Measuring Implied Volatility Surface Risk using Principal Components Analysis

The Black-Scholes formula Black and Scholes (1973) (BS hereafter) has remained a valuable tool for practitioners in pricing options as well as a precious benchmark for theoreticians. Indeed, the BS option valuation formula is a one-to-one function of the volatility parameter σ once the underlying stock level S _t , the strike price K and the remaining time to expiration τ are known and fixed. Using the quoted prices of frequently traded option contracts on the same underlier, one can work out the implied volatility σ by inverting numerically the BS formula. But it is notorious that instead of being constant as assumed by the BS model, implied volatility has a stylized U-shape as it varies across different maturities and strike prices. This pattern called the “smile effect” is the starting point of the implied theories which we concentrate on thereafter.