-1
$\begingroup$

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.

$\endgroup$
-1
$\begingroup$

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.

enter image description here

$\endgroup$
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.