I am not sure if this question is appropriate here, but I am just going to give a shot. It is related to an empirical observation about the price of a call option. Specifically, I was looking at a very out-of-the-money call on AMZN, namely AMZN Jan 15 '21 $3000 call. I noticed that it follows the price of the stock (AMZN) very closely.
I then plotted the correlation coefficient between the price of the call option and the price of the underlying stock and here is the result (according to trading platform I am using):
According to this plot, the correlation coefficient between these two fluctuates, but often it is very high, i.e. $\ge 0.9$. Current stock price (at close) is about \$2408 and option price is roughly \$104 (taking the midpoint between bid and ask). Again according the trading platform, the $\delta$ for this option is $0.286 \approx 0.3$ (with an IV of 33.6%). Assuming they are using the Black-Scholes model to price the option and they are doing the calculation correctly, it seems that there is a big mismatch between the model and the actual data (i.e., between 0.3 and 0.9)?
Is there an explanation for this? Am I missing something? I am not an expert in quantitative finance, but do people in the field care about these discrepancies between theory and empirical evidence? Are there any good papers on it to explore more?