Cap Implied Volatility

An implied volatility is the volatility implied by the market price of an option based on the Black-Scholes option pricing model. In cap market, a cap/floor is quoted by implied volatilities but not prices. An interest rate cap volatility surface is a three-dimensional plot of the implied volatility of a cap as a function of strike and maturity.

The term structures of implied volatilities which provide indications of the market’s near- and long-term uncertainty about future short- and long-term forward interest rates. A crucial property of the implied volatility surface is the absence of arbitrage.

In market, a cap/floor is quoted by implied volatilities rather than prices. An interest rate cap volatility surface is a three-dimensional plot of the implied volatility of a cap as a function of strike and maturity.

Vol skew or smile pattern is directly related to the conditional non-normality of the underlying returns. In particular, a smile reflects fat tails in the return distribution whereas a skew indicates asymmetry. A crucial property of the implied volatility surface is the absence of arbitrage.

To construct a reliable volatility surface, it is necessarily to apply robust interpolation methods to a set of discrete volatility data. Arbitrage free conditions may be implicitly or explicitly embedded in the procedure. Typical approaches are
 Local Volatility Model: a generalisation of the Blaack-Scholes model.  Stochastic Volatility Models: such as SABR, Heston, Levy  Parametric or Semi-Parametric Models: such as SVI, Omega  Market Volatility Model: directly modeling the implied volatility dynamics  Interpolation/Extrapolation Model: interpolating or extrapolating volatility data using specific function forms

Any volatility models must meet arbitrage free conditions. Typical arbitrage free conditions
• Static arbitrage free condition: Static arbitrage free condition makes it impossible to invest nothing today and receive positive return tomorrow. • Calendar arbitrage free condition: The cost of a calendar spread should be positive. • Vertical (spread) arbitrage free condition: The cost of a vertical spread should be positive. • Horizontal (butterfly) arbitrage free condition: The cost of a butterfly spread should be positive.

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