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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?

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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.

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Oh of course. The lower bound was kind of obvious. I'm surprised I overlooked that. Anyway, you made some other good points to take into consideration. Thank you! –  Good Guy Mike Feb 13 at 21:57
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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.

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Ditto on checking your forward/rates/dividends. Likely you are using the wrong discount curve. –  experquisite Feb 12 at 15:14
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Yes, that appeared to be the case. For deep ITM calls the spread was wide so even the average bid-ask was below the lower bound. Thank you –  Good Guy Mike Feb 13 at 21:58
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