Heath-Jarrow-Morton model ($df(t,T)=\sigma (t,T){\int }_t^T\sigma (t,u)dudt+\sigma (t,T)d{W}_t$)

A model evolving the entire instantaneous forward rate curve. In the displayed one-factor risk-neutral measure dynamics the drift restriction makes every suitably integrable discounted zero-coupon bond price a martingale. The multi-factor form replaces the product of volatilities by their dot product.

For a maturity-differentiable Heath-Jarrow-Morton model forward field with the integrability needed for stochastic differentiation, the short rate $r_t=f(t,t)$ has the displayed dynamics. Indeed changing observation time uses the forward-rate stochastic differential equation, while changing maturity adds ${\partial }_{T}f(t,t)dt$. Under a risk-neutral measure, $\alpha (t,T)=\sigma (t,T){\int }_t^T\sigma (t,u)du$, so $\alpha (t,t)=0$. The remaining drift depends on the diagonal slope of the whole forward curve, which need not be determined by $r_t$; a Markov process for the short rate is an additional modelling restriction. Without sufficient maturity regularity, the diagonal need not admit this classical Itô process formula.

For the Heath-Jarrow-Morton model with $\sigma (t,T)=T-t$, the risk-neutral drift is $\alpha (t,T)=(T-t{)}^3/2$. Its short rate is $r_t=f(0,t)+t^4/8+{\int }_0^t{W}_{s}ds$. The centered integrated Brownian component is not Markov: its past reveals its derivative $W_t$, which affects the conditional mean of future increments, whereas its current integral does not determine $W_t$. Indeed the conditional variance of $W_t$ given the current integral is $t/4$. A one-dimensional driving noise therefore does not force a one-dimensional Markov state.

With deterministic Hilbert-space-valued volatility, a deterministic initial curve and the Heath-Jarrow-Morton model drift restriction, the instantaneous forward rate is a Gaussian random field indexed by observation time and maturity. Put $\Sigma (t,T)={\int }_t^T\sigma (t,u)du$. Absence of arbitrage requires $\alpha (t,T)=⟨\sigma (t,T),\Sigma (t,T)⟩$. Indeed, Itô formula gives bond drift $r_t-{\int }_t^T\alpha (t,u)du+\Vert \Sigma (t,T){\Vert }^2/2$; equating it to the short rate and differentiating in maturity yields this restriction. The field covariance isFinite-dimensional volatility spaces give finite-factor models; an infinite-dimensional space permits much richer maturity correlations. Gaussian rates can be negative. Deterministic volatility gives explicit zero-coupon bond and bond-option formulas.

For a centered Gaussian forward-rate field with covariance $\mathrm{Cov}(X(s,u),X(r,w))=c(s\wedge r,u,w)$, integrate the short-rate past and the current forward curve together using the displayed mean-square integral. Its variance isFor fixed $T$, its increments are independent of the full earlier forward-rate filtration: their covariance with $X(z,w)$ at $z\le r\le s$ is the integral of $c(s\wedge u\wedge z,u,w)-c(r\wedge u\wedge z,u,w)=0$. Use the uncorrelated jointly Gaussian variables are independent principle first for finite collections and then for their generated sigma-algebra. With $c(0,u,w)=0$, this centered process starts at zero.

Suppose the instantaneous forward rate field has mean ${\mu }_{s,T}$ and covariance $c(s\wedge r,T,U)$, with deterministic initial curve and suitable integrability and maturity regularity. Its discounted zero-coupon bond prices are martingales exactly when the displayed mean identity holds. PutThe discounted bond price is $\exp [-A(s,T)-Y(s,T)]$, where $Y$ is the integrated Gaussian forward-rate process. Its conditional Gaussian exponential expectation gives the equivalent identity $A(s,T)-A(0,T)=v(s,T)/2$. Differentiating this in maturity gives the displayed restriction. Conversely, integrating that restriction on the diagonal and along the current curve recovers the half-variance identity, by symmetry of the covariance over the maturity square. The Gaussian exponential martingale with deterministic variance then verifies the full conditional martingale property. This formulation does not require a time derivative of $c$.

For time-to-maturity coordinates $g_t(x)=f(t,t+x)$ and time-homogeneous deterministic Heath-Jarrow-Morton model volatility, $A(x)=⟨\sigma (x),{\int }_0^x\sigma (u)du⟩$. The displayed evolution contains the maturity shift. Its mild solution is $g_t(x)=g_0(x+t)+{\int }_0^{t}A(x+s)ds+{\int }_0^t⟨\sigma (x+t-s),d{W}_s⟩$. The whole curve is a time-homogeneous Markov process under suitable function-space well-posedness conditions; the current short rate alone need not be a sufficient state.

If the displayed covariance is finite and the mean obeys $m^{\prime }(x)+A(x)=0$, initialize the Musiela forward-curve equation with the corresponding Gaussian random field, independently of future noise. The resulting curve is stationary in time. Indeed shifting the initial covariance contributes the integral over $[t,\infty )$ and fresh noise contributes the integral over $[0,t]$, so their sum is $K$. A deterministic initial curve generally is not stationary, and nondecaying volatility may make $K(x,x)$ infinite. Exponentially decaying finite-factor volatilities give stationary Ornstein-Uhlenbeck processes.

For a maturity-$T$ call on a unit maturity-$U$ zero-coupon bond, where $t<T<U$, deterministic forward-rate volatilities make the bond-price ratio lognormal under the T-forward measure. Its integrated variance is $V={\int }_t^T\Vert \Sigma (s,U)-\Sigma (s,T){\Vert }^{2}ds$. For $V>0$, put $d_1=[\log (P(t,U)/(KP(t,T)))+V/2]/\sqrt{V}$ and $d_2=d_1-\sqrt{V}$. Integrating the lognormal distribution above the strike gives the displayed price. At $V=0$, the price is $(P(t,U)-KP(t,T){)}^{+}$. The forward measure is essential: the raw bond price has a stochastic discount rate under the money-market measure.

Suppose $d{r}_t=a(r_t)dt+b(r_t)d{W}_t$ under a risk-neutral measure and $f_t(T)=F(T-t,r_t)$. By Itô formula, the forward-rate volatility is ${\sigma }_t(T)=b(r_t)F_r(T-t,r_t)$. The Heath-Jarrow-Morton model drift restriction is equivalent toTo verify it directly for bonds, put $G(\theta ,r)={\int }_0^{\theta }F(s,r)ds$. Integration gives $F-r=a{G}_r+\frac{1}{2}{b}^2{G}_{rr}-\frac{1}{2}{b}^2{G}_r^2$. The zero-coupon bond price $P(t,T)=e^{-G(T-t,r_t)}$ therefore satisfiesIts bank-account-discounted price is a positive local martingale. A common equivalent local martingale measure excludes admissible arbitrage for any finite collection of these traded maturities.

In the one-factor Heath-Jarrow-Morton model, put $b_t^T={\int }_t^T\sigma (t,u)du$. The Itô formula gives $dP/P=r_{t}dt-b_t^{T}d{W}_t$. Thus multiplying by the discount factor yields the displayed stochastic exponential. Bounded forward volatility on a finite maturity horizon implies the Novikov condition, so the discounted bond is a true martingale.