Why if we consider a not dividend paying stock and taking as numeraire the money market account the ratio is a martingale, but if we consider a stock paying a continuous dividend yield q the ratio wrt the mma is no more a martingale?dS/S=r*dt+sigma*dWdS/S=(r-q)*dt+sigma*dW

Because the dividend payment is not random.

QuoteOriginally posted by: gozzi84Why if we consider a not dividend paying stock and taking as numeraire the money market account the ratio is a martingale, but if we consider a stock paying a continuous dividend yield q the ratio wrt the mma is no more a martingale?Yes, but what really matters is that the gain process (the sum of stock and dividend) remains a martingale under the mmaP.

The point is well-made: all you really care about is the 'gains process.' Discounted gains processes are still q-martingales. If you were to consider only the stock/index/whatever that pays dividends and attempted to discount it by the MMA to make it a martingale under q-measure, you would be violating Girsanov. The whole idea of setting the MMA as the numeraire is to create a new measure where the underlying process is a zero-drift process (in terms of the numeraire). Under the p-measure (the world in which we actually live), the numeraire is money itself--dollars and cents. In the new measure, we're pricing everything in terms of their worth as multiples of the money market account.If you understand that a non-dividend-paying diffusion process is a martingale under Q, then you must see that a dividend-paying diffusion process cannot be. Why? Simply, when a stock pays dividends, it pays them out of the equity tied up in the stock. Ergo, the stock price falls after the payment of a dividend by exactly the value of the dividend payment. Of course this changes the drift of the entire process. This is what was meant by the comment stating that dividend payments are not random. Conceptually, if I know that my process will certainly drift downward at specified dates (the dividend payment dates), it's certainly not random. As such, its drift at those dates would not be matched appropriately by the MMA.In the end, if you consider the gains process (the stock plus the dividend), then you CAN discount by the MMA, and from the above discussion, I should hope you see why.

I can see your point which sounds to me quite logical.So if I consider the total return on a stock and I want to model the stock dynamics (dividends included) I can assume the stock price following a geometric Brownian motion under Q dS/S=r*dt+sigma_s*dW where r is the risk free interest rate and W is a Q - Brownian motion which is the same specification as the case of a stock doesn't pay dividend. Therefore we can assume the stock divided by the mma is a martingale under the Q measure.Am I right?

First off, please use MathType or LaTek when writing mathematical script.Secondly, that's not the diffusion process for a dividend-paying stock. We want to consider the entire gains process, and the one you just gave is strictly for the non-dividend paying stock. Look in Bjork for a little bit, and this should become clearer.