Black-Scholes in the real world

[OptionStudent]

This is a beginner's question about the relation between theory and practice. Read on only if you like to teach people who do not know very much. My question is about how to apply the Black-Scholes equation to E-mini S&P Futures prices. As will be obvious, I know something about the theory but nothing about how to read a trading chart. Today (4/13/2018), the S&P 500 is around 2650. On the CME Group website, I focused on end-of-the month options, which are evidently European options; hence, unmodified Black-Scholes calculations should be an ok starting point for understanding their prices. Options are mostly trading right at the money. For example, those that expire at the end of May (34 trading days left) have current prices of ~65 for a call and ~50 for a put for a strike price listed as 265000.0. These are close to Black-Scholes prices for an asset whose current price is 2650 with a strike price of 2650 (T = 0.135 yr, r = 0.017, sigma = 0.16 gives Black-Scholes prices of 64.90 for a call and 58.87 for a put). I have two questions:

1. Am I completely off track in relating theory to practice in this way? I know that the notional value of a E-mini contract is 50X the S&P 500 level. Hence, I was expecting the options to trade at 50X the above prices.

2. A minor point is the way the strike prices are listed. Are they in pennies and what happened to the factor of 50 when listing them?

Thanks for any help. I waded through many pages of CME Group educational material and looked at several other sources without getting any clue as to what how to read these basic charts and compare their entries to option-trading theory.

 

[Error Correction Funder]

You priced at 2650 not 265000.

Black scholes is only applicable to short term contracts. Variance gamma better prices long term options. No one to my knowledge has solved the Buffett complaint or the volatility smile etc.

 

[ajacobson]

So the core question is why are you modeling ? Generally the best use of the models is not to make money, but rather to capture the Greeks. As a student search out the Black Model which is an adaptation of the BS - used for futures options. You price off of the futures price so net carry comes from the forward price.

 

[stepandfetchit]

IMO: Your initial intuition is proper. BSM provides a relationship which balances the variables. BSM is a Garbage In - Garbage Out relationship. You provide incorrect inputs, you get what you deserve.
For example: Your simple reference lists T=0.135 yr (an indication you are not seeking precision) Just pointing out, that each input to the model should be taken seriously, if you REALLY want precision (time to the minute, is probably close enough for government work). For r, you may find that for expirations 6 months and under, using nearest term LIBOR rates may approximate correct results depending on your precision requirements for a no-touch derivation. The value for "volatility" is where most people screw up the relationship. For this, using the MID price of the contract, then use something like Newton Raphson method (with BSM) extract the proper IV for the specific strike! For the price of the underlying, you may wish to derive it from marshaling the option chain of the expiration of interest. I do not have first hand experience with ES for this step, but do for SPX. I think extracting the underlying from this method, except for expirations of the same date or perhaps 1 day out, will yield much tighter results. -- Using a process to derive the underlying must also take Call-Put parity into account. A final touch will be to use OTM & ATM only when deriving individual strike IV, so that for each strike the CALL and PUT IV are identical.

If you realize that "volatility" is really the only "unknown" input to the BSM, and resolve by using the actual option price data as mentioned above, then there is no difference between "theory and practice" for the purposes of option/Greek modeling for "present time" relationships. When you begin to speculate into the future, you will be force to "speculate" on what the "volatility" will be, which is uncertain.

I hope this helps.

 

[OptionStudent]

I appreciate the diverse inputs Error Correction Funder, ajacobson, and stepandfetchit provided to my question. I plan to pursue their suggestions for addressing limitations of the Black-Scholes model.

However, I do remain confused about the simpler point that initially motivated me to post it. My understanding of what one is doing when one buys or sells an E-Mini S&P future is making a bet about what the level of the S&P 500 will be on the expiration date. If I buy one call, I am betting that the index will be above the strike price. If I win, my counter party pays me 50X the difference between the index level and the strike price. If I lose, I just walk, losing the money I paid for the call. On this understanding, I expect all the numbers in my example to be ~50X higher than those on the CME Group's trading chart (with the added caveat that the strike prices are listed, for some odd reason, in pennies rather than dollars). Is some kind of shorthand at work in writing these charts or am I just confused about what the numbers mean. I am not a trader, just a "quant" who is attempting to learn to read actual trading charts. Traders may not realize how inscrutable these charts are to outsiders. I actually understand the statistical underpinnings of abstract-options-pricing models. I just don't understand what I would have been buying yesterday if I had paid $65 for an E-mini S&P call with a listed strike price of 265000.0 and expiry at the end of May.

 

[Aquarians]

>> Black scholes is only applicable to short term contracts. Variance gamma better prices long term options. No one to my knowledge has solved the Buffett complaint or the volatility smile etc.

I heard that before the crash of 1987, the option prices were pretty "flat" in volatility, thus very Black-Scholes. I don't have historical options data going back so far but the BS-formula was published in 1973, so this pricing went on for nearly 15 years. They couldn't have been all that retarded back then, not to realize there's a major problem with it.

So my reasoning is that the market is "pretty" Black-Scholes actually, meaning the distribution of stock returns at any maturity is well-approximated by a normal distribution. (You can actually verify this and roughly, this is the case).

Problem compared to 1987 is that the computing power is so much higher now that you just can't make money anymore by trading the classic "flat" vol. You could do if the spreads weren't so narrow (meaning the market so efficient). So now you're forced to look into the "edge" cases of Black-Scholes, meaning you have to model the distribution with way more precision than "normal", in order to have an edge.

 

[Error Correction Funder]

I appreciate the diverse inputs Error Correction Funder, ajacobson, and stepandfetchit provided to my question. I plan to pursue their suggestions for addressing limitations of the Black-Scholes model.

However, I do remain confused about the simpler point that initially motivated me to post it. My understanding of what one is doing when one buys or sells an E-Mini S&P future is making a bet about what the level of the S&P 500 will be on the expiration date. If I buy one call, I am betting that the index will be above the strike price. If I win, my counter party pays me 50X the difference between the index level and the strike price. If I lose, I just walk, losing the money I paid for the call. On this understanding, I expect all the numbers in my example to be ~50X higher than those on the CME Group's trading chart (with the added caveat that the strike prices are listed, for some odd reason, in pennies rather than dollars). Is some kind of shorthand at work in writing these charts or am I just confused about what the numbers mean. I am not a trader, just a "quant" who is attempting to learn to read actual trading charts. Traders may not realize how inscrutable these charts are to outsiders. I actually understand the statistical underpinnings of abstract-options-pricing models. I just don't understand what I would have been buying yesterday if I had paid $65 for an E-mini S&P call with a listed strike price of 265000.0 and expiry at the end of May.

Double check the multiplier.

 

[sle]

Black scholes is only applicable to short term contracts. Variance gamma better prices long term options.

What do you mean "better prices"? The price is coming from the market, you can use whatever model you want to risk manage your positions.

No one to my knowledge has solved the Buffett complaint or the volatility smile etc.

“You keep using that word, I do not think it means what you think it means”.

 

[Aquarians]

>> The price is coming from the market, you can use whatever model you want to risk manage your positions.

This statement implies you don't understand [much] about how options work. Which it's a good thing, those of us who do need this "market" if we're to make any money.

 

[sle]

This statement implies you don't understand [much] about how options work. Which it's a good thing, those of us who do need this "market" if we're to make any money.

I appreciate the greater fool sentiment, of course, but think of it this way. There roughly are 3 models of option market participants - there are price takers, liquid market price makers and illiquid market price makers.

The latter ones (illiquid market price makers) are the only ones who do not “care” about the market price. For example, a guy who makes a market on a 5 year call spread (a common companion structure for convertible bond issue) would probably go from the first principles to get his pricing parameters. I seriously doubt there are any OTC or otherwise bespoke MMs on this forum, so their modeling needs can be safely omitted.

The former ones, price takers and price makers are in one or another form linking their execution to the market price. A price taker looks at the current market and makes a decision if it looks attractive, either based on some outright break even or levels of implied volatility. A price maker looks at the current market based on the implied volatility and decides if it’s in his best interest to skew his own market.

The only one that would benefit from using an alternate model (such as variance gamma, SABR or whatever) is a liquid markets market maker that needs to risk mange his positions. A proper model ensures that his delta is consistent with the skew dynamics. Still, the MM uses the market price as a reference and makes his market around it and from that perspective “pricing” that comes from the model is not very important.

PS. Well, actually, the guy who warehouses the 5 year risk also uses “alternative” modes such as UVM, but that’s a different story