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Is there an accurate and strategic ratio between the price of the ES E-mini Futures and that of SPY? Despite the Quarterly rollovers, given the current quarter, I notice that typically ES is higher by around 0.9% Vs 10*SPY But this percentage will keep on reducing by a very minute amount as each trading day passes, until the day of the quarter expiry, at which point it may or may not be 1-1 still. And then the rollover into next Quarter starts off with a max discrepency again.

Is this something calculable? And also I notice the cash index, SPX is even different than both of them as well, which even complicates the matters even more!

Reason is, normally SPY is covered in most analysis, supports and resistances, etc. So in order to translate that to ES, this percentage offset is needed!

Thanks, Steve.

2 Answers 2

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The fair value of an equity future is just the future value of the current price, minus any expected dividends (which is an estimate since dividends are discretionary), which means the price if the money were instead invested at some continuously compounded rate of interest.

The future value would be

FV = PV * e^((r-d)*t)

where PV is the "present value" (current price), r is the rate of interest, d is the dividend yield, and t is the time in years until the future expires.

(note that r is not necessarily a constant for all values of t, but I don't think that's relative to your question)

So it's not a constant ratio, but an exponential ratio. The further out the future, the higher the future price relative to the current price.

The reason for that relationship is because in theory one could borrow money at rate r, buy the equity at the current price, and "sell" a futures contract. If the future price were higher that the amount that you'd have to pay back (borrowed amount plus interest) they you'd earn a risk-free profit. So the "future value" relationship is required to keep this "arbitrage" from happening.

This relationship is not true for commodity futures, where there is a cost to buy and store the commodity that is bought ("cost of carry") other then the interest on the borrowed funds. I can't just "buy" tons of gold and short gold futures - I have to have somewhere to put the gold, which costs money (either directly or indirectly) and/or the means to transport it.

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    You are missing dividends. Unless there are no dividends, you need to take them into account when computing theoretical future prices.
    – AKdemy
    Apr 17, 2023 at 20:41
  • @AKdemy good point - added.
    – D Stanley
    Apr 17, 2023 at 21:23
  • Thanks both! This is great...now I see why this is hard to calculate precisely due to all the moving parts, specifically dividends and interest rates. Unfortunately I cannot calculate that on a daily basis as I do not have an accurate measure of those 2 parameters. Or can I get them anywhere hourly or daily? P.S. I assume for Options pricing one would similarly use the Black-Scholes formula.
    – Steve237
    Apr 19, 2023 at 5:15
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    SPX is the ticker for the actual S&P 500 index, which is what the payoff of ES futures is based on. SPY is the ticker for a mutual fund that tracks the index but does not have the same value per unit - the relative returns for the fund are supposed to be the same as the returns for the index, but can be off due to fees and tracking error.
    – D Stanley
    Apr 19, 2023 at 18:09
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    SPY is an ETF, not a mutual fund. Despite mutual funds and ETFs both representing managed "baskets" or "pools" of individual securities, the actual structure is very different.
    – AKdemy
    Apr 19, 2023 at 21:03
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It is "simple" cost of carry. In words, the cost of carry relationship describes the relative cost of buying a stock with deferred delivery (the future) versus buying it in the spot market with immediate delivery and "carrying" it forward. If you buy stock now, you tie up your funds and incur a time value of money cost of r per period. On the other hand, you receive dividend payments (carry benefit) of d.

This advantage must be offset by a differential between the futures and the spot price. The future price is exactly offsetting this difference and there is no free money. It may be less obvious with equity but should be quite clear with FX (where the concept is identical, just with two interest rates). It is called Covered Interest Parity (CIP).

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No matter what you do, returns from investing domestically are equal to the returns from investing abroad. This works because you enter a forward and fix that rate that guarantees no arbitrage.

Actually computing this is tricky because future dividends are unknown until they are annouced. It is also quite sensitive to getting the exact daycount correct etc.

Edit

If r = 4% and d = 3%, and current SPX is 4150, a 1 year future will be priced at SPX*e^(r-d) = 4171.60, which is about 1%. If you have access to Bloomberg, you can look at FAIR to get the computations done accuratley without any additional work. There will be quite some work involved in getting reliable estimates for dividend return, as well as the interest rate (from swap curves).

For options, it's the exact same cost of carry problem, yes. You can see lots of details in this answer on quant stack exchange. Most equity options will be American style, in which case the closed form formula for Black Scholes doesn't work.

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  • Yes Sure, but this doesn't answer what percentage difference the SPY is with respect to ES on a given day/hour.
    – Steve237
    Apr 17, 2023 at 19:24
  • Thank you, this is great now! If Black.S won't work for Options, then what will?
    – Steve237
    Apr 19, 2023 at 14:34
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    Black scholes works, just not closed form (you need to solve eh PDE). However, these are price quoted listed options, meaning prices are anyhow readily available. Reversing that yourself (computing a vol surface, and solving the PDE) is very technical and a lot more complex than computing a fair future value.
    – AKdemy
    Apr 19, 2023 at 15:49

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