A Complete Markovian Stochastic Volatility Model in the HJM Framework

Asia-Pacific Financial Markets7: 293–304, 2000.© 2001 Kluwer Academic Publishers. Printed in the Netherlands.

293

A Complete Markovian Stochastic VolatilityModel in the HJM Framework

CARL CHIARELLA and OH KANG KWONSchool of Finance and Economics, University of Technology, Sydney, P.O. Box 123, Broadway,NSW 2007, Australia, e-mail: [email protected]; [email protected]

Abstract. This paper considers a stochastic volatility version of the Heath, Jarrow and and Morton(1992) term structure model. Market completeness is obtained by adapting the Hobson and Rogers(1998) complete stochastic volatility stock market model to the interest rate setting. Numerical sim-ulation for a special case is used to compare the stochastic volatility model against the traditionalVasicek (1977) model.

Key words: Health-Jarrow-Morton model, interest rate derivatives, stochastic volatility, termstructure of interest rates.

1. Introduction

The Heath, Jarrow and Morton (HJM) term structure model provides a consistentframework for the pricing of interest rate derivatives. The model is automaticallycalibrated to the currently observed yield curve, and is complete in the sense thatit does not involve the market price of interest rate risk, something which was afeature of the early generation of interest rate models, such as Vasicek (1977) andCox, Ingersoll and Ross (1985). The fundamental quantity driving the dynamicsof the HJM model is the forward rate volatility process, which, together with theinitial yield curve, is the main input to the model.

A great deal of research in this area focused on the different classes of interestrate models that arise from different assumptions about the form of the forward ratevolatility process. Originally the focus was on forward rate volatility processesthat depended on some function of time to maturity and the instantaneous spotrate of interest rate, as in Cheyette (1992), Carverhill (1994), Ritchken and Sank-arasubramanian (1995), Bhar and Chiarella (1997) and Inui and Kijima (1998).Subsequently, Chiarella and Kwon (1998c) and de Jong and Santa-Clara (1999)considered forward rate volatility processes dependent on time to maturity and avector of fixed tenor forward rates. The essential characteristic of all these modelsis that the form of the forward rate volatility processes allows them to be trans-formed to Markovian form, at the expense, however, of increasing the dimensionof the underlying state space.

294 CARL CHIARELLA AND OH KANG KWON

Chiarella and Kwon (1998b) have considered forms of the forward rate volat-ility process that yield some of the popular interest rate models, such as the Hulland White one and two factor models. All of the forward rate volatility processesreferred to above could be described as level dependent volatility processes. Itis also of interest to consider volatility processes that are themselves diffusionprocesses. Chiarella and Kwon (1998a) have investigated such a class of modelswhere the Wiener processes driving the diffusion process for the volatility processare independent of the Wiener processes driving the forward rate process. Thisclass of models also turns out to be Markovian, and so it is possible to generate, inthe HJM framework, a class of stochastic volatility models that are in some sensethe counterpart of the Hull and White (1987), Heston (1993) and Scott (1997)stochastic volatility models. In common with these models, they are incompleteas they involve the market price of risk that arises from the independent Wienerprocesses that drive the stochastic volatility process.

Hobson and Rogers (1998) introduced a special class of complete stochasticvolatility models in the standard Black and Scholes stock option framework. Theyobtained market completeness by setting up a class of diffusion processes which,ultimately, are driven by the same Wiener process that drives the underlying assetprice.

The aim in this paper is to obtain the counterpart to the Hobson and Ro-gers (1998) complete stochastic volatility model in the HJM framework. This isachieved by suitably adapting the offset processes, that ultimately depend on theWiener processes driving the forward rate process, and feed into the forward ratevolatility process.

The plan of the paper is as follows. Section 2 outlines the model and obtains aformula for the bond price in terms of the state variables of the model. Section 3considers a special case, analogous to the special case considered by Hobson andRogers, that is the basis of the numerical calculations undertaken in Section 4, andSection 5 concludes.

2. The Model

The stochastic volatility term structure model introduced in this paper is based onChiarella and Kwon (1998a), but with the volatility process modeled along thelines of Hobson and Rogers (1998) stochastic volatility stock price model.

Let M(n) be ann-dimensional risk-neutral HJM model on a complete filteredprobability space(�,F , {Ft},P), whereFt is generated by ann-dimensionalstandardP-Wiener process(Wi

t )16i6n. Let Pt,T be the price of aT -maturity zerocoupon bondat timet . Then the instantaneousforward rateft,T is defined as

ft,T = −∂ lnPt,T∂T

, (2.1)

and the instantaneousspot rateis defined asrt = ft,t . Themoney market accountrepresenting the value of initial unit investment at timet is defined byBt = e

∫ t0 rsds,

A COMPLETE MARKOVIAN STOCHASTIC VOLATILITY MODEL 295

and it was shown in Heath, Jarrow and Morton (1992) that thediscounted bondprice, Pt,T /Bt , is a P-martingale. It was shown in Brace, Gatarek and Musiela(1997) that if a different parametrisation,T = t + τ , is used for maturity, then thecorrespondingP-martingale is e−

∫ t0 (rs−fs,s+τ )dsPt,t+τ .

Although most of the ideas in this paper remain valid for the generalM(n), forsimplicity of notation, computations will be carried out only for the special caseM =M(1), and the 1-dimensional Wiener processW 1

t will be writtenWt .For 0< τ ∈ R andi ∈ N, define theoffset1 processS(i)t,τ by

S(i)t,τ =

∫ ∞0λe−λu(Zt,t+τ − Zt−u,t−u+τ )i du , (2.2)

whereZt,t+τ is the logarithm of theP-martingale processe−∫ t

0 (rs−fs,s+τ )dsPt,t+τ , sothat

Zt,t+τ = ln(e−∫ t

0 (rs−fs,s+τ )dsPt,t+τ ) . (2.3)

For eachi, the offset processS(i)t,τ is best regarded as an exponentially weightedhistorical moment of returns on aτ -maturity bond.

Fix n1, n2 ∈ N and 06 τ1 < τ2 < · · · < τn2 ∈ R, and define

St = {S(i)t,τj : 16 i 6 n1, 16 j 6 n2} . (2.4)

Assume now that the forward rate volatility,σt,T , becomes stochastic through itsdependence on the offset processes. Thus, inM, σt,T has the form

σt,T = ς(St )e−∫ Tt κ(u)du , (2.5)

whereς and κ are deterministic functions. Models similar toM, but with σt,Tdependent at most ont, T andr(t), were considered by Ritchken and Sankarasub-ramanian (1995), Inui and Kijima (1998) and Chiarella and Kwon (1998d), whoshowed that the respective HJM models transform to Markovian systems, andderived formulae for bond prices in terms of the Markovian state variables. It ispossible to show that a similar reduction to a Markovian form is possible forσt,Thaving the form (2.5).

The reduction to a Markovian form requires the introduction of a number ofsubsidiary quantities. For 06 τ ∈ R, define

αt,τ = e−∫ t+τt κ(u)du, βt,τ = αt,τ

∫ t+τ

t

αt,u−tdu

α̂t,τ =∫ t+τ

t

αt,u−tdu, β̂t,τ =∫ t+τ

t

βt,u−tdu ,(2.6)

ξt,τ =∫ t

0σ 2s,t+τds ,

ζt,τ =∫ t

0σs,t+τ

∫ t+τ

s

σs,ududs +∫ t

0σs,t+τdWs .

(2.7)


The Lemmas 2.1 and 2.2 show that the dynamics of the stochastic quantitiesξt,τandζt,τ can be expressed just in terms ofξt,0 ≡ ξt andζt,0 ≡ ζt .

LEMMA 2.1. Defineξt = ξt,0 andζt = ζt,0. Then for any06 τ ∈ R,

ξt,τ = α2t,τ ξt , (2.8)

ζt,τ = αt,τ ζt + βt,τ ξt . (2.9)

Proof. Note thatσs,t+τ = αt,τσs,t . So

ξt,τ =∫ t

0σ 2s,t+τds =

∫ t

0α2t,τσ

2s,tds = α2

t,τ

∫ t

0σ 2s,tds = α2

t,τ ξt ,

which is (2.8). Similarly,

ζt,τ =∫ t

0σs,t+τ

∫ t+τ

s

σs,ududs +∫ t

0σs,t+τdWs

=∫ t

0αt,τ σs,t

(∫ t

s

σs,ududs +∫ t+τ

t

σs,ududs

)+∫ t

0αt,τσs,tdWs

= αt,τ ζt + αt,τ∫ t

0σs,t

∫ t+τ

t

αt,u−tσs,tduds

= αt,τ ζt + βt,τ ξt ,which is (2.9). 2LEMMA 2.2. The variablesξt andζt satisfy the sde

dξt = [ς2(St )− 2κ(t)ξt ]dt , (2.10)

dζt = [ξt − κ(t)ζt ]dt + ς(St )dWt . (2.11)

Proof. First consider (2.10). Sinceσs,t = ς(Ss)e−∫ ts κ(u)du,

dξt = d∫ t

0σ 2s,tds =

[σ 2t,t +

∫ t

0

∂σ 2s,t

∂tds

]dt = [ς2(St )− 2κ(t)ξt ]dt ,

which is (2.10). Similarly,

dζt = d

[∫ t

0σs,t

∫ t

s

σs,ududs +∫ t

0σs,tdWs

]=[∫ t

0

∂

∂t

(σs,t

∫ t

s

σs,udu

)+∫ t

0

∂σs,t

∂tdWs

]dt + σt,tdWt

= [ξt − κ(t)ζt ]dt + ς(S)dWt ,

which is (2.11). 2


The following two propositions show that, under the stated assumptions, the for-ward rate curve and the bond price can be completely characterised in terms ofξtandζt .

PROPOSITION 2.3.The forward rate curve satisfies the equation

ft,t+τ = f0,t+τ + αt,τ ζt + βt,τ ξt . (2.12)

In particular, sinceαt,τ andβt,τ are deterministic, the entire forward rate curve isMarkovian with respect to the state variablesξt andζt .

Proof. From definitions offt,t+τ andζt,t+τ ,

ft,t+τ = f0,t+τ + ζt,t+τ = f0,t+τ + αt,τ ζt + βt,τ ξt ,where the last equality follows from (2.9).

PROPOSITION 2.4.The bond pricePt,t+τ is given by the formula

Pt,t+τ = P0,t+τP0,t

e−α̂t,τ ζt−12 α̂

2t,τ ξt . (2.13)

Proof. Only a sketch proof is given. For the details, see Ritchken and Sank-arasubramanian (1995, p. 60), Inui and Kijima (1998, p. 48), or Chiarella and Kwon(1998c) Theorem 4.5. Now, sincePt,t+τ = e−

∫ t+τt ft,udu, the integral

∫ t+τt

ft,udumust be computed. But from (2.12),ft,u = f0,t+u + αt,u−t ζt + βt,u−t ξt , and so theintegral reduces to integrating deterministic functionsαt,u−t andβt,u−t with respectto the variableu. The result of performing these integrals is (2.13). 2The dynamics of the bond price follows as an immediate corollary of the aboveresults.

COROLLARY 2.5. The bond pricePt,t+τ satisfies the sde

dPt,t+τ = µτ (t, ξt , ζt ,St )dt − στ(t, ξt , ζt ,St )dWt , (2.14)

whereµτ andστ are given by

µτ (t, ξt , ζt ,St ) = Pt,t+τ [(f0,t − f0,t+τ )+ (1− αt,τ )ζt − βt,τ ξt ] (2.15)

στ (t, ξt , ζt ,St ) = Pt,t+τ α̂t,τ ς(St ) . (2.16)

Proof. Consequence of Lemma 2.2, Proposition 2.4, Proposition 2.3 and Itô’sLemma. 2

COROLLARY 2.6. The state variableZt,t+τ satisfies the sde

dZt,t+τ = −1

2α̂2t,τ ς

2(St )dt − α̂t,τ ς(St )dWt , (2.17)


whereα̂t,τ is given by (2.6), andς(St ) is as defined in (2.5).Proof. From (3),Zt,t+τ = −

∫ t0(rs − fs,s+τ )ds + lnPt,t+τ , and so

dZt,t+τ = −rt + ft,t+τ + d lnPt,t+τ . (2.18)

Application of Itô’s Lemma to lnPt,t+τ , together with Corollary 2.5, yields thedesired result. 2The dependence of the volatility function,ξt andζton the offset process means thata stochastic differential equation for the offset process is required for the overalldynamics to be specified. The next lemma provides the required result.

LEMMA 2.7. The offset processS(i)t,τj satisfies the sde2

dS(i)t,τj = iS(i−1)t,τj dZt,t+τj +

i(i − 1)

2S(i−2)t,τj d〈Z〉t,t+τj − λS(i)t,τjdt . (2.19)

Proof. The details of the proof are given in Lemma 3.1 of Hobson and Rogers(1998), and only a sketch is given here. Following Hobson and Rogers (1998),consider

eλtS(i)t,τj =∫ t

−∞λeλu(Zt,t+τj − Zt−u,t−u+τj )idu

=i∑k=0

(i

k

)Zkt,t+τj

∫ t

−∞λeλu(−Zt−u,t−t+τj )i−kdu ,

and take the ‘differential’ of both sides. This results in the equation

λeλtS(i)t,τj + eλtdS(i)t,τj = eλt[iS(i−1)t,τj dZt,t+τj +

i(i − 1)

2S(i−2)t,τj d〈Z〉t,t+τj

],

which essentially is (2.19). 2PROPOSITION 2.8.The set{ξt , ζt , Zt,t+τj , S(i)t,τj : 16 i 6 n1, 16 j 6 n2} formsa Markovian system.

Proof. Follows from Lemma 2.2, Corollary 2.5, Corollary 2.6, and Lemma 2.7.23. A Special Case

In this section, considerM corresponding to the case in whichn1 = n2 = 1. Thisis the direct analogue of the special case considered by Hobson and Rogers (1998)for the stock price model.

For notational convenience, writeτ = τ1 andSt = S(1)t,τ . Then (2.19) simplifiesto

dSt = dZt,t+τ − λStdt . (3.1)


Substituting for dZt,t+τ from (2.17) yields

dSt = −[

1

2α̂2t,τ ς

2(St)+ λSt]

dt − α̂t,τς(St)dWt , (3.2)

whereα̂t,τ is given by (2.6), andς(St ) is as defined in (2.5). Note that, as in Hobsonand Rogers (1998),St is adapted to the filtrationFt of Wt .

LEMMA 3.1. The set{ξt, ζt , St} forms a Markovian system.Proof. Follows from Proposition 2.8, and Equations (2.10), (2.11) and (3.2),

since the dynamics of the system{ξt , ζt , St } does not depend onZt,t+τ . 2Since the discounted bond price processPt,T /Bt is aP-martinale, the bond price isgiven by the expectation

Pt,T = EP[e−∫ Tt r(u)duPT,T | Ft ] = EP[e−

∫ Tt r(u)du | Ft ] , (3.3)

and using the Feynman–Kac Theorem, the bond price must satisfy the pde

KPt,T − (f0,t + ζt )Pt,T + ∂Pt,T∂t= 0 , (3.4)

subject to the terminal conditionPT,T = 1, where

K = 1

2ς2(St)

∂2

∂ζ 2t

+ 1

2α̂2t,τ ς

2(St)∂2

∂S2t

− α̂t,τ ς2(St)∂2

∂ζt∂St+

+[ς2(St)− 2κ(t)ξt ] ∂∂ξt+ [ξt − κ(t)ζt ] ∂

∂ζt−[

1

2α̂2t,τς

2(St)+ λSt]∂

∂St.

The system of sdes underlying the pde are (2.10), (2.11) and (3.2). More generally,if Ct,TC is aTC-maturity European option onPt,T with payoff h(TC), whereTC <T , then

Ct,TC = EP[e−∫ TCt r(u)duh(TC) | Ft ] , (3.5)

andCt,TC must satisfy the pde

KCt,TC − (f0,t + ζt )Ct,TC +∂Ct,TC

∂t= 0 , (3.6)

subject to the terminal conditionCTC,TC = h(TC).The pdes (3.4) and (3.6) have three spatial variables, as well as the time variable,

and are further complicated by the absence of the second order term inξt . Thesepdes are rather difficult to tackle using the standard pde solution techniques, andso Monte Carlo simulation is used in the numerical results that follow.


4. Numerical Example

Let M be the special case considered in Section 3, and for this section, assumefurther thatκ is constant and

ς(s) = η√

1+ εs2 ∧N , (4.1)

whereη, ε andN are constants. This form of the volatilityς was introduced inHobson and Rogers (1998) for the stock price model.

Now, constantκ implies

αt,τ = e−κτ , α̂t,τ = 1

κ(1− e−κτ ), βt,τ = 1

κe−κτ (1− e−κτ ) , (4.2)

and the state variablesξt , ζt , andSt satisfy the sdes

dξt = [η2(1+ εS2t ) ∧N2 − 2κξt ]dt , (4.3)

dζt = [ξt − κζt ]dt + [η√

1+ εS2t ∧N]dWt , (4.4)

dSt = −[

1

2κ2(1− e−κτ )2[η2(1+ ε2

t ) ∧N2] + λSt]

dt (4.5)

− 1

κ(1− e−κτ )[η

√1+ εS2

t ∧N]dWt . (4.6)

This system of sdes was solved numerically using Monte Carlo simulation with an-tithetic variables. A flat initial term structure of 5% was assumed for the simulation,with parameter values

η = 0.025, N = 0.2, κ = 2, λ = 5, τ = 3 .

From the definition ofξt andζt in (2.7), it is clear that the initial values for thesestate variables areξ0 = 0 andζ0 = 0. The initial value ofSt was varied within therange –1 and 1.

The effect of varying the value ofε on the distribution of the bond price isshown in Figure 1. The figure shows an increasing variance in the bond price withincreasingε, which is expected. The figure also shows that asε increases, one isless likely to observe bond prices about the mean value. Sinceε = 0 corresponds tothe Vasicek model, this implies that the Vasicek model overvalues the deep in-the-money calls while undervaluing the deep out-of-the-money calls when comparedwith the stochastic volatility model of this paper.

For the remaining simulations,ε was set equal to 2. The price of a 3-year bondwas computed using (2.13), and the price of a 3-month call option on the bond wascomputed. To compare the price against the traditional Vasicek model, the Vasicek


Figure 1. Distribution of bond price with varyingε.

volatility implied by the call price was computed for various values ofS0 and thestrikeK. Under the Vasicek model, the forward rate volatility has the form

σ (s, t) = σ0e−κ(t−s), (4.7)

and the priceCt,TC of aTC-maturity call option on aTP maturity bond is

Ct,TC = Pt,TPN [h1(Pt,TP , t, TC)] −KPt,TCN [h2(Pt,TP , t, TC)] , (4.8)

where

h1(Pt,TP , t, TC) =logPt,TP − logPt,TC − logK + 1

2v2(t, TC)

v(t, TC)(4.9)

h2(Pt,TP , t, TC) = h1(Pt,TP , t, TC)− v(t, TC) , (4.10)

v2(t, TC) = σ 20

2κ3(e2λ(TC−t ) − 1)(e−κTP − e−κTC )2. (4.11)

The implied volatility surface is shown in Figure 2. The figure shows that for eachfixedS0 there is a skew in the implied volatility curve, as was the case in the stockoption case of Hobson and Rogers (1998).

In Figure 3, two implied volatility curves, forS0 = 0.2 andS0 = −0.2 areplotted to better illustrate their shape, and to illustrate the change in the directionof the skew with the change in the sign ofS0.


Figure 2. Implied Vasicek volatility surface. The strike increases from 0.855 to 0.8875 fromleft to right, andS0 increases from –1 to 1 from front to back. The implied volatility rangesfrom 0.025 to 0.032.

Figure 3. Implied volatility curves corresponding toS0 = 0.2 andS0 = −0.2.


5. Conclusion

This paper has considered the Hobson and Rogers (1998) technique for obtainingcomplete stochastic volatility models. In particular, the technique is used to obtaina complete stochastic model within the Heath, Jarrow and Morton (1992) interestrate framework. One of the main contributions of the paper has been to showhow the stochastic dynamics can be reduced to a Markovian form. This allowsthe bond price to be expressed in terms of the underlying state variables, thus con-siderably reducing the computational burden required for the calculation of interestrate derivative prices. The model has been simulated in the simplest case, and animplied volatility surface based on the Vasicek (1977) model has been generated.These results indicate that the model is able to capture important features, such asskewness, of the implied volatility surfaces.

Future research should take further the numerical simulations reported here,perhaps experimenting with a wider specification of the volatility function. Empir-ical research on an appropriate form for the the volatility functions also needs tobe undertaken.

Notes

1. This terminology is borrowed from Hobson and Rogers (1998).2. The reader may need to recall that the quadratic variation of the process forZt,t+τj , in the

current context, is given by

〈Z〉t,t+τj =∫ t

0σ2τ (s, ξs, ζs,Ss)ds .

References

Bhar, R. and Chiarella, C. (1997) Transformation of Heath–Jarrow–Morton models to Markoviansystems,European Journal of Finance3, 1–26.

Brace, A., Gatarek, D., and Musiela, M. (1997) The market model of interest rate dynamics,Math.Finance7(2), 127–155.

Carverhill, A. (1994) When is the short rate Markovian?Math. Finance4(4), 305–312.Cheyette, O. (1992) Term structure dynamics and mortgage valuation,Journal of Fixed Income1,

28–41.Chiarella, C. and Kwon, O. (1998a) A class of Heath–Jarrow–Morton term structure models with

stochastic volatility, Working Paper No. 34, School of Finance and Economics, University ofTechnology Sydney.

Chiarella, C. and Kwon, O. (1998b) Formulation of popular interest models under the HJM frame-work, Working Paper No. 13, School of Finance and Economics, University of TechnologySydney.

Chiarella, C. and Kwon, O. (1998c) Forward rate dependent Markovian transformations of theHeath–Jarrow–Morton term structure model, Working Paper No. 5, School of Finance andEconomics, University of Technology Sydney, to appear inFinance and Stochastics.

Chiarella, C. and Kwon, O (1998d) Square root affine transformations of the Heath–Jarrow–Mortonterm structure model and partial differential equations, Working Paper, School of Finance andEconomics, University of Technology Sydney.


Cox, J., Ingersoll, J., and Ross, S. (1985) An intertemporal general equilibrium model of asset prices,Econometrica53(2), 386–384.

de Jong, F. and Santa-Clara, P. (1999) The dynamics of the forward interest rate curve: A formulationwith state variables,Journal of Financial and Quantitative Analysis34(1), 131–157.

Heath, D., Jarrow, R., and Morton, A. (1992) Bond pricing and the term structure of interest rates: Anew methodology for contingent claim valuation,Econometrica60(1), 77–105.

Heston, S. (1993) A closed form solution for options with stochastic volatility with application tobond and currency options,Review of Financial Studies6(2), 327–343.

Hobson, G. and Rogers, L. (1998) Complete models with stochastic volatility,Math. Finance8(1),27–48.

Hull, J. and White, A. (1987) The pricing of options on assets with stochastic volatilities,Journal ofFinance42(2), 281–300.

Inui, K. and Kijima, M. (1998) A Markovian framework in multi-factor Heath–Jarrow–Mortonmodels,Journal of Financial and Quantitative Analysis33(3), 423–440.

Ritchken, P. and Sankarasubramanian, L. (1995) Volatility structures of forward rates and thedynamics of the term structure,Math. Finance5(1), 55–72.

Scott, L. (1997) Pricing stock options in a jump-diffusion model with stochastic volatility and interestrates: Applications of Fourier inversion methods,Math. Finance7(4), 413–426.

Vasicek, O. (1977) An equilibrium characterisation of the term structure,Journal of FinancialEconomics5, 177–188.

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A Complete Markovian Stochastic Volatility Model in the HJM Framework