Deriving the Futures–Forward Convexity Adjustment in One-Factor HJM
Summary
The document derives a futures rate from the expected value of a money-market rate in a one-factor Heath–Jarrow–Morton framework. Using the bond-price dynamics, it obtains a convexity term expressed as an integral involving bond volatilities. The adjustment raises the futures rate relative to the corresponding forward rate when the stated nonnegative convexity condition holds.
It also describes two sources of the futures–forward spread: daily mark-to-market cash flows and the difference between settlement dates for futures and forward agreements. The author asks how to separate these effects mathematically, but provides no derivation of that split. The formula is attributed to an interest-rate modeling text, and the document mentions that a cited earlier source contains numerical examples and futures-option analysis. Its discussion is limited to the one-factor HJM setup; it does not establish how the decomposition extends to other models or assumptions.
Key ideas
- The futures rate is calculated as an expectation of the rate fixed at the futures settlement date.
- In the one-factor HJM setup, bond-price volatility produces an integral convexity adjustment.
- Under the stated nonnegative convexity condition, the futures rate is at least as high as the forward rate.
- The document identifies mark-to-market and differing payment dates as two sources of the futures–forward spread.
- It poses the separation of those sources as an open derivation question.
Tags
Full text
# Components in HJM convexity adjustment
# Components in HJM convexity adjustment
I have the following derivation for the futures-forward convexity adjustment in the one-factor HJM framework:
Consider the implied futures rate $F(t,T,T+\tau)$: $$\begin{align} F(t,T,T+\tau) &= E^\mathbb{Q}[L(T,T,T+\tau)] \\ &= E^\mathbb{Q}\left[\frac{1}{\tau} \left( \frac{1}{P(T,T+\tau)} - 1 \right) \right] \end{align}$$
In the one-factor HJM framework we have: $$d_t P(t,T+\tau) = r_t P(t,T+\tau) \, dt - \Sigma(t,T+\tau) P(t,T+\tau) \, dW_t^\mathbb{Q} $$which leads to: $$\begin{align} P(T,T+\tau) &= P(t,T,T+\tau) \, \exp \left( - \frac{1}{2} \int_t^T [\Sigma(s,T+\tau)^2 - \Sigma(s,T)^2] \,ds - \int_t^T [\Sigma(s,T+\tau) - \Sigma(s,T)] \, dW_s^\mathbb{Q} \right) \end{align}$$
We can then derive that: $$ \boxed{ \begin{align} \\ \qquad F(t,T,T+\tau) &= \frac{1}{\tau} \left(\frac{1}{P(t,T,T+\tau)} e^{\Omega(t,T,T+\tau)} - 1 \right) \\ \\ \Omega(t,T,T+\tau) &= \int_t^T \Sigma(s,T+\tau) \cdot \left(\Sigma(s,T+\tau) - \Sigma(s,T) \right) \, ds \qquad \\ \\ \end{align}}$$
This is the same formula as in Andersen-Piterbarg: Interest rate modeling (Volume 1, p. 187). However, they note that:
In any rational model $\Omega(t,T) \geq 0$, such that $F(t,T,T+\tau) \geq L(t,T,T+\tau)$, consistent with the quantitative discussion in Section 4.1.2. As shown in Chapter 2 of Andersen[1996], the spread (also known as futures convexity) between futures and forward rates can be decomposed into two components:
- A term originating from the mark-to-market mechanism of a futures contract
- A term originating from the fact that a futures contract (unlike a regular FRA) pays out at the date it settles (At time T), rather than one period ahead (at time T+\tau). Andersen[1996], Chapter 2, additionalliy contains a number of numerical examples examining typical futures-forward spreads, and also investigates the pricing of options on futures rates.
However, I haven't found this source (Andersen[1996]), and I have been struggling to derive this split into two components myself. Does anyone have a derivation, showing which part of the formula is as a consequence of the mark-to-market mechanism, and which part is because of the payment mismatch?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.