# Determining the implied volatility for options with bid/ask prices below the intrinsic value

I need some help in understanding the Black-Scholes option pricing model. In my data there are several deep itm European index put options that have an ask price below the intrinsic value. Calculating the implied volatility using a built-in function in matlab leaves me with NaN as a result. I suspect there is an economic explanation for this, but since I do not fully understand the way option valuation works, I wonder if anyone could help me out.

To give you an example:

Suppose the Nasdaq 100 quotes 563.48, the strike price of the European put option is 670, an annualized interest rate of 5%, days to maturity is 44 and an option ask price of 101.375. Why would a calculation with the Black-Scholes model not result in a value for the implied volatility? Is my guess correct that it has to do with the option price being lower than the intrinsic value? If yes, why couldn't the intrinsic value of a European option be higher than the option price when there is still quite some time left until maturity?

• Hi Kevin Scheurwater, welcome to Quant.SE! I see you've made an excellent choice when choosing your education ;) Can you tell us where you get this data? – Bob Jansen Feb 1 '16 at 21:03
• If such a discrepancy really existed, one would want to buy lots of those options, short lots of futures of the expiry and wait for your risk-free profit. More likely, there's an error in your data somewhere though, either quotes from different times, using cash instead of forward prices, the wrong or missing implied dividends rates, etc. – experquisite Feb 2 '16 at 5:36
• Hi Bob Jansen, the data is from Option Metrics, I got it from Wharton Research Data Services – Kevin Scheurwater Feb 3 '16 at 9:55

## 1 Answer

Note that \begin{align*} (K-S_T)^+ \ge K-S_T. \end{align*} Then \begin{align*} p &\equiv E\Big(e^{-rT} (K-S_T)^+ \Big)\\ &\ge E\Big(e^{-rT} (K-S_T) \Big)\\ &=K\, e^{-rT} - S_0\\ &= 670 \times e^{-0.05 \times 55/365} - 563.48\\ &=102.49. \end{align*} However, the option price is 101.375, which is smaller. This is the reason that you have difficulty to obtain an implied volatility.

Note that $K\, e^{-rT} \approx K$ for a short maturity $T$. You basically need to have option value greater the intrinsic value.

• +1 can such contract be used to make risk free profit immediately at time zero? – Neeraj Feb 1 '16 at 16:57
• @Neeraj. Theoretically, you can, but the transaction costs need also to be considered. – Gordon Feb 1 '16 at 17:05