) ˆ 1D(0, t)

eh( t,x)‡A( t) (2:4a)

where

h(t, 0) 0, A(0) ˆ 0 (2:4b)

Then the valuation formula becomes

V (t, x) ˆ eh( t,x)‡A( t)bE V (T , X (T ))D(t, T ) e - h(T ,X(T )) - A(T )

‡…T

tC(t 9 , X (t 9 ))D(t, t 9 ) e - h(t 9 ,X( t 9 )) - A( t 9 ) dt 9 X (t) ˆ x (2:5)

The valuation formula (2.5), with the state variables X (t) determined by the stochastic process(2.2), de�nes a term structure model. As we shall see, if this model is consistent with the initialdiscount curve then it is arbitrage free. In particular, this model automatically satis�es the crucial‘forward rate drift restriction’ of HJM (1992) theory.3 We stress that apart from some mathematicalregularity conditions, the drift rates l i(t, x), the volatilities r i(t, x), and the function h(t, x) arearbitrary. The remaining function A(t) will be determined by requiring the valuation formula (2.5) tobe consistent with the initial discount curve.

To impose consistency, de�ne Z(t, x; T ) to be the value at t, x of a zero coupon bond which pays$1 at maturity T. From (2.5),

2 Equation 2.2 thus de�nes the risk neutral probability measure induced by the numeraire N (t, x), and does not represent ‘realworld’ probabilities.3 This is not surprising: (2.2) and (2.5) self-consistently de�ne the probability measure of the risk neutral-world under aspeci�c (if unknown) numeraire N( t, x); this then uniquely de�nes the risk neutral probability measure under every othernumeraire, including the money market numeraire; and HJM (1992) showed that for any self-consistent risk-neutral world, theinstantaneous forward rates must satisfy the forward drift restriction under the probability measure induced by the moneymarket numeraire.

Markov interest rate models 235

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

where

S(t, x; T ) bEfe - h(T ,X (T )) j X (t) ˆ xg (2:6b)

As we are taking X (0) ˆ 0, consistency with the initial discount curve requires Z(0, 0; T )D(0, T ). Since h(t, 0) ˆ A(0) ˆ 0, this requirement becomes

A(T ) ˆ log S(0, 0; T ) ˆ log bEfe - h(T ,X (T )) j X (0) ˆ 0g for all T . 0 (2:7)

Thus consistency with the initial discount curve requires choosing A(T ) so that the expectedvalue 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 themodel is consistent with the initial term structure is independent of the actual initial term structure.Once A(T ) has been selected according to (2.7), then if the initial term structure changes, all weneed do is replace the discount factors D(t, T ) with the new discount factors, and the model will beconsistent with the new initial term structure. This is useful during calibration since it separates thevolatility functions r i(t, X ), l i(t, X ), and h(t, X ) endogenous to the model from the currentdiscount factors D(t, T ), which are exogenous.

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

„ Tt f (t, x; T 9 )dT 9 g. From (2.6),

…T

tf (t, x; T 9 )dT 9 ˆ

…T

tf0(T 9 )dT 9 - h(t, x) - A(t) ‡ A(T ) - log S(t, x; T ) (2:8)

Hence the forward rate curve at t, x is

f (t, x; T ) ˆ f0(T ) ‡ A9 (T ) - ST (t, x; T )=S(t, x; T ) (2:9)

where we are using subscripts to denote partial derivatives. The short rate r(t, x) is de�ned asf (t, x; t), so we have

r(t, x) ˆ f0(t) ‡ A 9 (t) - ST (t, x; t)=S(t, x; t) (2:10a)

In Appendix A it is shown that this expression for the short rate is equivalent to

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

i

l i(t, x)hxi (t, x)

‡ 12

X

ij

rij(t)r i(t, x)r j(t, x)[hxi x j(t, x) - hxi

(t, x)hx j(t, x)] (2:10b)

With A(t) de�ned by (2.7), the valuation formula (2.5) and random processes (2.2) uniquelyde�ne a term structure model which is consistent with the initial discount curve. The instantaneousforward interest rates and short rate for this model are given by (2.9) and (2.10), respectively. At thispoint it would be natural to use Martingale theory (Harrison and Pliska, 1981; Harrison and Kreps,1979) to show that this model is arbitrage free. Instead, in Appendix A we show that this model �tswithin the HJM framework. There it is found that using the money market as a numeraire, theprocess for the instantaneous forward rate, F(t, T ) f (t, X (t); T ), satis�es


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dF(t, T ) ˆX

ij

rij(t)aiT (t, X ; T )a j(t, X ; T )dt -

X

i

aiT (t, X ; T )d eW i(t) (2:11a)

in the risk-neutral probability world, with

a i(t, x; T ) ˆ r i(t, x)Sxi

( t, x; T )S(t, x; T )

-Sxi

(t, x; t)S(t, x; t)

i ˆ 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 withinthe HJM framework, and provided it satis�es innocuous regularity conditions, the model isarbitrage free.4

Equation 2.5 can be used to directly value any contingent claim whose ‘payoff’ V (T , X(T )) andcash �ow C(t, X (t)) can be expressed as explicit functions of the state variables. Since (2.9)expresses the entire forward rate curve f (t, x; T ) explicitly, this includes all instruments whosepayoff and cash �ow depend only on the then current yield curve. This includes European,Bermudan, and American swaptions, caps, and most yield curve options. However, there is noguarantee that instruments whose payoff or cash �ow are path dependent (e.g. instruments withpayments determined by the average short rate over the preceding period) can be evaluated withoutadding 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 2to create a special class of one factor models. In Section 4 we will further specialize this classof models to obtain the b – g models.

Although one-factor models are usually expressed in terms of the money market numeraire usingthe short rate r as the state variable, any such ‘short rate model’ can be re-cast within theframework of Section 2 by using a zero coupon bond as the numeraire. Conversely, any one-factormodel within the framework of Section 2 can be transformed into a short rate model. Thesetransformations are carried out in Appendix B.

Within the framework of Section 2, a one-factor model is de�ned by the stochastic process for thestate variable,

dX (t) ˆ l (t, X )dt ‡ r (t, X )d bW (t) X (0) ˆ 0 (3:1)

and the numeraire

N (t, x) ˆ 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 innocuousmathematically, each points out a more stringent business condition. For example, the mathematical requirement ‘marketcompleteness’ 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 justi�ed in using both acomplicated function h(t, x) in the numeraire and a complicated stochastic process for X (t). Insteadwe should choose either a relatively simple function h(t, x) or a relatively simple stochastic process.We arbitrarily choose to use a simple numeraire

N (t, x) ˆ 1D(0, t)

ek ( t)x‡A( t) (3:3)

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

A(t) ˆ log bEfe - k ( t)X ( t) j X (0) ˆ 0g (3:4)

Now consider the drift term, l (t, X )dt. Suppose the model had, say, a positive drift so that X (t)increased on average. Then A(t) would be negative and would effectively cancel out the averageeffect of the drift term in the numeraire. Similarly, if the drift term was negative, A(t) would bepositive, again canceling out most of the drift. Indeed (3.4) can be written as

bEfe - k ( t)X (t) - A( t) j X (0) ˆ 0g ˆ 1 (3:5)

Since A(t) cancels out the main effects of the drift term, any residual in�uence of the drift terml (t, X )dt on the model would be quite subtle. Therefore, in the interests of simplicity, we takeour model to be driftless.

So consider the special class of one factor models

dX (t) ˆ r (t, X )d bW (t) X (0) ˆ 0 (3:6)

with the numeraire (3.3). As shown in Appendix C, this class of models includes the extendedHull–White, the extended CIR, and ‘Black–Karasinski like’ models. However, the Black–Karasinski 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) ˆ e k ( t)x‡A( t)bE V (T , X (T ))D(t, T ) e - k (T )X (T ) - A(T )

‡…T

tC(t 9 , X (t 9 ))D(t, t 9 ) e - k (t 9 )X ( t9 ) - A( t9 ) dt 9 X (t) ˆ x (3:7)

so the value of a zero coupon bond becomes

Z(t, x; T ) ˆ D(t, T )e - [k (T ) - k (t)]x - [A(T ) - A(t)]‡M( t,x;T ) (3:8)

Here we have de�ned

M (t, x; T ) ˆ log bEfe - k (T )[X (T ) - x] j X (t) ˆ xg ˆ k (T )x ‡ log S(t, x, T ) (3:9a)

and the consistency condition is now

A(T ) ˆ M (0, 0; T ) for all T (3:9b)

It is instructive to examine the forward rate curve. Since Z(t, x; T ) ˆ expf -„ T

t f (t, x; T 9 )dT 9 g,Equation 3.8 shows that if the economy is in state X (t) at time t, then the instantaneous forwardrate for date T would be


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f (t, X (t); T ) ˆ f0(T ) ‡ k 9 (T )X ‡ A9 (T ) - M T (t, X ; T ) (3:10a)

The corresponding short rate would be

r(t, X (t)) ˆ f0(t) ‡ k 9 (t)X ‡ A9 (t) - M T (t, X ; t) (3:10b)

For the moment, let us neglect the last two terms in (3.10a) and (3.10b). Then the forward ratecurve is just the curve k 9 (T )X added to the original forward curve f0(T ). Note that k 9 (T ) is a‘gearing’ factor: if the state variable X (t) is shifted by an amount D , then the short rate would shiftby d r(t) ˆ k 9 (t)D and the forward rates f (t, X ; T ) would shift by d f ˆ k 9 (T )D ˆ k 9 (T )d r=k 9 (t).Since we expect shocks to affect short maturities more than longer maturities, we expect k 9 (T ) to bea decreasing function of T .

Note that Equation 3.6 implies that X (t) is a Martingale, so for all times T . t, the expectedvalue of X (T ) is just X (t). Thus, if X (t) was shifted by an amount D , then for all later times T theexpected value of X (T ) would also shift by D . Consequently, if a shock shifted the short rate byd r(t) ˆ k 9 (t)D , then for all later times T the expected value of the short rate r(T , X (T )) would shiftby k 9 (T )d r= k 9 (t). As T increases, the expected value of the short rate returns to the initial forwardcurve f0(T ) at a rate determined by how rapidly k 9 (T ) decreases. So mean reversion of interest ratesis directly determined by the function k (T ).

With f (t, X ; T ) ˆ f0(T ) ‡ k 9 (T )X‡ . . . , the distribution of forward rates is determined by thedistribution of X (t). The width of this distribution is determined mainly by the overall magnitude ofr (t, x), and the shape of this distribution (i.e. the deviation from Gaussian or log normal behavior)is determined mainly by the functional form of the dependence of r (t, x) on x. As noted above,mean reversion of interest rates is determined mainly by the function k (T ). Since skews and smiles(the changes in an option’s implied volatility as the strike changes) depend mainly on the shape ofthe distribution, and since the change in the implied volatility as the duration of the underlyinginstrument changes depends mainly on mean reversion, this separation makes it easy to calibratethese models to both at-the-money and off-market instruments of differing durations. This makesthese models very useful for pricing, hedging, and understanding instruments in the presence ofimplied volatility skews and smiles.

The third term A9 (T ) in (3.10) embodies the consistency requirement, and is much smaller thanthe �rst two terms. Even though (3.6) implies that X (t) has mean zero, the convex relation betweenbond prices and interest rates would cause bond prices to drift, on average, as the distribution ofX (t) spreads out. The term A 9 (T ) cancels the expected value of this convexity effect. This is whythe ‘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 M T (t, x; T ) arises because as X (t) evolves away from X (0) ˆ 0, theexpected value of the convexity effect changes.

We can clarify the relation between the distribution of X (T ) and the convexity terms A(T ) andM(t, x; T ) by expanding (3.9) in powers of k . De�ne the moments

mk (t, x; T ) ˆ bEf[X (T ) - x]k j X (t) ˆ xg k ˆ 1, 2, . . . (3:11)

and note that m1(t, x; T ) ˆ 0. Expanding (3.9) yields

M(t, x; T ) ˆ 12k

2(T )m2(t, x; T ) - 16k

3(T )m3(t, x; T ) ‡ 124k

4(T )[m4(t, x; T ) - 3m22(t, x; T )] ‡ . . .

(3:12a)


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A(T ) ˆ 12k

2(T )m2(0, 0; T ) - 16k

3(T )m3(0, 0; T ) ‡ 124k

4(T )[m4(0, 0; T ) - 3m22(0, 0; T )] ‡ . . .

(3:12b)

Our experience calibrating one-factor models to US swaption and caplet markets shows that themagnitude of the quadratic term 1

2k2(T )m2(t, x; T ) is roughly (T - t)T 2 3 10 - 4, where time is

measured in years. The cubic and quartic terms, which arise from the skewness and kurtosis of thedistribution, are much smaller and have magnitudes of roughly (T - t)2T 3 3 10 - 8 and(T - t)2T 4 3 10 - 9, respectively.5 We conclude that it is safe to truncate after the quadratic termexcept for unusually sensitive instruments, unusually long durations, or unusually competitivemarkets.

Truncating after the quadratic terms yields the forward rates

f (t, x; T ) ˆ f0(T ) ‡ k 9 (T )x ‡ 12

@

@Tf k 2(T )[m2(0, 0; T ) - m2(t, x; T )]g ‡ . . . (3:13)

Consequently, the expected forward rate is

bEf f (t, X (t); T ) j X (0) ˆ 0g ˆ f0(T ) ‡ k 9 (T )k (T )m2(0, 0; t) ‡ . . . (3:14a)

Note that m2(0, 0; t) is the variance of X (t), not X (T ). The covariance between forward rates atdifferent maturities is

Covf f (t, X (t); T1), f (t, X (t); T2) j X (0) ˆ 0g ˆ k 9 (T1)k 9 (T2)m2(0, 0; t) ‡ . . . (3:14b)

showing that the two rates are perfectly correlated through this order.Before continuing, let us brie�y address the ‘unknown numeraire’ in (3.2). By construction,

N (t, x) is always a valid numeraire.6 Suppose we arbitrarily set k (T ref ) ˆ 0 for some date T ref . Then(3.9) would imply that A(T ref ) ˆ 0 and M (t, x; T ref ) ˆ 0, so the value of the zero coupon bond ofmaturity T ref would be D(t, T ref )e k (t)‡A( t) (see (3.8)). So if we made this choice, then the ‘unknownnumeraire’ N (t, x) would represent a zero coupon bond paying 1=D(0, T ref ) dollars at maturity T ref

– at least for times t < T ref . Although N (t, x) would remain a valid numeraire for times t . T ref , itwould no longer represent any simple security. In the same spirit, we could restrict the k (t) curve sothat k (t) !0 as t !1; then the numeraire would correspond to Flesaker and Hughston’s (1996)‘absolute terminal measure’.

4. The b –g models

We now specialize to volatilities of the form r (t, x) ˆ a (t)[1 ‡ b X (t)]g , where b and g areconstant, and de�ne the b – g models by

dX (t) ˆ a (t)[1 ‡ b X (t)]g d bW (t) X (0) ˆ 0 (4:1a)

5 This is equivalent to the change in interest rates over a period D t having a standard deviation of sizeO(100 bps [ D t=yrs]1=2), variance of size O(10- 4 D t=yrs), kurtosis of size O(1), and skewness of size O(10 - 2[ D t=yrs]1=2).6 SubstitutingV ( 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 thevalue of the interest rate option which has no cash �ow and has the ‘payoff ’ N(Tfinal, X(T final )) at T final.


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with the numeraire

N (t, x) ˆ1

D(0, t)ek ( t)x‡A( t) (4:1b)

Without loss of generality, we normalize k (t) so that k 9 (0) ˆ 1.A key advantage of b – g models is that there is a closed form solution for the transition density

p(t, x; t , x )dx ˆ Probf x , X ( t ) , x ‡ dx j X (t) ˆ xg (4:2)

The transition density p satis�es the backward equation

pt ‡ 12a

2(t)j1 ‡ b xj2g pxx ˆ 0 t , t (4:3a)

p ! d (x - x ) as t ! t (4:3b)

In terms of the new variables

s (t) ˆ… t

0a 2(t 9 )dt 9 y(x) ˆ j1 ‡ b xj1- g

b (1 - g )(4:4)

this becomes

ps ‡ 12

( pyy -2m - 1

ypy ) ˆ 0 s , s (4:5a)

p ![ b (1 - g ) y ] - 2m ‡1 d (y - y ) as s ! s (4:5b)

with m ˆ 1=(2 - 2g ). In particular, s (t) is roughly the variance of X (t), which is around 10- 4 t inUS dollar markets, where t is measured in years.

Equation 4.5 can be solved by a combination of Laplace transform and Green’s functiontechniques. For exponents 0 , g , 1=2 this yields

p(t, x; t , x ) ˆ (y= y )my

s - sf1

2I - m (y y=( s - s )) ‡ 12I m (y y=( s - s ))g e - ( y 2‡ y2)=2( s - s ) d y

dx

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

p(t, x; t , x ) ˆ sin p m

p(y= y )m

y

s - sK m (y y=( s - s )) e - ( y 2‡ y2)=2( s - s ) d y

dxfor x , - 1=b

(4:6b)

Here

d ydx

ˆ [(1 - g )b y ] - 2m ‡1 (4:7)

and K m and I m are the modi�ed Bessel functions.The b – g models have an arti�cial barrier at x ˆ - 1= b , where the volatility j1 ‡ b xjg goes to

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 re�ecting boundarycondition at y ˆ 0; this would eliminate (4.6b) and change the Bessel functions in (4.6a) from12(I - m ‡ I m ) to I - m .

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


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small to affect the pricing of realistic products. Since b and g can be chosen to reproduce thevolatility smile reasonably well, indicating that the model is reproducing the right shape of the risk-neutral probability distribution near the distribution’s centre, we argue that the bene�t of increasedaccuracy near the distribution’s centre far outweighs the liability of having an arti�cial barrier in theextreme tails of the distribution. In fact, we do not expect any model to be valid in the extreme tailsof the distribution where there is no relevant market experience.

For exponents 1=2 < g , 1 we have

p(t, x; t , x ) ˆ (y= y )my

s - sI m (y y=( s - s ))e - ( y 2‡ y2)=2( s - s ) d y

dx‡ C (m , y2=2( s - s ))

C (m )d (x ‡ 1= b )

(4:8)

where C (m , h ) is the incomplete gamma function. For these exponents the volatility declines fastenough to prevent y from becoming negative, but it doesn’t prevent y from reaching the origin.This causes a very small, but �nite, probability of y being exactly zero. As before, thisprobability 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 ) ˆ(y= y )m - 1=2

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

dx1 -

4m 2 - 18y y

( s - s ) ‡ . . .

( )

(4:9)

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

y(x) ˆ b - 1 log (1 ‡ b x) (4:10a)

Then the transition density is simply

p(t, x; t , x ) ˆe - b y

[2p ( s - s )]1=2e - ( y - y‡ b ( s - s )=2)2=2( s - s ) (4:10b)

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

M (t, x; T ) ˆ log…

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

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 ) (4:11b)

where

S ˆ b 2( s - s )=(1 ‡ b x)2(1- g ) H ˆ k (T )(1 ‡ b x)= b (4:11c)

This is useful since for each exponent g , the function M (S, H ) can be calculated and tabulatedonce, eliminating most of the computational burden of the b – g model. Once these tables havebeen calculated, the values Z(t, x; T ) of zero coupons bonds, the forward rate curve f (t, x; T ),and the short rate r(t, x) can be obtained directly from (3.8)–(3.10).

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


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dX (t) ˆ a (t)[1 ‡ b (t)X (t)]g d bW (t) (4:12)

Here we quote only the results with b constant.When g ˆ 0, the b –g model becomes the extended Hull–White model. In this case A(T ) and

M(t, x; T ) are given by

A(t) ˆ 12k

2(T ) s M(t, x; T ) ˆ 12k

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

where

s ˆ…T

0a 2(t 9 )dt 9 s ˆ

… t

0a 2(t 9 )dt 9 (4:14)

Equations 3.8–3.10 now yield explicit expressions for the value of zero coupon bonds, theforward rate curve, and the short rate.

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

A(T ) ˆ k 2(T ) s2 ‡ b k (T ) s

M (t, x; T ) ˆ(1 ‡ b x)k 2(T )( s - s )

2 ‡ b k (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 beobtained from (3.8)–(3.10). Besides deriving the corresponding formulas when b ˆ b (t), inAppendix C we �nd that b (t) can be chosen so that this model is exactly the CIR model.Alternatively, one can use the extra freedom of choosing b (t) to calibrate the model to a seriesof off-market instruments as well as a series of at-the-money instruments, and thus reproduce thevolatility smile.

The exponent g ˆ 1 (Black–Karasinski like models) is more dif�cult. Although we have beenunable to derive explicit expressions for A(T ) and M(t, x; T ), the moments mk (t, x; T ) can bewritten explicitly. When b is constant,

m2(t, x; T ) ˆ [(1 ‡ b x)= b ]2[e b 2( s - s ) - 1] (4:16a)

m3(t, x; T ) ˆ [(1 ‡ b x)= b ]3[e3b 2( s - s ) - 3e b 2( s - s ) ‡ 2] (4:16b)

m4(t, x; T ) ˆ [(1 ‡ b x)= b ]4[e6b 2( s - s ) - 4e3b 2( s - s ) ‡ 6e b 2( 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 theseexpansions into (3.8) and (3.10) to obtain Z(t, x; T ), f (t, x; T ), and r(t, x). Even if only thesecond moment is used, this should be accurate enough to price all but the most sensitiveinstruments.

4.1. European option prices

A vanilla receiver swaption is a European option which gives the holder the right to receive apredetermined series of cash payments C i on dates t i, i ˆ 1, 2, . . . , n, in return for essentially7

paying K on the settlement date ts. A payor swaption is a European option which gives the

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


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holder the right to receive the strike K at settlement in return for making a predetermined seriesof payments. Similarly, a caplet or �oorlet is a European option which gives the holder the rightto exchange speci�c payments with the issuer.

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

V (0, 0) ˆ D(0, te )…

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

Here

Q(te, xe ) ˆX

i

C i Z(te, xe ; t i) - KZ(te, xe ; ts)

( )‡

(4:17b)

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

V (0, 0) ˆ…

p(0, 0; te, xe )X

i

C i D(0, t i) e - k i xe - Ai‡M(te,xe ;t i )

(

- KD(0, ts) e - k s xe - As‡M( te ,xe ; ts )‡

dxe (4:18)

where k i ˆ k (t i), k s ˆ k (ts), etc. Since the transition densities p(0, 0; te, xe ) are known and thefunction M (t, x; T ) is pre-tabulated, these options can be valued exactly with a single integration.

Let F0 be today’s forward value of the cash �ow at settlement,

F0 ˆX

i

C i D(ts, t i ) (4:19)

The implied price vol of the option8 is de�ned as the volatility r B for which Black’s formula,

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

d1,2 ˆlog F0=K 1

2r2B te

r B��t

pe

(4:20b)

yields the correct price V0 given by (4.18).In Hagan and Woodward (in preparation) singular perturbation techniques are used to obtain

accurate approximations to the option price (4.18), with the results stated in terms of implied pricevols. There it is found that to leading order the option price is

r B ˆ 1te

… te

0a 2(t 9 )dt 9

1=2

K 1(1 ‡ b x0)g ‡ . . . (4:21a)

where x0 is de�ned implicitly by

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


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12(F0 ‡ K ) ˆ

X

i

C i D(ts, t i) e - ( k i - k s )x0 (4:21b)

Here

K 1

PiC i D(ts, t i)( k i - k s) e - k i x0

PiC i D(ts, t i) e - k i x0

(4:21c)

A much more accurate formula is quoted in Appendix F.Equation 4.21a illustrates the roles played by local volatility, mean reversion, and skew in

pricing. The local volatility a (t) only enters (4.21a), which shows that the implied price vol r B isproportional to the mean-square-average of the local volatility between today and the expirationdate.

Recall that k 9 (t) is a decreasing function, and that ‘mean reversion’ refers to how rapidly k 9 (t)decreases (see (3.10) et seq). As mean reversion increases, k (t i) - k (ts ) k i - k s decreases,which decreases the value of K 1. Thus, as mean reversion increases, the price vol r B of theswaption decreases, as expected. Increasing the mean reversion also increases the importance ofthe earlier payments C i relative to the later payments.

Equations 4.19 and 4.21b imply that x0 ˆ 0 when the option is struck at-the-money (K ˆ F0), andthat x0 is a decreasing function of the strike K . Thus, the price of an at-the-money swaption isindependent of b and g , at least to within the approximations used to obtain (4.21). If b or g is zero,the price vol is independent of the strike K ; as b and g increase from zero, the implied price volbecomes a decreasing function of the strike K .

Besides giving insight into how the model parameters affect option prices, Equation 4.21 canbe used effectively in model calibration. Calibration is the process of choosing the modelparameters a (t) and k (t) to minimize or eliminate the errors between the model’s predictedprices and the actual prices of a set of standard market instruments. This is typically aniterative process, with most of the computational time spent calculating derivatives of thepredicted prices with respect to variations in the model parameters. By using the approximateformula (4.21) to obtain a shrewd initial guess for the model parameters, and then using thederivatives of (4.21) in place of the actual derivatives, one can greatly speed up the calibrationprocess.

5. Conclusions.

We used HJM theory to develop a framework for creating arbitrage-free Markovian term structuremodels. This framework makes designing arbitrage-free interest rate models which match theinitial discount curve a straightforward task, freeing the designer to concentrate on imbuing themodel with less fundamental attributes. Not only are the resulting models Markovian, but theyhave no more state variables than are needed to handle the random factors. This parsimony opensup new methods for evaluating and using multi-factor models: trees, explicit and implicit �nitedifference schemes, and Green’s function techniques can be used as well as Monte Carlomethods.

All arbitrage-free Markovian term structure models we are aware of �t within this framework.


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Appendices B and C explicitly demonstrate that the extended Hull–White (1990a, b), extended CIR(Cox et al. 1985), Black–Karasinski (1991), and Jamshidian’s (1991) Brownian path independentmodels �t within this framework, and Appendix D shows that Flesaker and Hughston’s (1996)rational interest rate models also �t within the framework. The separable n-factor models are moredif�cult since they require n(n ‡ 3)=2 state variables. Appendix E shows that these models can beconstructed as degenerate n(n ‡ 3)=2 factor models. There we also create n-factor models which areanalogues of the separable models and are asymptotically equivalent to their separable counterpartsin the limit of small volatilities.

In Section 4 we used this framework to create the b –g models, a class of one-factor modelsde�ned by the process

dX (t) ˆ a (t)[1 ‡ b X (t)]g d bW (t) X (0) ˆ 0 (5:1)

and the numeraire

N (t, x) ˆ 1D(0, t)

ek ( t)x‡A( t) (5:2)

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

Describing local volatility, mean reversion, and skew is as much as can be expected from a one-factor model. However, some products are sensitive to risks which can not be described by a one-factor model. The two most common risks are sensitivity to rate decorrelation/curve �exing andforward/stochastic volatilities. Decorrelation risk occurs because forward rates become progressivelyless correlated as the difference between the maturity dates (and start dates) increases; thisdecorrelation tends to make the forward rate curve ‘�ex’. In contrast, all forward rates in a one-factor model are essentially perfectly correlated. Forward volatility risk occurs in deals whichcontain an ‘option on an option’. Examples are captions (options on caps), and Bermudan andAmerican swaptions. In each case exercising the option involves either receiving or giving up anoption, so the exercise decision must balance the intrinsic value of the payoff against the value ofthe option received or lost, which is determined by the volatility environment that pertains at theexercise date.

The theoretical framework makes it easy to extend popular one-factor models to include theserisks. For example, we can convert the b – g model to a stochastic volatility model by changing theequation for the state variable to

dX (t) ˆ a (t)[1 ‡ b (t)X (t)]g Y (t)d bW1(t) X (0) ˆ 0 (5:3a)

dY (t) ˆ c (t)Y (t)d bW2(t) Y (0) ˆ 1 (5:3b)

and using the same numeraire as before,

N (t, x) ˆ 1D(0, t)

ek ( t)x‡A( t) (5:3c)

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


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

Acknowledgements

We gratefully acknowledge the assistance of A. Berner, Olivier Van Eyseren, our colleagues onParibas’ New York Swaps desk and Paribas’ London FIRST team. The views presented in thisreport do not necessarily re�ect the views of NumeriX, The Bank of Tokyo-Mitsubishi, or any oftheir af�liates.

References

Black, F. and Karasinski, P. (1991) Bond and option pricing when short rates are lognormal, FinancialAnalysts Journal, 46 52–59.

Cheyette, O. (1992) Term structure dynamics and mortgage valuation, Journal of Fixed Income 28–41.Cox, J.C., Ingersoll, J.E. and Ross, S.A. (1985) A theory of the term structure of interest rates, Economics 53

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

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 of

Economic Stochastic Processes and Theory, 20 381–408.Harrison, J.M. and Pliska, S.R. (1981) Martingales and stochastic integrals in the theory of continuous trading,

Stochastic Processes and Applications, 11 215–260.Heath, D., Jarrow. R. and Morton, A. (1990) Bond pricing and the term structure of interest rates: A discrete

time approximation, Journal of Finance and Quantitative Analysis, 25 419–440.Heath, D., Jarrow. R. and Morton, A. (1992) Bond pricing and the term structure of interest rates: A new

methodology, Econometrica, 60 77–105.Ho, T.S.Y. and Lee, S.B. (1986) Term structure movements and pricing interest rate contingent claims, Journal

of Finance, 41 1011–1028.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, 3 573–592.Hull, J. and White, A. (1990b) Ef�cient procedures for valuing European and American path-dependent

securities, Journal of Finance and Quantitative Analysis, 25 87–100.Jamshidian. F. (1991) Forward induction and constructin of yield curve diffusion models, Journal of Fixed

Income, 62–74.Li, A., Ritchken, P. and Sankarasubramanian, L. (1995) Lattice models for pricing American interest rate

claims, Journal of Finance, 50 719–737.Longstaff, F.A. and Schwartz, E.S. (1992) Interest rate volatility and the term structure: a two-factor general

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 the

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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 ) ˆ f0(T ) ‡ A 9 (T ) - ST (t, X (t); T )=S(t, X (t); T ) (A:1)

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

S t ‡X

i

l i(t, x)Sxi‡ 1

2

X

ij

rij(t)r i(t, x)r j(t, x)Sxi x j ˆ 0 0 , t , T (A:2a)

S(T , x; T ) ˆ e - h(T ,x) (A:2b)

Applying Ito’s lemma to (A.1), and using (A.2) to simplify the result, then yields

dF(t; T ) ˆ 12

X

ij

rij(t)r i(t, X )r j(t, X )(Sxi Sx j =S2)T dt -X

i

r i(t, X )(Sxi =S )T d bW i(t) (A:3)

Equation A.3 appears to violate the forward drift restriction of HJM (1992) theory. However, thisrestriction is phrased in terms of the risk-neutral probability measure in the money marketnumeraire. To switch to the money market numeraire, we apply the backwards Kolmogorov equationto the valuation formula

V (t, x) ˆ eh( t,x)‡A( t)bE V (T , X (T ))D(t, T ) e - h(T ,X(T )) - A(T )

‡…T

tC(t 9 , X (t 9 ))D(t, t 9 ) e - h(t 9 ,X( t 9 )) - A( t 9 ) dt 9 X (t) ˆ x (A:4)

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) ˆ V (t, x) e -„ t

0f0(T 9 ) dT 9 - h( t,x) - A( t) (A:5a)

satis�es the partial differential equation

bV t ‡X

i

l i(t, x)bVxi‡ 1

2

X

ij

rij(t)r i(t, x)r j(t, x)bVxi x j ˆ - C(t, x)e -„ t

0f0(T 9 ) dT 9 - h( t,x) - A( t)

(A:5b)Consequently, V (t, x) satis�es the partial differential equation

V t ‡X

i

~l i(t, x)Vxi‡ 1

2

X

ij

rij(t)r i(t, x)r j(t, x)Vxi x j - r(t, x)V ˆ - C(t, x) (A:6a)

where

~l i(t, x) ˆ l i(t, x) - r i(t, x)X

j

rij(t)r j(t, x)hx j(t, x) (A:6b)


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and

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

i

l i(t, x)hxi(t, x)

‡ 12

X

ij

rij(t)r i(t, x)r j(t, x)[hxi x j(t, x) - hxi

(t, x)hx j(t, x)]: (A:6c)

Expression (A.6c) for the short rate appears to be different from (2.10a). We can show that thesetwo expressions are equivalent by applying Ito’s lemma to e - h(T ,X (T )) in (2.6b), taking the expectedvalue, and letting T ! t. This yields

-ST (t, x; t)S(t, x; t)

ˆ h t ‡X

i

l i(t, x)hxi‡ 1

2

X

ij

rij(t)r i(t, x)r j(t, x)[hxi x j - hxi hx j ] (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)

ˆ hxi (t, x) (A:7b)

De�ne the money market numeraire B(t) by

dB(t)B(t)

ˆ r(t, X (t)) dt (A:8)

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

V (t, x) ˆ eE B(t)B(T )

V (T , X (T )) ‡…T

t

B(t)B(t 9 )

C(t 9 , X (t 9 )) dt 9 X (t) ˆ x

( )

(A:9)

where eE refers to the expected value in a probability measure for which

dX i(t) ˆ ~l i(t, X (t)) dt ‡ r i(t, X (t)) d eW i(t) i ˆ 1, 2, . . . , (A:10a)

Here eW1(t), eW2(t), . . . , eW n(t) are Brownian motions with the same correlation structure asbefore:

d eW i(t)d eW j(t) ˆ rij(t)dt (A:10b)

Clearly, (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 is

equivalent to using Girsanov’s theorem to change the probability measure so that

d eW i(t) ˆ d bW i(t) ‡X

j

rij(t)r j( t, X )hx j (t, X )dt i ˆ 1, 2, . . . , n, (A:11)

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

dF(t, T ) ˆX

ij

rij(t)aiT (t, X ; T )a j(t, X ; T )dt -

X

i

aiT (t, X ; T )d eW i(t) (A:12a)


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with

a i(t, x; T ) ˆ r i(t, x)Sxi(t, x; T )S(t, x; T )

-Sxi (t, x; t)S(t, x; t)

(A:12b)

where we have used (A.7b) to replace hxi (t, x). Inspection of the drift term in (A.12a) shows thatthe forward rate process satis�es 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) ˆ eE e -„ T

tR(t 9 ) dt9 V (T , R(T )) ‡

…T

tC(t 9 , R(t 9 )) e -

„ t 9

tR( t 0 ) dt 0 dt 9 R(t) ˆ r

( )

(B:1a)

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

dR(t) ˆ [ h (t) ‡ l (t, R)] dt ‡ s(t, R)d eW (t) (B:1b)

under this numeraire. Here the function h (t) must be selected to match the initial discount curve.With some ingenuity, h (t) can be found analytically for some models; otherwise a forwardinduction 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 asthe numeraire. Equations B.1a and B.1b imply that V (t, r) satis�es

V t ‡ [ h (t) ‡ l (t, r)]V r ‡ 12s

2(t, r)V rr - rV ˆ - C(t, r) (B:2)

Let us select a maturity T ref , and de�ne the zero coupon bond

Z(t, r; T ref ) ˆ eEfe -„ Tref

tR( t 9 ) dt9 j R(t) ˆ rg (B:3)

Then Z solves the partial differential equation

Z t ‡ [h (t) ‡ l (t, r)]Z r ‡ 12s

2(t, r)Z rr - rZ ˆ 0 t , T ref (B:4a)

Z(T ref , r; T ref ) ˆ 1 (B:4b)

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

bV (t, r) ˆV (t, r)

Z(t, r; T ref )(B:5)

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

bV t ‡ [ h (t) ‡ ^l (t, r)]bV r ‡ 12s

2(t, r)bV rr ˆ -C(t, r)

Z(t, r; T ref )(B:6a†

where


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^l (t, r) ˆ l (t, r) ‡ s2(t, r)Z r(t, r; T ref )Z(t, r; T ref )

(B:6b)

This is equivalent to

V (t, r) ˆ Z(t, r; T ref )bE V (T , R(T ))Z(T , R(T ); T ref )

‡…T

t

C(t 9 , R(t 9 ))Z(t 9 , R(t 9 ); T ref )

dt 9 R(t) ˆ r

( )(B:7a)

where R(t) evolves according to

dR(t) ˆ [ h (t) ‡ ^l (t, r)] dt ‡ s(t, r)d bW (t) (B:7b)

as can be seen by applying the backwards Kolmogorov equation to (B.7a). Note that thisargument represents an extension of Jamshidian’s (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) ˆ l (t, X (t)) dt ‡ r (t, X (t)) d bW (t) (B:8a)

under the numeraire

N (t, x) ˆ 1D(0, t)

eh( t,x)‡A( t) (B:8b)

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

A(t) ˆ log bEfe - h( t,X (t)) j X (0) ˆ 0g (B:8c)

Once this is done, the short rate is given by

r(t, x) ˆ f0(t) ‡ A9 (t) ‡ h t(t, x) ‡ l (t, x)hx(t, x) ‡ 12r

2(t, x)[hxx(t, x) - h2x(t, x)] (B:9)

See (A.6c).To re-write this as a short-rate model requires switching to the money market numeraire. Using

(A.11), the risk-neutral process for X (t) is

dX (t) ˆ [ l (t, X ) - r 2(t, X )hx(t, X )] dt ‡ r (t, X )d eW (t): (B:10)

under the money market numeraire. Applying Ito’s lemma to (B.9) now shows that the processfor the short rate, R(t) r(t, X (t)), is

dR(t) ˆ [rt ‡ ( l - r 2hx )rx ‡ 12r

2rxx ]dt ‡ r rx d eW (t): (B:11)

We see that expressing an arbitrary one-factor model (B.8a)–(B.8c) as a short rate model requiresinverting the relationship R ˆ r(t, x) given in (B.9) to obtain x ˆ x(t, R), and then substituting thisinto (B.11) to obtain the drift and volatility terms as functions of t and R. This is a tedious (andneedless) process, but it simpli�es signi�cantly for several classes of models. For example, considerthe class of models examined in Section 3. For these models l (t, x) 0 and h(t, x) k (t)x, so thecorresponding short rate models are


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dR(t) ˆ [rt - k (t)r 2(t, x)rx ‡ 12r

2(t, x)rxx] dt ‡ r (t, x)rx d eW (t) (B:12a)

where

r(t, x) ˆ f0(t) ‡ A9 (t) ‡ k 9 (t)x - 12k

2(t)r 2(t, x) (B:12b)

C Special exponents

The b – g models are de�ned by the numeraire

N (t, x) ˆ1

D(0, t)ek ( t)x‡A( t) (C:1a)

and the process

dX (t) ˆ a (t)[1 ‡ b (t)X (t)]g d bW (t) (C:1b)

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

M (t, x; T ) ˆ log bEfe - k (T )[X (T ) - x] j X (t) ˆ xg (C:2)

The value Z(t, x; T ) of zero coupon bonds, the instantaneous forward rates f (t, x; T ), and theshort rate r(t, x) can then be read off from (3.8) and (3.10).

C.1. Extended Hull-White ( g ˆ 0)

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

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

[2p ( s - s )]1=2(C:3)

where

s ˆ… t

0a 2(t 9 )dt 9 s ˆ

… t

0a 2(t 9 )dt 9 (C:4)

Integrating to compute the expected value in (C.2) yields

M (t, x; T ) ˆ 12k

2(T )( s - s ) A(T ) ˆ 12k

2(T ) s (C:5)

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

r(t, x) ˆ f0(t) ‡ k 9 (t)x ‡ k 9 (t)k (t)s (t) (C:6)

Equation B.12 now shows that the equivalent short rate model is

dR(t) ˆ [ h (t) - l (t)R(t)] dt ‡ s(t)d eW (t) (C:7a)

under the money market numeraire, where


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l (t) ˆ - k 0 (t)= k 9 (t) s(t) ˆ k 9 (t)a (t) h (t) ˆ f 90(t) ‡ l (t) f0(t) ‡ [ k 9 (t)]2 s (t) (C:7b)

The extended Hull–White model is usually expressed as the short rate process (C.7a), where l (t)and s(t) are arbitrary model parameters and h (t) is chosen as

h (t) ˆ f 90(t) ‡ l (t) f0(t) ‡… t

0s2(t 9 ) e - 2

„ t

t 9l ( t 0 ) dt 0 dt 9 (C:8)

to match the initial discount curve. Note that mean reversion is expressed through the parameterl (t) in the extended Hull–White model, while it is expressed through the ‘gearing factor’ k (t) inthe equivalent b –g model.

C.2. CIR-like models ( g ˆ 12 )

When g ˆ 12 the state variable obeys

dX (t) ˆ a (t)[1 ‡ b (t)X (t)]1=2 d bW (t) (C:9)

Consequently, the transition density p(t, x; t , x ) satis�es the forward Kolmogorov equation

p t ˆ 12a

2( t )([1 ‡ b ( t )x ]p) x x t . t (C:10a)

p ! 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

- 1e - k x p(t, x; t , x )dx (C:11)

which yields

P t ˆ 12a

2( t )k 2fP - b ( t )Pk g t . t (C:12a)

P ˆ e - k x at t ˆ t (C:12b)

This is a �rst-order hyperbolic equation which can be solved by the method of characteristics.We obtain

log P(t, x; t , k ) ˆ - k x ‡ k 2 x[B( t ) - B(t)]2 ‡ k [B( t ) - B(t)]

‡… t

t

2a 2(t 9 )dt 9(2 ‡ k [B( t ) - B(t 9 )])2

( )

(C:13a)

where

B(t) ˆ… t

0a 2(t 9 )b (t 9 )dt 9 (C:13b)

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

M (t, x; T ) ˆ k 2(T )x[B(T ) - B(t)]

2 ‡ k (T )[B(T ) - B(t)]‡

…T

t

2a 2(t 9 )dt 9(2 ‡ k (T )[B(T ) - B(t 9 )])2

( )

(C:14a)


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A(T ) ˆ k 2(T )…T

0

2a 2(t 9 )dt 9(2 ‡ k (T )[B(T ) - B(t 9 )])2

(C:14:b)

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

r(t, x) ˆ f0(t) ‡ K(t)[x ‡ C(t)] (C:15a)

with

K(t) ˆ k 9 (t) - 12k

2(t)a 2(t)b (t) C(t) ˆ k (t)… t

0

a 2(t 9 )dt 9(1 ‡ k (t)[B(t) - B(t 9 )]=2)3

(C:15b)

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

dR(t) ˆ [ h (t) - l (t)R(t)] dt ‡ s(t)��d(t) ‡ R(t)

pd eW (t) (C:16a)

under the money market numeraire, where

l (t) ˆ - [K 9 (t) - k (t)a 2(t)b (t)]=K(t) (C:16:b)

h (t) ˆ f 90(t) ‡ l (t) f0(t) ‡ K(t)C 9 (t) - k (t)a 2(t)K(t)[1 - b (t)C(t)] (C:16c)

s(t) ˆ a (t)��K(t)b (t)

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

The extended CIR model is usually expressed as

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

pd eW (t) (C:17)

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

Z(t, r; T ) ˆ eEfe -„ T

tR( t9 ) dt 9 j R(t) ˆ rg (C:18)

Equivalently, Z(t, r; T ) is the solution of the backwards equation

Z t ‡ [h (t) - l (t)r]Z r ‡ 12s

2(t)rZ rr - rZ ˆ 0 t , T (C:19a)

with

Z(T , r; T ) ˆ 1 (C:19b)

The solution of (C.19) is

Z(t, r; T ) ˆ e - B(t,T )r - A( t,T ) (C:20a)

where B(t, T ) satis�es the Ricatti equation

B t ˆ l (t)B ‡ 12s2(t)B2 - 1 t , T (C:20b)

with the boundary condition B(T , T ) ˆ 0, and

A(t, T ) ˆ…T

th (t 9 )B(t 9 , T )dt 9 (C:20c)


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

0f0(T 9 ) dT 9 , which requires

h (t) to be chosen so that…T

0h (t 9 )B(t 9 , T )dt 9 ˆ

…T

0f0(T 9 )dT 9 - B(0, T ) f0(0) (C:21)

We see that matching the extended CIR model to the initial discount curve requires solving aRicatti equation to obtain the zero coupon bond values. This can be done ef�ciently, but it isdif�cult 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 b – gmodel with g ˆ 1=2 and with b (t) chosen so that d(t) ˆ 0. Unsurprisingly, making this choice ofb (t) also requires solving a Ricatti equation.

A simpler alternative is to use the g ˆ 1=2 model without requiring d(t) 0. The g ˆ 1=2 modelthen gives the zero coupon bond values directly, without solving a Ricatti equation. In addition, b (t)can be used to match the implied volatility smile by calibrating the model to the prices of off-market instruments. Although the CIR model is more aesthetically appealing, with its natural barrieroccuring at R(t) ˆ 0 and its simple relation between the volatility and the short rate, we stress thatneither model can be expected to be valid near R(t) ˆ 0. Since the barrier in the g ˆ 1=2 model isirrelevant for pricing commonly traded US instruments, it seems better to use b (t) to match theimplied volatility smile.

C.3. Black–Karasinski-like models (g ˆ 1)

When g ˆ 1 we have been unable to �nd the Laplace transform of the transition density, andthus M (t, x; T ). However, we can �nd the moments mk (t, x; T ) explicitly, and then use theexpansions

M (t, x; T ) ˆ 12k

2(T )m2(t, x; T ) - 16k

3(T )m3(t, x; T ) ‡ . . . (C:22a)

A(T ) ˆ 12k

2(T )m2(0, 0; T ) - 16k

3(T )m3(0, 0; T ) ‡ . . . (C:22b)

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

With the exponent g ˆ 1,

dX (T ) ˆ a (T )[1 ‡ b (T )X (T )] d bW (T ) (C:23)

and applying Ito’s lemma yields

d[X (T ) - x]k ˆ 12k(k - 1)a 2(T )[1 ‡ b (T )X ]2[X - x]k - 2 dT

‡ k a (T )[1 ‡ b (T )X ][X - x]k - 1 d bW (T ) (C:24)

Taking the expected value then yields equations for the moments mk (t, x; T ):

dmk

dTˆ 1

2k(k - 1)a 2(T )f(1 ‡ b x)2m k - 2 ‡ 2b (1 ‡ b x)mk - 1 ‡ b 2mkg (C:25)


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

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

m2(t, x; T ) ˆ…T

ta 2(t 9 )[1 ‡ b (t 9 )x]2exp

…T

t 9a 2(T 9 )b 2(T 9 )dT 9

( )

dt 9 (C:26b)

m3(t, x; T ) ˆ 6…T

t

… t 9

ta 2(t 9 )a 2(t 0 )b (t 9 )[1 ‡ b (t 9 )x][1 ‡ b (t 0 )x]2

. exp 3…T

t 9a 2(T 9 )b 2(T 9 )dT 9 ‡

… t 9

t 0a 2(T 9 )b 2(T 9 ) dT 9

( )

dt 0 dt 9 (C:26c)

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

D Rational interest rate models

Consider the interest rate model de�ned by the valuation formula

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

‡…T

t

C(t 9 , X(t 9 ))N (t 9 , X (t 9 ))

dt 9 X (t) ˆ x

( )

(D:1a)

with the numeraire

N (t, x) ˆ 1D(0, t) ‡ b(t) . x

(D:1b)

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

dX i(t) ˆ r i(t, X )d bW i(t) X i(0) ˆ 0 i ˆ 1, 2, . . . , n (D:1c)

under this numeraire in the risk-neutral world. Noting that X (t) is a Martingale, we �nd that thezero coupon bond prices are

Z(t, x; T ) ˆ bE D(0, T ) ‡ b(T ) . X (T )D(0, t) ‡ b(t) . x

X (t) ˆ x

( )ˆ D(0, T ) ‡ b(T ) . x

D(0, t) ‡ b(t) . x(D:2)

Consequently, the instantaneous forward rate curve is

f (t, x; T ) ˆ -DT (0, T ) ‡ b9 (T ) . xD(0, T ) ‡ b(T ) . x

(D:3)

Flesaker and Hughston’s (1996) rational lognormal model is the special caser i(t, x) ˆ a i(t)(1 ‡ xi). These models have the advantage that all forward rates f (t, x; T ) areguaranteed to remain positive provided that b 9i(T ) < 0 for each i and DT (0, T ) ,

Pb 9i(T ) for all T.


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

E Separable models

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

dF(t, x (t); T ) ˆX

ij

rij(t)aiT (t, x (t); T )a j(t, x (t); T )dt -

X

i

a iT (t, x (t); T )dfW i(t) (E:1)

in the risk-neutral world under the money market numeraire:

V (t) ˆ eE e -„ T

tR( t 0 ,x 0 ) dt 0 V (T , x (T )) ‡

…T

tC(t 9 , x (t 9 )) e -

„ t 9

tR( t 0 ,x 0 ) dt 0 dt 9 F t

( )

(E:2)

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

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 ) ˆ k 9i(T )r i(t, x (t)) i ˆ 1, 2, . . . , n (E:3a)

so that

a i(t, x (t); T ) ˆ [ k i(T ) - k i(t)]r i(t, x (t)) i ˆ 1, 2 . . . , n (E:3b)

Then the stochastic process for the forward rates becomes

dF(t, x (t); T ) ˆX

ij

k 9i(T )[k j(T ) - k j(t)]rij(t)r i(t, x (t))r j(t, x (t)) dt

-X

i

k 9i(T )r i(t, x (t)) d eW i(t) (E:4)

Integrating over t now yields

F(t, x ; T ) ˆ f0(T ) ‡X

i

k 9i(T )X i(t, x ) ‡X

ij

k 9i(T )k j(T )Yij(t, x ) (E:5)

where

X i(t, x ) ˆ -… t

0r i(t 9 , x 9 )d eW i(t 9 ) -

X

j

… t

0k j(t 9 )rij(t 9 )r i(t 9 , x 9 )r j(t 9 , x 9 )dt 9 i ˆ 1, . . . , n

(E:6a)

Yij(t, x ) ˆ… t

0rij(t 9 )r i(t 9 , x 9 )r j(t 9 , x 9 )dt 9 i, j ˆ 1, . . . , n (E:6b)

One now assumes that the coef�cients r i(t, x (t)) depend on the path taken by the Brownian


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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.5)expresses the entire term structure in terms of these variables. With this assumption, (E.6) becomes

dX i(t) ˆ - r i(t, X , Y ) d eW i(t) ‡X

j

k j(t)rij(t)r j(t, X , Y )dt

( )X i(0) ˆ 0 i ˆ 1, . . . , n

(E:7a)

dYij(t) ˆ rij(t)r i(t, X , Y )r j(t, X , Y )dt Yij(0) ˆ 0 i, j ˆ 1, . . . , n (E:7b)

and the instantaneous forward rate curve is

f (t, x, y; T ) ˆ f0(T ) ‡X

i

k 9i(T )xi ‡X

ij

k 9i(T )k j(T )yij (E:8)

where xi ˆ X i(t) and yij ˆ Yij(t) are the values of the state variables at time t.Under the money market numeraire, the general separable model is de�ned by the forward rate

curve (E.8) and the stochastic processes (E.7). We can put this model into the context of Section 2by switching to the numeraire

N (t, x, y) ˆ 1D(0, t)

expX

i

k i(t)xi ‡ 12

X

ij

k i(t)k j(t)yij

( )(E:9)

Under this numeraire the value of a tradable instrument is

V (t, x, y) ˆ

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

‡…T

t

C(t 9 , X (t 9 ), Y (t 9 ))N (t 9 , X (t 9 ), Y (t 9 ))

dt 9 X (t) ˆ x, Y (t) ˆ y

( )

(E:10)

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

dX i(t) ˆ - r i(t, X , Y )d bW i(t) X i(0) ˆ 0 i ˆ 1, . . . , n (E:11a)

dYij(t) ˆ rij(t)r i(t, X , Y )r j(t, X , Y )dt Yij(0) ˆ 0 i, j ˆ 1, . . . , n (E:11b)

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

Z(t, x, y; T ) ˆ D(t, T )exp -X

i

[k i(T ) - k i(t)]xi -12

X

ij

[ k i(T )k j(T ) - k i(t)k j(t)]yij

( )

(E:12)

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

Since Y ij(t) ˆ Y ji(t), the n-factor separable model generally requires n(n ‡ 3)=2 state variables.As the random processes Yij(t) are not being driven directly by Brownian motions, we can only viewthe separable models within the framework of Section 2 by considering (E.11) as a highlydegenerate n(n ‡ 3)=2 factor model.

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


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account for convexity effects in an arbitrage-free manner. Since convexity effects are quite small,and since (E.11b) shows that the Yij(t) are only driven indirectly by the stochastic processes dcW i(t),the distributions of Yij(t) are highly peaked about their means, which are near zero. Speci�cally, asthe volatilities r i decrease, the means of the variables Yij(t) scale like r 2 t and the variances ofYij(t) scale like r 6t3. This suggests that the convexity effects can be accounted for by replacing thedistributions of Yij(t) by d -functions, selecting the position of the d -functions to make the theoryarbitrage free. Carrying this idea through leads to the arbitrage-free n-state variable model de�nedby the valuation formula

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

‡…T

t

C(t 9 , X(t 9 ))N (t 9 , X (t 9 ))

dt 9 X (t) ˆ x

( )

the numeraire

N (t, x) ˆ 1D(0, t)

ek ( t).x‡A( t)

and the risk neutral processes

dX i(t) ˆ - r i(t, X )d bW i(t) X i(0) ˆ 0 i ˆ 1, . . . , n

under this numeraire. Here A(T ) is de�ned by

A(T ) ˆ log bEfe - k(T ).X(T )jX (0) ˆ 0g

This model is the multi-factor generalization of the one-factor models in Section 3. Following theanalysis there shows that the zero coupon bond prices are

Z(t, x; T ) ˆ D(t, T )e - [k(T ) - k( t)].x- A(T )‡A( t)‡M( t,x;T )

where

M (t, x; T ) ˆ log bEfe - k(T ).[X(T ) - x] j X (t) ˆ xg

Consequently the instantaneous forward rates are

f (t, x; T ) ˆ f0(T ) ‡ k9 (T ) . x ‡ A 9 (T ) - M T (t, x; T )

Typically k 9i(T )r i is around 10 - 2 in US dollar markets. At these volatilities the n-state variablemodel (E.13) is virtually identical to the analogous separable model in (E.9)–(E.11). Although then-state variable model has the disadvantage of requiring computation of M(t, x; T ) to obtain thezero coupon bond prices, this is usually outweighed by having many fewer state variables than theseparable 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 �owsC i on dates t i, i ˆ 1, 2, . . . , n in return for paying the strike K on the settlement date ts. Alsode�ne


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F0 ˆX

i

C i D(ts, t i ) s e

… te

0a 2(t 9 )dt 9 (F:1)

as before. To express the option prices succinctly, de�ne t0 by the relation… t0

0a 2(t 9 )dt 9 ˆ s e=2 (F:2)

de�ne x by the implicit relation

12(F0 ‡ K ) ˆ

X

i

C i D(ts, t i) e - [ k i - k s]x - Ai‡As‡M(t0,x ;t i) - M( t0,x ;t s ) (F:3)

and de�ne K k by

K k ˆP

iC i D(ts, t i)[k i - k s - M x(t0, x ; t i) ‡ M x(t0, x ; ts )]k e - k i x - A i‡M( t0,x ; t i )P

iC i D(ts, t i) e - k i x - Ai‡M( t0,x ;t i)(F:4)

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

r B ˆ K 1( s e= te )1=2(1 ‡ b x )g 1 ‡ 124(1 ‡ b x )2g s e[ K 2

1 -g (2 - g )b 2

(1 ‡ b x )2‡ 2

K 3

K 1- 3

K 22

K 21

]

(

‡ (F0 - K )2

6K 21(F0 ‡ K )2

[2K 21 -

g (1 ‡ g )b 2

(1 ‡ b x )2‡ 3g b K 2

(1 ‡ b x )K 1‡ K 3

K 1- 3

K 22

K 21

] ‡ . . . (F:5)

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


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markov interest rate models

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