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Two of the conditions for an asset price to have a lognormal distribution are:

  1. The volatility of the asset is constant.
  2. The price of the asset changes smoothly with no jumps.

In practice, neither of these conditions is satisfied for an exchange rate, but I don't understand why these reason determine the form of a "smile" for implied volatility?

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    $\begingroup$ nothing in financial markets is lognormally distributed, that is just a convenient toy model $\endgroup$
    – vonjd
    Commented Jun 16, 2017 at 11:07
  • $\begingroup$ You are right! I reformulated the question to be clearer! $\endgroup$
    – Mike9
    Commented Jun 16, 2017 at 11:23
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    $\begingroup$ The conditions you mention in your question are sufficient but not necessary. You can still have normally distributed log-returns when e.g. the volatility is a deterministic function of time or there is a normally distributed jump with a deterministic time of occurrence. $\endgroup$
    – LocalVolatility
    Commented Jun 16, 2017 at 11:58
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    $\begingroup$ Also, there are a few closely related questions with answers already e.g. quant.stackexchange.com/questions/29914/…, quant.stackexchange.com/questions/4503/…. $\endgroup$
    – LocalVolatility
    Commented Jun 16, 2017 at 12:03
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    $\begingroup$ By the way: it is called "implied volatility" and not implicit .. with the correct expression you can look it up better. $\endgroup$
    – Richi Wa
    Commented Jun 16, 2017 at 13:32

2 Answers 2

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There are plenty of theories but no one is 100% certain.

Theory 1: Options which have strike prices increasingly far away from the spot price are including extreme movements in the market such as black swan events into the price. When an event like this happens there is a sharp change in price of the option and extreme change in volatility.

Theory 2: Investors may have the same demand in terms of expiration date but not strike price. Investors tend to prefer in-the-money and out-of-the-money options over ATM. Since demand pushes up the prices of options, implied volatility also becomes higher.

Theory 3: Similar to 1, implied volatility can include jump predictions, but this theory includes all types of jumps, small and medium ones as well as large ones. These jumps are accompanied by large kurtosis as stated in the link @LocalVolatility commented.

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The smile is there exactly because the model is wrong.

The reason it's used though (despite being wrong) is that it provides a convenient space to look at the underlying - the vol*

The (undiscounted) value of an option is given by:

$$ \int_0^\infty \mathrm{PDF}(s) (s-K)^+ \mathrm{d}s $$

where $\mathrm{PDF}(x)$ is the real probability distribution of the underlying. This is model independent.

Under BS, the value is the following:

$$ \int_0^\infty \mathrm{PDF_{LN}}(F,\sigma)(s) (s-K)^+ \mathrm{d}s $$

where $\mathrm{PDF_{LN}}(F,\sigma)$ is the pdf of a lognormal variable with an expected value of $F$ and vol of $\sigma$.

So now, we just need to make them match - and we have one parameter to solve for - $\sigma$. This gives us the BS vol. If the distribution truly were lognormal, you'd obtain the same vol everywhere. Since it's not, you get a changing vol.

*BS vol.

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