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>> So far we've discussed option pricing and derivatives pricing in the binomial

Â model, but we've made no mention of dividends.

Â In this module we're going to discuss dividends and point out that it's actually

Â include, easy to include dividends, in the binomial model.

Â So here's our one period binomial model again.

Â We start off with t equals 0, the security price of S0.

Â We have a probability of p, a true probability of p, and a true probability

Â of 1 minus p of an up move and down move respectively.

Â We're going to assume that if the security goes up, the value of the stock goes to

Â uS0 and if it goes down it goes to dS0. But we're also going to assume that the

Â security pays a proportional dividend of C times S0.

Â And at C times S0's what get pays out, paid out as a dividend at time t equals 1,

Â regardless of whether the stock has gone up or down.

Â So you'll see that CS0's here and CS0's down here as well.

Â So this is the dividend that gets paid out, to the shareholders.

Â Okay, as before we have a cash account which pays out a gross risk-free rate of

Â four, we assume the cash account is worth one dollar or one euro or one whatever at

Â time t equals 0 and its worth R at time t equals 1.

Â So it's easy to check again that there are no-arbitrage conditions for this one

Â period binomial model and in particular the no-arbitrage conditions are given to

Â us by here. And this is the direct analog of the

Â no-arbitrage conditions we saw when there were no dividends.

Â So d plus c is the total return of the stock if the stock falls from S0 to dS0.

Â And u plus c is the total return of the stock, if the stock raises, or rises, from

Â S0 to uS0. So, clearly, R cannot be less than d plus

Â c, otherwise there would be an arbitrage. You could buy the stock and sell the cash

Â account. Likewise, if R was greater than u plus c,

Â then you could short sell the stock and buy the cash account and again create an

Â arbitrage. Okay.

Â So what we can do is that suppose we want to price the derivative security with

Â value C1 of S1 at time 1. It pays off Cu in this upper state here,

Â and it pays off Cd in this lower state over here.

Â So what we can do is we can again construct our one period replicating

Â strategy to replicate the payoff of this option.

Â So, in particular, we're going to let x be the number of units of the stock that we

Â purchase at time 0, and y be the number of units of the cash account that we purchase

Â at time 0. Well then the replicating aspect of this

Â portfolio means that we must have these two linear equations.

Â In particular, the value of the portfolio here on the left-hand side if the stock

Â goes up is uS0 times x plus cS0 times x. So this is the dividend piece, this is the

Â value of the stock. Plus R times y, this is the position of

Â the cash account. And that must be equal to C0, if the stock

Â price goes up. If the stock price goes down, then we have

Â dS0x plus cS0x plus Ry equals Cd. So these are the two equations that we

Â have. If we can find x and y so that these two

Â equations are satisfied, then the portfolio xy, which is purchased at time

Â 0, replicates the payoff C0 and Cd. So therefore the value of this portfolio

Â at time 0 must be the arbitrage-free value of this derivative security.

Â So we can solve these two equations and two unknowns.

Â If we do that, we'll find that C0 can be written as follows.

Â The fair value C0 is equal to xS0 plus y, and we can write it like this.

Â Again, we can write it in the form of having risk-neutral probabilities and now

Â Q is equal to this quantity here. Which is the same as our original one

Â period risk-neutral probability, except now, we have this additional minus c.

Â 1 minus q is equal to u plus c minus R divided by u minus d, and that's that term

Â there. So, again, when the security pays a

Â proportional dividend, we can still construct a replicating strategy.

Â And we can compute the price of a derivative security as the discounted

Â expected term or payoff of the derivative security using these risk-neutral

Â probabilities. Okay.

Â In the multi-period binomial model we can assume a proportional dividend is paid in

Â each period. In particular we would assume a

Â proportinal dividend of cSi is paid at t equals i plus 1.

Â Then each embedded one period model has identical risk-neutral probabilities.

Â They're given to us by these quantities here, and derivative securities are priced

Â as before. We can work backwards in the binomial

Â lattice using these risk-neutral probabilities, to calculate the va-, fair

Â value, or arbitrage-free value, of the derivative at each time period.

Â Until we get back to t equals 0, which gives us the initial price, arbitrage-free

Â price, of the security. In practice dividends are not paid in

Â every di-, period. They're often paid every six months or

Â every year, and so they're a little more awkward to handle in practice.

Â We might discuss that later in the course, but for now we'll, we'll, we'll leave it

Â at that point. Another point to keep in mind is, is the

Â following. Suppose the underlying security does not

Â pay dividends. Then it's easy to check that this is true.

Â The initial value of the stock is equal to the expected value, using the risk-neutral

Â probabilities of the terminal value of the stock discounted.

Â And of course, that follows easily from our risk-neutral pricing, because we know

Â the stock price Is equal to Sn minus 0, 0. Take the max of these two.

Â So in fact this is just a call option with K equal to 0.

Â So the stock price is equal to a call option on this stock with its strike K

Â equal to 0, and so this just follows from the pricing of call options.

Â But of course, every security must be priced like this.

Â Okay. Suppose now that the underlying security

Â pays dividends in each time period. Then you can check that eight no longer

Â holds. Instead what holds actually is the

Â following, the initial value of the stock, S0 is equal to the expected value at time

Â 0 using the risk-neutral probabilities of Sn divided by R to the n, which is what we

Â have here, plus the sum of the dividends, which we'll call Di, discounted

Â appropriately. Okay.

Â We're assuming Sn is the x dividend security price at time n.

Â So Sn is the price at time n immediately after the dividend at time n is paid.

Â And you don't need any new theory to prove this.

Â We can just use what we've seen already. It follows from this mutual pricing, and

Â observing the dividends and Sn may be viewed as a portfolio of securities.

Â So, to see this we can view the ith dividend as a separate security with value

Â equal to this quantity here. So if we view the ith dividend which is

Â paid at time i as a security that pays at cash flow only at time i, then our

Â risk-neutral pricing tells us that the value of this dividend is Pi.

Â And it's equal to this quantity here. Now we can view the owner of the

Â underlying security as owning a portfolio of securities at times 0.

Â The value of this portfolio is the value of the securities within the portfolio.

Â Those securities are each of the n dividends.

Â Okay, and each of those n dividends is value Pi for i equals 1 to n, plus the

Â terminal value of the stock, or if you like, the, the value at time n.

Â And that's Sn discounted by R to the n taking its expectation with respect to the

Â risk-neutral probabilities. But of course we also note the value of

Â the underlying security is S0, so S0 must be equal to this quantity here.

Â Okay. And so we get S0 equals to the sum of the

Â Pi's plus this quantity and that is just equation nine.

Â