No-Arbitrage Bounds and Data Quality in Deep In-the-Money Call Implied Volatility
Summary
The document examines why a deep in-the-money European call may have a quoted price below the Black–Scholes lower bound, leaving no implied volatility solution for that quote. For a call, the lower bound is the spot price minus the discounted strike. The discussion emphasizes that this is a general no-arbitrage bound, not merely a feature of the Black–Scholes formula: a genuine tradable price below it would imply an arbitrage opportunity. A quote or midpoint below the bound, however, may simply be unreliable or untradeable.
The responses point to wide spreads and poor liquidity in in-the-money options as practical sources of misleading prices. They recommend comparing theoretical implied volatility with that of the corresponding out-of-the-money put, while checking the forward price, rates, and dividends. A price exactly at the bound corresponds to zero time value and zero implied volatility. The document also warns that minimum price increments can flatten out-of-the-money put prices near the minimum quote, creating an artificial rise in computed implied volatility. These observations caution that apparent smile patterns in sparse or coarse market data need not reflect genuine volatility dynamics.
Key ideas
- An in-the-money call's lower price bound follows from no-arbitrage and equals spot less the discounted strike.
- A quote below the bound has no implied volatility solution, but a poor midpoint may not represent an executable market price.
- Wide spreads and weak liquidity can make in-the-money option prices unreliable.
- The corresponding out-of-the-money put can provide a better price comparison, provided forwards, rates, and dividends are correct.
- Minimum quote increments can distort implied volatility and create a misleading apparent smile.
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# Lower bound of ITM Calls when computing Implied Volatility
# Lower bound of ITM Calls when computing Implied Volatility
Assuming the Black Scholes model and pricing formula of a European call option. Then, if the call is ITM, i.e. if $ln(\frac{S}{K})>0$, the $d_1$-term will go towards infinity as $\sigma$ goes to zero. This also implies that the $d_2$-term will go to infinity and the normal cdfs will both approach 1. This creates a lower bound $S-e^{-r(T-t)}K$ for the option price.
Now let's assume I want to compute the implied volatility for the ITM call option, but the price of call is smaller than lower bound of the B.S. pricing formula. Then the equation I'm trying to solve for the IV does not have a solution. Precisely this is happening as I'm trying to compute the IV of deep ITM calls. However, usually one talks about the volatility smile, where deep ITM calls has larger volatility than ATM calls. Is there any reasonable interpretation of this?
## Answer by sashkello (score 6, accepted)
https://quant.stackexchange.com/a/10212
The lower bound is not just a BS-specific bound. It is a no-arbitrage bound and so if the price is lower than this, you have an arbitrage opportunity (some good explanation here). It doesn't mean it is present in the market necessarily, because mid price is not necessarily the price you can trade and when you take spread into account this is likely to go away. It is quite often the case for ITM options because data for them is of lower quality (low liquidity).
When the price is exactly on that border (zero time value), it actually would mean that the implied volatility is exactly zero, because you are essentially stating that there is zero probability of price going higher than the strike. Volatility smile is a topic on its own, and there are books written about this phenomenon. From the practical point of view, however, you again need to think about the quality of the data - while in theory you should have some sort of nice smooth volatility smile, when you are working with real data you can get some weird results. First thing is that using ITM options is a bad idea because price quality is more likely bad than good, use only OTM's. Implied volatilities "in theory" should match, but OTM put price is a better estimate of a fair price than ITM call price. Also keep in mind that "smile" doesn't mean that it is symmetric, in fact it can take different kinds of shapes (sometimes quite weird).
Just as a final note, since you are talking about deep ITM calls (or OTM puts), the volatility can go up due to the fact that the prices are quantized. That is, all OTM puts from some point will cost exactly \$0.01, simply because there is nothing below that. Obviously, the implied volatility for higher strike will be higher in such case, with this price fixed and you'll see it going straight up from some point. This can be mistaken for the genuine volatility smile, while it is not it - in theory, option price should go below \$0.01 to fractions and the implied volatility would be completely different.
## Answer by derenik (score 3)
https://quant.stackexchange.com/a/10213
Most likely you are looking at bid prices which are lower that fair (theoretical) price. It is very common that bid price of an ITM option is below the lower bound as bid-ask spreads are wide. The IV of ITM call at theoretical price should match IV of OTM put at corresponding strike. If this does not happen then check your forward price, rates and dividends.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.