Deriving the Multi-Factor HJM Forward Rate Drift
Summary
The document examines how the Heath-Jarrow-Morton framework extends from one source of rate uncertainty to multiple Brownian factors. It sketches a one-factor derivation by starting with the zero-coupon bond price process, expressing forward rates through bond prices, and taking a short-maturity limit. It then attempts a two-factor extension with correlated risk drivers.
The central issue is the drift restriction under the risk-neutral measure. The derivation in the question produces a term involving the combined magnitude of factor volatilities, while the canonical HJM expression sums each factor’s volatility times its own maturity integral. The document poses this mismatch as a question but contains no answer or resolution, so it is most useful for identifying the derivation step that needs scrutiny. Its displayed formulas also use different integration bounds in places, making careful notation and assumptions important before applying the equations.
Key ideas
- HJM specifies forward rate dynamics consistent with the observed yield curve.
- The risk-neutral drift is constrained by the volatility structure to prevent arbitrage.
- In a multi-factor model, the stated canonical drift sums contributions factor by factor.
- The proposed derivation instead combines factor volatilities, and the document leaves this discrepancy unresolved.
Tags
Full text
# Extension of HJM to multiple factors
# Extension of HJM to multiple factors
The HJM model calibrates the entire forward curve using the existing yield curve data and this results in the following expression for its instantaneous forward rate-
$$df(t,T)=\sigma(t,T)\int_0^T\sigma(t,\tau)d\tau+\sigma(t,T)dW^Q$$
The above can be derived using the following expression for the zero coupon bond-
$$\frac{dZ(t,T)}{Z(t,T)}=r_tdt+\sigma_Z(t,T)dW^Q$$
$$f(t,T,T+\tau)=\frac{\ln Z(t,T)-\ln Z(t,T+\tau)}{\tau}\implies\\ df(t,T,T+\tau)=\frac{\sigma_Z^2(t,T+\tau)-\sigma_Z^2(t,T)}{2\tau}dt+\frac{\sigma_Z(t,T)-\sigma_Z(t,T+\tau)}{\tau}dW^Q$$
Taking limit $\tau\rightarrow0$, we get-
$$df(t,T)=\sigma_Z(t,T)\frac{\partial \sigma_Z(t,T)}{\partial T}dt-\frac{\partial \sigma_Z(t,T)}{\partial T}dW^Q$$
Setting $\frac{\partial \sigma_Z(t,T)}{\partial T}=\sigma(t,T)$ gives us the first expression. We can extend the HJM similarly for two independent factors.
$$df(t,T)=m(t,T)dt+\sigma_1dW_1^Q+\sigma_2dW_2^Q$$
The evolution of the zero price process would be-
$$\frac{dZ(t,T)}{Z(t,T)}=r_tdt+\rho\sigma_Z(t,T)dW_1^Q+\sqrt{1-\rho^2}\sigma_Z(t,T)dW_2^Q$$
After manipulating the above similar to the 1 factor case, we get-
$$df(t,T)=\sigma_Z(t,T)\frac{\partial \sigma_Z(t,T)}{\partial T}dt-\rho\frac{\partial \sigma_Z(t,T)}{\partial T}dW_1^Q-\sqrt{1-\rho^2}\frac{\partial \sigma_Z(t,T)}{\partial T}dW_2^Q$$
Setting the loadings on the Brownian motion increments equal to $\sigma_i(t,T)$ gives us-
$$\frac{\partial \sigma_Z(t,T)}{\partial T}=\sqrt{\sigma_1^2+\sigma_2^2}$$
Thus the drift of the instantaneous forward rate should be given by-
$$\sqrt{\sigma_1^2(t,T)+\sigma_2^2(t,T)}\int_t^T\sqrt{\sigma_1^2(t,s)+\sigma_2^2(t,s)}ds$$
However that's not the case. The canonical expressions for the drift are-
$$\sum_k\sigma_k(t,T)\int_t^T\sigma_k(t,s)ds$$
The risk-neutral drift is a function of the the partial standard deviations rather than the total standard deviation as was implied by my derivation. Can someone please explain why this is the case?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.