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I hope someone can clarify my ideas about the constant implied volatility in the classical Black Scholes framework.

As well known, market practitioners quote the prices of vanilla call and put options in terms of implied volatilities. For inputs $K$, $S$, $r$, $T$ and the price of the option $V$, one can determine the implied volatility $\sigma$ such that

$V = BS(K,S,r,T,\sigma)$ (1)

When the market quoted implied volatilities are plotted against different strike prices for a fixed maturity $T$, the graph would tipically exhibit a 'smile' shape and hence the name volatility smile.

Theory says that this implies a deficiency in the Black Scholes model since it assumes a constant volatility parameter, not depending on $K$ nor $T$. Hence the volatility smile would be flat.

Here my ideas get confused. Assuming that $S$, $r$ and $T$ remain constant, for a fixed market price $V$ of a vanilla option the implied volatility will vary depending on the value of strike $K$ under the Black Scholes model (1). Hence, if the implied volatility is plotted against different strikes for a fixed $V$ it will indeed show a smile behaviour, which is in contrast to what theory states.

Furthermore, do the market quoted implied volatilities that form the volatility smile according to the theory correspond to a fixed vanilla option price $V$ and with varying $K$?

I think I am making a mistake in my reasoning but I do not understand where. I would be glad if someone can point me in the right thinking direction.

Thanks in advance.

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  • $\begingroup$ The so-called volatility smile comes from market prices. That is, you fix an exercise date and look at all the options that trade which expire on that date, take the market price of each and use B-S to back out an implied volatility. In the early days, these implied vols were all over the map...since B-S they tend to be quite constant near the ATM strike, only showing significant variation for significantly high or low strikes. $\endgroup$ – lulu Feb 3 '16 at 11:23
  • $\begingroup$ Thank you for your answer. So by plugging in the market values in de BS formula and retrieving the implied volatilities gives a graph that is nearly constant near ATM values and exremes at lower/higher strikes. This sounds to me as a volatility smile/skew, but theory says that the implied volatility in the BS model has to be constant. Where do I make a mistake in my reasoning? $\endgroup$ – Tinkerbell Feb 4 '16 at 8:37
  • $\begingroup$ In addition to this, currently I am doing a research on the Libor Market Model. I price swaptions by simulating the forward Libor rates trough a Monte Carlo routine and by means of the Rebonato swaption pricing approximation formula. After calibration I would like to compare the two results, does anyone have an idea on how? Would it make sense to compare these results in a graph where different strikes are plotted (ATM + x) against the implied volatility for a given set of calibrated parameters? Or has this no meaning since the implied volatility in Black's formula must be constant? $\endgroup$ – Tinkerbell Feb 4 '16 at 8:45
  • $\begingroup$ Remark: The Libor Market Model I am analyzing is a plain vanilla model meaning that no stochastic volatility is involved. $\endgroup$ – Tinkerbell Feb 4 '16 at 9:00
  • $\begingroup$ Not sure I understand the question. As you understand, there are multiple models for option prices. B-S is a good, simple one. Given a collection of caps and swaptions, and picking an expiration date, you could try to use B-S to explain the prices, though this would be fairly crude. You would, presumably, get a different volatility for 3MLibor, 2YSwap, 3YSwap, and so on. This is fine, but somewhat unsatisfying (as these are obviously very dependent objects). Libor Market Models attempt to coordinate all of these into a single model. $\endgroup$ – lulu Feb 4 '16 at 11:50

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