Why Deep In-the-Money European Puts Can Lack Implied Volatility
Summary
The document explains why a Black–Scholes implied volatility solver can fail when a deep in-the-money European put is quoted below a model pricing bound. A put’s payoff at expiry is at least the strike minus the underlying price. Discounting that inequality and taking expectations gives a lower bound on the put’s current value: the present value of the strike minus the current underlying price. If a market quote is below this bound, no volatility input can make the model price match the quote, so a numerical routine may return no solution.
The example compares the stated option ask with the calculated bound and finds the ask below it. The discussion notes that for short maturities the discounted strike can be close to the strike, but the relevant comparison uses discounted intrinsic value. This is a model consistency bound, not an explanation of why a market quote might violate it; quote quality, conventions, or inputs would need separate investigation.
Key ideas
- A European put’s price cannot be below the discounted payoff bound based on the strike and current underlying price.
- An ask below that bound cannot be matched by any Black–Scholes volatility, so implied volatility may be undefined.
- For short maturities, the discounted strike may be close to the strike, but discounting still matters.
- A violated bound signals a pricing or data inconsistency to investigate; the post does not identify its market cause.
Tags
Full text
# Determining the implied volatility for options with bid/ask prices below the intrinsic value
# 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?
## Answer by Gordon (score 3, accepted)
https://quant.stackexchange.com/a/23038
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.