# The SABR model - Theory and Implementation

Specialeforsvar ved Paul Ferninand Popp

Titel: The SABR model - Theory and Implementation

Abstract: In the original Black-Scholes model exists a one-to-one relationship between the price of an option and the important parameter volatility. By solving the Black-Scholes formula, we can calculate the implied volatilities from options with the same maturity, but different strike prices. Opposite to the constantly assumed volatility in the Black & Scholes model, we would observe different volatility levels of the options in order to match their market prices. In fact, the implied volatility as a function of strike price will typically have the shape of a smile for currency and fixed income markets (”volatility smile“) and is downward sloping for equity markets ( ”volatility skew“). Local volatility models treat volatility as a function of the current underlying asset and time and model the instantaneous volatility well. However, it predicts the wrong dynamics of the volatility function when the underlying varies and causes unstable hedges. Several stochastic volatility models have been developed to overcome these shortcomings of Black & Scholes and local volatility models. One of them is the SABR model, which we will present in this paper. More importantly, we implement the SABR model and show that it predicts the correct dynamics of the volatility smile. We also analyze how the parameters α, β, ρ and ν influence the shape of the implied volatility function

Vejleder: Rolf Poulsen
Censor: Nina Lange, DTU