FX Carry Adjustments and Intraday Correlation with Equity Indices
Summary
This note asks how to estimate intraday co-movement between an FX pair such as EUR/USD and an equity index under the real-world measure. It treats interest-rate differentials as a source of forward-price drift and proposes adjusting FX changes around the settlement window before measuring covariance with index changes. It also asks whether spot-index changes should be adjusted for local interest accrual over short intervals.
The proposed procedure is a question, not a validated method: the document supplies no answer, empirical comparison, or derivation. Its equations and timing assumptions concern close-price observations, a stated futures trading session, settlement during that session, and no dividend payment in the interval. Those conventions affect how synchronized returns and carry adjustments should be defined. The note also characterizes FX behavior as mean reverting with a changing level, but offers no evidence for that model. Readers should treat the proposed covariance construction as an unresolved modeling idea rather than an established way to isolate market action.
Key ideas
- The note frames FX forward carry as a component to distinguish from market-driven intraday changes.
- It proposes adjusting FX changes at settlement before calculating covariance with index changes.
- It asks whether local interest accrual should adjust spot-index changes at minute frequency.
- The proposal is not supported by an answer, derivation, or empirical test in the supplied document.
- Return synchronization, settlement conventions, and dividends can affect the resulting covariance.
Tags
Full text
# FX modelling in the real-world measure, $\mathbb{P}$ and its correlation to indices
# FX modelling in the real-world measure, $\mathbb{P}$ and its correlation to indices
So I am looking to model FX pairs in the real world measure $\mathbb{P}$ and their relationship to indices. From what I understand, if we have an FX, say EURUSD, the price will trend with the forward rate. So if $r_d < r_f$, EURUSD will rise at the difference of their rates, $(r_f - r_d)/252$ on $T+1$ days, whilst $3*(r_f - r_d)/252$ on their $T+2$ day. And so any movement outside of this price change is market action by traders.
Now from historical observations, we see that major FX pairs (and EURUSD in this case) are typically not martingales, and are like a CIR model with a time-varying mean-reversion level, which in this case, would be the difference in $r_f$ and $r_d$.
If we wanted to measure the intraday correlation of the FX pair, $F_t$ and an index, $S_t$, where index has a local interest-rate $r_l$, would it make sense to then remove(discount) the forward rate, which "isolates" and only leaves the impact of the market's perception on the FX pair, then measure the correlation? So if we had $t=1$-minute OHLC data points (using close prices), the index future trades for 20-hour days, it is a $T+1$ day, the FX settlement time was during the 20 hour period and no dividend payout during the period, then the measurement of the correlation would be:
$$dS_{t+1}=S_{t+1} - S_t$$ $$dF_{t+1}=F_{t+1} - F_t, \quad \text{if}\quad t\neq t_{\text{settlement}}$$ $$dF_{t+1}=e^{\frac{r_d-r_f}{252}}(F_{t+1} - F_t), \quad \text{if}\quad t= t_{\text{settlement}}$$ $$\therefore \Sigma = cov(dS_t, dF_t)$$
Which is affectively removing the price adjustment during the settlement window at 10pm UK time.
Extra: If $S_t$ was instead the spot index (I understand the spot technically doesn't exist), but then would $dS_{t+1}$ be (assuming a 6.5 hour trading day) given by: $$dS_{t+1}=e^{\frac{r_l}{252\cdot 6.5 \cdot 60}}(S_{t+1} - S_t)?$$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.