In this screen time your typical model would be again a binomial tree model, it would be your interest rate moving. So first we modeled interest rates when you would move it up by multiplying by u, by u, by u. You move it down by multiplying by d, and then at the end, then you want to compute bond prices from the interest rates. For the bond prices, so you think of a bond prices as an option really on on the interest rate. So bond price pays 1, 1, 1 at the end, and then you go backwards in the tree by discounting, and you compute bond prices at the time 2 and go backwards. You compute bond prices at a time 1 and finally going backwards to compute the bond prices at a time 0. And once you have the bond prices, maybe you can compute prices of an option on the bond. All right, that's the idea, I'm not going to really do much about this. You can read this in, you can find in different books, including ours more models in discrete time. I'm just going to do one simple example in this discrete time then we'll go to continuous time. Here is the example. We have a one-year interest rates, which is today 4% and we are going to assume here two periods one and two years. And so the interest rate is going to be 4% for the first year. And then it's a Binomial model then it can go up to 5% in the second year or it can go down to 3% during the second year. And there is a two-year zero-coupon bonds, let's say it pays $100, it's just one and it sells for $92.278. And we want to price European call option on the bonds with maturity of one year. So the bond is a two-year bond, the maturity of the option, it would be one year, it might have been better to call this Tau to be consistent with the notation from the previous slide. That's the maturity of the option and the strike price is 96, okay, strike price is 96. All right, I want to price this option, notice, I didn't give you the p or p star or q, didn't give you the risk neutral probability. But in a way I gave you enough information similarly as before, because usually for stocks I give you up and down factor. And then you can compute the risk middle probability from the up and down factor. Here I give you also up and down values for the interest rate. And I give you the price of a bond rather than initial price of the stock, which was the case before, when we were pricing options and stocks. So it should be enough information to actually compute the risk neutral probability of up and down moves. How do we do that? Well, the idea is, this is the idea, and so we have the tree like these two periods. And the idea is I know the values of the bond here, it's 1, 1, 1 and I know what the interest rate will be if I'm up here or if I'm down here is either 5% or 3%. So using that, I should be able to compute the prices of the bond here after one year. And once I have these since I know the price today, then I will be able to find q and 1 minus q. By knowing this price today and knowing what the price in the future should be. And then there is a connection between those prices and q and 1- q, right? But that's the idea, so let's do that. Okay, so first in this picture I first have to find the bond price here and the bond price here after one year. And if the interest rate goes up to 5% then my price of the bond is going to be 100 over 1.05 which is 95.238, right, so it would be here 95.238. On the other hand, if the interest rate goes down to 3% and the price of the bond is 100 prices of bonds one year from now would be 100 over 1.23 which is 97.087. So here you will have 97.087 in your model. These are the bond prices after one year, possible bond prices after one year. Well, that's it, now I have these two numbers and if they are weighted and discounted back to today with q and 1- q they you they have to give me 92.278. So I'm going to do that in the next slide. Okay, so in the next slide, 92.278 which is the today's bond price is discounted by today's interest rate which is 4%. So 1 over 1.04, I'm calling it p here rather than q as I said in the first slide, I'm just going to call everything p. And so it's p times the upper value which was 95.238. Well, it's lower for the bond price but upper value for the interest rate + 1- p, the value corresponding to the lower interest rate 97.087, okay? So this is the equation for the price, for p, for the risk to the probability. All right, so if you compute p from here then you get 0.605 and now you can price whatever you want. Once you have the risk neutral probabilities you can price whatever you want. So let's price the call option. The price of the call option which matures one year from now. It's discounting by today's interest rate for the first period which is 4%, that's discounting, p times 95.238 which is the bonds if the interest rate goes up- the strike price. So this section can be 0, this part is 0, + (1- p) bond prices in the lower note 97.087- the strike price post part. So only the second term gives us 1- 0.605 times 1, this is 1.087 discounted, you get 0.413, right? And that's pricing the bond option in the simplest possible binomial tree model for bonds or for interest rates really. Okay, that's it for this set of slides. Then in the next set of slides, we'll start doing continuous time models for interest rates.