Ho–Lee Convexity Adjustment Between FRA and Futures Rates

Summary

The document asks for a derivation of the convexity adjustment between forward rate agreements and futures rates under the Ho–Lee short-rate model. It specifies that the short rate evolves with a time-dependent drift and constant volatility, and states a relationship in which the forward rate equals the futures rate minus a term involving volatility and the two contract dates.

The question gives the intuition that daily marking to market makes the futures rate higher than the FRA rate, with the adjustment depending on short-rate volatility. It does not provide the requested proof or explain the intermediate pricing steps, so the result should be treated as a stated target rather than a derivation established by the document. Its contribution is to frame the model, formula, and intuition for investigating the rate difference.

Key ideas

• The question studies the FRA–futures rate relationship under Ho–Lee short-rate dynamics.
• It states a convexity adjustment that depends on short-rate volatility and both contract dates.
• The proposed intuition links the futures premium to daily marking to market.
• The document asks for a rigorous derivation but does not provide one.

Tags

# Relationship between FRA rate and Futures Rate


# Relationship between FRA rate and Futures Rate












I am trying to understand the derivation of the convexity adjustment between FRAs and futures rates under the Ho–Lee short rate model (Page 15 of Modelling Single-name and Multi-name Credit Derivatives by Dominic O’Kane).

Assume the short rate follows the Ho–Lee dynamics:

$dr(t) = \theta(t)dt+\sigma dW(t)$

The result I am trying to derive is the following relationship:

$f(0,T_1,T_2) = F(0,T_1,T_2) - \dfrac{\sigma^2T_1T_2}{2}$

I understand the intuition that the futures rate should be higher than the FRA rate due to daily marking-to-market and the convexity adjustment depends on the volatility of the short-rate process. But I am having some difficulties where to begin the rigorous proof.

Thank you in advance!

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