How Spot-Rate Correlation Affects Futures Convexity Adjustments

Summary

The document examines a cash-flow explanation for the convexity adjustment between an interest-rate futures contract and a forward rate. Daily variation margin can create an investment or financing effect: a short futures position receives funds when rates rise and faces losses when rates fall. The question is whether this explanation assumes that the spot rate used to invest or borrow margin moves with the futures reference rate.

The answer says the adjustment depends on the correlation between the futures rate and the relevant spot rate, defined as the rate from the present to futures expiry. That correlation is described as usually positive in practice, but not necessarily near perfect. Simplified explanations often leave the correlation dependence unstated. The discussion offers a conceptual clarification rather than a formula, calibration procedure, or empirical evidence, so it does not quantify the adjustment for a particular contract or market.

Key ideas

• Variation-margin cash flows create an investment or financing effect behind futures convexity adjustments.
• The adjustment depends on correlation between the futures rate and the relevant spot rate.
• The relevant spot rate runs from the present to the futures contract's expiry.
• The correlation is said to be usually positive but can be materially below perfect correlation.
• The explanation is conceptual and does not quantify the adjustment for a specific market.

Tags

# Convexity adjustment--Assume sport and futures rates move together?


# Convexity adjustment--Assume sport and futures rates move together?












A cash flow argument I typically see for why a convexity adjustment is necessary is the following (taken loosely from Hull 9/e, p. 143):

> Say I am short an interest rate futures contract (e.g. Eurodollars). If the futures rate (the rate referenced in the contract) rises, then I am credited funds in my margin account and am able to invest these proceeds at a higher rate. Conversely, if the futures rate falls, then I have a loss in my margin account and must finance this loss but at a lower rate.

Therefore the market will set the futures rate higher since the long side of this transaction will experience the opposite effect of investing/financing the daily settlements (finance at higher rates, invest at lower rates).

What is not clear to me is that the spot rate at which I am able to invest/borrow the balance of my margin account changes in the same direction as the futures rate. The type of argument given above seems to implicitly assume the two rates always move in the same direction--is this correct? If so, does this assumption always hold?

## Answer by dm63 (score 1, accepted)

https://quant.stackexchange.com/a/42989

You are absolutely right. The convexity adjustment is proportional to the correlation between the spot rate (actually , the rate from today to the expiration of the futures) and the futures rate. In practice the correlation is nearly always positive, although not always close to 100%. Many simplified explanations ignore this effect.

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.