Delta hedging is a method of hedging options against fluctuations in the price of the underlying asset. Delta hedging allows you to earn through rehedging if gamma and vega are positive.

# Basic definitions

Put-call parity shows a mathematical relationship between the prices of a European put option and a European call option with same expiry date, strike price and underlying asset.

**Definition.** Option is a derivative financial instrument, it is a contract sold by one counterparty (option writer) to another counterparty (option buyer).

In the option market, historical volatility is calculated as the standard deviation of daily returns on the underlying asset over a specified period of time. In practice, traders and analysts use an annual expression of volatility, just as interest rates are always measured in annual terms.

Realized volatility is a standard deviation of daily price fluctuations of an underlying asset, expressed as a percentage. Historical volatility measures the size of price fluctuations over a certain period of time. As a rule, the value of volatility is translated into an annual expression. For example, on July 31, 2014 the volatility is translated into an annual expression.

The option price consists of time value and intrinsic value. For call options, the intrinsic value is defined as the maximum of two numbers - zero and forward price of the underlying minus strike, for put options - zero and strike minus forward price of the asset.

Volatility smile is a graphical image that reflects the dependence of implied option volatility (with one expiry date) on the strike price and looks like a smile. This shape of the curve is observed in the Japanese stock market, i.e. Japanese investors do not demand a higher premium for put options than for call options.

**American option**

Option price = Intrinsic value + Time value

**Time value**