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Modeling the Implied Volatility Smile-The Sticky-Delta Smile Approximation

This study develops the sticky-delta rule presented by Derman (1999) into a model that is believed to be reasonably efficient for option pricing with a smile. The model enables pricing with respect to different strikes and maturities where the observed implied volatility is not constant over strikes. The smile in implied volatility, using the stickydelta rule, is a function of the delta of a particular strike. This implies that the volatility is stuck to the delta of that particular strike. The sticky-delta rule is here developed into a model that is used for approximating the volatility smile on stock indices on three European markets. It is suggested that the shape of the smile is only dependent on the call delta and thereby on time and volatility and moves with the underlying, leaving atthe-money volatility unchanged. The result is a model that is computationally simple and more accurate than a constant volatility. The model fails to model the smile for lower strikes as the sensitivity of the delta to the underlying for options far-away-fromthe-money is minimal. This problem is in practice overcome by extrapolation.