Currently, I am working on my thesis (MSc. Finance) and I run into an interesting “phenomenon”. I have option data for a non-dividend paying stock. In class I have learned, how to calculate the implied volatility of options but in this case, the data provider quoted their implied volatility. So, I thought it would be possible to calculate the “implied risk-free rate”.
I know that a European call and put option with the same maturity and strike price should have the same implied volatility (also described in Option, Futures and Other Derivatives by John C. Hull) and it makes sense that it is the same for the “implied interest rate”. So, I rearranged the put-call parity as follows, to calculate the implied risk-free rate:
$$r= -\frac{ln \left(\frac{S_t-C_t+P_t}{K} \right)}{T-t} $$
$S_t$ is equal to 102.05. When I plot the option chain for date $x$, I get the following graph:
See also the example below for the data overview (it is summarized)
I expected a flat line because the risk-free interest rate should not be affected by any factor but as you can see, the line is not flat. I know that the put-call parity assumes European options and my data contains American options. This does not matter for the Call options (European call and American call are equal to each other). However, my question is, is there a name for such a phenomenon? Or is there a paper written about this? I like to learn more about this. Thank you in advance and if you have any further questions, please let me know!
UPDATE 1 @Andrew mentioned I made an error by rearranging the put-call parity, I have adjusted the formula, graph and print screen of the spreadsheet in this post.
UPDATE 2 @Magic is in the chain asked if I have checked everyday, below is a graph with different strike dates on date x-1 (also pick some other random dates and the results are similar). I also checked this for another stock and get a similar curvature. Furthermore the line smooths when maturity is further away (see also the picture below). Also the longer the maturity, the lower the difference between the minimum and maximum. One last remark, Galapgos xxxx indicates the maturity month and year.