>> So far we've discussed option pricing and derivatives pricing in the binomial model, but we've made no mention of dividends. In this module we're going to discuss dividends and point out that it's actually include, easy to include dividends, in the binomial model. So here's our one period binomial model again. We start off with t equals 0, the security price of S0. We have a probability of p, a true probability of p, and a true probability of 1 minus p of an up move and down move respectively. We're going to assume that if the security goes up, the value of the stock goes to uS0 and if it goes down it goes to dS0. But we're also going to assume that the security pays a proportional dividend of C times S0. And at C times S0's what get pays out, paid out as a dividend at time t equals 1, regardless of whether the stock has gone up or down. So you'll see that CS0's here and CS0's down here as well. So this is the dividend that gets paid out, to the shareholders. Okay, as before we have a cash account which pays out a gross risk-free rate of four, we assume the cash account is worth one dollar or one euro or one whatever at time t equals 0 and its worth R at time t equals 1. So it's easy to check again that there are no-arbitrage conditions for this one period binomial model and in particular the no-arbitrage conditions are given to us by here. And this is the direct analog of the no-arbitrage conditions we saw when there were no dividends. So d plus c is the total return of the stock if the stock falls from S0 to dS0. And u plus c is the total return of the stock, if the stock raises, or rises, from S0 to uS0. So, clearly, R cannot be less than d plus c, otherwise there would be an arbitrage. You could buy the stock and sell the cash account. Likewise, if R was greater than u plus c, then you could short sell the stock and buy the cash account and again create an arbitrage. Okay. So what we can do is that suppose we want to price the derivative security with value C1 of S1 at time 1. It pays off Cu in this upper state here, and it pays off Cd in this lower state over here. So what we can do is we can again construct our one period replicating strategy to replicate the payoff of this option. So, in particular, we're going to let x be the number of units of the stock that we purchase at time 0, and y be the number of units of the cash account that we purchase at time 0. Well then the replicating aspect of this portfolio means that we must have these two linear equations. In particular, the value of the portfolio here on the left-hand side if the stock goes up is uS0 times x plus cS0 times x. So this is the dividend piece, this is the value of the stock. Plus R times y, this is the position of the cash account. And that must be equal to C0, if the stock price goes up. If the stock price goes down, then we have dS0x plus cS0x plus Ry equals Cd. So these are the two equations that we have. If we can find x and y so that these two equations are satisfied, then the portfolio xy, which is purchased at time 0, replicates the payoff C0 and Cd. So therefore the value of this portfolio at time 0 must be the arbitrage-free value of this derivative security. So we can solve these two equations and two unknowns. If we do that, we'll find that C0 can be written as follows. The fair value C0 is equal to xS0 plus y, and we can write it like this. Again, we can write it in the form of having risk-neutral probabilities and now Q is equal to this quantity here. Which is the same as our original one period risk-neutral probability, except now, we have this additional minus c. 1 minus q is equal to u plus c minus R divided by u minus d, and that's that term there. So, again, when the security pays a proportional dividend, we can still construct a replicating strategy. And we can compute the price of a derivative security as the discounted expected term or payoff of the derivative security using these risk-neutral probabilities. Okay. In the multi-period binomial model we can assume a proportional dividend is paid in each period. In particular we would assume a proportinal dividend of cSi is paid at t equals i plus 1. Then each embedded one period model has identical risk-neutral probabilities. They're given to us by these quantities here, and derivative securities are priced as before. We can work backwards in the binomial lattice using these risk-neutral probabilities, to calculate the va-, fair value, or arbitrage-free value, of the derivative at each time period. Until we get back to t equals 0, which gives us the initial price, arbitrage-free price, of the security. In practice dividends are not paid in every di-, period. They're often paid every six months or every year, and so they're a little more awkward to handle in practice. We might discuss that later in the course, but for now we'll, we'll, we'll leave it at that point. Another point to keep in mind is, is the following. Suppose the underlying security does not pay dividends. Then it's easy to check that this is true. The initial value of the stock is equal to the expected value, using the risk-neutral probabilities of the terminal value of the stock discounted. And of course, that follows easily from our risk-neutral pricing, because we know the stock price Is equal to Sn minus 0, 0. Take the max of these two. So in fact this is just a call option with K equal to 0. So the stock price is equal to a call option on this stock with its strike K equal to 0, and so this just follows from the pricing of call options. But of course, every security must be priced like this. Okay. Suppose now that the underlying security pays dividends in each time period. Then you can check that eight no longer holds. Instead what holds actually is the following, the initial value of the stock, S0 is equal to the expected value at time 0 using the risk-neutral probabilities of Sn divided by R to the n, which is what we have here, plus the sum of the dividends, which we'll call Di, discounted appropriately. Okay. We're assuming Sn is the x dividend security price at time n. So Sn is the price at time n immediately after the dividend at time n is paid. And you don't need any new theory to prove this. We can just use what we've seen already. It follows from this mutual pricing, and observing the dividends and Sn may be viewed as a portfolio of securities. So, to see this we can view the ith dividend as a separate security with value equal to this quantity here. So if we view the ith dividend which is paid at time i as a security that pays at cash flow only at time i, then our risk-neutral pricing tells us that the value of this dividend is Pi. And it's equal to this quantity here. Now we can view the owner of the underlying security as owning a portfolio of securities at times 0. The value of this portfolio is the value of the securities within the portfolio. Those securities are each of the n dividends. Okay, and each of those n dividends is value Pi for i equals 1 to n, plus the terminal value of the stock, or if you like, the, the value at time n. And that's Sn discounted by R to the n taking its expectation with respect to the risk-neutral probabilities. But of course we also note the value of the underlying security is S0, so S0 must be equal to this quantity here. Okay. And so we get S0 equals to the sum of the Pi's plus this quantity and that is just equation nine.
