Hence my confusion, because if you agree with cash and carry no arbitrage pricing isn't 2 totally incorrect (except for the minor exception I listed)?
Index future contract's relationship to underlying index
- Thread starter TD877
- Start date
Differences you see between the ES and the SPX are due to...
1. Short term anticipation in the ES which will soon be corrected back to approximately "fair value".
2. The ES price gets reflected sooner than the SPX on the tape, and therefore appears to "lead". On the surface, it seems one might be able to arb the difference. I tried that years ago but with nil results overall.
For practical trading, ultimately they are effectively the same with the inclusion of "time" consideration for the ES.
Bottom line.... suggest you trade either the SPY with real-time data or trade the ES. Whichever you do, ignore the other.
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So you're believe is that the market thinks the index at the expiry of ES is where ES future trades now..... interesting...
So how does that work in relation to ES options? Wouldn't you get filthy rich by slamming straddles all the time towards zero?
Also, if you believe in the risk free arb pricing... (which IMO is a fact, since if that parity doesn't add up there's 100% chance of profit in the index vs future arb.)... then you should know the interest rate used in the pricing of the futures is basically the risk free rate (treasury etc) and if there are no dividends, the market expects the index to only rise by the risk free rate?
Than would be silly, since stocks (even broad index) isn't risk free... and therefore expected returns shouldn't be only based on risk free rate.. The risk free rate is only to ensure a no arbitrage pricing...
You're erroneously confounding volatility and expectations here. The fact that the price of futures reflects expectations at any given moment doesn't mean that those expectations remain constant over time.
You're confused and are contradicting yourself somewhat.
Firstly, the "risk-free" (I hate this terribly misleading term, BTW) rate which is commonly used nowadays in the mkt is actually the FedFunds OIS. This is an imperfect but practical approximation, since in order to truly ensure no arbitrage pricing, the rate used should be your specific funding rate.
Secondly, what do you think is the "true" expected return offered by the equity index, in the absence of dividends? If you know this number is different from the "risk-free" rate, why do you think this knowledge is not available to other mkt participants? And, if everyone is aware, why isn't this rate used in the calculation of the forward?
No, not confused and neither contradicting.
An index/stock is basically future expected cashflows discounted to now.... Those cashflows are more uncertain than the risk free rate what you would get on riskless (or so assumed) assets like deposits/treasury/etc. So those cashflows are more volatile... and that reflects in a generally speaking higher return on stocks than the risk free rate...
If you would only get the risk free rate, why would you invest in more risky stocks?
A forward or future is nothing more than a swap product, where you take delivery of the index on a future date. But the index can be traded now, with no changes... the only cost for holding the index is interest (compared to say wheat or another commodity, where delivery in one year means higher holding costs/different quality at maturity/etc etc...).
So if expected return of an index is say 10% in a year... and the future trades at 110 to reflect this as you say it should... then you can arb that by borrowing cash at (close to) riskfree rate + buying the index components now... assuming risk-free below 10% means you make money by holding the index vs short future...
That's why a "true" expected return is not in the calculation of the forward/future, but the risk free rate, because you can get funds to buy the spot at risk free rate...
And the expected return of the index basically already sets the current index price level.
You stated "...there are no dividends". What cashflows are you referring to here?
You are very confused here... Again, in the absence of dividends, why would the return of the index be 10% and higher than the "risk-free" rate?
At any rate, just think about it, you'll figure it out yourself, I'm sure. Or maybe someone else wants to chime in and help.
Oh this sounds like fun....let me play. 
So the return of the index over the long term usually has 3 components: risk free rate, div cash flows, risk premium. I like to add the long term inflation rate to the risk free rate since I need to be compensated for that long term as an opportunity cost assuming substituting bonds will compensate me for the inflation rate via long term rates.
The knowns: risk free rate (or as Marty says, our real internal borrowing costs), div cash flows (give or take), inflation rate. That leaves us with the risk premium. The risk premium is known long term. Long term, equities have historically provided about a 2% risk premium to bonds. This is actually pretty constant. So if all these factors are relatively known, we can use them as one large discount rate.
So the value of the S&P 500 today is equal to the present value of the future cash flows of ES (dividends). And this PV is derived from taking the FV of the dividends divided by the discount rate (risk free rate + inflation rate). The inflation rate incorporates the risk premium.
Am I getting warm?
So the return of the index over the long term usually has 3 components: risk free rate, div cash flows, risk premium. I like to add the long term inflation rate to the risk free rate since I need to be compensated for that long term as an opportunity cost assuming substituting bonds will compensate me for the inflation rate via long term rates.
The knowns: risk free rate (or as Marty says, our real internal borrowing costs), div cash flows (give or take), inflation rate. That leaves us with the risk premium. The risk premium is known long term. Long term, equities have historically provided about a 2% risk premium to bonds. This is actually pretty constant. So if all these factors are relatively known, we can use them as one large discount rate.
So the value of the S&P 500 today is equal to the present value of the future cash flows of ES (dividends). And this PV is derived from taking the FV of the dividends divided by the discount rate (risk free rate + inflation rate). The inflation rate incorporates the risk premium.
Am I getting warm?
It should be pointed out that the stocks that make up the s&p 500 are also properly discounted to reflect their future cash flows. This might be where the confusion lies. The price of AAPL today is reflecting it's future growth and cash flows. So when we add AAPL to the index along with all the other stocks, all the other stocks have been properly discounted as well. If AAPL should be 10% higher then it should be 10% higher immediately or there is free money available. So everything checks out except for as Marty pointed out, the long term expectations. As expectations change, so then should the discount rate. Both for the index and for the individual components.
I'll start with an intuitive look. Right now the Dec ES futures are trading at about 7 points lower than the spot SPX. Assuming 0% interest, which is essentially the case right now, that 7 point delta will gradually decay as component stocks of the S&P 500 go ex-dividend until they match on the day the Dec ES futures expire. This is a monotonic decrease in the delta, I can be sure it will never go to an 8 point delta between now and then (keeping interest rates out of it). If you follow the ES/SPX delta you will see this happens without fail every time.
Now let's add interest rates. If you buy Dec futures you're paying less to get the Dec price than someone who bought all the stocks in the index at the right ratios (or SPY), because you just have to post margin, not the full purchase amount. This acts in the opposite direction of dividends, so if interest rates were 10%, for example, the Dec ES futures would trade at a premium to SPX. Because they're so low right now, they don't have much impact, but if interest rates went up dramatically tomorrow, for example, the delta between SPX and ES would decrease because of it, and if it were possible for them to go down dramatically the delta between the two would increase.
At no time will this delta move simply because the market feels that the S&P 500 will be higher or lower in Dec than today, it is purely a mechanical relationship based on dividends and interest. This is a required relationship based on the math. The only variables are those dividends and interest, if they don't pan out during the interim period like the market thought they would, i.e. dividends change appreciably from historic levels or interest rates change, then the delta between SPX and ES will change.
Now let's assume that somehow the Dec futures started to show a higher price than would be warranted by the equation, supposedly because the market thought it should be higher. At this point, I can short the Dec ES futures and buy SPY and I'd be sure to capture a risk free profit by at least the time the ES future expired in Dec. Since this particular market is highly efficient and arbitrageurs step in the microsecond it gets out of balance, the market's perception of what the future price should be will never be reflected in an ES price that varies from the no arbitrage formula.
One last intuitive point, the current price of any security in a freely trading market is always the current market consensus of its discounted net present value. If the market thought the S&P 500 would be up 200 point in Dec, it would push the S&P 500 up nearly 200 points now. It wouldn't show up in the futures and not the spot. The thought experiment is to think of a thing everyone thought would be worth $100 on Dec 16th, what would they collectively agree to pay for it Dec 15th? Probably just under $100 right, accounting for 1 day of interest to compensate for paying for it early? What about on Dec 14th? Again just under $100 right? And Dec 1st, Nov 15th.... As you can see adjusted for carrying costs if the market feels the value of something in the future is X, the value of that something will be X today.