An Analytic Solution For Hedge Ratios On Stocks With Dividends And The Accuracy Of Black-Scholes Hedge Ratios

Abstract

The Black-Scholes model continues to be the standard option pricing model discussed in virtually all corporate finance and investments texts and continues to be widely used in practice.  The model’s associated hedge ratio has also been widely used for hedging purposes.  The associated hedge ratio (or delta) is determined as part of calculating the Black-Scholes option value.  However, the original model assumes no dividends on the underlying stock.  The model has been modified to allow for dividends, but the modification does not lead to values as precise as other models, such as the Roll-Geske-Whaley model that specifically account for dividends.  Empirical research has shown that the RGW model values are closer to actual market prices than the modified Black-Scholes values. This paper is primarily concerned with the hedge ratio. We derive an analytic solution for a more accurate hedge ratio based on the RGW model. The paper is then concerned with how large the errors are associated with using the BS approximation rather than the more complicated model that specifically accounts for dividends. We find that although there are times when the BS approximation can be accurate, at other times the differences can be significant.  These differences are related to the size of the dividend, the difference between the time to expiration and the time to ex-dividend, the rate of interest, the stock volatility, and the degree to which the option is in-the-money.