In this module, we're going to discuss the cash account and pricing zero-coupon bonds in the context of the binomial model for the short rate. The cash account and zero-coupon bonds are extremely important securities and derivatives pricing in general and so we're going to spend some time now figuring out how to price them and understand the mechanics of these securities. So let's get started. If you recall, this is our binomial lattice model for the short rate, ri is the short rate that applies for lending between times i up to i plus 1. In general, it's a random variable because it can take on any of these values for example at time 2. So time t equals to the short rate r2 could take on this value at state zero, this value at state 1 or this value at state 2. We also have our risk-neutral probabilities qu and qd and of course, qu plus qd must sum to one. These are risk-neutral probabilities so they're strictly positive. What we will do is we price everything with these risk-neutral probabilities. In particular for example, if we want to price a non-coupon paying security and I use the term non-coupon here loosely. So a coupon could refer to any cash flow. So if we want to price a non-coupon paying security at time i state j, well, we just do our usual discounted risk-neutral pricing. So Z_ij, the value of the security at time i state j is 1 over 1 plus the short rate at time i state j times the expected value of the security one period ahead where we take that expected value with respect to the risk-neutral probabilities qu and qd. Now, as we said as well before there can be no arbitrage when we price using three, and the reason for that is as follows. If you recall our definition of a type A arbitrage for example. So a type A arbitrage was a security of the form V_0 being less than zero and its value at time 1, V_1 must be greater than or equal to zero. So we said any security like this in a one-period model constituted a type A arbitrage. Well, this is not possible over here if we price everything according to three. The reason is so Z will take the place of our V here. We see that it is not possible for this to be greater than or equal to zero and this to be greater than or equal to zero and yet have this being less than or equal to zero. This is not possible because the q's and r are all strictly positive and if the zeds are greater than or equal to zero then this must be greater than or equal to zero as well. So we actually cannot get a type A arbitrage, that's not possible. It's the same for type B. If you recall a type B arbitrage assumed a security of the following form V_0 less than or equal to zero, V_1 greater than or equal to zero but V_1 not equal to zero which means that V_1 is greater than or equal to zero and all states and is at least one state where it's actually strictly greater than zero. Well, again, the exact same argument over here would show that that is not possible as well. So there can be no arbitrage when we price according to three. I now want to talk a little bit about the cash account. The cash account is a particular security that in each period earns interest at the short rate. We're going to use B_t to denote the value of the cash account at time t and we will be assuming without loss of generality that it starts off with a value of one. So B_0 is equal to one. The cash account is not risk-free and the reason it's not risk free is because interest rates are uncertain, they're stochastic. In particular, the value of the cash account at time t plus s say, is not known at time t for any value of s greater than one. However, it is locally risk-free because I do know the value of B_t plus 1 at time t. In fact, B_t plus 1 will always be equal to B_t times 1 plus the short rate. I'm going to know the short rate at time t and therefore I will know B_t plus 1 at time t. So again, think of your bank account analogy. If I deposit money today for one month, I know what rate would apply for that one month period and so I will know how much I will get at the end of the month, but I will not know what interest rate will prevail in one month's time and therefore will not know future values of the cash account beyond one month. So a quick thing to notice here so B_t therefore has this expression here based on the argument I just gave you I can look at B_t plus 1 and B_t and divide 1 by the other and see that I get 1 over 1 plus rt. The reason I want that expression is down here, I want to derive equation 4 here. So how do I derive equation 4? Well, again, for a non-coupon paying security, Z at time t state j is equal to 1 over 1 plus r-tj times the expected value of the security 1 period from now. So this is our familiar risk neutral pricing expression from the previous slide. I can actually rewrite this expression as the expected value under q. Remember q is equal to qu and qd the risk-neutral probabilities. I can replace my 1 over 1 plus r_tj with this expression here B_t over B_t plus 1. So therefore, I can write the value of the non-coupon paying security at time t as being the expected value at time t under q of Z_t plus 1 multiplied by B_t over B_t plus 1. So rewriting equation 4, I can just bring the B_t over to the left-hand side and I get this expression here. This is an important expression but I can go a little bit further. I can actually iterate to get the following. So for example, I can write Z_t over B_t is equal to E_t under q of, well, we know it's Z_t plus 1 over B_t plus 1, but I can actually use this equation 5 again to write Z_t plus 1 over B_t plus 1 as the expected value under q. Conditional on time t plus 1 information of Z_t plus 1 over B_t plus 2. Using the law of iterated expectations this is equal to the expected value conditional on time t information of Z_t plus 2 over B_t plus 2. I can repeat the same trick again and again and so it's easy to see that this condition is so. So this is our risk-neutral pricing condition for a non coupon or non-dividend paying security. In particular, it's the pricing equation that we use for any security that does not pay any intermediate cash flows between times t and t plus s. If I want to use risk-neutral pricing for a security that pays a coupon, then it just takes the following form. So Z_tj equals 1 over 1 plus r_tj times the expected value of the value of the security plus the coupon or cash flow in the next period. So I can rewrite this as the expected value under q of Z_t plus 1 plus C_t plus 1 divided by 1 plus r_tj. When we're doing risk-neutral pricing for a coupon paying security, we use the exact same idea. So Z_tj equals 1 over 1 plus r_tj times the expected value under q of the value of the security plus the cash flow at time t plus 1. So that just gives us this expression here. For the same reason as before, we can see as long as we price any coupon paying security this way there cannot be an arbitrage. There is no way that this quantity can be greater than or equal to zero and yet to have Z_tj being less than zero. So you couldn't have a type A arbitrage because if this is greater than or equal to zero, the risk-neutral probabilities we know are strictly greater than zero and this is greater than zero and so all of this expectation must be greater than or equal to zero. So in particular this is not possible. So you couldn't get a type A arbitrage and for the same reason, you couldn't get a type B arbitrage as well. All right. So we have seven. Well, it's easy to rewrite seven using the same ideas we used in the previous slide. I can replace 1 over 1 plus r_tj with B_t over B_t plus 1, bring the B_t over to the other side and I get expression 8. Now I can iterate, I can substitute in for example, if I substitute in the following, I know that Z_t plus 1 over B_t plus 1 is equal to the expected value under q conditional on time t plus 1 information. C_t plus 2 over B_t plus 2 plus Z_t plus 2 over B_t plus 2. So if I substitute that in down here, I'm going to get this expression here when s equals two. It is easier to see that this expression holds more generally for general values of s or for integer values of s greater than t. So this is an extremely important expression. By using this expression to price every security in our model, we're going to ensure that our model is arbitrage free and we will use this throughout this section on term-structure modeling and pricing of fixed income derivatives.
