Inferring Interest Rates from Option Prices and Put-Call Parity

# Why do I get a curved line when I plot "implied interest rate" on the strike price?


# Why do I get a curved line when I plot "implied interest rate" on the strike price?












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.

## Answer by Andrew (score 1)

https://quant.stackexchange.com/a/45985

Those numbers should indicate that something went wrong.

The Put-call parity for non dividend paying stocks is given by $C_t-P_t = S_t -e^{-r(T-t)}K$ . Solving this for r gives $r=\frac{-\ln(\frac{S_t-C_t+P_t}{K})}{(T-t)}$ . When using this formula you get more reasonable results, e.g. $r=-0.00151$ for $K=50$.

## Answer by D Stanley (score 1)

https://quant.stackexchange.com/a/45993

> I know that a European call and put option with the same maturity and strike price should have the same implied volatility

They should, but that is not observed in the market. I don't know what source data you're using , but if I look at deep away-from-the-money options for AAPL, for example (which is incredibly liquid), I see different implied vols for the same strike. Sure, that may mean that there is an arbitrage opportunity, but my guess is that either the arb potential is too small to be worth it or the liquidity for one side or the other is insufficient to monetize the difference.

There could also be some problems with data precision. To see if that's a possibility, what implied rates do you get if you use prices +/- half a cent to account for rounding?

Even at-the-money options can have different implied vols in the market, probably due to different supply/demand for puts/calls, but the discrepancy is more evident the further away you get from at-the-money, again probably due to lower liquidity.