We have now the basic black Scholes formula for the vanilla call options and put option. Also, if you use put call parity, let's see how we can extend this to some other situation still in the Black Scholes Merton model. So the first thing we are going to do is dividends. And we are going to look at two cases, none of which is necessarily very realistic, maybe in some situations, but they're very tractable and they can give you an approximate answer to the question. The first case is going to be dividends paid at a continuous rate. So basically it's as if the stock in addition to having its own new return rate, it also has a kind of continues to compounded rate of return, which would be dividends being paid at that rate, okay? Which means that the total value, total gains that you have from holding the stock called G(t), is how much you can sell the stock for which is S(t) and plus the total value of cumulative value of dividends, which is going to be assumed to be of this form, okay? So we assume that the total value of dividends at the time T from time zero. And we are assuming that dividends are reinvested is equal to the integral Qs of (u)du, okay? That's what it means that they are paid at a continuous rate q, constant continues rate q. All right, so what is going to happen here turns out that in this case the under the risk neutral pricing probability, this kind of stock price will not be a marking here because of the extra dividends. When we discount the stock price, there will be an extra term in the drift and the drift even under the pricing probability the drift will not be zero. And therefore this kind stock price is not a martingale. So what we have to do, we have to make the discounted wealth process and nothing else. We know that always works if we can do that, that's our pricing probability. Let's look at the wealth process. The wealth process is going to be the dynamics of the wealth process is here. The exchange in your wealth process is, this is the bank counter which we had before. Money in the bank, X minus pi, pi's the money in the stock. So X minus pi is money in the bank over B would be number of shares if you want in quotation marks, in the bank times the change of the bank account's value DB plus here before we had pie over s, which was the number of shares in the stock. And we had the S but now it's going to be DG because our gains and losses come both from the stock price and from dividends, okay? So it's going to be dG not dS before we had dS, now it's going to be dG. So I just substitute dB over B is these are this dG. Well, what is dG. Okay, let's just compute that dG, is going to be dS, right? And then Dq's the integral, differentiation q's the integral, so I'm just going to have plus Q. As the thing and that's what my dG is, it's the S plus QSdT. So you plug that in, substitute for it and we get this which looks the same as before except there is an extra we add to the new to the return rate of the stock, we add the rates of the great dividend rate, which is intuitive. If you're holding the pie the amount in the stock, not only am I getting the benefit of the expected return rate of the stock price but also the dividends, okay? And now we can see that we want to this kind of wealth process to be a martingale which means we want this by the same logic we did for the stock. You want to have rXdT term here because when you discount then you're going to get minus RxdT term which will cancel this one and you will have zero in the dT term and therefore martingale, okay? So what we want is rXdT plus pi(t)sigmadW superscript queue for some Q, okay? If you make this equal to this, it's an easy computation to check that your WQ has to be W plus T mew plus q minus r over sigma. First you get dWQ, then you integrate and you get this. This is what we did before also when we were talking about Castano theorem. So the same thing here, the only thing is instead of mew we'll have mew plus q. Okay, so this is kind of an adjusted Sharpe ratio if you want. It's a X return of the stock plus the return of the dividends minus the risk free return over the standard deviation of the love returns over the volatility of the stock, okay? That's very similar to before everywhere where we had mew I have mew plus Q now. Fine, so I know now how my WQ looks like, but that means that I can also see how the stock looks like under the pricing priority, okay? So how the stock looks like under the pricing probability. And again, it's a simple algebra to show that the stock looks like this, okay? So if you are in the Black Scholes model, so if the S is equal to s mewdT plus the W, then from the previous slide, you can see that the W is equal to the W superscript Q minus mew is a mess here as a minus mew plus Q minus R Or sigma dT, all right? So if I do that and replace the W here by this guy mew and mew will cancel but I will have r minus Q sigma symbol. There is a sigma missing here. Sigma and sigma will cancel, mew and mew will cancel, but I will get r minus Q here, okay? So the black Scholes model for the stock under the pricing probability becomes this, it's r mew gets replaced by r minus Q. So before we had just are instead of new now we have r minus Q. This is just because in theory, the stock when it pays dividends to have no arbitrage, the stock value should immediately go down by the value of the paid dividends. Since here the dividends are paid at the time the t the dividend is exactly S times QdT, we are just subtracting the dividends here, okay? When they are paid, you subtract them from the stock value. That's the intuition. But the mathematics is as what I explained here, okay? So, we have this sort of stock and then you can either do the partial differential equation method or you can do the martingale pricing, this pricing method. We know now that everywhere where we had r in the formulas, for s we have r minus Q, okay? I'm going to show only the partial differential equation way, but it would be similar as before using the marketing and pricing risk, pricing, all right? So before we had in the partial differential equation, like shows partial differential equation we had our r in two places they had r multiplying sCs, the delta of the option. And also we have minus RC. Now this r is not going to change because this was from discounting when computing the price is still discount by r but this r will become r minus Q. Why, because that r came from here. It came from the dynamics of the stock. When you do rule and the first derivative multiplies the drift of the stock, okay? So that's why this r becomes r minus q, all right? So now we have the partial differential equation. Or we could also get the formula for s of capital T from this equation and then we could compute the expected value under the Q probability, under the risk neutral probability. And if you do the computations, what happens is you get this formula down here for Coral option. You don't even have to look at it carefully. There is only one change, wherever you have s before, which was the current stock price, now you have S times E to the minus Q times time to maturity, okay? So wherever we had s now, it's discounted by the dividend rate, okay? The value of the option s for the forward contract we talked about before, the value of the option co option is going to be lower because when the stock pays dividends, because we are not going to be getting those dividends until the maturity of the option when we exercise and get the stock. But foregoing we are missing on those students before. And therefore the value of the option is actually lower if the stock pays dividends, if compared to when it doesn't, the way it's lower, you reduce the stock price by discounting it with h minus Q, capital T minus lowercase t, okay? So otherwise the Black Scholes formula looks the same. And okay here, I had asked now when I do s to this exponential log will kill the exponential and I get this minus QT minus T in here, okay? So there is an additional minor q here with this r but not with this r because these r came from this context, okay? That's all that happens, easy enough. We compute the dividends, the option pricing formula with continuously paid dividends. Let me also tell you about discreetly paid dividend. I'll just tell you the way the formula is obtained without proving it. Assume now the other extreme in which the dividends are paid discreetly at known times with known values, which means that it's completely deterministic, we know the dividend payments in the future. Therefore we can compute the present value or the discounted evidence and we denoted by D bar of T, okay? So you still use the same black Scholes formula except you replace s, the stock price, by s minus discounted evidence, okay? This is consistent with the Black Scholes model, but in a slightly different form. In this case, if you look at the process SG, which is stock minus the present value of dividends, if you assume that that process follows Black Scholes model. Then the way to price the options in this model is by simply putting in instead of s into the option pricing formula, just put s minus discounted dividends minus the present value of dividends, okay? You still reduce the stock price, but you reduce it by subtracting the present value of dividends rather than by multiplying by the exponential to the minus rate of dividends, okay? And that's how it works with dividends. Easy enough, slight modifications.
