So I'll do that in your numerical example, but the proof, which just goes the same way, general proof. Just a little bit easier to do it with numbers maybe. So let's assume that we have a stock which costs $100 today and we know that it's going to pay one dividend in the next six months. And that dividend is going to be, well, in the next year, there's going to be only one dividend paid and it's going to be 5.65 paid in six months, okay? And we also know that the one year continues, the compound interest rate is 10%. This is very high for today's market environment, but let's say it's 10%. And we also know the six month continuous, annualized rate is 7.41%, all right? And then we can compute using the formula from the previous slide or the price of the one year forward contract should be on this stock, okay? So I have to multiply by the one year factor, how much $1 will be in one year. By my assumption, it's 10% interest rate, so it's going to be exponential function to the 0.1 to 10%. And then here I have to have the stock price minus the present value of the dividends. The stock price today is 100, dividends is 5.65. Well, this is going to be paid in six months, so the present value I obtained by discounting by the six month rate. So it's e to the -0.0741, except this is annualized rate. So I have to divide by 2 because six months is one half of a year, okay? So it's times one half, 7.41% times one half. So this is my discounting factor for six months payments of $5.65, okay? Use that formula from the previous slide and I get $104.5 as the no arbitrage price for this forward contract in this market environment, okay? So I'm going to just show that one case that there is arbitrage. Let's say that the price is cheaper than this for example, let's take the price to be 104. So that means that the forward contract is cheap, relative to what my formula, theoretical formula tells me. So it means I should go along, I should buy the fourth contract along in the fourth of contract and sell the underlying. So let's do that, it's a pretty much the same proof as before. Except, I have to worry about the dividends. Let's see how that goes. So at small t today I go along in the forward as I said because it looks cheap. Go long the forward, I sell short one share. Okay, here's the difference, if I sell short one share of the stock and that stock pays dividends, I have to pay the dividends to the holder of the stock, okay? When you're selling a stock, you're selling all the rights that go with the stock, including the right to receive dividends. So when you sell one share shorter or otherwise, if you sell one share of the stock, you need also to pay the dividends. So you have to deliver the dividends when they come. So I want to cancel that. I want to have enough money to do that in six months. I know it's going to be 5.65, okay? So what I'm going to do here, I'm going to buy the six month bonds in an amount which is the present value of 5.65. So that I have exactly 5.65 in six months. So since the six month rate, annualized it's 7.41%. I'm using the same discount factor here, like here. All right, I'm discounting 5.65 by the six month factor rate and I get something which is 5.4445. So, this is how much I will invest in the six month bonds. Okay, I pay six months, the six month bonds, zero coupon six months one in this amount, okay? That means that in six months I would have, that bond will deliver to me 5.65, which is what I want. And I will use that to pay the dividends, all right. So, I still have selling one share, I got $100, I pay you 5.4445 to buy this six month bonds. So I have extra money, I put that money in the bank while in the one year bond, let's say. Whatever it is, but at the rate of 10%. So I put 100-5.4445 in the remaining amounts of money that I have, 94.5555 in let's say, in the 1-year bond, okay? So I already told you what's going to happen in six months. In six months, I will have to pay the dividend of 5.65, but I will receive 5.65 from my position in the six month bond. That's exactly how I constructed that position to have 5.65 in 6 months, okay? At 1 year from today and use capital T, I will receive, that goes with 10% my one year bond. I invested 94.5555 into that bond that's going to go up to that times c to the 0.1 which you can compute, is exactly 104.5, okay? That's going to be exactly 104.5. [COUGH] But I only have to pay, so I will have 104.5 in the bank or the bond, but I only had to pay 104, okay? In the forward contract, I get my one share, I can deliver that share to cover my short position and close it. And I still have the difference of 0.5, is profit. And if I, instead of doing this for one forward and one share, do it for 1000 forwards and 1000 shares, then I have 0.5 times 1000, whatever I can do, right? So this is my arbitrage profit. The same logic as without dividends, you just have to make sure that in your arbitrary strategy you have enough money to pay the dividends when you are short the underlying. All right, so that's that. And let's move next. So, here's an example that I told you was, that there was going to be an example which is kind of similar to dividends. It's for, it's a very much used type of forward contracts, although it's usually futures, we'll talk about future soon, follow contract on foreign currency. So it's a contract in which, you know, you will need one Euro. And you have a pre specified exchange rate in the formal contract so that you can buy that Euro in whatever number of months for that amount. Okay, so I'm going here to use the same notation, I'm going to use S(t) to denote the current price in dollars of one unit Euro. So let's say, one unit of the foreign currency. So as it is actually the exchange rate, right? It's how many dollars they have to pay today for one unit of the foreign currency. So the exchange rate, I'm using the notation s, which is usually for stock just to keep the notation same, but it's the exchange rate. I'm going to have two risk free rates here. I'm going to denote by rf the foreign risk free rate and by r, the domestic risk free rate. Let's say continuous compounding, just to be specific, okay? So here is the intuition of what I'm going to do here. I'm going to think of the foreign currency as a risky asset. It's like a stock, it's something risk from the point of view of the dollar investor, Euro is a risky asset. Okay, it's fluctuating in a risky way, you cannot predict what's going to do. So from the point of view of the dollar investor, the foreign currency, Euro is risky, but it's an asset which pays dividends. Because if you hold Euro, you get in a Euro bank, you will get rfr, if you will get the interest, you can get some interest in that figure. So it's really like a risky assets which pays dividends at the continuous rate. Here I assume continuous compounding. So, it's really like an asset that base continues to compound dividends. Which really means that it should be the same formula as the one from two slides ago. This one for dividends, for an asset, which base continuously compounded dividends, but instead of q, I should have rf. I should have the risk free foreign rate and then this is going to be in the continuously component domestic rate. This is going to be e to r, T-t, right? So I'm going to have (t,T)e to minus rf, T-t. That's intuitively what should happen. And that's what I'm claiming here to slides next. That's exactly what's happening. That the forward price should be equal to the e(r-rf)(T-t), where rf is discounting like, by the dividend rate. Whereby by the dividend rate here, is the interest rate on Euro. And when I do the multiplication, I get e(r-rf(T-t), times the today's price or today's exchange rate. All right, so I'm going to just show one side. For example, if the forward is cheap, relatively speaking. So, if F(t), let's assume that F(t) is strictly less than this product, I want to show that there is going to be, you can construct average rate. Again, the forward looks cheap. The underlying which is Euro looks expensive. So I'm going to go today long in the forward contract on Euro. I want to sell short the underlying asset. Selling Euro is borrowing, okay, selling short is really like you borrow the stock and then you sell it. So selling short Euro, it just means borrowing. So I'm going to borrow Euro, but here I'm going to have to be a little bit more careful. I'm not going to borrow one unit of Euro, because in the forward contract I will receive one Euro, a couple of t for the price of 40. So I want to borrow that many units of Euro so that I actually have to pay one Euro at maturity to the foreign bank. Or whether it's in foreign bank or exchange dollars, doesn't matter. But I'm going to borrow this many discounted one Euro by the risk free rate discounted. I'm going to borrow discounted one Euro, so that it will go to one Euro by capital T, okay? So if I, [COUGH] do that and I can invest its value in dollars. Which is just multiplying this by the exchange rate today into the, so I can put that into the domestic bank at rate r, that's what I do. All right, so let's see what happens at maturity. Well, at maturity, this thing in the US bank would go up by the factor into the (r-r)(T-t) because it's in the US bank after this interest rate. So I will have this much in dollars. And that was by assumption more than what I have to pay in the forward contract. So I will have more than enough money to pay the forward price for the Euro. I will get my Euro. What do I do with that Euro? Well, that's the Euro that I borrowed here and I have to return. Okay, I borrowed less than, I borrow this kind of Euro but now it's going to be exactly one Euro what I have to pay back. So I'm going to receive that Euro from my forward contract. I effectively closed the short position in Euro in the European bank, give back that Euro and then I have zero balance in the European bank. And I still have this extra money that dollars I had, sorry here, I had more dollars than I needed to pay in the forward contract, okay? So, arbitrage. And I'm leaving to you as an exercise to do the other case, if it's strictly less, if this product is strictly less than the forward price of the Euro, you do exactly opposite positions. You can go shorter forward and you would deposit money in the Euro bank, okay? So, I'll leave that to you. All right, so we now have a version of the forward price on foreign currency here assuming that the continuously compound interest rates. But there is a similar formula if the interest rate is compounded in a different way.
