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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.

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Both formulas (generalised Black Scholes) are correct. In the first one, you have the BS model with dividend. In the second, you have a non-zero cost of carry. It's really the same formula. Note that cost of carry = risk free rate - dividend. You can think cost of carry like the negative growth rate.

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  • $\begingroup$ I can't see these are the same at all. Suppose r = 2.5% g = 5% (given by the forecasting model) q = 3% (assumption about div yield) If you substitute these in, you get completely different numbers. $\endgroup$ – quis est ille Nov 9 '15 at 8:51
  • $\begingroup$ @quisestille Black Scholes is built on no-arbitrage hedging, so your argument using BS to price arbitrage-violated option makes no sense. Are you sure you haven't confused the generalised Black Scholes? $\endgroup$ – SmallChess Nov 10 '15 at 1:23
  • $\begingroup$ People use Black Scholes with real world assumptions, no? Indeed they do. $\endgroup$ – quis est ille Nov 11 '15 at 18:19
  • $\begingroup$ @quisestille Who are they? Do you know the classical BS is already based on real-world assumptions? Are you really sure you know what you're talking about? Who are they? Do you mean yourself? $\endgroup$ – SmallChess Nov 12 '15 at 0:13
  • $\begingroup$ Not me. Actuaries. They plug in forecast growth into the forward rate of the Black 76 model. Yes I know what I am talking about. I wanted someone to say this is very very wrong. $\endgroup$ – quis est ille Nov 13 '15 at 17:35

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