When trading options, a trader is constantly faced with the choice between the gamma and theta. In simple words, if a trader has bought options, the gamma will be positive and the price for positive gamma will be theta. Conversely, if a trader has sold options, the trader will receive theta for negative gamma.
Option’s speed measures the change in gamma with respect to the change in price of the underlying asset, all other things being equal. Mathematically, a dGamma/dSpot is the 1st derivative of the gamma of the price of the underlying asset or a 3rd order derivative of the price of an option with respect to the price of the underlying asset.
According to Black-Scholes model assumptions, all options with different strikes and exercise dates have the same implied volatility, i.e. implied volatility does not depend on either the option's moneyness or the expiry date.
Theta measures the sensitivity of an option's price to changes in time, all other parameters being equal. Theta usually has a negative value if the trader owns options, and a positive value if the trader has a short option position. Figure 1 illustrates how the call option price changes as the time passes.
Vanna or dVega/dSpot or dDelta/dVol is a derivative of option vega with respect to the price of the underlying asset, which is equivalent to a derivative of the delta with respect to volatility. According to the laws of mathematics, it also indicates the sensitivity of the option delta to changes in implied volatility.
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.
Futures contract - a standardized agreement between two counterparties to buy or sell an underlying asset (security, goods) at a price agreed in advance at a specified future date. The contract specifies the quality and quantity of the underlying asset. There are two types of futures: delivery and cash-settled.
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.
The volatility index is a set of implied volatilities of a whole series of call and put options for a particular stock index. As a rule, a volatility index includes implied volatility of a stock index, but more complex volatility indices are calculated based on the implied volatility of individual shares included in the index.