Let's start pricing our first derivatives. First, starting with the easiest ones to price, which are linear functions of the underlying assets, forwards, futures and swaps. Then we are going to see that we can price those instruments no matter what our mathematical model is for the underlying price. We will only need to know the value of the underlying price and not the mathematical model for it so that's why we call this model independent pricing relations. Let's move to the next slide, and start with pricing forward contracts. Just to remind you what a forward contract is on an Asset S, I'm going to call the initial time lowercase t, small t for today and the payoff is that capital T, the maturity. The payoff of our forward if you're long forward you get your asset value at time capital T, your Google stock or whatever S&P 500 index, you pay for it, the forward price. The notation is f of small t, just to indicate that this forward price is known, it's decided at time small t. Although it's actually paid at capital T, it's already known as small t so that's why I called f of small t. I think I told you that the forward contracts, when you enter them today at small t, there is no exchange of money then, buyer and the seller don't exchange any funds, any money at initial time, there is only pay off positive or negative at capital T. Here's the question. What is the value for f of t for the forward price that would make time t value today's time value when we enter the contract, make equal to 0? Make the valid contract equal to 0 when we enter it. We want that because we want in a forward contract that there is no exchange of money today. I will need only one assumption here. The assumption will be that there is a risk-free asset at which we can invest, and that risk-free asset to be general, I'm going to denote the payoff at capital T of investing $1 in the risk-free asset. At time small t, I'm going to denoted by B of t capital T. B for a bank account or B for bonds, either way it works, it's a risk-free asset which pays something at time capital T. For example, we had examples of B of t, T if it's a continuous, the compounded rate it would be e to the r, T minus t, capital T minus lowercase t. That's one example or it could be 1 plus r to the T or if there is compounding, then you divide r by the number of compounding periods. These are the examples, but that's what we had in mind. We have in mind by B of t capital T. Let's move to the next slide so that's the only assumption is we have something like that. We have a risk-free asset there. Here's our first result here. The claim is that, there is another assumption, the other assumption is that there is no arbitrage in the market. If there is no arbitrage and we have a perfect market, then the default price has to be equal to the product B of t, T times S of t. What does this mean? This means that for example, if it's a forward contract on a stock, it means this value is exactly the same. If I sell the stock today for which I will get S of small t and I put it in the bank, I will have exactly this product at capital T. Sell the stock today, you get S of t, put it in the bank per each dollar you get B of t, T so for S of t dollars, I'm going to get this product. The claim is, this is what the forward price has to be in order not to have arbitrage in the market. It has to be as if you sell that asset right now and put the value in the bank, put the money in the bank. If there is no arbitrage, we claim this is the unique price. The unique value for the forward contracts so that the contract has time zero value equal to 0. All right. We are going to have a lot of similar proofs now of these type of relationships, and they're all going to be based on the same idea, which is, let's assume that something like this is not true, and then let's find arbitrage, and then therefore it has to be true. By contradiction, if it's not true, there is arbitrage, therefore it has to be true. That's how we are going to do it. In this case, we have equality, so I'm going to do it. I'm going to prove two things, I'm first going to prove that it cannot be strictly higher than this product. I will find arbitrage if it's strictly higher. Then later I will find arbitrage if it's strictly lower. In either case, I will find arbitrage and I will conclude, well, it has to be equal, otherwise there is arbitration. The proof is, in this case, quite short. Suppose first that the forward price is strictly larger than this product here. We are always going to go with the logic, buy cheap and sell expensive. Here, since this is larger than this, relatively speaking, it seems that the forward contract is expensive and the stock is cheap. I'm going to buy the stock, I'm going to buy cheap, and I'm going to sell expensive. Which means in this case, I will go long in the forward contract. Let's do that. I'm going to buy one share. I'm assuming I'm starting with let say zero money, so I have to borrow S(t) to do that. I borrow S(t), today, it's not D. I buy one share and I go short in the forward contract, selling the forward contract in this score, I'm going short in the forward contract. What does that mean for the future, a T? Well, if I'm short, the forward contract means I have to deliver one share of the underlying. But I have that share because I bought it initially. I deliver that share, I receive F(t), which is this left-hand side, I receive F(t). There is that in the bank I have to cover because I borrowed S(t), so I owe something to the bank. Am I negative in the bank? How much do I owe? Am I short in the bank? Well, it's exactly the right-hand side. I borrowed initially, S(t), that's going to increase to S(t) times B(t,T) when I pay interest to the bank. This is the total amount, borrowed amount at time T. But because of this assumption, I received, from the forward contract, more than what I need to pay to the bank, so I'm fine. Not only fine, I have extra positive money after I covered the debts of this, equal to this product. That's arbitrage. I'm sure to make money by this strategy, borrowing S(t) to buy one share, borrow that from the bank, it's going to increase to this much borrowed money at T. But since I'm going to go short in the forward contract, I will receive this much from the forward contract for my stock that I will deliver at T. But that's higher than this product, so there you go, we created arbitrage. Okay? All right. Similar logic, if the forward price is less than this product, I'm just going to reverse all my positions. When I was buying, I would be selling, when I was selling, I would be buying. But the logic is the same, buy cheap. Now the forward contract is cheap and sell expensive, now the stock is expensive. Let me sell short one share. When I sell short one share, I'm going to receive S(t), so I have to do something with it. Let's put it in the bank and let's go long in the forward contract. It costs me nothing today. That's exactly the opposite position than before. At T, I just check that everything works and that I have extra money. At T, I am going to have to pay F(t) for one share of the stock. Okay? But this is the amount of money I have in the bank because I put S(t) in the bank, initially when I sold the stock. This is how much it's going to increase by the factor of B. There you go. I have more in the bank, then I have to pay for the forward, in the forward contract for one share of the stock. I do pay F(t), I get one share. What do I do with it? I close my short position. I borrowed that share effectively by going short in that share, so I have to return that share sometimes. I return it now, in the forward contract, when I get that share by paying F(t), and I'm fine. I close my short position, I have more than enough money in the bank to pay the share in the forward contract, and there is extra money that is arbitrage, arbitrage profits. In both cases, F(t) is strictly larger than this product. F(t) is strictly Lower than this product, I construct arbitrages and if you assume there is no arbitrage, the price has to be like this. Now, in reality, the price may not be exactly following the formulas that we are going to see here, including this one because there are some imperfections in the market, there are tax issues. there are transaction costs, selling short has some costs going with it, you have to have a margin accounts, so selling short is not as easy as going along for stocks, let's say. There are these imperfections, transaction costs in the trade. There are these imperfections that sometimes will make these formulas not quite be true in reality, but in this course we are not taking those imperfections into account. We're assuming we have perfect markets. We can borrow and lend at the same rate, there is zero transaction costs, there's no taxes, selling short is free. All these things we are assuming here. It's called perfect markets. We now know how to price a forward contract. Let's move next. Well, this was fine if nothing happens between today and cap T but if there are payments in between, we have to take those into account. Hopefully you didn't look too much to carefully this slide. Here's a question for you. Suppose now you have forward contract and a stock that pays dividends. Intuitively, E is the forward contracts for our price going to be lower or higher. Is the forward contract be more expensive or less expensive in terms of its forward price for the stock that pays dividends relative to forward contract on an equivalent stock which doesn't pay dividends? Let's think about it. If the stock pays dividends, then if you're holding that stock, it's not just the profit or loss that you will get from increase or decrease of the stock in the future, you are also getting the dividends in the meantime. Now with the foreign contract, you are only going to get that stock three months from now, let's say, not today. Which means if there are dividends between today and three months from now, it's not really holding the stock, you are losing those dividends. Forward contract is not as good as holding the stock. We might guess here that what, that forward contract on a stock which pays dividends, the forward price is going to be lower than the forward price for an equivalent stock which doesn't pay dividends. That's the next formula. There are two cases, maybe not completely realistic either one of them, but there are two easy cases. The first case, if I assume that S pays deterministic dividends, meaning I know sitting today at time small t, I know what the dividends will be. Then I claim that the formula is like this. It's the same formula as before, except I subtract the present value of dividends from the current value of the stock, so D of t is the notation for the present value of the dividends. It's lower as we guessed. The forward price on stock which does pay dividends is lower. You just reduce the value of the stock by the present value of the dividends. There is another elegant way of modeling dividends. If S&P 500 an index on average of many stocks, then dividends are coming pretty frequently. If you model those, dividends is coming continuously at the continuously compounded rate called q. Then the formula is actually like this. You don't subtract from S, you discount S by the rate of dividends. Here dividends are not deterministic because they are going to be proportional to how much S of t as it moves through time, the payments are proportional. It's like qS of t that you are getting each small time dt. It's not subtracting something because we don't know exactly the value of dividends. We just know it's at constant rate. Instead of subtracting, you discount the stock by the rate of dividends. I'm going to show one of these next, the first one, and then the second one I'm going to do something similar for foreign currencies, so I'm not going to show this one. Let's do that next.
