Why a smile?

Why should there be a smile? 

 

BS is not the true process followed by the underlying.

Therefore using BS formula to back out the volatility given prices in the market will not give a constant number. 

 

What causes the volatility smile? 

 

2 ways of looking at this: 

 

1) Supply and Demand of vanilla options 

 

FX markets

 

Volatility smiles in FX markets are often fairly symmetrical, ex: EURUSD.

This is because euro investors see the market as the inverse of the way dollar investors see it. This explains the symmetry. 

 

But why do far OTM/ITM ptions have higher IV than ATM options? 

Well, investors usually want to protect themselves from adverse moves in the FX rate. So those people for whom a drop in the FX rate would be bad buy OTM  put options (low strikes) to protect themselves. Meanwhile, those for whom an increase in the FX rate would hurt buy OTM  call options (high strikes) to protect themselves.

Since there is greater demand from buyers than sellers, the prices are a little higher than you would otherwise expect, and as price increases with volatility, this means the volatilities at low and high strikes are higher. 

 

Equity markets

A similar argument shows why it is that in equity markets the volatility smile is heavily skewed. Typically, investors need to protect against decreases rather than increases in the index. 

 

The skew is often explained as this concept of insurance. It is a market where operators cover their downside risk. This phenomenon (skew) emerged after the 1987 market crash. 

 

2) The true underlying dynamics are not BS. 

 

It expresses the fact that market participants are well aware that the returns are not Gaussian. 

 

What could they be? 

 

If S drops lower than one would normally expect, or goes higher than expected, this is probably because something has happened in the market and so volatility has increased. 

 

A. Local Volatility = LV

One could construct a model in which the volatility depends on the spot so that large moves away from today's spot cause the volatility to go higher.     

 

B. Stochastic Volatility = SV

Another idea is that volatility itself could be random. 

If we make the volatility stochastic, we obtain a classic smile for the volatility surface. 

If we also make the stochastic process for the volatility correlated to the stochastic process for the spot, we obtain a skewed smile. 

 

C. Local Stochastic Volatility = LSV

It is also possible to construct models that are some way between local and stochastic volatility --> local stochastic volatility models. 

 

 

Both suggestions are good ways of understanding why the volatility smile is as it is.

Investors' view of the market affects the way they trade in the spot as well as the vanilla hedges they put on.

By no-arbitrage arguments, if it seems clear that volatilities should be higher at certain strikes for supply and demand reasons, then the true market dynamics must reflect this, and vice versa. 

 

The reason behind the skew becomes apparent when thinking in terms of realised gamma losses as a result of rebalancing the delta of the option in order to be delta hedged. 

In downward spiraling market, gamma on lower strike increases, which combined with a higher realised volatility causes the option seller to rebalance his delta more frequently, resulting in higher losses for the option seller. 

Option sellers want to get compensated for this and charge the option buyers a higher IV of these options. 

 

 

One thing that can certainly be said is that volatility frowns (curves that take the shape of an inverted smile) are rare. 

Since the payout of a put option increases as strike increases it must be true that the value of a put option increases with strike, and the opposite is true for  calls. This puts a constraint on the shape of smile curve that is possible without arbitrage.

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