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I'm trying to use the Black & Scholes to calculate the price for some options on the CBOE, but I'm having a hard time matching what I calculate with what i see on the market.

As an example I took the following screenshot (from this page):

enter image description here

At the time of writing, the numbers are as follows:

  • S&P500: 2441.32
  • VIX (from here): 15.51

Together with a risk free interest rate of 5% I calculated the price for the option with the strike price of $2450 using an online Black&Scholes calculator. The price I got for the call option is $2429.325:

enter image description here

$2429.325 is nowhere near the going rate of about $36 listed on the first screenshot above. Even if I multiply the rate of $36 with the usual multiplier of 100 I get $3600, which is also nowhere near the $2429.325 I got from the Black&Scholes.

Does anybody know what I'm doing wrong here? All tips are welcome!

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    $\begingroup$ Read the asterisk footnote. A volatility of 15% needs to be entered a 0.15 not 15.0 $\endgroup$ – Alex C Aug 12 '17 at 19:43
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    $\begingroup$ Although it is immaterial for such short expiries, you should also use an interest rate that is appropriate for the expiry and currency. A rate between 1.0% and 1.2% for short term USD would be good starting point $\endgroup$ – James Spencer-Lavan Aug 13 '17 at 7:29
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Also, your input for volatility into the BS Calculator is 1551%. I am assuming you want to input the volatility as 15.51%, which would be 0.1551.

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The Black & Scholes model is exactly that: a model. And it is wrong. Stock price returns are not lognormal distributed with a constant vol.

If your function is CallPrice(spot, strike, time, r, q, vol) and you give me all of the arguments, I can give you the price according to the black Scholes model. If you give me the price, and all the inputs except one, then I can work out (essentially by trial and error) the possible values of the missing parameter that give the same price. If we do this for the volatility, given option to prices, we can work out the Black & Scholes implied volatility for each option. And you'll get a different vol for each option (ie clearing the BS model is inconsistent).

So then you have the question why use this model and work out the implied volatility? It is simply a mapping from option price to something else, you could just as easily decide you want to look at your option prices as their value above intrinsic (ie time value). Is makes it easier to compare a stock to itself in the past. To see where options are overpriced vs historical levels, etc.

That is problem 1. Problem two is that you're using vix. Vix is the "30 day option implied varswap par rate", it is calculated using g the varswap static replication portfolio, which is essentially a weighted sum over otm options. It's okay as an approximation, but it will be higher than the atm vol (which is approx. what you need).

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Option prices are not determined by a model.

Option prices are determined by an exchange matching two traders' bid and ask.

The model is simply an attempt to explain why that specific bid and ask was reached. The real answer is "because both sides of the trade thought it was acceptable."

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