 So in last lecture, I had in mind that I would like to cover these three topics, trinomial trees, some exotic options, and interest rate models. And we were able to get through these two, but we were not able to finish off the interest rate models, and that's what I will complete today. And as a point of interest, next week is your term test, right? October 19th, and it will be conducted in this room. And it will be probably a 90-minute term test, almost certainly, and the structure is more or less the same as it has been in the previous years, so I've posted a couple of term tests for you. You can take a look at that. And the topics that it covers will be everything up to including what we do today. So interest rate models will be one of the last topics that we'll cover for the term test. And after the term test, we start with continuous time modeling. So far, we've been dealing with everything in discrete time, and today we'll continue in discrete time, and whenever we did something as a continuous time, we took a limit. But next week, we'll start working directly in continuous time and how we can deal, or sorry, not next week, but two weeks from today, we'll deal directly with continuous time and start doing some stochastic analysis, working on learning about what are Brownian motions, what are Ito processes and Ito's lemma, and dynamic hedging in continuous time setting. So that's where we'll be heading. But for today, we're still going to stick to trees, and we're going to cover interest rate models. As always, I'd like to just do a quick reminder of what we did in the last lecture. So today, I'm going to do that reminder fairly quickly because it's a little bit disconnected from the main topic of today. So trinomial tree is one of the things we discussed, and one of the points that I made for you was, if you looked at the binomial tree and you took two steps with half the time steps, then that is essentially equivalent to taking three outcomes after one single time step. And we went ahead and looked at how we could calibrate this, and this fixed a couple of sort of little oddities that we saw in some numerical analysis. I posted up the correct solution, by the way, of that linear system. I kind of mucked up when I did the linear system in class, so I posted the update of that. And we went on and said that if you solve this type of trinomial tree backwards in time, and in fact, you can solve for evaluations not just for the single starting point in the tree, which is what you normally did. Normally we sort of viewed the tree as starting in a single point and we just focused on, say, something in this region, right? And that ended up with a particular starting point for the tree. Instead I pointed out that you can, in fact, build a tree where you can find valuations for all initial starting points, right? All initial starting points on your grid. And one way to do this was in the trinomial tree setting was to build a lattice of this kind. And here you have the point that at every point in the lattice you have three points to go to, but the same methodology, the basic principle that valuations of any assets are given as discounted expectations under the risk-neutral measure for one time step and then repeat that through the whole process still holds here. There's nothing fundamentally new except for the fact that you're now viewing the tree as starting, or sorry, now viewing it not just as a tree but as an entire lattice, which allows you to do valuation for all initial starting points of the underlying asset. Okay, so that was one of the main points of that little analysis. And we talked a little bit about this boundary condition. And I believe then we went on and started discussing exotic options. And there were mainly one main class of options that we looked at were these barrier options. And we talked about, for example, the knock-in option. And to remember what that is, it's an option which only comes into existence once the asset rises above a certain critical level or drops below a certain critical level or drops above and below a certain critical level. In other words, it's a regular European option which comes into existence upon entering some region. In general, that's what it is. You just enter some region and at that point in time is when you receive a European option. So here was an example where you had the double knock-in case. And if you look at this green sample path, once you hit that point there that I'm just highlighting now, you end up with a European option or the blue sample path. Once you hit this point here, you end up with the European option. If on the other hand your asset dynamics never touched the upper or lower barrier in this particular case, then you would receive nothing because the option never entered the region. So this is a basic idea of barrier options. And they actually are fairly simple to value. The idea is first of all, imagine what happens when the option becomes knocked in. Once it's knocked in, it is actually just a European option. So all you need to do is value a European option but you need to value that European option along all points in the barrier boundary. So in other words, you just value European option and you find it along all points along the barrier. And then you insert those boundary conditions together with the terminal boundary condition and work backwards, discounted expectation as always. So the only trick in this is first to solve the sub-problem, the European option that's embedded. Once you solve the sub-problem, the solution of that sub-problem feeds in as a boundary condition for the real problem, for the barrier problem. And that's a little bit of a tricky twist in the way they're thinking and it's a little bit unusual and often causes a bit of confusion. Is there anything that I can clarify about this particular type of option valuation? In specific orders? Oh, well, this example is this particular type of barrier option is in whatever order. This particular barrier option. So in fact, let me write on a new sheet because it'll start to look a little bit too muddled on there. So the option that I'm imagining is the following thing. You define a time, it's called the hitting time, right? This is how we did it. We said you look at the first time that the underlying asset price is no longer in the region lower to upper. So L and U are my upper and lower boundaries. So in terms of a picture, this could happen either I exit the region from above or I exit the region from below first. It doesn't really matter. I can enter over here, over here, first or second. If I hit there, I'm going to get my European option. And if I do this and I hit there, I'm still going to get the European option. So this is independent of the order. Hit the lower and upper. Oh, hit the lower and the upper prior to maturity. So this is something quite different. It's not an option on the minima of the two times. It's an option on, well, one thing you can do is you can say, let's define tau 1, which is the first time that S is, say, greater than or equal to U. And then tau 2, or maybe I might as well call this tau U and tau L. So that's the first time that I hit the upper barrier. This is the first time that I hit the lower barrier. And your option, actually, by the way, in this case, what is it? In the first case, what is my payoff? It's 1 if tau is less than or equal to the maturity times the regular European option. So if we had a call, for example, it would be this. This new option that I'm being asked about is really an option on the minimum of these two things. So now what I do is I define tau to be tau U wedge tau L. So what does that wedge mean? It means just the minimum of the two things. This is sort of stochastic process notation. So if I define this to be the minimum of these two, my payoff is the indicator of the event that tau is less than or equal to T. It's the same payoff function, but tau represents something else here. That's what you mean. Is that correct? The first one is a maximum. No, that's in. I should want the maximum. Thank you for what you're asking for is the maximum of these two. So you want the maximum. Let's draw this diagram, capital T here. And what you want is you want to make sure that I do this and then I do that and that will be a path that actually has a positive payment, right? And the order doesn't matter, so I could have done that and then that and the order doesn't matter again, right? So both of these would have a positive value in your... Well, it depends on where the strike is, but both of these would have been knocked in, right? In this situation, I get knocked in here and in the first situation, I would have been knocked in there. Is that right? And you want to know how to do the valuation for something like this. I mean, it's following more or less the same kind of principle, but perhaps this is a little bit too subtle to talk about here and maybe what you can do is talk to me during my office hours because I can talk about the solution, but it'll be something I'm not going to test you on and it might be too confusing, okay? But it's an interesting kind of option. In terms of simulation, how would you simulate that if you were to do a Monte Carlo simulation? How would you simulate it? No ideas? Well, I mean, you won't be able to simulate it exactly, correct? Because this is a continuous process. You're looking for continuous... when the instant that it occurs, but you can at least approximate it, right? You can perhaps put some small steps here, right? You can take some small time steps and what you could do is generate the asset price at each one of those little steps. So call that asset price at step N. And how do you get that? You don't use the CRR model because these steps are not necessarily small enough, but you combine it with the distributional property. Remember, what's the distributional property for the asset price under the risk-neutral measure? S at any point in time, T, for example, equals in distribution, the exponential of R minus a half sigma squared capital T plus sigma squared T times the normal. So we can use this basic idea here to generate a sample path, and here this is ZN. So the Z, one, Z two, these are i, i, d. Under Q, these are normal zero, one. I should say here this is normal zero one under the measure Q. So what does this do? Let me blow up this little diagram that I just drew. So this is a small portion of the diagram. That's my S zero. I generate S at step one by using this formula, right? I know what S zero is. I take the exponential. I take this little drift. I add in a random number generated from a standard normal. That gives me S at time one. Then I generate another S at time two. To get that, I have to draw another standard normal. I get another S at time three. I get another S at time four. I get another S at time five. I get another S at time six. So I would have generated a sample path, at least at these discrete points in time, and what you can do is simply check this condition. Is the maximum of the first hitting time to those boundaries less than capital T? In other words, do you hit both barriers before T? But that's an easy numerical calculation, right? All you have to do is just find, you can do this. You can just say, what's the maximum of these S's? What's the minimum of these S's? Right? And if the maximum is bigger than you or bigger than or equal to, and the minimum is less than or equal to L, if both of those is true, then my path hit the top barrier and hit the bottom barrier. Do you agree? So the indicator of the event how is less than T, that is equivalent to, let's call, intersect, that's a set, let's call that set omega. That's equivalent to, completely equivalent to, the indicator of the event omega. If omega is true, then it's also true that tau is less than or equal to T, capital T. Do you agree? Approximately, right? In this approximation, in this discrete time approximation, and of course, the idea would be to do as many steps as you can. So this would give me a sample path of S. And then how would I approximate a sample path of S generated under the risk-neutral measure? It's a sample path generated under the risk-neutral measure. And then what my goal is, my goal is to compute this expectation, right? Expectation under Q, this is the value of the option, of the indicator that tau is less than or equal to T times S capital T minus K plus. That's my goal. And what's the approximation of this? The approximation, your standard Monte Carlo approximation, is to say this is approximately 1 over N. N is going to be, N is not the same as the number of steps. Okay, let me use M here, capital M. I generate capital M sample paths, a total of capital M. In fact, I'll do this exact experiment for you. And I check to see whether omega in sample M has occurred, and if it is, then I simply calculate this average. Okay, the expectation is identical to the expectation of a random variable. This is something that maybe, I'm expecting that you know and maybe you don't know yet. So tell me if this looks familiar. If you want to estimate the expectation of a random variable X, how can you estimate it? It's 1 over sample mean, right? So I can create samples of the random variable. These are IID draws from the distribution of X. Is this a familiar concept? Yes or no? Who has seen this before? So some of you have and those of you, so has everyone actually seen it before? Come on, just be honest, let me know. Don't waste our time. Who has not seen this before? There's a few of you. Okay, well, I mean, it's really just the law of large numbers, isn't it? It's effectively the law of large numbers that tells you this. It means that all you're doing, if I want to generate, if I want to compute the mean of something and it has some funky distribution, I can equivalently compute that mean just by taking a random number generator and drawing from that mean. So I draw samples from that mean and wherever this distribution is higher, it's going to draw more samples. Wherever it's lower, it'll draw fewer. And if I compute the average of those samples, that's approximately the actual average of the distribution. So that's the intuition for it. I'm not going to go and do a class on law of large numbers now, but that's the intuition that you should take from it. And it's that intuition or that representation that I'm using here. We're computing an expectation of some random variable. The random variable is a little more complex now. It's the indicator of tau less than or equal to t times s minus k plus. But this is the analog of the random variable x. It's just a random variable. So all I need to do is draw samples of that random variable and compute their average in order for me to estimate this expectation. And how do I compute a draw from that random variable? Well, that random variable so happens to depend on the entire path that my asset price actually takes. So I have to draw one entire sample path to give me one outcome for x. And then I'd make another sample path that would give me another outcome for x. Another sample path, another outcome for x. And I just compute the average of those outcomes. So that's my approximation. And this would actually give me a fairly good estimate of what that price is. So maybe to help with this example, let's run an experiment. And what shall I do this in? Let's do this in Excel, okay? Why not? So let's take a situation. You have some underlying asset. You have some volatility. Let's say it starts at 100. It's volatility is 20%. The risk-free rate is 5%. And let's take a maturity of one year. Okay, that's our maturity. Let's take a strike of 100. And we have an upper barrier and a lower barrier. Let's say 100 and, I don't know, 110 and 90. Okay, so it has to go up by at least $10 and down by at least $10. At some point in time. We don't care about which order. Okay? So that's my upper boundary. That's my lower boundary. Okay? So I've pretty much got... These are all of the contract parameters and the model parameters. But in order to do the simulation, I need to know how many time steps do I take in one year to approximate this. So why don't we do, I don't know, 52 once a week? Okay? So we'll take 52 steps in a year. And that tells me that my time steps are equal to this divided by n steps. That's the size of my time steps. And what I need to do is I need to draw a sample path for the underlying asset. That's what we need. We need one sample path for the... We need more than one. We need many sample paths. So here this will be little m. m is one, two, three, four, five, six, and so on. And in this direction here, we'll have s. Okay? Or we'll have time in this direction here. So... It'll be time one, two, or one, twelve... Well, I'll write it in terms of weeks. One, two, three, four, five, six, and so on. We'll fill it in afterwards, eight, nine, ten. And what I want to do is along here, I want to generate the sample path that makes my asset grow according to... According to this kind of scenario generation. I'm basically going to use the formula on the top of the screen to iterate through. So let's put in that formula and then put in... Then we also need some random numbers for it to run. But they all started at zero. So in fact, let's put in a time zero here. So the asset price is all started at zero. And then we're supposed to take our previous value, multiply it by the exponential of the risk-free rate minus half the volatility squared times dT plus the volatility times the square root of dT times some standard normal random number. So the way I'm going to do this is I'm going to put on the other sheet right behind here, I'm going to put a whole bunch of random numbers. And that's in B14. Okay, so B14. It'll be there. Okay. And that's that. Okay, so I hope you all... Whoops, what did I just do? Somehow I deleted my formula by mistake and I can't get it back. That equals risk-free rate minus a half vol to be squared times dT plus vol times the square root of dT times sheet two B14. And it's giving me a funny... What doesn't it like? It doesn't like dT. I didn't define dT. There. See, I noticed that because it didn't give dT a color. It was just black. So that's how it always highlights, you know, tries to match up the colors and it's very visual. Okay, so I need to copy that formula across all the way here. Now, of course, because... And I didn't do the previous cell times that, the cell times that. There we go. Okay, so because I didn't put any random numbers in that previous sheet, it's just growing at the risk-free rate, basically, at r minus half sigma squared. So let's put in some random numbers. So we need to put them in there. And I think you can do this from data, data analysis, random number generator. And we want 52 of them because we want 52 random numbers. And let's do, I don't know, 5,000 sample pods is what we'll do at the end of the day. And this is coming from a standard normal, normal zero one, and the output range starts there. Okay, so it's generating it. And Excel is not terribly fast. Okay, we'll let that run in the background. Oh, there we go, it's done. Okay, so now, so all of these here are just standard normal random variables, zero one. They're all independent of one another. And so this here is going to represent my sample path. Okay, one sample path. So we need 52 of these things because we want to get out to year 52, or sorry, time step 52, which is the end of one year. So I'm going to just stretch it all the way out until that hits 52. All right. And I can just copy this formula into there. Okay, so if I made a plot, it's because of my labeling, it's not picking out the correct cell. So I just have to tell it which ones to use. And where to go. There it is. Okay, so that's a particular sample path of my underlying asset. And again, let's just put some reasonable bounds around this, say from 80. Okay, so this is time on the horizontal axis and this is a particular sample path of that underlying asset. Now if I copy that formula, that same formula, and just copy that entire row in fact there, this is another sample path that's generated by the other random numbers on the other line. And so we can in fact just add to this. If I click this, I think I can extend it, and then there we go. That's a different sample path. It's another outcome of the underlying asset. And if I made another one, and I drag this down, that's a third sample path. So these are just all outcomes of my random process, right? This is a sample path for the underlying asset. And since I'm interested in computing the maximum and the minimum of this, in order to see whether the barrier has been breached, I can, we might as well just do it right here. So I'm going to shift this over a few cells, and I'm going to have the max here, the min here, and then knocked in. So I'll put a little knocked in question mark there. So this is a maximum, and this is a minimum, okay? And then we can say, if this is bigger than U boundary, now I forget how you do a compounding in Excel. Is it double and? Does anyone remember? It's an and-and. And this is smaller than L boundary than true, otherwise false. Oh, it doesn't like that and-and. So that's the one thing I don't remember in Excel, but, can we do that? There we go, I think that might work. Okay, just a single and. So if I copy that formula there, we again see, did we rise above and below? I'm going to add onto this chart here, select data, I'm going to add X value of 0 and 52, and the Y value will be U boundary, U boundary. I don't know if it can do a formula in there. Nope. What was my upper boundary? 120, 110, 110, 110. So I think that should just give me a straight, well, okay, I'll make it give me a line in the second here. And another one, which is going to be from 0 to 52, and 90 was the other one. Okay, good. Sorry, this is taking a little bit longer than I wanted to do, but it's useful I think, line style. So it does look as if those paths did not hit both boundaries, right, the first two. And we can see that here, right, in their max and min. Maybe this next one. Oh, it didn't either. It looked like it did, but it didn't. So let's go to another scenario. There we go. Oh man, I'm not going to get too many of them, I know. Maybe these are too wide. But in any case, I think you get the picture, do you? Do you get the picture of what I want to do? So at the end, what I'm going to do is we're going to take this and we're actually going to copy it down 5,000 cells. So we'll get 5,000 of these sample paths. And if you're actually doing this in real world, I would not use Excel because it's fairly slow. There are ways to do this more efficiently. Some people like using Excel because it's easy and intuitive. Personally, I'd use Matlab or something like that or programming it in C. And then we need to copy this all the way down as well. Okay, and do I get any truth? Okay, let's see. Let's look at the max of the maxes. Actually, that's not going to help me. I wonder if this if statement is correct. That's all. It's not clear if it is. Look, this should have been true. Right? This should have been true. So somehow Excel is not accepting my if statement in the correct way. So what I could do is if that's greater than that, then embed another if. Okay? Then it's true. Otherwise that's false and then false. Okay, now we got some hits. Okay, good. So that's true. That's true. Looks correct. Okay? Good. So we have all of those scenarios and then what we should have here is the payoff. Right? Because that only tells us whether the option is knocked in. So what's the payoff? It's the maximum of the terminal asset price, which is right there, minus my strike, or zero. But it's only that if this thing is true, right? If that, then one, otherwise zero. Okay? So I multiply by something that's zero or one. That if statement there is like my indicator, that's mathematically this indicator of omega m. Okay? Enter. And in this case it's false. So we get zero, which is correct. And okay, that's true. And that's good. Okay, so you notice that there's some cases here in which we have a true, but the payoff is zero. Why is that? The terminal asset price is too low. Okay? So if we look at this scenario here, we see the terminal asset price was $88 approximately. And that's below my strike. So even though the option was knocked in, I get nothing. And now in order to compute the price, I would take just the average, the expectation. This is the estimate of the expectation. So maybe I will, let's put this maybe over there. Huh? Okay? Expectation payoff. Okay? That's the approximate value. And then the option value is then what? Approximately. It's this times the exponential of negative rf times t. All right? We have the discount factor as well. I guess I didn't give this a name. Oh, Matt. Okay. So that's the approximate option price in this case. It's about $1. All right? So the specific number is not anything that I really want you to get, but this idea, this concept that you can generate approximations to prices simply by using effectively, law of large number, but embedded in that means that you have to simulate a sample path. And you can come up with all sorts of complex options that can easily be answered in this way. All right? That question that I had asked you to do in a simple 12-step tree or a five-step tree or something like this in the quiz last week, you could answer it here. All right? We can actually just ask, well, does it hit two before it hits one? All right? Does it hit you before it hits L? You can answer that. You can just put a different if statement there. And then that's the only thing that would change. The indicator of the event is the only thing that changes. Do you understand? Does that make sense? Okay? So obviously questions of this nature, I cannot ask you on a term test, right? I can't ask you to do a simulation on a term test. So this is not what I'm at. This is not why I'm doing it. I'm doing it because the hope is that it's helping you understand some of the theory that's going on behind there. And as well, in the real world, particularly in insurance companies, they tend to resort to simulations of this kind in order to value almost anything. They tend not to use a lot of sophisticated mathematics to get to the answer. Instead, they would rather do a simulation because you can understand that, right? Everyone can look at that and understand, okay, these are the sample paths that are there and some of them are making, some of them are paying you and some of them are not. If you work through the analysis, the mathematical analysis, to derive the formula for this case, it's actually not very easy. You can do it, but it's not very easy. But the simulation is. Okay, so I hope that answers partially at least the question that you asked. Is there anything else that I can address regarding the previous lecture? Because I'd like to move on. But if there's anything else that I can address. So what else did we talk about? Well, we had some other examples. And, right, and the last important, or the last analysis that we did was this idea of a forward-starting option. So remind you again how that worked. The forward-starting option was to generate, you get a payoff of a call in this particular case, the forward-starting call, but the strike is not set until a future date, until t0. And as well, we talked about how you could simulate that. If we wanted to, I could also build up an Excel spreadsheet for this. And I urge you, try it. Just do it on your own, just to try it, to see, make sure you really understand what's going on. But the basic principle is the same. I would generate s at time t0 and I'd generate s at time t1 using that distributional property that we already have. And I do that many, many times, and I just compute the payoff in each scenario, and I compute an average. And that will give me an approximate price. And you can check to see whether or not the analytical formula that I derived here actually is correct. It's kind of a fun little experiment for you to do. Okay? Any questions about that example that we did last time? All right, great. So then let's get on with the new material. And as I mentioned, what I want to discuss are interest rate models now. Models of stochastic interest rates. Okay, now, interest rate models are quite distinct from models of underlying asset prices. And the reason is actually quite simple. When we drew, before, when we drew a tree for an underlying asset, what does A0, AU, and AD represent? It's a price, right? It is the price of asset A. Right now, that's A0, and the price of asset A if the world enters into the U state and the price of asset A if the world enters into the D state. So those are actual prices. Now, if I try to imagine drawing a similar tree but what I draw instead of prices, so I put interest rates, there's a big distinction. Now, if these are interest rates, this object, these objects here are not prices. These are rates. And so when I draw that tree, that does not represent the price of a traded asset that I can use to replicate payoffs and so on, like we did before. Do you see that difference? There's that one fundamental difference. You don't trade 5%. If interest rates are 5%, that makes no sense. You don't trade 5%, you don't trade 6%. That's not an underlying asset. So this is not a tree associated with a traded asset unlike what we've had before. And because of that, we're going to run into a little bit of a problem. So let's see if we can probe the question a little bit more then. This is not the price of a traded asset. It's not the tree associated with such price. But what would be the closest thing to interest rates that are prices? T-bos, bonds in other words, right? Zero coupon bonds. That would be our basic building blocks. That's our natural atoms or our natural fundamental assets that we can use in the context of interest rate modeling. So let's go ahead and ask questions about the bonds in the market then. Let's take a look at, say, excuse me, at the one period bond. And because there's going to be a whole sequence of bonds, I'm going to write them all like this. The subscript is going to denote time or the time step. And the parameter that shows up in the bracket there represents the maturity of the contract, the time at which it matures. So this here is the price of a one-year bond at time zero. That's what P01 would represent. And what must that price be? Well, that price has to be the discounted version of whatever that interest rate tree, the discounted version of $100 using whatever interest rates that interest rate tree implies. So now we have to go back to the interest rate tree and understand a little more details about what it actually represents. So when you draw an interest rate tree, these are particular kinds of interest rates. They're what are called short rates of interest. Someone know what I mean by short rate of interest? It's the interest that you receive just for that one period. R0 represents the interest that I would receive right now if I invested money. And I only get that interest over the next period. So if I put $100 into the money market account, which is what the interest rate tree is representing the dynamics of, then I will get 100 times 1 plus R0 equals 1. And if I then take that same money, so suppose I took that, I put it into the bank, into the money market. I get 100 times 1 plus R0 at time 1, and then I take that and I put it into the bank again. I reinvest it. What I will receive is 100 times 1 plus R0, which is what I started with, times 1 plus R, whatever R is at that point in time. Now remember, in this model, this can be RU or RD. It's a random variable. We don't know what it is. Times 0, you don't know what it is. So when you put money into a bank account in the stochastic interest rate models, the bank account is now something that's also random. Before, we always had it as something that wasn't random. It was just growing at a constant risk-free rate. But in principle, and in real world, you can imagine, yeah, you know what the overnight lending rate is. You know what you can get for very short-term borrowing or lending, and big institutions can do this between themselves, in fact. As individuals, perhaps you can borrow and lend for the shortest time of three months, 90 days, the 90-day T-bow. But institutions can actually do it overnight. And that rate will change constantly. And that's what this rate is. And so on. So if I go two days, if I invest it for two periods, I'd have 100 times 1 plus R0 times 1 plus R1 times 1 plus R2. And R1 and R2 are random. You don't know what they are. So these rates here, although they're shown at the node and they represent the rate that you receive over that next period. Okay? So, given that information, how would you correctly value this one-year bond in terms of the tree above? Well, regardless of what interest rates become at time one, regardless of what they are, I'm getting $100, right? So the bond has to be the discounted version of $100. But what discount factor do I use? I should be using R0, because whatever I put into the bank now, it will earn at the rate of R0. So since my bond is maturing in one period from now, and it has to become $100 in one period from now, the bond's worth has to be 100 divided by 1 plus R0. And this is exactly consistent with this risk-neutral pricing idea. Notice I didn't have to use probabilities here because the two outcomes were identical. 100 and 100. So no matter what probability measure I use, the expectation is still 100. But we could write this in terms of this is the expectation under Q of my payoff, which is 100, over 1 plus R0. It's that formula. Okay, so in principle, if you were told to build an interest rate tree that is consistent with one-year bond prices, you can do that, right? You can say, if I told you the one-year bond was worth $95, well, you can easily calculate R0 from that expression. So this can be used to what's called calibrate to one-year bond prices. So when I say calibrate here, I'm talking about the calibrating the interest rate tree. Because at this point, we don't know what interest rate should I really be using. You don't have... Before we had an underlying asset, and we can just go to the market and say, A0 is this, it's $30. Historically, it had this volatility. So I can use that volatility to build my tree. So I know where to start it, and I know what states to make it move into. But for the interest rates, you need to have something in order for you to start it out at. And this is the something. This is the analog of knowing what the current price is. The analog for interest rates is knowing what is the current price of the bond, of the one-year bond, or one period bond. So let's see if we can continue this idea and build the rest of the tree. Our goal is now build a model that's going to be consistent with market prices for bonds. So what about P02? So again, I have 100 here, 100 in that state, and 100 there, and 100 there. So this is representing, maybe I will... This is P1U2, P1D2, that's P0. So I'm going to write below the P's here, the corresponding R's. That's R, 0, that's RU, that's RD. Right, from every level there are two branches. So that's why I've drawn four in the end. Now of course this tree recombines in a natural way, so I could have actually put those two together. So if you were to use risk-neutral valuation, how would I compute P02? What do you need to do? You need P1U2, right? We need to certainly find the value of the bond in this state of the world and the value of the bond in this state of the world. So let's go ahead and proceed with that. What is P1U2 going to be equal to? Well, according to risk-neutral expectations, it's the expectation under the measure Q and at this point it's not clear what the measure Q is. We haven't actually specified it. We're trying to kind of come up with it in some way. Of our payoff, which is always $100, divided by 1 plus what? RU. Because from that node... Remember, we're interested in this node here. From that node, the interest rates are RU over that node. Okay, well, in this case, that's pretty simple. That's just 100 all over 1 plus RU. What about P1D2? Again, that's an expectation under Q. You notice I'm wasting a lot of time by writing an expectation of 100, but I'm writing it like that just so that you see it follows directly from our standard risk-neutral expectation. Later on you, we won't bother doing it. But for now, for the first few times, we'll write it in. And divided by 1 plus RD. Because again, interest rates associated with this node is RD. So we end up with 100 all over 1 plus RD. Okay, now we have to calculate P02. Now we have a problem. We still need to do expectation under Q of my payoff, which is the two-year maturity bond at time step one. And I'm dividing by 1 plus R0, because again, I'm looking for valuation here. The discount factor associated with that node is R0. And here you realize that P12 is no longer constant. Before, whenever I took an expectation, what it was that I was taking the expectation of was a constant, so the measure didn't matter. Now the probability measure does matter. And let's call this probability measure here, let's call that Q as little Q as usual, the branching probability. And then the bottom node is by definition 1 minus little Q. So I would have Q, so sorry, first of all, let's put the discount factor up front. I'll have little Q 100 all over 1 plus RU plus 1 minus little Q 100 all over 1 plus RD. Agree? Okay, so now, if I tell you what the price of a two-year bond is, can you imply the model? Can you tell me what is the correct interest rate model to use? It should be fairly evident the answer is no. You cannot do it uniquely. We have already identified R0 from the one-year bond, right? We've got that. So we've got the starting point. So this is no longer a degree of freedom, R0, so I'll just check that off. Okay, R0 is done, we can't use it to change anything. But we do have three parameters. We have Q, we have RU, and we have RD. And we have to ensure that Q, in fact, is a probability. We have to ensure that our risk-neutral measure is truly a measure, which means that all branching probabilities are in the range 0 to 1, excluding 0 and 1. So here you have a couple of degrees of freedom. And then the question is, now it's a modeling issue as to how do we go about and come up with an appropriate model for valuing this. Now imagine, because now we have three parameters and only one equation, right? Three parameters, one equation is the one in front of you. The market is going to tell you this price, P02, market tells you that. So you have two degrees of freedom. That's not a great thing, right? I mean, it's sometimes good to have more degrees of freedom than your model tells you, but here we're trying to pin a model down. Does anyone have an idea of how could we possibly pin it down? Can we borrow ideas from what we've already used for the asset price dynamic somehow? Okay, sure. Go ahead. Yeah, so one idea, one sort of kind of natural idea, is why not choose RU and RD to somehow be some sort of spread around R0? Plus or minus something, right? Just like before. Before we had the asset prices in log space, they grew with plus or minus something, right? The natural model, let me just remind you, what is the natural model that we had over here? It was e to the sigma squared delta t and e to the minus sigma squared delta t. So in other words, there was a little Bernoulli that told us whether we go up or down, right? So we can use that same principle here. We could either make R grow geometrically as the asset price does to make sure it stays positive, or we can make it grow arithmetically. Either approaches can work. Geometric is kind of nice, so what do I mean by geometric? I mean this. I mean R0 e to the sum sort of sigma square root delta t, and here R0 e to the minus sigma square root delta t. So this would be a geometric row. Arithmetic, I just removed the exponential. So I have R0 plus sigma squared delta t or R0 minus. So the arithmetic is bad why? It's one sort of obvious fact. Suppose I did this over many steps. What could happen to R? It could become negative. Interest rates aren't negative. Interest rates are positive, right? So that's why the arithmetic model is not a good model. Sometimes people use it nonetheless because it's simple, and it has a nice, simple, continuous time limit. And we may in fact investigate that, maybe not today. But so I'm going to put a comma here, or R0 plus sigma square root delta t, R0 minus sigma square root delta t. These are two perfectly valid types of approaches. Okay, so if we do that, now we have a model for RU and RD. So this is reduced to how many parameters now? In fact, let's write out the arithmetic version because it's easier to write. I don't have to keep writing the exponential factor everywhere. You'll see that the analysis for the other case is identical. You just replace with exponential factor. So how many free parameters do I now have? I have two, right? Q and sigma. R0 is fixed. Sigma is now my free parameter, and Q is a free parameter. How could I fix sigma, potentially? Yeah, so one of the things that I could go back in history, we can see what are... Sorry, let me slide over here. What is the historical short rate of interest through time and do exactly what we did for the asset price dynamic matching? You might remember... I guess it might have been lecture two. I'm just sliding back here. Lecture two. Yeah. Do you remember this idea here in lecture two? We had a discrete time model for the underlying asset. We said... Let me go back. Okay, I guess I don't have it written there. Yeah, here we are. We had a discrete time model for the underlying asset, and what we wanted is we said we go through some historical data and we estimate the mean and the variance of the return of that asset from the data. And then we make our model match that. Yeah, that's what we did in the past. So why not use the exact same idea here? So I'm not going to repeat the analysis because it's more or less the same thing. Instead of A, you have little r's. R's are interest rates now. So in principle, can be calibrated from historical data. And all that represents is the volatility of the moves of the little interest rates, of the short rate of interest. So now, we're down to one free parameter. Q. We have one equation, one unknown, therefore we can solve it. Okay? Seems great, right? There's a problem with it still. And we'll talk about the problem when we get back from the break. So let's take a little break. So I shouldn't say why it doesn't work. I mean, it works for now, right? At this point, it looks like it works. We've got a sigma for calibrated from historical data. We are given the current price of the two-year bond. We have a formula that connects the price of the two-year bond to all of our model parameters. There is only one degree of freedom, Q. Therefore, we can find Q. And if prices are reasonable, we will find that Q is in the interval zero to one. So that's not the problem, right? It looks like it works perfectly. Where the problem shows up is, suppose we go one more step, not just for the two-year bond, but for the three-year bond. So let's draw a tree again now for the three-year bond. And we know we're going to get things recombining here, so I'll simply draw it in that manner. So we have the two-step. We have P2UU for the three-year bond, P2UD for the three-year bond, P2DD for the three-year. It becomes P1U for the three-year, P1D for the three-year, and P03. And again, just so that we have the interest rates below it, we'll write our little notations here. RUD, RUU, RUD, RDD. And remember, the model that we're currently trying for interest rates is that R times step N is equal to R times step N minus 1 plus or minus sigma square root delta t. And we can do the plus or minus by putting in a little Bernoulli random variable. Or, alternatively, I said we could also use the exponential version of things. Okay? So if we went ahead and tried to use this procedure, this model and this approach to figure out what are our branching probabilities? Let's see where we get stuck. First of all, what Qs, what defines our risk-neutral measure? It's the branching probabilities on this whole tree, correct? It's not just a one-point. We cannot assume that Q is the same everywhere, a priority. We have to somehow let the model tell us that. We've already computed the little Q for that first step via this calibration, right? That's what we've done here. Now, there's a couple more Qs. Let's call that Qu and Qd, representing the probability of going up once we've already gone up and the probability of going up once we've already gone down. I mean, it doesn't matter what the labels are. Maybe you want to call it, let's call it Q1, Q2, Q3, actually. Make it easier, I think, rather than confusing you with the U's and D's. So these are our probabilities. What have we got so far in our model? In terms of our model, we know what R0 is. We know what Ru is. We know what Rd is. We know Ru, we know Rd, we know Rd. We have all of those. We've got Q1. Do we agree? At this stage, once we've done the one-year bond, we've done the two-year or the one-period bond, the two-period bond. We're now in a three-period bond. So what are our unknowns? Q2 and Q3. These are the two things that are unknown. We're also going to be given the current price. This will be given. So it doesn't make sense for me to write down the formula here for you. You all know how it would work. I would get P2Uu to be 100 divided by 1 plus RUu. P2DD to be 100 divided by 1 plus RDD. That gives us all of the bond prices here. So we can easily compute those guys just by discounting. Because the branching probabilities on this part of my tree don't matter because I always get $100. So the expectation is always 100. But when I compute the discounted expectation here, I have Q2 coming into play. That's one variable. When I compute the discounted expectation coming from here, I have Q3 coming into play. That's a second variable. When I do this discounting, that's fine. Q1 comes into play, but we've got that. That's not a parameter. And then we'd be left with matching the market price One equation to unknown. Again, we're stuck. So even though our model, this simple model, was able to fit the one-year or the one-period and the two-period bond prices, we're unable to now come up with a unique way of getting the three-period bond price matched. We will have some degree of freedom. We'll have a choice in how we play with Q2 and Q3. It's theoretically not a problem, but practically it becomes a problem. Because how do you consistently get your model to work? How do you pick a particular risk-neutral measure? This has the problems much like what we talked about in incomplete market settings when the risk-neutral probability is not unique. We talked about that already. The fact that the three-period model, for example, or the three-period model, it does not have a unique risk-neutral measure when you have two traded assets. And if you try to, and you cannot find a unique one, so instead you have a series of price ranges that are valid. Here, it turns out that, well, you don't get a series of prices for the bonds. They'll all give me the same price because that's our constraint. But it gives us a collection of risk-neutral measures and if we then value something else, such as an option on a bond, then we will have a range of prices. So we are not able to pick a unique one using this methodology and using this particular model. So there's another approach that we can take. And the other approach requires reminding you about the connection between probability measures and drifts. And this is one of the, oh, by the way, thank you all for handing in those little information sheets about what it is that you understand from the course so far and what it is that you're having trouble with. And one of the things that kept coming up in particular was this concept of connections between risk-neutral measures or probabilities and drifts. That was one thing. And as well, why risk-neutral measure shows up and what's the difference between P and Q. So I'm going to try to work that into this lecture. So you might remember when we were dealing with just the asset price tree and we calibrated our model. When we calibrated our model, what we found is that P was approximately this. The real-world probabilities were that. Now, remember, what does real-world probability mean? The real-world probability means this is how the asset actually behaves. It actually moves with these probabilities. These are true probabilities. These are historical. These are statistical. These are how the asset actually moves. And that is to distinguish, that is quite distinct from risk-neutral probabilities, which are completely fictitious objects. They do not represent how the price of the asset actually moves through time. So I want to make sure that you understand that concept very clearly. These probabilities, the historical ones, the ones that you actually get by calibrating the model to data, by calibrating the historical movements of data, P probabilities, these are real-world probabilities. And that's why I call them that. These are real-world probabilities. And they represent the actual outcomes of the data, how the asset is actually going to move. So when we did this calibration, we found that the branching probabilities for such a model have to be given by this. Then we went ahead and we said, okay, in order for us to do risk-neutral valuation, even though asset prices move like this, if we are to calculate a price based on expectation, the expectation that we have to compute is a risk-neutral expectation. Notice how that's a different statement than saying assets move with risk-neutral probabilities. Assets move with real-world probabilities, not risk-neutral, but prices are computed as expectations under risk-neutral probabilities. So we said that, and the reason why they're computed that way is to avoid arbitrage. We showed there's an equivalent between the absence of arbitrage and the existence of this probability measure, this fictitious probability measure. It's something that happens to hold in order for us to avoid arbitrage opportunities in our market. And what we showed in this simple model was that Q, the branching probabilities, was a half, 1 plus r minus a half sigma squared over sigma squared delta t. And then there's some small corrections here. Okay, so we now understand these are risk-neutral probabilities. They are not the way that asset prices actually move, but there is a connection somehow to these two probabilities. One is that the only difference between them is in one place you have the real-world drift, mu, and the other you have a risk-free rate r. And if we compute expectations under the risk-neutral measure, the asset grows at a risk-free rate. So this statement here simply says the expected value of the asset at one time step is its current value plus a risk-free interest rate on its current value. That's all it is. So it actually has a drift equal to r. And in the continuous time limits and so on, when we found the distributional properties, we found that if you computed the expected value of the asset at some future date under the risk-neutral measure, this is given by e to the r little t, r0, I'm sorry, a0, while under the p measure, under the actual outcomes of the asset, it's mu. It's a real-world drift. So we have this nice relationship and we also had the fact that the variance on the other hand, and let's write it in terms of the log prices is easier to write in terms of log prices. The variance under p and under q happens to be the same. This was sort of a very interesting magical fact. We'd shown that fact. Now, why am I bringing this up in this context? I'm bringing this up in this context for what I said just prior to it. The fact that we're going to try to fix this model by using somehow the equivalence of probabilities and drift. So this expression here, the fact that you can see when I change the probabilities, what I really did, one way to think about it is, all I'm doing is in fact changing the drift. That's all it really does. Changing the probabilities is equivalent under certain assumptions and et cetera, et cetera, and I'm going to get into all of the details, but under certain assumptions it's equivalent to changing the drift of the asset. The only difference between expectation under q and under p is probabilities, and the right-hand side is telling us the effect of that change in the probabilities is simply to change the drift of the underlying asset. So that gives us an idea. In this interest rate model, instead of trying to find a risk-neutral measure, q with the probabilities that, so sliding back up here, instead of trying to find a collection of risk-neutral probabilities which defines my measures, q1, q2, q3, that make my market price be matched by my model price, why don't I simply pick the risk-neutral measure? I have a degree of freedom. I can do it as long as it's a true measure. I'm going to avoid arbitrage, the fundamental theorem of asset pricing says there exists a q such that discounted expectations are, if I take an expectation of the asset in the future, it equals its current value grown at risk-free rate. Discounted expectations are martingales, that's what it means, fair game. So I can find such a q. If I can find such a q, then I know that I don't have arbitrage in my market. So rather than using this to construct the q from prices, I'm going to instead impose q and change my model to make the model match the market price. And how am I going to change the model? I'm going to add drift into the model. So again, this is the analog of the fact that under p and under q, when I change measures, all I really am doing is I'm changing drift. So let me write the model down and you'll see what I mean. And hopefully you'll see what I mean. So instead of rn equals rn minus 1 plus minus sigma squared delta t and finding q's, or let's say I'm finding script q, in other words q1, q2, q3, all of the branching probabilities, right? Remember that measure, that script q represents the probabilities of my entire tree, not just one part of it, probabilities of the entire tree. Instead of this, let's put, let's fix q's. And in particular, I'm going to make it really simple because I can choose, I have a degree of choice. I can choose a measure as long as it is a valid measure and it makes the discounted expectation of prices be equal to their current value, right? That's all I need. So what's a nice neat choice for q? Exactly. I'm going to choose q to be half. And if we go back here, we can see, yeah, you know, these probabilities are kind of almost a half anyhow. They're just a little bit above or a little bit below a half. So why don't I fix them to be a half and instead I'm going to add drift into my process. So what does it mean to add drift? It means very simply this, okay? Yes, we do add and subtract sigma squared delta t, but at every stage I'm going to add some constant times delta t, right? This is effectively adding a drift. This is saying after every step, instead of if I draw it, if I make my vertical spacing actually represent, not just outcomes, but actually represent the level, if I started at 5%, for example, before we would have had this going to 4% or 6%, as an example, and then we would have had it going to 7%, 5%, 3%. These would have been equally spaced, right? Before, in our old model, let me, in fact, why don't I do this, slide it in between these two things. So before we had that and what we were trying, and what our object was, was to find all of these little probabilities, right? That's what we were doing before. So now what I'm going to do instead is, our tree will start, for example, at 5%, but it's going to go up to 7% and 6%, as an example, and then from here it's going to go, maybe it goes flat, it goes like that, and then maybe it again goes down, why not? So all I'm doing is, I'm adding a drift into the tree itself, and here all of the branching probabilities are half, and the goal is to find what is the theta, what is the drift, which makes my model match my market. And you'll see that now, at every single time step, there is only one new parameter. When we view it from the perspective of the tree with this tree, and we're looking for branching probabilities, every time we add a time step, we actually add quite a bit of new parameters, all of those queues, all of those branching probabilities keep getting added in. Here, whenever I add a new branch or add a new time step, I only have one more drift term, one more theta. So if I have n bond prices, I'm only going to have, in fact I'll have n-1, theta's to calibrate, plus r-0. It gives me n variables, and I'll be able to do it analytically. Sorry, not analytically, numerically. So again, the key point that I want you to take away here is that there are two possibilities for introducing bias in your tree. One is, choose the probabilities. Pick the probability so that your tree is going to be drifting in a certain direction. So as an example, suppose this probability was 1 was 9 over 10, and this was 6, this was 1 over 10, and this was 9 over 10, and this is 1 over 10, and this is 1 over 10, and this is 9 over 10. Suppose that turned out to be my calibration. What is the typical path look like for this tree? Do you notice that you're kind of getting pushed toward the middle most of the time? The probabilities for going away from the middle are kind of low. Maybe to really make it symmetric, I should make this one a half. And maybe this one too. Then it really looks much closer to that. So you can see, when I'm above, the probability of going up is low. And below, the probability of going up is high. So it's likely that I stay toward the center. This is adding some sort of drift. Alternatively, we can make the tree something like this. Have it equally likely, but make the tree itself represent the tendency of interest rates. Let the tree states represent that. So let's see how we can use this procedure to actually do a calibration. I'm going to describe it sort of algebraically, and then we'll actually implement an example. So P01, what is that? Based on, let me write the model down here one more time. Rn equals Rn minus 1. And maybe I'll put the theta n minus 1 first. It doesn't really matter which order I do it in. I have that. So in other words, we have R0 going to R0 plus theta 0 delta t plus sigma square root delta t minus sigma square root delta t. And the tree recombines, by the way. It's easy to check to see that it recombines theta 0 plus theta 1 to sigma square root delta t minus 2 sigma square root delta t, et cetera. Now I've drawn them as if they're equally spaced and that there is no drift in there, but that's just because it's easier to draw it. So if you're actually to make this correspond to the correct levels as opposed to states, then you'd see the tree drifting according to what theta is. Much like it is, the tree should really look something like what's drawn in blue there, but it's a mess to draw it like that. So I'll keep it flat. Okay, so what's the one-year bond price in terms of a tree like this? It's just 100 over 1 plus R0 as before. So this, I can compute R0. If you tell me market prices, I can compute R0. What about the two-period bond? Well, it's the same expression that we had before. There's nothing special about this tree except its structure. In terms of the discounted expectations, it's identical to before. So I have 1 over 1 plus R0 times 100 over 1 plus RU, which happens to be this big junk of stuff here, and that's Rd. And this happens. All of these branches in risk-neutral probabilities occur with probability a half. So the Q becomes a half. And what will this give me? What's my degree of freedom here? What's my unknown? The only thing that's not known is theta 0. So this gives me theta 0. So in this case, you can algebraically solve it, but there's no particular insight into algebraically solving that equation. So I'll just leave it like this. And then you can imagine P03, well, you're going to get something that's a function of theta 1. That will be the only unknown. Why? Because theta 3 will depend on this interest rate, this interest rate, and this interest rate. We've already fixed, by this point in time, we fixed R0, we fixed this, and we fixed that. So that means we've actually fixed all of those. Everything highlighted in yellow is known. So the only unknown is theta 1. So this would give me a way of computing theta 1, et cetera. If you do this for many time steps, it's tedious to do on paper. On a term test or anything like that, as you can see from last year, I don't ask you to do many steps. In fact, just one. One, maybe two. At most is how I might ask. But I'd like to actually show you how this works in practice. So I'd like to do this kind of procedure in Excel for you, unless the class would rather not see that. Would you like to see an example in Excel? Yep. Okay. So let's go through. By the way, I will post this as well for you to save. Let's go through the process. So we're going to have a collection of interest rates. We're going to have an interest rate tree here. We need our zero is going to be one of our degrees of freedom. And above it, we'll have, let's put time just so that we can see the index for time. We'll put theta here. So this is time zero, one, two, three, four, five. And in terms of what time actually represents here, monthly time steps, why not? Okay. So that's our time step, DT. R zero, we're going to have to try to figure it out. Right? For now, I'm going to just guess it to be 2%. So my interest rate tree, that's R zero, and theta, for now, we'll just start them all out at zero. Okay. We don't know what they are. The whole process will be find the theta's, and the R zero's such that this interest rate tree matches a collection of bond prices. And we'll specify some prices. Okay. In fact, we might as well specify a collection of a term structure of interest rates. So this is R zero. And I'm going to use the same principle as before. Going across the row represents going up in my tree. Okay. So that means take this tree and kind of rotate it 45 degrees. Okay. So this is supposed to be equal to that point plus sigma, oh yeah, we need a sigma here. Let's say interest rates are not terribly volatile, so we're going to use 1%. Sigma times square root of DT. And then we're going to add to it that theta. And when I copy this throughout, I always want to refer to this, to the one theta at a particular time, so I want to pin the row. Okay. So that's how you pin a row, right? C dollar sign. And then the row number. Okay. And I copy that across. So this is what my tree would be for all those upstates. And here as I go diagonally down, that's going down. So that's supposed to be equal to, in the previous level, minus sigma times square root of DT. And again, plus the same theta. Oh, and I didn't, I forgot to multiply by DT above, didn't I? Multiply by DT. So let me just put in multiply by DT. Okay. So this formula here, just so that you match it with what I've got here, it's just that, at the top of the screen. Okay, Rn plus theta, plus sigma squared delta T for all the way along the top branches of the tree. And as I go along the bottom branches of the tree, it's minus sigma squared DT. And this is the bottom branches of my tree. And then we can just copy these cells over. Okay. So just going across is always up. So this is my interest rate tree under this particular set of modeling assumptions. Let's make it, let's actually put interest rates there. Okay. And that's, yeah, that's enough. One decimal is enough. I don't need to see anything more. So of course, if I put some number here, say 0.1, that's going to change the whole tree, right? Because it changes the first two points, which then changes the next, et cetera, et cetera. And if I put 0.2, you know, it gives me a collection of interest rates. So this is my interest rate tree dynamics. And from that, I want to get bond prices. So let's have here the one year, in fact, I'm going to put the label over here. And over here, it's going to be our bond price. So 100, 100. And this has to be equal to, I'm going to just put the generic formula. 1 divided by 1 plus my interest rate times dt. In fact, let's use continuous discounting. Okay. e to the negative of the interest rate times dt times 1 half times what it is in the upstate plus 1 half times the outcome in the downstate. Okay. So all I've done is I'm using that formula there, but I'm explicitly writing the expect, instead of writing 100, I'm computing the expectation of 100, which is trivial in this case, but just so the formula looks symmetric everywhere. Okay. So that's our current price according to this, according to this modeling assumption. Now suppose the market is actually at 99.5. Okay. So this is the market on this column, and in this column will be model. So there's an error over here, which I'll write below it, and that equals the market minus the model, and let's compute a squared error. My degree of freedom is r0. I know I can analytically find it. In this case, it's easy to analytically find it, but we can get Excel to try to figure it out. Or I can just do it by hand. I can just keep raising this until I see that it goes below. So it's 6.5-ish. It's 6.2, 6.1, 6.05, 6.025, 6.0125, 6.0... Now I can't do it. 6725, is that right? Something like that. Okay, so that's kind of close, right? So that's the initial r0. I can put in the explicit formula. It should be 100 divided by this minus 1. Or we can just get Excel to do this for us, Solver. We say that error is my target cell, and I want it to minimize that target cell, and I want it to change r0 and solve. And it's found it. So if I percentage, and let's just put, I don't know, four decimals for that first one. So okay, it's kind of close to where I had it. Okay, so the error is really tiny. It's been able to, it can find it exactly more or less. Okay, so the one year is done. Two year. 100, 100, 100. And all I have to do is more or less, well, maybe I'll do the discounted expectation explicit here. So let's write the formula inexplicitly. See, it's in the negative. The corresponding interest rate for that node, times dt, times one-half, times my payoff, plus one-half times the other payoff. Okay, and I copy that to those two cells, copy that there, and that's my model price. Now, again, I'd like, I really wanted to match the market. Let's say that the market is actually 98.5. Actually, let's say it's 99. Let's decrease by half a cent. So what's my degree of freedom now? Theta zero, right? I shouldn't change R zero, because if I change R zero, I will not match the one year price. So I keep R zero fixed, and I change theta zero. So theta zero is too high, it looks like. Is it too high, or it's too low? 0.01. Okay, yeah, it's too high. Okay, so it's probably going to be about that, and let's make this a percentage as well. Okay, again, we can just get Excel to do this for us. Solver, oh, I need to put an error again, right? So we need to compute an error. That's my error from the model in the market. Solver, that's my target now. I'm allowed to minimize theta zero, and I solve. Okay, so that's the exact answer found, or well, more or less exact, right? The error is very tiny. And now we have our interest rate tree consistent to the two-year asset. And now we can do the three-year, four-year, et cetera. Okay, let's, I don't know. How many of these do you really want to see? All right, negative corresponding interest rate in my tree times dt times 0.5 times the payoff plus 0.5 times the other payoff. Okay, that's the market price for this guy. Suppose it's 98.5, play the same game again. Compute the error. Okay, and this time I'll just get Excel to do it outright. Solver, I want to minimize this error here and by changing now theta one. Okay, that's my degree of freedom. Solve. Okay, it found the parameter. Here's 10 to negative 22, extremely small. Not as accurate as these two. These are actually zero according to machine error, basically. Actually, even smaller than machine error, zero according to whatever Excel rounds to zero. Well, this one's 10 to negative 22. So there you go. I mean, that's the procedure, right? So, so far we've calibrated our tree out to two, out to the three-year term, which tells us r zero, theta zero, and theta one. And we can continue on. Do you feel, is that good enough example? Yeah, because if the price of the two... But it is different. This is $99 and this is $98.5. The price is different. Yeah, the whole idea is that the one tree has to correctly price all the bonds. So you have to do this in a consistent manner. You do it in like a bootstrapping manner. You know, much like when you want to do valuation for a coupon-bearing bond, what do you do? You start with the shortest-term bond. You find the interest right there. Then you go to the second-year bond. You keep the first-year constant. You find the two-year. You go to the three-year. Keep the first and second-year constant. Find the three-year, et cetera. You bootstrap forward. You don't change what you've already found because then you will miscalibrate. You calibrate from the shortest-term out to the longest-term. Actually, and this shouldn't say year. This is one month, two months, three months, because my DT is a month. Okay, so here, yeah. Okay, yeah, so that's what we're going to talk about next. Now we've got a model that is consistently calibrated with the current prices of bonds. It behaves in the historical manner. It has the correct volatility. So what's your next object then? What do you want to do? What's the usual question? How do you value an option on this thing? Right? That's the natural next question. So suppose now I want to calculate an option on the bond. An option on, let's say we want to calculate the price of a one-year option on the three-year bond as an example, okay? Three-year bond with a strike equal to, let me see what's a reasonable strike. Volatility is so low here. It's, yeah, let's say 99.98, oh my gosh. 99.99 with a strike of 99.99. Okay, call option on three-year bond with that strike. Sorry, three period, three months. I keep mixing up year and month here. With that strike. And the maturity of this call option is in one month. Okay, so how would I do that? Well, this is now my underlying asset. This is my underlying asset. All I need to do is find what does my call option pay on that underlying asset? Since the maturity of the call is in one year, in one month, sorry, I need to put a pay off here. And that's supposed to be the maximum of whatever this asset pays minus 99.99 and zero. Oh, shit. I'm going to get zero in both places, right? Let's say 98.99. Okay, 98.99. So these are my payoffs for the two outcomes. And what's the value of the option? Well, same principles as before. Discounted expectation under the risk-neutral measure. What is the expectation under? What is the discount factor? E to the negative. We have to use a corresponding point in my interest rate tree and times a half times the two outcomes. So the principle stays the same. Okay, I calculate the three-year bond prices at those two nodes. And then this gives me my call option pay off, which is P1U, three years minus K plus and P1D, the three-year minus K plus. And then I calculate my C zero just by the usual discounted expectation of C1. So that's all I've implemented. Nothing more. There's sort of nothing new here fundamentally. Let's try this with the, instead let's do the call on the T equals, on the three-year, three maturity bond with the same strike. But the maturity of the option is at two months. So how about call K equals 98.99. Maturity of the call is two years and the bond maturity is three years. Or two months and three months. Goodness gracious, I keep saying months, years instead of months. Okay, so how would I do that? Well, again, same principle, right? The maximum of this, 98.99 and zero. And in this case I get actual payments in every state of the world, okay? How would I compute the value of this node? It's the exponential, negative. I have to use a corresponding interest rate. I have to use the interest rate from that interest rate tree, which is not R zero. It's whatever we found from that calibration procedure. Times DT times 0.5 times my two outcomes. And I just repeat, whoops, and I just repeat that formula, okay? So you notice that here, at that node, the interest rate that's being used is 6%. At this node, the interest rate that's being used for discounting is 5.8%. Because that's the corresponding node in my tree. This node, the interest rate that's being used is 6.3%. So this is how the discounted expectation works exactly as before over every little time step and in every part of your tree. But you can't take multiple steps at once. You can't just forget the fact that the interest rate is dynamic. So even though this is my underlying asset price here, I still have to go back to the interest rate tree to give me information about how to discount. Okay? Questions? Yeah? Second period? So this one here? The first period. This is zero, this is now. This is now. And you're being told the market price is at these levels here? That's the observed. And so you need to make sure your model replicates the observed. No, it's not a two-month bond. It's still, it's the price of the three-month bond. Well, okay, okay. So it's the price of the bond that matures at T equals 3, from there forward is two months from now. That's correct. Yeah, and these are the two possible prices that the bond can take in your model. Yes, that's right. So when the wording is like I said here, it's the call option on the three-month bond. So that means it's the three-month bond viewed from today. So I wouldn't try to trick you like this in the wording. I don't do tricks like that. Word play is just silly. It's more about the concepts. So yes, the call on the three-month bond is the call for the object that will deliver in three months from zero. Just like over here. Okay, so there's, again, you could ask any of the interest rate problem, any of the options that we talked about before, we could talk about it in the context of these bond options as well. I can get you to do a barrier option here. Your underlying asset happens to be, the underlying asset tree happens to be this, and when you do the discounting as before, the discounted expectation over each node, you just have to use the appropriate tree up there, the appropriate discount rate in that tree. Any other questions? No? I guess this is a good point to give you your quiz. Okay, so the idea is you're just basically supposed to take discounted expectations in order for you to get the bond price, but don't forget to incorporate the coupon. So this equals exponential, negative, the corresponding interest rate times dt, times 0.5 times up value, times 0.5 times the down value, and you need to add in the coupon, plus 5. You immediately get the coupon. So if you use discrete discounting instead of continuous, that's fine, but if you didn't include the factor of dt, that's a little bit of a mistake, because you didn't account for the fact that you're analyzing. Okay, so this gives me these two values in the two states, and again, I can repeat that calculation here, except like I said, when you buy the bond, you're not going to immediately get a coupon. That's kind of silly, it's ridiculous to think of a contract of that kind. There is no coupon here, so there's no plus 5, just discount it. So you all should have gotten something like this, 113.35 or maybe 113.4. Okay, now what about the call option? How can you be handed it in late? Well, okay, I have to deduct something for that. We can't keep allowing that. Okay, for the call option, that equals maximum of the value of the coupon-bearing bond minus the strike, which is 113.910. Pays nothing there, it pays about 7 cents there, and again, we just do the discounted expectation times dt, times 0.5, times whatever it is there, plus 0.5 times whatever it is there, and we get about 3.5 cents. Okay, maybe 3.5 cents, because if you did discounting in a different way, and this is using the exact numbers, so the numbers might be a little bit off, but I want to see that approach. Okay, so next week, sorry, just a second before you all scatter out, next week, remember, it is your term test. Come here on time, okay? We start sharp. Ten after two, sharp. You're late, too bad. And I will hold additional office hours. Hold on a second, okay? I'm going to hold additional office hours on Monday, okay, so that you have some time to ask questions before the test. What about Friday? Does Friday work for, would people like to have some office hours on Friday as well?