# European option on a dividend paying stock, limits to arbitrage?

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?

 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. 