Comparing ATM Skew in ES and SPX Options
Summary
The document asks whether at-the-money implied-volatility skew should match between ES futures options and SPX index options for the same maturity. It defines skew as the slope of implied volatility with respect to log-moneyness, where log-moneyness depends on the forward price. The author compares forward estimation through put-call parity, using a futures-based relation for ES and spot, rates, and dividends for SPX.
The reported exploration checks whether inferred forwards are similar across strikes and whether parity can reproduce option prices. The author says ES parity appears to hold using a zero rate, while SPX parity comes closer using a SOFR-based rate; the resulting skew estimates differ, and using a zero rate for SPX makes them look more similar despite parity not holding. No answer resolves whether the skews should theoretically agree. The plots are absent from the text, and details needed to assess the calculations—such as data conventions, dividend treatment, and exact rate inputs—are not provided, so the observed comparison remains inconclusive.
Key ideas
- ATM skew is defined as the implied-volatility slope at zero log-moneyness.
- The forward price used in log-moneyness depends on put-call parity and the instrument’s carry assumptions.
- The author compares ES and SPX skew estimates using different rate assumptions for their forward calculations.
- The reported parity checks and skew comparisons do not establish whether the two products should have equal skew.
- Missing plots and calculation details limit independent assessment of the observed differences.
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Full text
# Should ES and SPX options have the same ATM skew
# Should ES and SPX options have the same ATM skew
According to my understanding, ES options are based on futures, whereas SPX options are based on the underlying S&P 500 directly. However, how does this impact their ATM skew. These should still be equal right?
The ATM skew is given by $$ \psi(T) \;=\; \left.\frac{\partial \sigma_{\mathrm{imp}}(k,T)}{\partial k}\right|_{k=0}, \qquad k \;=\;\ln\!\bigl(\tfrac{K}{F}\bigr) $$
where $k \;=\;\ln\!\bigl(\tfrac{K}{F}\bigr)$ is the log-moneyness of the forward. Now my understanding on how to derive the forward is as follows:
For futures contracts, so ES, this is simply $F_t =C_t \;-\ P_t\; \;+\; K$, since we are already in the forward measure (?).
Then, for the SPX, we have the spot multiplied by the cost of carry ($r-q$). However, with $q$ being hard to find, I believe we can again find $F$ through the put-call parity as $$ C_t \;-\; P_t \;=\; S_t\,e^{-q\,(T-t)} \;-\; K\,e^{-r\,(T-t)} \;\;\Longleftrightarrow\;\; C_t - P_t = e^{-r(T-t)}\bigl(F - K\bigr) , \quad F = e^{r(T-t)}\bigl(C_t - P_t\bigr) + K= S\,e^{(r-q)(T-t)}$$ I've tried to verify this as follows. For the SPX, I calculated the forward through the put-call parity by either setting $r=0$, or using the $3-$month sofr futures rate for a given strike at a given maturity.
Then, if this is the correct forward value it should (i) be roughly equal for every strike at a given maturity (ii) be able to recover $C$ at a given maturity for a given set of strikes and $P$.
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I believe these graphs further conclude that this is indeed the correct idea, as ES put-call parity seems to hold when using $r=0$ (graph on the left), whereas SPX put-call parity seems to come pretty close with $r=sofr$.
However, when I now compute the ATM skew using $r=0$ for ES and $r=sofr$ for SPX, I find the following
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We can see that they are kinda off. But in principle, they should be equal for a given maturity, correct?
However, setting $r=0$ also for SPX gives:
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Now they seem more equal, especially for the call option? Even though this is obviously wrong? (e.g the put call parity does not hold for $SPX$)?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.