Cheyette (1992) Markovian HJM Model Lab

1. The Heath-Jarrow-Morton (1992) Path Dependency Dilemma

In the continuous Heath-Jarrow-Morton (1992) framework, the instantaneous forward rate for maturity \(T\) follows the stochastic differential equation:

Under the risk-neutral martingale measure \(\mathbb{Q}\), no-arbitrage uniquely fixes the drift \(\alpha(t, T)\) to:

Integrating over time reveals that the forward rate \(f(t, T) = f(0, T) + \int_0^t \alpha(s, T) ds + \int_0^t \sigma(s, T) dW(s)\) depends on the entire historical path of Brownian increments across the continuum of maturities. Consequently, general HJM models are infinite-dimensional and non-Markovian: evaluating path-dependent options (such as Bermudan swaptions) requires simulating the entire forward curve at every time step, creating severe computational bottlenecks.

2. The Cheyette (1992) Separable Volatility Breakthrough

Oren Cheyette (1992) discovered that if the volatility structure \(\sigma(t, T)\) factorizes into an exponential separable kernel:

where \(\kappa > 0\) is the constant mean-reversion parameter and \(\sigma(t) = \sigma\), the continuous continuum of forward rates collapses into an exact 2-dimensional Markov diffusion process governed by state variables \((x(t), y(t))\):

Here, \(x(t)\) represents the stochastic deviation of the short rate from the initial market forward rate, while \(y(t)\) represents the deterministic accumulated variance and convexity adjustment:

3. Exact Zero-Coupon Bond Pricing & Instantaneous Forward Rates

With the state variables \((x(t), y(t))\) established, zero-coupon bond prices \(P(t, T)\) are given in exact closed form without numerical integration:

where \(G(t, T) = \frac{1 - e^{-\kappa(T - t)}}{\kappa}\). Notice that at time \(t = 0\), since \(x(0) = 0\) and \(y(0) = 0\), the exponential equals 1, meaning that Cheyette identically fits any arbitrary market yield curve \(P^M(0, T)\) by construction.

The corresponding instantaneous forward rate curve is:

4. Jamshidian (1989) Exact European Swaption Decomposition

Because zero-coupon bond prices \(P(T_0, T_i; x(T_0), y(T_0))\) are strictly monotonic decreasing functions of the single random factor \(x(T_0)\), Farshid Jamshidian's (1989) decomposition theorem applies. At swaption expiry \(T_0\), the payer swaption payoff is:

where \(c_i = K_{\text{swap}} \Delta t\) for \(i < n\) and \(c_n = 1 + K_{\text{swap}} \Delta t\). By solving for the unique root \(x^*\) such that:

each bond strike is defined as \(K_i = P(T_0, T_i; x^*, y(T_0))\). The swaption decomposes into an exact portfolio of European zero-coupon bond put options: