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. • Hi 10uss. As you've noted you are dealing with American options, so $C-P$ becomes vol dependent, contrary to the European option case. It is thus only normal that by doing what you do (there is something wrong in your formula anyway), the IV smile effect transpires somehow. I'm not aware of any name for this phenomenon. Jun 7 '19 at 11:06
• REM: when I say something wrong I mean that for European options, C-P parity writes: $C-P = DF(F-K)$, with $DF=e^{-rT}$. However, even when there are no dividends, you cannot reasonably assume the implied carry rate for the equity position is simply $r$, rather the forward price reads $F=S_0e^{(r+s)T}$ where $s$ reflects the additional funding spread over the risk-free rate (e.g. $s = -\text{repo margin}$). As such for any given pair at $K$, you have 2 unknowns: $r$ and $s$. Jun 7 '19 at 11:08
• Maybe I wasn't clear. The funding spread does not depend on the strike indeed. Still, it is potentially wrong not to account for it (from an arbitrage free perspective, the full equity carry cost $r+s$ should be the risk-neutral drift). @BrownianBread, most of what you say is correct in terms of interpretation, however you need both $r$ and $s$ to price the options, so I don't agree: pricing a call and put in a solver does not give you $r+s$, you still have two unknowns, unless you are willing to fix $r$ someway like you say and try to imply $s$ thereafter. Jun 7 '19 at 14:34
• @10uss say we look at selling options far above the atm. The call is going to be worth nearly zero, and be largely independent of the underlying price, so will have very low margin requirement, while the put will be worth very nearly (k-s) and will be almost delta 1, this will give it a higher margin requirement. Since selling the put will cost me more (since posting margin ties up cash/collateral) , I'm going to ask for a higher price for it. The same is true for low strike calls, and I think one of the reasons you often see a skew in the put/call implied fwd.
– will
Jun 8 '19 at 9:13
• I think this phenomenon is coming from the American exercise nature of listed options. In fact when interest rates are negative, it is not true that American call= European call. Hull’s book assumes interest rates are positive. When rates are negative, it may be optimal to exercise early an American call. This is giving the deep in the money calls a bit of extra value, which is causing the phenomenon.
– dm63
Jun 8 '19 at 10:30

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 $$K=50$$.

• thank you for correction! I adjusted the formula and spreadsheet! Jun 7 '19 at 11:36

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.

• Stanley thanks for your comment! I am aware that this doesn't hold in practice. I also observe this in my data. About your point about rounding, I have three variables which could be affected by rounding errors (put, call and underlying), I don't think personally it is that informative to just add or subtract 0.005. I see the same shapes and paterns for other option chains. Lower liquidity is not the case, this are just quotes given by Euronext (not prices observed in the market). The data is from (free to download): euronext.com/en/reports-statistics/derivatives/daily-statistics Jun 8 '19 at 17:11
• By the way your point about rounding could explain the smoother line for longer maturities (in this case the put and call are more valuable compared to shorter maturities, so less sensitive for rounding errors) Jun 8 '19 at 17:18