Question

Explain in your own words why volatility "smiles". If the volatility of a stock grows indefinitely, what will happen to the delta of the call option? What will happen to the delta of the put option? Justify.

Answer 1

Volatility smiles occur because demand is greater for options which are ITM (in-the-money) or OTM (out-of-the-money). Implied volatility assumes that stock price is driven by Geometric Brownian motion & hence moves in continuous fashion. But we observe that stocks might jump (more downwards than upwards). This possibility of jumps result in prices of Out of the Money options to be very high, and to account for such high price we need relatively high implied vol.

If the volatility of the stock grows indefinitely, the call option will behave like a stock and hence delta of call option will be equal to 1. This happens because call option has a much higher chance of becoming/remaining in the money and there is no downside as premium has already been paid.

If the volatility of the stock grows indefinitely, the put option will have delta equal to zero. Because put will become out of the money.