Getting non-sense values for certain inputs using "Implied Volatility with the Newton-Raphson Method"

[ewmailing]

Hello, I was just curious as to why I sometimes get non-sense values using your:
2021-03-29 Option Implied Volatility using Newton_s Method in Python

https://github.com/TheQuantPy/youtube-tutorials/blob/main/2021/001%20Jan-Mar/2021-03-29%20Option%20Implied%20Volatility%20using%20Newton_s%20Method%20in%20Python.ipynb

It usually gives pretty good values, but sometimes I do get non-sense values. So I thought I would provide a legitimate real-world value that is outputting non-sense.

I was looking at the TLT today, and stumbled upon some values that broke this.

The underlying price is $85.37.
The strike price is $75.00.
The time to expiration is .6767 (247 days / 365 days).
The interest rate is 0.043.
The market price is $11.73 for a call option.

According to the options chain from my brokerage, the implied volatility should be about 20% for this.

So modifying your workbook:

S0, K, T, r = 85.37, 75.00, 0.6767, 0.043
market_price = 11.73
implied_vol_est = implied_vol(S0, K, T, r, market_price, flag='c')
print("Implied Volatility is : ", round(implied_vol_est,2)*100, "%")

Produces the output:

Implied Volatility is :  nan %

<ipython-input-5-76ce81ec5ffe>:17: RuntimeWarning: divide by zero encountered in scalar divide
  vol_new = vol_old - C/Cprime

Tweaking the values a little bit to avoid the divide by zero, I still get non-sense values.

S0, K, T, r = 85.00, 75.00, 0.6767, 0.04
market_price = 11.73
implied_vol_est = implied_vol(S0, K, T, r, market_price, flag='c')
print("Implied Volatility is : ", round(implied_vol_est,2)*100, "%")

Produces the output:

Implied Volatility is :  -286.0 %

I'm curious if this is a bug, or if there are just some (legitimate) values that cause this technique to break-down and not work. And is there a fix for this?

Thank you