What is the price C of a European call option on a dividend paying stock? I believe it is:
C = U. N(d1) - exp(-rt).K.N(d2)
d1 = [ ln(U/K) + (r + v^2/2).t ]/[ v.sqrt(t) ]
d2 = d1 - v.sqrt(t)
U = S.exp(-qt)
Where S is the spot price of the stock, q the dividend yield, K the strike, r the risk free rate, t time to expiry, v implied vol, and N the cumulative normal distribution function. All yields/rates continuous basis.
However, I have been told that because there is no long-dated forward price for stocks, the no arbitrage principle fails to apply, and so in this case the correct formula for U is
U = S.exp( (g – r)t )
where g is the forecast growth rate of the stock. Since q is typically 2-3%, and since g is assumed to be 5%, this leads to a considerable difference in the two methods. Which is correct?
[Edit] In the interests of clarity (see the first answer below, which claims the formulas are really the same) note the inconsistent assumptions about growth in the original question. If q = 3%, and r = 2.5%, this would imply a negative growth rate g of -0.5% in a no arbitrage world. However, it has been argued that where no arbitrage does not apply, we can use a forecasting model that predicts 5% growth in equities in the long term, and so g-r <> -q.
