I'm working on the S&P500 European index options data(call options). On 2017-10-23, we have the closing price as 2564.98, and risk free rate is 1.09%(3 months treasury bill). If I choose the option that strikes at 1000 on 2017-11-17(25 days), it's current price is 1561.65.
If I use BS formula, I will get an implied volatility close to 0. But even if I use the volatility that close to 0 and substitute backward to the BS formula, I still can't get that option price (i.e., 1561.65). The calculated result will be a bit larger.
My questions are:
How do we get this option price when strikes are very low in the real market? There's no volume at all.
If I draw the volatility smile, it seems the implied volatility first increase and then decrease when strikes are getting smaller (call options in the money part). Could somebody explain it to me?
Thanks for your help.