# Volatility smiles

A plot of the implied volatility of an option as a function  of its strike price is known as a volatility smile.

Describe volatility smiles in different way.

## Put call parity

Put-call parity provides a good starting point for understanding volatility smiles. It is an important relationship between the price,c, of a European call and the price,p,of a European put .

$P+Soe^{-qt}$ = $C+k^{-rt}$

The call and the put have the same strike price ,k, time to maturity,T,

So= The variable is the price of the underlying assets today

r= Risk free interest rate

q= the yield on the assets

A key feature of the put-call parity relationship is that it is based on a relatively simple no-arbitrage argument. It does not require any assumptions about the future probability distribution of the assets price. It is true the when the assets price distribution is log normal and when it is not log normal.

Suppose that for a particular value of the volatility, PBs and CBs are the value of European put and call option calculate using the Black-Schloes model suppose further that Pmkt and Cmkt are the market values of these option.Because put-call parity holds for the market value of these option.Because put-call parity holds for the Black-Schloes model, we must have.

$P^{_{Bs}^{}$+$Soe^{-qt}=C^{_{Bs}$+$Ke^{-rt}$

Because it also holds  for the market prices we have

$P^{_{mkt}}$+$Soe^{-qt}$ =$C^{_{mkt}}+Ke^{-rt}$

Substracting these two equation gives

$P^{_{Bs}}$$P^{_{mkt}}$ = $C^{_{Bs}}$$C^{_{mkt}}$

### Foreign currency option

The volatility smile used by traders to price foreign currency options. The volatility is relatively low for at-the-money options.It becomes progressively higher as an option moves either in the money or out of the money.

We show how to determine the risk-neutral probability distribution for an assets price at future time from the volatility smile given by options maturing at that time. We refer to this as the implied distribution.

A log normal distribution with the same mean and standard deviation as the implied distribution has heavier tails than the log normal distribution.

Consistent with each other consider first a drop-out of the money call option with a higher strike price of K2. This option pays off only if the exchange rate proves to be above K2.

The probability of this is higher for the implied probability distribution than for the implied probability distribution. We therefore expect the implied distributed to give a relatively high price for the option. A relatively high price leads to a relatively high implied volatility.

#### Equity options

The volatility smile used by traders to price equity options (both those an individual stocks indices) This is some times referred to as a volatility skew. The volatility decrease as the price increase. The volatility used to price increase.

The volatility use in  price a low strike price option.(deep-out-of the money put on a drop-in-the money call) is significantly higher strike price option (deep-in-the-money put on a deep-out- of -the money call.

The volatility smile for the equity option corresponds to the implied probability distribution given by the solid  line. A log normal distribution with same mean and standard rather than as the relationship between the implied volatility and K. The smile is then usually much less dependent on the time to maturity.