Forward Rate Dynamics¶
The Heath–Jarrow–Morton (HJM) framework models the entire forward rate curve directly. Instead of specifying a short rate, HJM postulates dynamics for instantaneous forward rates.
Instantaneous forward rates¶
Recall the instantaneous forward rate
so that
In HJM, \(f(t,T)\) for all maturities \(T\ge t\) is the state variable.
Stochastic dynamics¶
Under the risk-neutral measure, HJM postulates
where: - \(\sigma_i(t,T)\) are volatility functions, - \(W^i\) are Brownian motions, - \(\alpha(t,T)\) is the drift.
Crucially, the drift is not arbitrary.
QuantPie Derivation: From Forward Rates to Bond Prices¶
Forward Rate Definition¶
For \(t < S < T\), the forward rate is defined as:
where the forward rate is:
Instantaneous Forward Rate¶
Taking the limit as \(S \to T\):
This shows the fundamental relationship between instantaneous forward rates and the log of bond prices.
Forward Rate Dynamics¶
The instantaneous forward rate follows the stochastic differential equation:
where: - \(\mu^{\mathbb{Q}}(t,T)\) is the drift under the risk-neutral measure - \(\sigma(t,T)\) is the volatility of the forward rate - \(dW^{\mathbb{Q}}(t)\) is a standard Brownian motion increment
Interpretation¶
- Volatility structures determine how different maturities move together.
- The model is infinite-dimensional because \(T\) is continuous.
- Short-rate and market models arise as special cases.
Advantages of forward modeling¶
- Exact fit to the initial yield curve by construction.
- Transparent no-arbitrage conditions.
- Direct modeling of curve dynamics.
Key takeaways¶
- HJM models forward rates directly.
- The entire yield curve is the state variable.
- Drift restrictions ensure no-arbitrage.
Further reading¶
- Heath, Jarrow & Morton (1992).
- Filipović, Term-Structure Models.
Exercises¶
Exercise 1. Given the initial forward rate curve \(f(0, T) = 0.04 + 0.005\,T - 0.0002\,T^2\) for \(T \in [0, 30]\), compute the initial zero-coupon bond prices \(P(0, 5)\), \(P(0, 10)\), and \(P(0, 20)\) using the relation
Verify that \(P(0, T)\) is a decreasing function of \(T\).
Solution to Exercise 1
We are given the initial forward rate curve \(f(0, T) = 0.04 + 0.005\,T - 0.0002\,T^2\) and need to compute
Step 1: Compute the integral.
Step 2: Evaluate at \(T = 5\).
Step 3: Evaluate at \(T = 10\).
Step 4: Evaluate at \(T = 20\).
Step 5: Verify that \(P(0, T)\) is decreasing.
We have \(P(0, 5) \approx 0.7757 > P(0, 10) \approx 0.5580 > P(0, 20) \approx 0.2817\). More generally,
Since \(f(0, T) = 0.04 + 0.005\,T - 0.0002\,T^2 > 0\) for \(T \in [0, 30]\) (the discriminant of the quadratic shows \(f(0, T) > 0\) on this interval) and \(P(0, T) > 0\), we have \(\frac{d}{dT} P(0, T) < 0\), confirming that \(P(0, T)\) is strictly decreasing.
Exercise 2. In the one-factor HJM framework with constant volatility \(\sigma(t, T) = \sigma_0\), write down the explicit dynamics \(df(t, T)\) under the risk-neutral measure (including the drift determined by the no-arbitrage condition). Integrate the SDE to obtain \(f(t, T)\) as a function of \(f(0, T)\), \(\sigma_0\), and the Brownian motion. Identify the resulting short-rate model.
Solution to Exercise 2
Step 1: Determine the drift from the HJM no-arbitrage condition.
With constant volatility \(\sigma(t, T) = \sigma_0\), the HJM drift condition gives
Step 2: Write the forward rate SDE.
Step 3: Integrate from 0 to \(t\).
Evaluating the deterministic integral:
Therefore:
Step 4: Identify the short-rate model.
Setting \(T = t\):
The short rate is Gaussian with a deterministic drift \(\theta(t) = f'(0, t) + \sigma_0^2\,t\) and diffusion \(\sigma_0\), i.e., \(dr_t = \theta(t)\,dt + \sigma_0\,dW_t\). This is the Ho--Lee model, the simplest HJM model with no mean reversion.
Exercise 3. Consider the forward rate volatility \(\sigma(t, T) = \sigma_0\,e^{-\kappa(T-t)}\). Show that the short rate \(r_t = f(t, t)\) satisfies a mean-reverting SDE and identify the mean-reversion speed \(\kappa\). What well-known short-rate model does this volatility specification correspond to?
Solution to Exercise 3
Step 1: Compute the HJM drift.
With \(\sigma(t, T) = \sigma_0\,e^{-\kappa(T-t)}\), the drift condition gives
Step 2: Write and integrate the forward rate SDE.
Step 3: Derive the short rate dynamics. Setting \(T = t\) in \(df(t, T)\):
Using the integrated form and differentiating (via Leibniz), or equivalently, applying the Musiela parameterization, one obtains:
More compactly, this can be written in the Hull--White form:
where \(\theta(t)\) is a deterministic function of the initial curve \(f(0, \cdot)\) and the parameters \(\sigma_0, \kappa\).
The mean-reversion speed is \(\kappa\), which is the same parameter appearing in the exponential decay of volatility. This model is the Hull--White (extended Vasicek) model. The exponential decay in the forward rate volatility directly translates to mean reversion in the short rate: faster decay (\(\kappa\) larger) means stronger mean reversion.
Exercise 4. Explain why the HJM framework is said to be "infinite-dimensional," whereas short-rate models like Vasicek or CIR are finite-dimensional. In practical terms, what does infinite-dimensionality mean for the computational complexity of Monte Carlo simulation in HJM?
Solution to Exercise 4
Infinite-dimensionality of HJM:
In the HJM framework, the state variable at each time \(t\) is the entire forward rate curve \(T \mapsto f(t, T)\) for all \(T \geq t\). This is a function, which lives in a function space (typically \(L^2[0, T_{\max}]\) or a weighted Sobolev space). A function has infinitely many degrees of freedom: specifying \(f(t, T)\) requires its value at every maturity \(T\) on a continuum. Hence the state space is infinite-dimensional.
By contrast, short-rate models like Vasicek or CIR have a single state variable \(r_t \in \mathbb{R}\). Given \(r_t\), the entire forward curve \(f(t, T)\) is determined by the closed-form bond pricing formula \(P(t, T) = e^{A(T-t) - B(T-t)r_t}\). The state space is \(\mathbb{R}\) (one-dimensional). Two-factor models live in \(\mathbb{R}^2\).
Computational implications for Monte Carlo:
In a Monte Carlo simulation of HJM, at each time step one must evolve the forward rate \(f(t_k, T_j)\) at every maturity grid point \(T_j\). If the maturity axis is discretized into \(N\) points:
- Each time step requires \(O(N)\) random updates (one for each maturity).
- The drift at each maturity involves an integral \(\int_t^{T_j} \sigma(t, u)\,du\), which costs \(O(N)\) operations per maturity point, for a total of \(O(N^2)\) per time step (or \(O(N)\) if cumulative sums are used efficiently).
- Total cost for \(M\) time steps and \(P\) paths: \(O(P \cdot M \cdot N)\) at best, \(O(P \cdot M \cdot N^2)\) at worst.
For a short-rate model, each time step involves evolving a single scalar, costing \(O(1)\) per step per path. This makes short-rate Monte Carlo dramatically cheaper, typically \(O(P \cdot M)\).
The infinite-dimensional nature of HJM therefore imposes a significant computational burden, which motivates the use of finite-factor approximations in practice.
Exercise 5. Starting from \(P(t, T) = \exp\bigl(-\int_t^T f(t, u)\,du\bigr)\), verify that
and that the short rate satisfies \(r_t = f(t, t)\). If \(P(t, T) = \exp(-a(T-t) - b(T-t)\,r_t)\) for some functions \(a(\cdot)\) and \(b(\cdot)\), express \(f(t, T)\) in terms of \(a'\), \(b'\), and \(r_t\).
Solution to Exercise 5
Part 1: Verify \(f(t, T) = -\partial_T \log P(t, T)\).
Starting from
take the logarithm:
Differentiate with respect to \(T\):
by the fundamental theorem of calculus. Therefore \(f(t, T) = -\frac{\partial}{\partial T}\log P(t, T)\). \(\square\)
Part 2: Verify \(r_t = f(t, t)\).
By definition, the short rate is \(r_t = \lim_{\Delta \to 0} \frac{-\log P(t, t+\Delta)}{\Delta}\). From the formula above,
Since \(\log P(t, T) = -\int_t^T f(t, u)\,du\), evaluating at \(T = t\) gives \(\log P(t, t) = 0\), and
This is the instantaneous rate of return on the shortest bond, which is exactly the short rate \(r_t\). \(\square\)
Part 3: Express \(f(t, T)\) for affine bond prices.
Given \(P(t, T) = \exp\bigl(-a(T-t) - b(T-t)\,r_t\bigr)\), let \(\tau = T - t\). Then:
Differentiating with respect to \(T\) (noting \(\partial \tau / \partial T = 1\)):
where \(a'\) and \(b'\) denote derivatives with respect to \(\tau = T - t\). The forward rate is an affine function of the short rate, with slope \(b'(\tau)\) and intercept \(a'(\tau)\).
Exercise 6. A two-factor HJM model has volatilities \(\sigma_1(t, T) = 0.01\) and \(\sigma_2(t, T) = 0.008\,e^{-0.3(T-t)}\). The first factor represents parallel shifts and the second represents slope changes. Compute the instantaneous variance of the forward rate \(f(t, T)\) as a function of \(T - t\) and sketch its term structure. At what time-to-maturity is the variance maximized?
Solution to Exercise 6
Step 1: Compute the instantaneous variance.
For a two-factor model, the instantaneous variance of \(df(t, T)\) is
With \(\sigma_1(t, T) = 0.01\) and \(\sigma_2(t, T) = 0.008\,e^{-0.3(T-t)}\), let \(\tau = T - t\):
Step 2: Analyze the term structure.
At \(\tau = 0\): \(v(0) = 10^{-4} + 6.4 \times 10^{-5} = 1.64 \times 10^{-4}\)
As \(\tau \to \infty\): \(v(\tau) \to 10^{-4}\)
The function \(v(\tau) = 10^{-4} + 6.4 \times 10^{-5}\,e^{-0.6\tau}\) is strictly decreasing in \(\tau\) since \(\frac{dv}{d\tau} = -0.6 \times 6.4 \times 10^{-5}\,e^{-0.6\tau} < 0\).
Step 3: Identify the maximum.
Since \(v(\tau)\) is strictly decreasing, the variance is maximized at \(\tau = 0\) (the shortest maturity), with maximum value \(v(0) = 1.64 \times 10^{-4}\).
Step 4: Sketch description.
The term structure of instantaneous variance starts at \(1.64 \times 10^{-4}\) for the shortest maturity and decays exponentially toward the asymptotic value \(10^{-4}\) for long maturities. The decay is governed by \(e^{-0.6\tau}\), so the second factor's contribution effectively vanishes beyond \(\tau \approx 5\)--\(8\) years. The first factor (parallel shifts) contributes a constant floor \(10^{-4}\) at all maturities.
Exercise 7. The forward rate \(R(t, S, T)\) for the period \([S, T]\) is defined by \(P(t, T) = P(t, S)\,e^{-R(t,S,T)(T-S)}\). Show that in the limit \(S \to T\), \(R(t, S, T)\) converges to the instantaneous forward rate \(f(t, T)\). For finite \(T - S\), express \(R(t, S, T)\) as an average of instantaneous forward rates and discuss how a parallel shift in \(f(t, \cdot)\) affects \(R(t, S, T)\).
Solution to Exercise 7
Part 1: Show \(R(t, S, T) \to f(t, T)\) as \(S \to T\).
From \(P(t, T) = P(t, S)\,e^{-R(t,S,T)(T-S)}\), take logarithms:
So
In the limit \(S \to T\):
by the definition of the derivative. \(\square\)
Part 2: Express \(R(t, S, T)\) as an average.
Since \(\log P(t, T) = -\int_t^T f(t, u)\,du\) and \(\log P(t, S) = -\int_t^S f(t, u)\,du\):
This shows that \(R(t, S, T)\) is the arithmetic average of instantaneous forward rates over the period \([S, T]\).
Part 3: Effect of a parallel shift.
Suppose \(f(t, u)\) is shifted to \(f(t, u) + c\) for all \(u\), where \(c\) is a constant. Then:
A parallel shift of \(c\) in the instantaneous forward curve shifts every forward rate \(R(t, S, T)\) by exactly \(c\), regardless of the interval \([S, T]\). This is consistent with the "parallel shift" interpretation: all rates, whether instantaneous or discrete-tenor, move by the same amount.