4
The Volatility Surface

4.1 GENERAL DEFINITIONS

In Chapter 3 we defined the volatility smile; we now extend this notion by defining the volatility surface. This is a fundamental building block of market-making activity and risk management, so we devote an in-depth analysis to the building of a consistent volatility surface.55
 
Definition 4.1.1. Volatility surface. The volatility surface, or matrix (we will use the two terms without any distinction), is the map of the implied volatilities quoted by the market for plain vanilla options struck at different levels and expiring at different dates. Implied volatility is the parameter σ to plug into the Black-Scholes formula to calculate the price of an option.
 
The volatility smile refers to a single expiry, whereas the volatility surface refers to a set of maturities. In practice, the matrix is built according to three main conventions, each prevailing as a standard in the market according to the traded underlying: the sticky strike, the sticky Delta, and finally the sticky absolute. These are simple rules used to conveniently quote and trade options written on different assets and, as such, are not intended to model the evolution of the volatility surface.
 
Definition 4.1.2. Sticky strike rule. When the sticky strike rule is effective, implied volatilities are mapped, for each expiry, with respect to the strike prices; this is the rule usually adopted in official markets (e.g., equity options and futures options).
 
The name

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