In this module, we'll introduce the concept of no-arbitrage and work through a very simple example of the application of the no-arbitrage principle to pricing. Consider a contract that pays ck dollars at time t equal to k, where k takes values one, two, three up to T. And let's say that one has to pay a price p at time t equal to zero in order to receive this contract, in order to get this cash flow. So the story is, here is time t equal to zero, where you pay a price p. Here is time t is equal to one, and all the way up to time t equal to capital T. And you, you get cash flows c one, c two, and so on, up to c capital T. And the goal of this module, is to introduce ideas of no-arbitrage that allow us to, fix this price p. It'll turn out that in certain circumstances we will be able to fix this price exactly. And in certain other circumstances, we can only be able to bound the price. We can tell you the price cannot be larger than a certain quantity, and cannot be smaller than a certain quantity. There are two ideas of arbitrage that are used in financial engineering. One idea is called weak no-arbitrage, and the second idea is called strong no-arbitrage. Both of these ideas essentially eliminate the possibility of a free lunch. Let's first consider weak no-arbitrage. What this condition says is, suppose there is a contract such that the cash flows associated with this contract are non negative for all times K greater than equal to one. C1, C2, C3 and so on up through C capital T are all greater than equal to zero. If there is such a contract then the price for this contract must be greater than equal to zero. The stronger arbitrage condition says that suppose there is a contract for which the cash flows are non negative for all times in the future, and there exists some time, cl in the future, such that the cash flow is strictly positive. Then the price for such a contract must be strictly positive. Both of these conditions are motivated by the fact that in a market if there are contracts for which you get something for nothing, then just by supply and demand that price, that contract will be priced to a point where you had to pay a fair price. Let's walk through the rationale for the weak no-arbitrage condition. Suppose there exists a contract for which CK is greater than equal to zero. That means you get non negative cash flows for all times in the future, but the price for such a contract is less than zero. So, weak no-arbitrage says that price must be greater than equal to zero. Here, I'm assuming that the price is less than zero, and let's see what happens. Since CK is greater than equal to zero, you do not owe anything in the future. And the buyer of such a contract receives minus P, because P is less than zero, which means you have to pay a negative amount or receive it. Which is the same thing as saying you receive a positive amount at time T equal to zero, the buyer never loses anything. And at the same time, gets money at time t equal to zero. This is a free lunch, weak no-arbitrage condition says this cannot happen. Why cannot it not, what, why can't this happen? This cannot happen because if there is such a contract such that p is less than zero, then the seller of such a contract will start increasing the price. It's a bad deal for the seller. The seller will keep increasing the price, but for any price p less than zero this is a very good deal for the buyer, so the buyers are still going to be there. And this price, the seller will keep increasing the price until p hits equal to zero. At least she might be able to increase the price to be something greater than zero, but the weak no-arbitrage condition does and, doesn't allow us to figure out how exactly this price is going to work once the price becomes greater than zero. We can only say using a weak no-arbitrage condition, that the price will be increased until p becomes greater than equal to zero. You can make the same argument from the buyer's perspective, since it is a good deal for any price less than zero, the buyers will be willing to pay a higher price in order to compete. They will compete with each other until the speed less than zero cannot be sustained in the market. Recall, in the other module, I talked about the fact that how prices get set by supply and demand. You again see here that the rationale for weak no-arbitrage is built on supply and demand. We are building it on the fact that there are many buyers, many s-, sellers. We are also using the fact that the information about the details of the contract are publicly available, are uniformly available to buyers and sellers. This is going to be important and we'll emphasize this again on the next slide. Here, I'm going to work through the rationale for a strong no-arbitrage condition. Suppose p is less than equal to zero. Strong no-arbitrage conditions says that p must be strictly greater than zero for a contract for which the future cash flows are greater than equal to zero, and that exists sometime where the cash flow is strictly positive. Since CL is greater than zero for some L greater than equal to one, even if p is equal to zero, this is a free lunch. It's a free lunch as long as p is less than equal to zero. Again using the same arguments that we used before, the seller of such a contract will have an incentive to increase the price. The buyers will still be around, because it's still a good deal at something positive. We cannot guarantee what the precise value of the price is going to be. But for some price strictly positive it'll still remain a good deal because there exists a cash flow cl which is strictly positive for some time, L, in the future. And again, buyers and sellers will compete in order to set the price p. The implicit assumptions that are underlying the no-arbitrage conditions are the markets are liquid, which means there are sufficient numbers of buyers and sellers. If the markets are il-liquid, then no-arbitrage condition is not valid, and the bounds that we generate using the no-arbitrage argument will no longer be valid. We also assume that price information is available to all buyers and sellers. The price information here basically means, what are the cash flows? If there are buyers and sellers which are ignorant about the cash flows, then the price formation process will not happen. We also assume that the competition in supply and demand will correct any deviation from the no-arbitrage prices. And this again presupposes that there is a market, these markets are liquid. The price information goes to every buyer and every seller, so that they can decide how to set the price efficiently. The rest of this module I'll just walk you though a very simple example in how to use no-arbitrage condition to price a very simple fixed income instrument. In the later modules we will go over more complicated examples where you use no-arbitrage condition to set prices for more complex derivatives. So consider a very simple bond. What this bond gives you is 8 dollars in one year, and we want to set the price for this bond. Suppose in the market, one is able to borrow and lend unlimited amounts at an interest rate r per year. So, the bond pays $8 in one year. We can borrow and lend unlimited amounts at the interest rate of r per year. And we want to figure out, what is a fair price, or an arbitrage free price for a contract that pays $8 in one year. We'll do it by constructing two different portfolios. So let's construct the following portfolio. You buy the contract at price p, and you borrow a divided one plus r dollars at an interest rate of r. Consider the cash flows associated with this portfolio. In the future, in one year, the contract will pay A dollars and you've borrowed A divided by one plus r at the interest rate of r, so you have to pay A dollars. These two cancel each other. So the cash flow in one year is A minus A equal to zero. So c1 is equal to zero. And what is the price of this portfolio? The price of this portfolio is p, the amount that you had to pay to get the contract, minus A divided by one plus r, because this was the cash flow that you received at time T equal to zero. So the net price that you paid for the portfolio is p minus A over one plus r. Now, let's use the weak no-arbitrage condition. The weak no-arbitrage condition says that if the cash flows in the future, and in this particular case there's only one time in which there is a cash flow in the future in one year. If the cash flows in the future are greater than equal to zero, then the price must be greater than equal to zero. C1 greater than equal to zero implies that the price of the portfolio Z must be greater than equal to zero. Price Z is equal to P minus A divided by one plus R. This must be greater than equal to zero, which means that P must be greater than equal to A divided by one plus R. So we now get a lower bound for the price. No-arbitrage weak no-arbitrage condition gives me a lower bound on the price. And we got this lower bound by constructing an appropriate portfolio. Now, in the next slide, we'll get an upper bound for the price by constructing a different portfolio. So construct a portfolio, we sell the contract at price B, and lend A divided by one plus r at the interest rate r. Now what happens? The cash flow is again zero in the future, because you've lent an amount A divided by one plus r at interest rate r. So in one year this becomes, this returns your value A. Since you have sold the contract, you are now responsible to pay A dollars to the buyer of the contract in one year. These two cash flows cancel each other, and therefore C1 again is greater than equal to zero. In fact C1 is equal to zero, which is the same thing as greater than equal to zero. It satisfies the condition that greater than, C1 is greater than equal to zero. What about the price of this portfolio? The price of the portfolio is how much did you have to pay, in order construct this portfolio. Since you sold the contracted price p you receive that amount, so that's minus p, because you had to lend A divided by one plus r, the interest rate r. This is the out flow, so A divided by one plus r is the total amount that you had to pay. The difference between these two, A divided by one plus r minus p, tells you the price of the portfolio. A weak no-arbitrage condition tells me that since C1 is greater than equal to zero, the price that I paid for this contract must also be greater than equal to zero. So, z is equal to A divided by one plus r minus p. This must be greater than equal to zero, which means that A divided by one plus r must be greater than equal to p. If you combine this upper bound with the lower bound, that we constructed in the slide just before, you get that p must be exactly equal to one plus r. Could this be a surprise? Not at all. All we are doing is constructing the net present value calculation using a no-arbitrage condition. The advantage of doing this net present value calculation using the no-arbitrage condition is that it clearly shows what are the assumptions that are needed in order for the net present value to exist. It relied heavily on the ability to borrow and lend at the interest rate r, an unlimited amount. If we are a, if these assumptions are not true, then the net present value is not the correct price. And we would have to use some other techniques to figure out what the correct price would be. And no-arbitrage conditions would still be valid, except that we will have to use them with different portfolios perhaps. So what happens if the borrowing and then the lending rates are different? I can't use net present value, but I can use no-arbitrage. What if the borrowing and lending markets are elastic, which means that if you want to borrow or lend. The interest rate that you're going to be charged depends on the amount that you're borrowing or lending. What happens in that case? Again the net present value calculation is not valid, but we can construct no-arbitrage conditions which would give us, if not the exact price, at least bounds on what such a contract can cost.
