The Heath–Jarrow–Morton (HJM) framework is a general framework to model the evolution of

interest rate An interest rate is the amount of interest due per period, as a proportion of the amount lent, deposited, or borrowed (called the principal sum). The total interest on an amount lent or borrowed depends on the principal sum, the interest rate, t ...

curves – instantaneous forward rate curves in particular (as opposed to simple

forward rate The forward rate is the future yield on a bond. It is calculated using the yield curve. For example, the yield on a three-month Treasury bill six months from now is a ''forward rate''.. Forward rate calculation To extract the forward rate, we n ...

s). When the volatility and drift of the instantaneous forward rate are assumed to be

deterministic Determinism is a philosophical view, where all events are determined completely by previously existing causes. Deterministic theories throughout the history of philosophy have developed from diverse and sometimes overlapping motives and cons ...

, this is known as the Gaussian Heath–Jarrow–Morton (HJM) model of forward rates. For direct modeling of simple forward rates the

Brace–Gatarek–Musiela model The LIBOR market model, also known as the BGM Model (Brace Gatarek Musiela Model, in reference to the names of some of the inventors) is a financial model of interest rates. It is used for pricing interest rate derivatives, especially exotic deriva ...

represents an example. The HJM framework originates from the work of David Heath, Robert A. Jarrow, and Andrew Morton in the late 1980s, especially ''Bond pricing and the term structure of interest rates: a new methodology'' (1987) – working paper,

Cornell University Cornell University is a private statutory land-grant research university based in Ithaca, New York. It is a member of the Ivy League. Founded in 1865 by Ezra Cornell and Andrew Dickson White, Cornell was founded with the intention to tea ...

, and ''Bond pricing and the term structure of interest rates: a new methodology'' (1989) – working paper (revised ed.), Cornell University. It has its critics, however, with

Paul Wilmott Paul Wilmott (born 8 November 1959) is an English researcher, consultant and lecturer in quantitative finance.istakesto be swept under".''Newsweek'' 2009
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Framework

The key to these techniques is the recognition that the drifts of the no-arbitrage evolution of certain variables can be expressed as functions of their volatilities and the correlations among themselves. In other words, no drift estimation is needed. Models developed according to the HJM framework are different from the so-called

short-rate model A short-rate model, in the context of interest rate derivatives, is a mathematical model that describes the future evolution of interest rates by describing the future evolution of the short rate, usually written r_t \,. The short rate Under a s ...

s in the sense that HJM-type models capture the full dynamics of the entire forward rate curve, while the short-rate models only capture the dynamics of a point on the curve (the short rate). However, models developed according to the general HJM framework are often non- Markovian and can even have infinite dimensions. A number of researchers have made great contributions to tackle this problem. They show that if the volatility structure of the forward rates satisfy certain conditions, then an HJM model can be expressed entirely by a finite state Markovian system, making it computationally feasible. Examples include a one-factor, two state model (O. Cheyette, "Term Structure Dynamics and Mortgage Valuation", ''Journal of Fixed Income,'' 1, 1992; P. Ritchken and L. Sankarasubramanian in "Volatility Structures of Forward Rates and the Dynamics of Term Structure", ''Mathematical Finance'', 5, No. 1, Jan 1995), and later multi-factor versions.

Mathematical formulation

The class of models developed by Heath, Jarrow and Morton (1992) is based on modelling the forward rates. The model begins by introducing the instantaneous forward rate

\textstyle f(t,T)

\textstyle t \leq T

, which is defined as the continuous compounding rate available at time

\textstyle T

as seen from time

\textstyle t

. The relation between bond prices and the forward rate is also provided in the following way: :

P(t,T) = e^

Here

\textstyle P(t,T)

is the price at time

\textstyle t

of a zero-coupon bond paying $1 at maturity

\textstyle T\geq t

. The risk-free money market account is also defined as :

\beta(t) = e^

This last equation lets us define

\textstyle f(t,t) \triangleq r(t)

, the risk free short rate. The HJM framework assumes that the dynamics of

\textstyle f(t,s)

under a risk-neutral pricing measure

\textstyle \mathbb Q

are the following: :

df(t,s) = \mu(t,s)dt + \boldsymbol \sigma(t,s) dW_t

Where

\textstyle W_t

is a

\textstyle d

-dimensional

Wiener process In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is ...

and

\textstyle \mu(u,s)

\textstyle \boldsymbol \sigma(u,s)

are

\textstyle \mathcal F_u

adapted process In the study of stochastic processes, an adapted process (also referred to as a non-anticipating or non-anticipative process) is one that cannot "see into the future". An informal interpretation is that ''X'' is adapted if and only if, for every rea ...

es. Now based on these dynamics for

\textstyle f

, we'll attempt to find the dynamics for

\textstyle P(t,s)

and find the conditions that need to be satisfied under risk-neutral pricing rules. Let's define the following process: :

Y_t \triangleq \log P(t,s) = -\int_t^s f(t,u) du

The dynamics of

\textstyle Y_t

can be obtained through Leibniz's rule: :

\begin dY_t &= f(t,t) dt - \int_t^s df(t,u) du \\ &= r_t dt - \int_t^s \mu(t,u) dt du - \int_t^s \boldsymbol \sigma(t,u) dW_t du \end

If we define

\textstyle \mu(t,s)^* = \int_t^s \mu(t,u) du

\textstyle \boldsymbol \sigma(t,s)^* = \int_t^s \boldsymbol \sigma(t,u) du

and assume that the conditions for

Fubini's Theorem In mathematical analysis Fubini's theorem is a result that gives conditions under which it is possible to compute a double integral by using an iterated integral, introduced by Guido Fubini in 1907. One may switch the order of integration if th ...

are satisfied in the formula for the dynamics of

\textstyle Y_t

, we get: :

dY_t = \left(  r_t - \mu(t,s)^* \right)dt - \boldsymbol \sigma(t,s)^* dW_t

By Itō's lemma, the dynamics of

\textstyle P(t,T)

are then: :

\frac = \left( r_t - \mu(t,s)^* + \frac \boldsymbol \sigma(t,s)^* \boldsymbol \sigma(t,s)^ \right)dt - \boldsymbol \sigma(t,s)^* dW_t

But

\textstyle \frac

must be a martingale under the pricing measure

\textstyle \mathbb Q

, so we require that

\textstyle  \mu(t,s)^* = \frac \boldsymbol \sigma(t,s)^* \boldsymbol \sigma(t,s)^

. Differentiating this with respect to

\textstyle s

we get: :

\mu(t,u) = \boldsymbol \sigma(t,u) \int_t^u \boldsymbol \sigma(t,s)^ ds

Which finally tells us that the dynamics of

\textstyle f

must be of the following form: :

df(t,u) =  \left( \boldsymbol \sigma(t,u) \int_t^u \boldsymbol \sigma(t,s)^ ds \right) dt + \boldsymbol \sigma(t,u) dW_t

Which allows us to price bonds and interest rate derivatives based on our choice of

\textstyle \boldsymbol \sigma

References

Notes

Sources

* Heath, D., Jarrow, R. and Morton, A. (1990)
Bond Pricing and the Term Structure of Interest Rates: A Discrete Time Approximation
''

Journal of Financial and Quantitative Analysis The ''Journal of Financial and Quantitative Analysis'' is a peer-reviewed bimonthly academic journal published by the Michael G. Foster School of Business at the University of Washington in cooperation with the W. P. Carey School of Business at ...

'', 25:419-440. * Heath, D., Jarrow, R. and Morton, A. (1991)
Contingent Claims Valuation with a Random Evolution of Interest Rates
. '' Review of Futures Markets'', 9:54-76. * Heath, D., Jarrow, R. and Morton, A. (1992)
Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claims Valuation
''

Econometrica ''Econometrica'' is a peer-reviewed academic journal of economics, publishing articles in many areas of economics, especially econometrics. It is published by Wiley-Blackwell on behalf of the Econometric Society. The current editor-in-chief is ...

'', 60(1):77-105. * Robert Jarrow (2002). ''Modelling Fixed Income Securities and Interest Rate Options'' (2nd ed.). Stanford Economics and Finance.

Framework

Mathematical formulation

See also

References

Notes

Sources

Further reading