Swaption Implied Volatility

An implied volatility is the volatility implied by the market price of an option based on the Black-Scholes option pricing model. An interest rate swaption volatility surface is a four-dimensional plot of the implied volatility of a swaption as a function of strike and expiry and tenor.

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

An swaption volatility surface is a four-dimensional plot of the implied volatility of a swaption as a function of strike and expiry and tenor. The term structures of implied volatilities provide indications of the market’s near- and long-term uncertainty about future short- and long-term swap rates.

Vol skew or smile pattern is directly related to the conditional non-nomality of the underlying return risk-neutral distribution. In particular, a smile reflects fat tails in the return distribution whereas a skew indicates return distribution 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.

Vertical arbitrage free and horizontal arbitrage free conditions for swaption volatility surfaces depend on different strikes.

There is no calendar arbitrage in swaption volatility surfaces as swaptions with different expiries and tenors have different underlying swaps and are associated with different indices. In other words, they can be treated independently.

SABR stands for “stochastic alpha, beta, rho” referring to the parameters of the model. The SABR model is a stochastic volatility model for the evolution of the forward price of an asset, which attempts to capture the volatility smile/skew in derivative markets.

There is a closed-form approximation of the implied volatility of the SABR model. In the swaption volatility case, the underlying asset is the forward swap rate.

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