 All right, so today is our 10th lecture. We've got three including this one, two more and two more excluding this one. And today we're going to cover one of one of the most fundamental results shall we say in derivative pricing theory in continuous time. And we're connecting back now from the abstract material that we've been studying so far. We're coming back to finance and what we'll be doing by the end of today or by the end of actually this first lecture is deriving something called the Black-Scholes partial differential equation. And once we've derived this equation I'm going to show you eventually how practitioners actually use one of the key results from this equation to dynamically what's called dynamically hedge options in continuous time. And we'll look at some simulations and so on. Okay, so what is the key idea in the Black-Scholes partial differential equation for? It's basically for the valuation of contingent claims in continuous time directly. So if you remember the big framework of how we've presented this course, we started off in the first few weeks looking at discrete time models or in the first lecture looking at the discrete time model and then eventually took a continuous time limit and got a distribution of an asset price at a fixed point in time. So we looked at distributional properties of the asset price dynamics. And then once we've got that we went on and we did some valuation for options by computing expectations and we realized that if we were to do that we have to introduce a so-called risk-neutral measure in order to compute expectations. And we looked at interest rate models in discrete time. We looked perhaps I think at the continuous time limit of one of them. And then after all of that discrete time material we went on to continuous time modeling but without finance really in mind. We were just looking at how do you model stochastic processes in continuous time using Brownian motions as an underlying driving source. And so we talked about this idea of first what is a Brownian motion and then took Brownian motion and derived new processes from that and those were what we called the Ito processes and defined, required us to define the so-called stochastic integral. And that led to the concept of how do you change variables via Ito's lemma and as well another key concept mathematically, Ito's isometry. So that's where we are at this stage. Now I'm going to use that continuous time tool, those continuous time tools and connect it back directly to the financial question that we basically answered in the first lecture in discrete time but now directly in continuous. That's where we're going. So the idea is that we want to value a contingent claim directly in continuous time. So the framework that we're going to use is the Black-Scholes model. So remember, by the way, do you remember what is the difference between the Black-Scholes model and log normal distribution? Are they the same thing? They're not, okay? Remember keep that in mind. The Black-Scholes model is not equivalent to log normal. They're not interchangeable statements. The Black-Scholes model is a dynamical model that tells you how asset prices move. A log normal distribution, on the other hand, is simply a distributional property at a fixed point in time. So whenever you say the Black-Scholes model, what you really mean is this underlying assumption on the stochastic dynamics of the stock price. And we've introduced this model before where the asset price, the return, instantaneously has an expected return of mu and has a low volatility of sigma. And as well, to complete the model, we really need to say something about the interest rates. And for now, we'll just assume the interest rates are constant, okay? So this is the Black-Scholes model. The consequence of that model is at a fixed point in time, asset prices are log normal distribution. So the log normal is a consequence of Black-Scholes. And we saw already using Ido's lemma that you could write ST directly in terms of the underlying Brownian motion by solving the stochastic differential equation. And we found that little solution there. And that solution has one surprising component to it. It's the minus a half sigma squared T term. If you did standard calculus, you wouldn't have it. And we need stochastic calculus in order for us to figure out that there's that extra correction term. Okay? And like I said already, this is in distribution, you can write this as mu minus a half sigma squared T plus sigma square root T times the standard normal, okay? And Z here is under the real world measure by the way, I should be a little bit more specific here and tell you that this is a P Brownian motion. So it's a Brownian motion under historical probabilities, okay? So we know that in distribution, you have a distributional property in the second line and an actual path-wise property in the first line. There was a question to me in office hours about the difference between these two and somehow maybe I'm not convinced that the difference is clear to everyone. So what I'm going to try to do here is remind you of how these two things are different. Suppose you were to draw that sample path as a function of T, what would it look like? Well, we all know that you get these kinds of diffusive shape curves that look like that, right? And this is a path-wise behavior. If, on the other hand, you drew this, what would it look like as a function of T? So first of all, I have to pick the random number Z, right? There's a random number in there. So let's say we went to our random number generator, our normally distributed random number generator, picked a number, this would be fixed, and then T would vary and this here is just an exponentially growing factor. And what we would likely get is something that looks like that. If Z was negative, then you might get something that looks like that. Right? If Z was actually the curvature would be the other way. Z were negative, it would look like this. So clearly the red curves look nothing like the blue curves. There are very different objects. You cannot ever say that this equals that. It is not equal. It is only equal in distribution. And again, what does equal in distribution mean? What it means is if I take a fixed point in time, T, and I ask myself, and I say here's a random variable, so that little purple dot, there's a random variable representing the asset price at time T, the distribution of that asset price at time T is given by the expression in the second line on the right-hand side. Okay? That's the difference. One is a fixed random variable, or one is just simply a random variable while the other is an entire path. Random variables Z are not stochastic processes. That's simply a random variable. While WT is an entire, is a stochastic process. It's a collection of n-threads list. Okay? That's the distinction between the two. Now what we're gonna do is we're gonna base our argument on trading these two assets and trading them in such a way that we replicate the value of a contingent claim. Remember in discrete time, that's what we did. Let me just remind you of how we were able to value and why did we come up with this risk-neutral measure and why are prices given by an expectation under the risk-neutral measure. So you might remember in discrete time setting, these are our, this is our risky asset and our risk-free asset. And what we wanted to do was value some contingent claim which paid off these two values. And I could always view this as a function of SU and this is a function of SD. All right? There's no loss in generality by doing that. And what we did in order to figure out how to value this claim is we said, let's choose a position, alpha units in this claim, beta units, sorry, alpha units in the risky asset, beta units of the risk-free asset. That portfolio has a position of value on the right-hand side here. And all we simply do is make that portfolio equal in value to our contingent claim. And once it equal value in value of the contingent claim, then there's a no arbitrage argument, right? We have to say buy no arbitrage then. This has to be the value, the value of the claim must be the value of the replicating portfolio. Because if it wasn't, I can just buy the thing that's cheap, sell the thing that's expensive, and eventually I will owe nothing. I would have gained some money now. I would have owed nothing. And that is a financial interpretation of what's an arbitrage. To convert it to a strict arbitrage, I have to push the money that I have right now forward in time so that I actually start with nothing and I gain something. But at the end of the day, by no arbitrage, we must have that relationship. And this relationship, once we rewrote this a little bit, once we said, okay, let's take, look at the little bit of algebra that goes in there, we realize that that algebra had the interpretation or could be rewritten, let's say, as an expectation under the risk-neutral measure of the claim at time one. We were able to do that just by straightforward algebra. So the whole finance component is really in the first line, the first step, set up the replicating portfolio to replicate your claim. Therefore, the value now is the value of the replicating portfolio. And then the second line was a mathematical equivalence between the expression of the replicating portfolio and an interpretation in terms of probabilities, okay? So what we'd like to do now is essentially the same kind of argument, but in continuous time. So here's how the argument goes. So I'm gonna slightly change the order in which something is down here, not really the order, but change the view rather than saying replicate C by trading alpha and beta. Instead what I'm gonna do is I'm gonna put all three of them together. So I'm gonna put alpha units of the ASC, the risky asset, beta units of the money market, and I'm gonna sell the claim, C. So what should happen here is I should end up with a portfolio that ends up with zero, zero. All right, you can see if I just change the equality to a minus sign, I'm equal zero. So it's a slight change, but not significant. And the reason that I do this is just that in continuous time, the argument is a little more solid when you work through it in this direction. So first thing we're gonna do is we're gonna set up a portfolio or strategy, let's call it a strategy because it's actually gonna be dynamic, where we have alpha T units of the risky asset, beta T units of the money market account, and minus one units of the claim, okay? So what's the value of that strategy at any point in time? It's alpha times S plus beta times the money market account, minus the value of the claim. So let's just give this claim a label, let's call it GT, so its value is given by that. Okay, now what we're gonna do as a sub stage here is we're gonna make that value of this, make the strategy start with zero dollars. So basically we're gonna try our best to create an arbitrage and then we're gonna say at one point in order for us to avoid arbitrage, we'll have a consequence. And that consequence is gonna be something fairly non-trivial but that will be the idea. So I'm gonna set it up to initially cost nothing. Okay, so that means I can create this portfolio with zero dollars in my hand. I may have to borrow some from the money market in order to pay for this but that's okay, it's incorporated in that equation, beta maybe negative. Okay, now what I'm gonna do, so given that we've started there, let's investigate how that price changes, how the value of the portfolio changes over an instant. The dynamics of the value function. Now here's where the little bit of finance comes into play. If I just take the differential of that whole thing on the right hand side, I should have the D of this product, the D of this product and the D of G. The differential operator, it's a linear operator, it'll lock linearly on any linear combinations. And if I was using standard calculus, what would my next line be? I would have D alpha S plus alpha D S, correct? Plus D beta M plus beta D M, this is standard 11, it's a product of two functions, minus D G. There's not much I can do yet with the G, eventually we're gonna do something with it but for now, just that. Okay, so that's what you'd have from standard calculus. Now stochastic calculus, we know that S is a risky asset, it's a stochastic thing. It's driven by a Brownian motion. Alpha is our strategy that we're trying to figure out and in principle it has to be stochastic as well. So we have to modify this to account for the stochasticity in alpha and in S in principle. So that means there's actually one extra term here and the extra term, if you work out just Ito's lemma on the product of alpha times S, what you'll find is that you could rewrite that as the D of the quadratic of the, sorry, covariation of the process alpha with S. Remember I defined for you the concept of covariation. Covariation, this is a reminder, the covariation of two stochastic processes is by definition, the limit. You put a partition down, you take the limit in which the partition goes down to zero. You sum over the increment for the X process times the increment for the Y process. So it's exactly the same idea as quadratic variation except you're taking two different processes and not absolute values, it's just the process itself. Okay, that's just a reminder. So you have a term that looks like that. As well, data in principle is a stochastic process as well. So there should be this extra covariation term. Think of this as the Ito correction term. Think of it as the Ito correction term that you have. Now, if you look at this expression, three terms have some sound financial sense to it and sound financial interpretation. This one, this one and this one. They actually have a nice interpretation financial age. These, on the other hand, don't. I'm going to color them separately and eventually what I'm going to do is, in fact, I'm going to get rid of them via financial arguments. So if you were to interpret, sorry, it's my mistake. I highlighted the wrong ones, darn it. I meant to write it with the other term, no, write it. Now it doesn't line up nicely. I wanted it to be lined up nicely. I want those three terms, okay? So the terms that are underlying are the terms that I want to keep and they have a nice financial interpretation. If you look at it financially, let's take the first one that I've underlined, alpha ds. What does that mean? It means that one instant in time I'm holding alpha units of the asset. That's currently worth alpha times s now. Let's see if we can think of this term financially. Basically it's alpha t times the increment in s, isn't it? And what does that increment in s, what does this financially represent? It represents that you're holding, you buy an asset now, you hold it for an instant and then you sell it, right? And the amount that you hold is alpha. So this is really just the change in the portfolio value via induced by the changes in the asset price itself. When you throw away the other remaining terms there, that equality is called a self-financing equality. So if you write down just that dv equals you get rid of all the other terms and just keep those three terms. This is called a self-financing constraint. And what it represents is the fact that the change, the change in your portfolios value is due only to the change in the underlying asset values and not due to instantaneous changes in your position. The other terms, the other four terms that are highlighted in green there, those terms are sort of awkward and strange. They're because of the weird behavior of stochastic processes. Effectively that's why they're there. And all you're doing is, again financially you can think of it as simply you're holding a position now over a little incident time and you're looking at how much the portfolio has changed in value. And that little incident time you don't change the positions. You only change them at the end. Okay? Just like what you did in discrete time. Let's take a look at this little example here. If you hold alpha units of s and beta units of the money market count over the little incident time, what's the change in the portfolio value? It's alpha times s1 minus s0 plus beta e to the r delta minus one. Right? This is alpha ds plus beta dm. There are no other funny terms in there, right? There is no term corresponding to sd alpha, okay? There is no term like this. There is no term like md beta when you look at the change in the portfolio value before and afterwards here. So the self financing constraint is inducing a financial meeting into what the strategy is. You could even think of it as a constraint if you like on the system. You're constraining your trading strategies such that they are self financing. Okay? Are there any questions about that? Because it's an important point. And it really simplifies our work as well. Not only is it a nice financial piece of information that goes in, but it also simplifies our work. Here's why. Let's now just go ahead and use the expression for what ds actually is equal to. Well, remember that ds is s times mu dt plus s times sigma dwt. What is dm? dm is m times r dt and there's nothing else. And then the last line, dg. What can we do with dg? We don't know what the claim value actually is. That's what we're trying to find. But I can make an assumption and I can assume that the value of the claim is or at least we can hope that we can assume that the value of the claim is and we'll see if it's self consistent. And if it is, then we're happy. And if it's not, then our assumption has to change. I could assume that it's simply some function of two arguments, time and the asset price. And that kind of makes sense, right? If you think of an option, what does it depend on? It depends on what time you're at. Certainly depends on parameters, but in terms of dynamical variables, it's going to depend on what time you're at and what the current level of the asset price is. So it depends on those two things. And furthermore, I'm going to assume that this is once differentiable in T and twice differentiable in S. And you can see why, I hope. If you recall what we did last lecture about Itto's Lama, there was that requirement. So if I then go ahead and just apply Itto's Lama to the d of g, what will I get here? You have to write down all the entire Itto's Lama in order for you to get this. So let's just go ahead and write. I'm just going to shorthand notation it like this. Partial derivative of g with respect of g with respect to t plus the drift of S, which in this case is mu times S. Partial derivative with respect to S plus one half the volatility of S, two derivatives with respect to g. That's my dt term. And then I've got the volatility of S, partial derivative of g with respect to S, dw. This is straight from Itto's Lama. Everything in the square bracket there, this is Itto's Lama on g, okay? Remember, you don't even have to memorize this formula. I'm going to be giving it to you on the final exam. Okay, so now that we're at this stage, let's remind ourselves what we've done. We've taken a portfolio, we've taken a strategy, we've started off at zero, we've made it self-financing, and now we're at this stage. Let's collect a couple of the terms because there's a couple of terms here that are similar in nature, and I want to underline them, so let's pick. So there's terms that have dT's in them. Okay, there are these, in fact, I'll just underline the dT. There's these dT terms, and then there are these dw terms. Right? So there are those two terms. What does the dT term, what does the dw term represent? It represents the uncertainty, right? Represents the noise, the fact that you don't know what the asset price will do over the next instance in time. If you did know the function g, then the dT terms tell you essentially where the drift is, right? Where it's going to tend toward, and the dW tells you the fluctuations around that, right? That was the interpretation that we had for an Ito process. The coefficients of dT tells you where the process is tending toward, and the coefficient of dW tells you the size of the fluctuations. So if I drew this sort of pictorially over an instant in time where some are there, like this, dT kind of tells me where I'm tending towards, and the coefficient of the omega tells me the size of that distribution. And since these are Ito processes, if I have the information of time t, then I actually know everything about the coefficient of dT, and I know everything about the coefficient of dW. I actually know it. What I don't know is what random number is going to get drawn over that little instant in time, and therefore where on this little Gaussian I will end up. I don't know that. But I do know the variance of that Gaussian, and I do know the drift. I do know it's mean. Okay? So here's the little, here's the insight. The insight is, suppose you made the dW coefficient zero. Remember, you have a choice. You are allowed, you have basically three degrees of freedom here. Alpha, positions in units and stock, beta, what's in the money market account, and what's your third one? It's not obvious. G, the claim price itself, because you're trying to find that. So far we've put one constraint on the system, self-finance. Still have two degrees, two degrees of freedom. Think of it as, you know, there are three variables who fixed some combination of them, only one combination of them. Now we're going to fix a second combination somehow. And we're going to fix it to remove all risk. That is, make the dW coefficient be identically zero. What is that the analog of in the discrete time? So let me just write that down first of all. So we're going to make, let me put it in quotes, the instantaneous volatility equals zero. I.e, remove the risk, remove local risks. Okay, why do I say local risk? What I mean by local risk is you can't remove the risk off of the entire thing all at once, because I'm sitting here at current point in time. And all I know is where the process is going to go and it's volatility here. So if I can reduce that volatility to nothing, I can know exactly where it's going to go to, right? And that's local. I'm not trying to, I'm not being able to fix where it goes to eventually, but just local. It's kind of like when you're driving a car, right? You just locally fix where it's pointing. But here you're actually locally fixing the risk. Okay, what is it? I was trying to say, I wanted to connect this back to the discrete time setting. Let's just go back again. The dW coefficient is basically the same thing as whether the asset goes up or down, okay? dW is telling me go up or go down, okay? Making the coefficient of the dW term zero is equivalent to making the contingent claim have the same value, sorry, making the replicating portfolio have the same value as the contingent claim. It says no matter where you go, the fluctuations are zero. The difference between your traded portfolio and the thing that you're hedging, that you're replicating or hedging, have exactly the same variability. You're making the fluctuations in your portfolio fluctuate in tandem with the contingent claim rather than slightly mismatched, okay? That's the picture. No matter where you end up, you're gonna replicate the claim. This is the idea of replication or the analog of replication in discrete time. Okay, so if you make, how can I make the coefficient of dW equal to zero? What's the consequence of that? You can see there's only one variable there. There's the alpha sitting out front. Alpha times s times sigma must be sigma times s times the partial derivative of g. Therefore, alpha, and there's a minus sign between the two things. So to make the dW term go away, I must choose alpha to be the partial derivative of g with respect to s. You see that? So I choose alpha to be this partial derivative. Very important quantity, and we're gonna have to a lot of interpretation around that eventually. For now, it's just a mathematical formula. If we choose that, then you can see that, there you go, that's dV there. Then you can see that dV has no dW term whatsoever. It only has terms in dT. So let's go ahead and just write down that big mess. It's a little bit of a mess, but it's not too bad. So we've got mu s times alpha, we'll put the dT outside, plus r times m times data minus, and then here's the, all of the dT coefficient of the Ito's lemma basically. It shows up there. Minus partial with respect to g plus mu times s partial s with respect to t plus one half sigma squared s squared, two derivatives g, and all of that is times dT, plus zero dW. I'll explicitly put in dW and put the zero there so that you realize by this choice we've made that coefficient zero. Okay, does anyone notice another cancellation? There's something that cancels here. Alpha is that expression, partial with respect to s of g. There's also partial with respect to s of g here. The coefficient is exactly mu times s. There's a minus sign. Those two terms actually exactly cancel each other. So this can be simplified further. This is just r times m data, minus dT plus zero, okay? Now, there's something, oh, there's a minus here. Yeah, there are both minuses, it's all one bracket. Okay, now here's one more bit of finance coming into the picture. This coefficient here, this is some horrible function, right? It's something messy, but a time little t, you know that, don't you? You know what it is. It's measurable, you can observe it at time little t. So what does it say? This equation is saying that my portfolio that I've just set up instantaneously grows at whatever rate I've underlined. Guaranteed, there is no uncertainty. That's why this coefficient t in zero is important. I'm guaranteed this drift, right? It's a guaranteed drift. Now if it's a guaranteed drift and my portfolio started with zero, what would this correspond to? No, well, it's not quite yet, but suppose this number, you know, I compute this number and I get three, okay? Started portfolio with nothing. I'm guaranteed three times dT, right? Instantaneous little, little dT is really tiny. It's an arbitrage, right? It would be an arbitrage, and if it was negative four, it's also an arbitrage. I just take the short position of everything and I create an arbitrage. So this guaranteed drift, if it's positive or it's negative, there is an arbitrage in the market. The only thing that it can possibly be is zero to avoid arbitrage, okay? So the argument goes, the logical argument goes like this. Since V zero equals zero and dVT equals a bunch of junk dT plus zero dW, then to avoid arbitrage, we must have whatever is in front of the dT being equal to zero as well. It's kind of odd, isn't it? The only constraint that we put was we made the risk go away. By making the risk go away, it also forces us to make the drift go away. Arbitrage forces us to do that. That's kind of nice. And it's only true because V zero was zero. So you see how setting up the initial V zero to V zero is important, okay? So you must have that coefficient of V zero. Therefore, and this has to be true regardless of all paths of the asset price. Doesn't matter what path the asset price took. It has to be true at every point. Okay, there's another consequence of this. Can someone else give me another implied consequence? So we've got V zero. So let's, so we have V zero equals zero we have V zero equals zero and dV equals zero. Therefore, V is in fact always zero. It's not just zero at time, it's got to be zero everywhere. Otherwise, there's an arbitrage again, right? That's an important, that's a powerful tool because remember what I said, we had three degrees of freedom, right? Self-financing fixes one combination of those things. Choosing alpha, choosing the coefficients to make the dW coefficient go away, that's a second constraint. We need a third one in order for us to actually get everything to work out, right? For us to have at least the same number of constraints as we have degrees of freedom. This is our third one, V always is zero. Remember, we started it off not making that assumption at all, that wasn't an assumption. We simply said V is the value of the portfolio, alpha, beta, minus one. Now we realize in fact it has to be zero as well once we've enforced the other things. So what's the consequence of this? Therefore, let's just write down what is VT? VT was equal to alpha times S plus beta times N minus G. This has to be zero. So let's use this equation to tell me what beta is. We've basically already found alpha, we'll use this to tell me beta. And in fact it's always convenient not just to find beta but to find beta times N. So beta times N is G minus alpha times S. But what is alpha? Now I'm going to insert that expression into this requirement. Insert this into there. So remember, what's in the round bracket with the dot in the middle? It's everything that I've underlined, the guaranteed drift. The guaranteed drift is to what's showing up there. So to enforce the quality of zero that's in the purple square, all we need to do is write down what's in the round bracket above and we can see the consequence quite easily. So notice the first term there, it's R times N beta. That's why I said it's better to just solve for M times beta. So we'll get R times G minus S partial S G. Subtract partial T G. Subtract one half sigma squared S squared to derivatives of G. And that has to be zero. So this is what I get after inserting that expression for beta into the boxed expression above. Now I just do a little bit of simplification. Okay, first of all, let's just do a little bit of a simplification and let's just do a little bit and we've got a bunch of terms with minus signs. Let's put those all on one side of the equation and the term with a plus sign, leave it on its own side. Okay, so we've got that equation. Now here's one more bit of mathematical argument. It's not a very sophisticated one, but it's a simple one. This has to hold for every T and ST, right? It has to hold no matter where I am in that path for S. So in fact, I can actually replace all of the dynamical variables, all of the stochastic processes. I can just remove them and just replace them by a real variable. So in other words, this equation is in fact a partial differential equation in the half plane. So this tells me that as a function only of its arguments, so g, t, s plus r, s, partial g, t, s, this is not a process here. These are simply describing for you a partial differential equation of a usual function. It's r, g, t, s and this has to hold on the entire r plus cross r plus. R plus, okay, actually the quarter plane, because s is strictly positive, right? It's a strictly positive thing. So this is t, s. That PDE has to be satisfied in this entire domain almost. There's one critical maturity, one critical point, the maturity of our contingent claim on which the PDE cannot or won't be satisfied. Instead, that puts our boundary condition. We know that g has to be equal to something at maturity. We know it's value. We know that g at t, s has to be the payoff. That's the options payoff. So what we find is that the PDE has to be satisfied there in that domain and at the maturity the payoff of the claim shows up there. Now I'm gonna try to emphasize this difference again between the half and, okay, so that there is a sample path and this PDE here is representing the various partial derivatives at each point on that sample path. But that equation has to hold for all paths, for all outcomes of the asset. So that's why you can go away from the PDE which talks about the process to one that just simply talks about a function. That's the difference between those two things. One is describing along the path itself or along a given path while the second one, which is what I'm boxing in now, is describing the entire strip. So let me change, so I modified this, right? I said, well, there's actually a maturity date here. So this is valid on zero T cross R plus. This equation and that partial differential equation, this is the block shows partial differential and black and shows won the Nobel Prize not for that model, not for the fact that they said, let's assume that the stock price is a geometric Brownian motion, ds over s equals mu dt plus sigma dw. That is not the Nobel Prize winning result. That's a kind of, just a model, it's a basic model. What they did win it for or, and neither did they win it for the fact that the consequence of this is log normal that was well known by much more complicated stochastic differential equations were known. But not that, what they won the Nobel Prize for is an argument not identical to what I've outlined here, but similar, and their argument has a few holes in it. This is solid. They've mathematically had a few holes in it. So they wanted for this idea, this basic idea here, that's the one. Instantaneously make the portfolio riskless. Remove all risk in the portfolio by trading in the underlier allows you to value the claim. That is a Nobel Prize winning result and insight. The fact that you can somehow trade in underlying assets and value a claim, a contingent claim that's written on those underlying assets. And the way you trade them is that you hold enough positions of the underlier so that the fluctuations in the underlier match the fluctuations of your claim. That's the key result. That's the real genius, financial genius that went into it. And it really is. The way that you all learn this stuff nowadays when you start with the discrete, and the way I learned it in fact as well, starting the discrete time model and then there it's completely obvious in some sense. When you write this down, this tree, and you say, okay, let's replicate it, therefore it has to be that value, otherwise you have an ARP, it makes complete sense. But this was not known at the time. People didn't think about a discrete time model like this at all. The discrete time model was even introduced only, I don't know how many years later, it was like five or six years afterwards. All of this risk neutral measures and all of those things that were developed afterwards. The PDE via this dynamic hedging argument. So that's what this is also called, sometimes called the dynamic hedging. So let me put this terminology down here. The dynamic hedging argument is what came up first. And the result is this partial differential equation. And then there's still work to be done, right? Because what we have at this stage is you now know that the value of a contingent claim must satisfy that partial differential equation. And you now have a mathematical problem. How do you solve that partial differential equation? There's no longer a financial question anymore. Okay. Okay, questions about this methodology? Would you like me to outline again the key steps that we went through? So let's slide back to the start. So the key steps are, we set up a strategy that consists of trading in the risky asset, the risk-free asset, and shorting the contingent claim. That's our strategy. We also make sure that we start with nothing. Then we impose, we look at the dynamics of that portfolio value. We impose a self-financing constraint so that the increments in the portfolio value is given solely by the changes in the value of the assets that you're holding. That's your self-financing. Then you locally remove the risk and you get rid of the volatility. You remove the DW term. Consequence of that is that the portfolio value has a deterministic or instantaneously deterministic, a guaranteed drift over an instant. You don't know what it would be right at the start, but once you're here, you know what it will be over a little instant. And if that return isn't zero, there is an arbitrage, okay? That's a very important key insight. Given that you now know that the instantaneous drift must be zero and you start it with nothing, you always have nothing. That's what this equation is telling us. We plug that in, algebraically simplify. We end up with partial differential equation along the path of the asset price. But this has to hold for every path. So in fact, you end up with a partial differential equation in the entire strip. Know this procedure well. It's very important, fundamental, as I said at the start of the lecture, key concept in derivative pricing. Okay, so maybe we take a little break and when we get back, I'm going to show you a few examples of solutions of this. Then I'm going to give you a general solution of how a general way that you can solve this using stochastic representation. And we'll talk about some more surprising features of it and then we'll do some computer implementations, okay? All right, so welcome back. So before going on to some examples, I want to point out a couple of interesting things about the equation. Actually, rather than point it out, maybe I can get you to suggest some interesting things. Do you notice anything peculiar about this equation? What was the original assumption on the stock price dynamics? So let me remind you, under the Black-Scholes assumption, we had this, right? Yeah, exactly. This mu is completely absent from that partial differential equation. Doesn't show up. In fact, in one of these, there was sort of a cancellation that I mentioned, right? One point, I'm trying to remember where it is now. Yeah, it was basically right there and going from the top equality to this equality here, we saw that this term and this term actually exactly canceled one another and those were the terms of mu. So anyway, that's a mathematical result, right? That they canceled. Financially, that has an interesting interpretation. So financially, it says that although the asset price has some drift in it, some drift mu, the contingent claim, on the other hand, cannot depend on what that return actually is. So no matter how strong and how much of a positive growth that contingent claim may have, I'm sorry, the underlying asset may have, the contingent claim cannot depend on that drift. So even if you get it wrong, the contingent claim can still be priced correctly. And the reason is essentially the fact that you're making the locally the risk go away. So it doesn't matter what the tendency, doesn't matter what the drift of the underlying asset is, all that matters is that you matched the volatility. And that automatically takes gets rid of the drift. So you locally match when you get rid of the drift. You might remember in discrete time, there's a strong connection between the branching probability P and the drift mu. You might recall in, for example, in the C-R-R model, it depends on specifically what model you're using, but in the basic C-R-R model, where S went to S e to the sigma squared delta T, S e to the minus sigma squared delta T. When you calibrate this to real world, made the expected growth of that asset become equal to mu, you might recall that P was equal to one half, one plus mu minus a half sigma squared over sigma squared delta T, right, for small delta T. It had that form, sorry, this form. And that's where mu shows up. But when you calculate risk-neutral probabilities, which is how you compute contingent claim values, under the C-R-R model, again, same model, you end up with an expression of this kind. And instead of mu, there's this R. Mu is no longer in the equation, only R is. So we've actually seen that happen in the discrete time setting before. That the real world return does not show up in the price of the contingent claim. And here it's showing up again directly in the continuous time setting. So that's kind of nice, and that's surprising. But what does show up is the volatility sigma. You might remember in discrete time setting, when we took the continuous time limits, the variance of the process did not change, right? The variance, sorry, the variance of the distribution did not change. They're both log normal with the same variance of sigma squared t, under p and under q. Here, you see the sigma that shows up in our original equations showing up again in the equation there. It's not obvious yet that that sigma corresponds to a volatility of any kind, but you'll see how it does in a minute. Okay, so those are the two, that's basically the interesting thing that I wanted to point out. There's something else that we could do, and that is, this has to be satisfied for all contingent claims, right? Any claim. So is the underlying asset a contingent claim? It is, can be viewed as a contingent claim. I can view a contract in which at capital T, I will give you one unit of the asset. That's a contingent claim whose value has to be equal to the asset right now, right? Agreed? So in fact, the asset itself had better satisfy that black-shell's partial differential equation. So let's check and see if that actually holds. So phi with equals s, this claim is the asset itself, and its value must be st. In other words, the function g t s has to be equal to s. It has to be. So we know that just from pure financial argument, so now let's check, is that actually a solution of the black-shell's partial differential equation? So in order for us to check, in order for us to check that it is a solution of black-shell's pvd, we need to compute the partial derivative of g with respect to t, what is that? Good. G with respect to s is one, and two derivatives is zero, right? We need those three things. What is the black-shell's pvd state? Well, okay, I'm gonna just write it here for you. Okay, we need to check. So this is a question mark here. Is that equal to r times g? And I think it's fairly easy to see it as, right? This is zero, this is one, this is zero. So you just get r times s and g is s. So check, it works. It had better. If it didn't, then I've written something wrong. Okay, the question of how to solve it in general is something I want to address, but before solving the general case, let's take a look at another example. Suppose, let's take a look at, I want a simple example. I don't want anything too complicated. Let's take another example of a digital option. 5s equals indicator that s is bigger than k. So I'm gonna tell you the answer and we're gonna check to see that it's true. Now, before telling you the answer, what do you expect the answer to actually be? What do you expect the value to be equal to? Based on what you've already learned, right? Not based on trying to figure out how to solve the black-shell's PE, but what you've learned in discrete time taking continuous time limits because you know in distribution they match. So in value, they should match. So if you were to do this by distributional calculation, what would you compute? You compute the discounted expectation. Let's say, I'm gonna put a question one equals, okay? We expect the discounted expectation under Q of the indicator that s bigger than T, s of capital T is bigger than k given my initial price, right? And s capital T would be equal in distribution. Because the black-shell's model, you know the solution is geometric Brownian motion, which equals in distribution to s e. And under the measure Q, we have to get rid of mu under place of i r plus sigma. And since we're talking about little t, sorry, should put s little t there. Square T z and z under the measure Q is normal zero one. So we haven't proven that this is a correct result yet. All we know is that if we took the discrete time model and took the continuous time limit, that's the answer that we would naturally write down. Again, so the indicator that s is bigger than t, the expectation, that's simply the same thing as the risk-neutral probability, that s is bigger than k, and that equals the risk-neutral probability. Just take logarithms, right? Put that expression in for s capital T, because it's a probability, I can use a distributional properties. Put it in, take logs, put everything the other side, you get z is bigger than, z is bigger than log k over s minus r minus a half sigma squared, big t minus little t, all over sigma squared, big t minus little t. And in terms of the standard normal CDF, okay, I did make one small typo in this whole thing. I should be discounted the time t. The factor out front shouldn't be just e to the negative r capital T, it should be e to the negative r capital T minus little t. And this is just phi d minus, because that same thing is a phi of negative of what's on the right-hand side. It's the upper teleprobability, so it's the CDF of negative at the lower tail. And d minus is log s over k plus r minus a half sigma squared. Okay, so this is a slightly more complicated example now, but it's gonna illustrate to you how we get the solutions. Okay, so this is our guess. Let's see, does it actually satisfy the Black-Scholes PDE together with the boundary conditions? So partial derivative of g with respect to t, what is that equal to? Well, little t shows up in a number of places, so we've got to be careful. So there's a little t in first of all, in the factor out front. So that's gonna pull on a factor of r times g itself. You agree? When the partial time hits the exponential factor, then I get this factor times phi time, d minus times the partial derivative of d minus with respect to t. And let's just do it all in one. So the partial derivative of d minus with respect to t, there's two terms. So I can think of the log over sigma times the denominator, there's a square root of big t minus little t. So when I take a partial, I'll get a half, and I'll get big t minus little t to the three halves in the denominator. Okay. It'll be minus a half, but there's a minus t there. And then the second term is r minus a half sigma squared all over sigma. And this gives me the square root of big t minus little t in the numerator when I collect it with the other big t minus little t. Take a partial derivative, and I'll have negative one over this to the power half. Okay, do we agree? Yes, no? All right, partial with respect to s. Partial with respect to s. We'll just get the discount factor out front. We'll get five prime d minus times the partial derivative of d minus with respect to s. That one's easier. That's just one over s sigma square root of t. Square root of t minus t minus. Sorry, square root of t minus little t. Okay, two derivatives with respect to s. That's that equal to. So that's e to the minus this guy. Derivative when it hits the five prime. I'll get five double prime, t minus one over s sigma, big t minus little t squared. And then there's a term when the derivative hits the one over s. So that'll be plus minus one over s squared, one over sigma, big t minus little t. Okay, looks a little bit messy, doesn't it? But it's going to simplify. Watch. Okay, so here's our first simplification. First simplification is rewrite the second derivative of the CDF in terms of the first derivative of the CDF. So what do I mean by that? Well, remember, what is pi of anything? That's the integral from minus infinity up to x, e to the minus one half x squared dx over root two pi. So five prime is e to the, oh sorry, that should be say another dummy variable here. Let's call it y. That's e to negative one half x squared over root two pi. That's obvious. That's just the density of a standard normal. And if you take one more derivative, what happens? You get negative x e to the negative one half x squared. But that's just negative x pi prime x. Okay, so that's the first simplification. Because now I'm going to realize that there is no more dependence on the pi prime. Actually, you know what? I don't know if I really want to keep going through with this algebra because time is running short. So let me, do I want to point out what else? Okay, so let me just mention what you need to do and try it on your own. You'll see, it gives you the correct answer. All you need to do is you need to take that first term multiplied by s times r, the second term, right? You have to add, so take this term times s times r. You'll notice there's a cancellation there. The s is canceled, okay? The pi prime is occurring in both of those terms so they can just come out front. Agree? I don't want to write down every single little line here because it's really algebra at this point. Pi prime will come out out front. S times r, you'll cancel one s. This last term is going to be multiplied by one half sigma squared s squared, right? So pi double prime, the way that you're going to write that is you'll have a d minus in there times pi prime, okay? And you'll see that the pi double, the pi prime again comes out out front and the s squared kills this. The d minus is going to be collecting with that term and when you add them together, you'll see that the four terms that remain there all go to zero. All go to, sorry, r time goes back to r times g. Just try it, try it on your own. Don't believe me, in fact. But I don't want to waste your time. It'll take some time to write it out loud. Okay, let me see if there's another example. Problem with the examples for this is they're never very neat unless I give you something that's a bit too... Okay, yeah, let's do this. Let's do one more example and then go on. What about a claim? Which is s to the power one half. So it's a claim that's a square root of s. So what do we expect, again, from what we've known before? Excuse me. This one will be much, much shuff cleaner, okay? So if you put those terms together, you see you've got an s, you've got e to the r minus e to the blah. There's a half there, so it's e to the minus r over two, big t minus little t. So I'm putting this s to the power half, so there's an s to the half out front. I have the minus r over two coming out there as well because there's a factor of r there. There's a minus one half, which becomes a quarter when I raise it to half, sigma squared, big t minus little t. And then we have the expected value of one half of e to the one half sigma squared, big t minus little t times the standard normal. And I made it, not a mistake, but a, don't forget, equal distribution. And if you write it down without the d in there on the exam, you get minus the half. This is not true. It's not, it's only equal in distribution. As well, make sure you specify the objects that you introduce, such as this. Okay, so then we have to multiply by the expected value of e to the one half sigma squared, big t minus little t. So that's supposed to be e to the one half, one half sigma squared, big t minus little t squared. I'm just using the moment generator function. And now if you collect terms, we get e to the minus r over two, big t minus little t. And we have one half squared, which is one quarter times a half, which is one eighth. That's a positive sign, and we're subtracting a quarter. So does that minus, that's what minus one eighth, right? So it becomes minus sigma squared over eight, or I can just collect this all together. Minus r over two over eight times that. Okay, so we can go ahead and try to compute, do the same thing. Partial g with respect to t, what's it gonna be equal to? Minus r over two plus sigma squared over eight. I should go with a plus sign times g, right? Because it's minus two negative signs on t. Partial sg, that's gonna give me s to the minus a half times a half times maybe I wanna call it g divided by s to the half. That's confusing you, let me just put the actual factor. Okay, two derivatives, I'm gonna get one half times minus a half, s to the minus three halves. And again, the same factor. Okay, what's the black-shell equation? What do I need to do? I need to take this. I need to add partial tg, multiply this by s, add that, multiply this by r times s, multiply this by half sigma squared s squared. And hopefully that sums up to become r times g itself. Okay, so there's gonna be an overall factor of g. You can see when you multiply that second line by s, the s to the minus a half changes to s to plus a half, which I can collect back together with the discount factor and that gives me g. This times this is just r times a half times g. Yes, no? Yes or no? This is correct? Okay, good. Then minus, here we've got one quarter because of the minus a quarter there. There's a one over half here that makes it one eighth, sigma squared. And you can see the s squared times s in negative three halves gives me s to the half, which you collect with the discount factor again and that also gives me g. So maybe I shouldn't put this big bracket here. Just leave it there. So it's just that sigma squared g. And what does that simplify to? We have an r over two there, plus that r over two that gives me r times g and the sigma squared over eight is canceled. So, that's it. That's a block shows PD. So that's correct. Whenever you check the block shows equation, you always have to check the boundary condition as well. GTST as T goes to capital T is what? That's the function g there. The little t approaches big t, the exponential factor there goes to zero up in the exponent. So that just becomes s to the one half. And that is my payoff. So therefore, this does satisfy the block shows equation with the correct boundary condition. So it is the correct price. So we can see that our guess based on this expectation result is actually correct. Okay. I hope that's convinced me. Yeah. We don't know because so far in continuous time, what we've proven is that the price of the contingent claim has to satisfy that PDE. That's all we've shown. We have not shown that a solution of that PDE has a representation as an expectation. We've not done that. But what we know from our earlier work is that we expect the prices. We know in the discrete model when we take the limit as the number of steps goes to infinity, we get a log normal distribution and that log normal distribution is identical to the distribution of the black-shoulds model. So we expect that the results we had in discrete time by taking the limit, it should be the same as solving the PDE. We expect that, but we haven't proven that. And so what I'm doing is taking the result that we have from the continuous time limit of the discrete model and then checking, does it actually satisfy our PDE? And the answer is yes. At least for these two examples plus the third one, which you're gonna convince yourself on your own. I think even in the notes, there might be a solution from previous year that has something to do with homework assignment. Okay, so what I would like to do now is not show you the general solution yet. I wanna go back to the PDE and give some interpretation and explain to you how we can use this thing. Okay, so suppose we have a solution to the PDE. Suppose you actually have a solution to the PDE, which means you have a price. And then you ask, then a client of yours comes to you and says, okay, they want to buy an option from you. And that option is something different than what's traded on the market, such as the square root option, something unusual. And then you have to decide whether you wanna sell it to them. And it's suppose you've made the decision already. And you therefore need, since you've sold this option, you want to somehow trade in a self-financing manner in order for you to replicate that option payoff. All right, that's what you want to do. And so how are you gonna do that? Well, this dynamic strategy actually tells you how to do it. It tells you that at each instant in time, you have to hold this many units of the underlying asset. And if you're gonna do it in a self-financing manner, it automatically tells you what you have to put in the bank account. You'll see there's an immediate consequence. And that's what I'm gonna try to describe for you. And we'll implement it and see what the output looks like, okay? So let's go, we're gonna call this dynamic hedging in discrete time. So what do I mean by that? Because as a trader or as somebody who's sold another individual an option, there's no way that you can trade continuously. That's what the Black-Scholes argument would require us to do, right? At each instant in time, instantaneously make your portfolio riskless. But that's practically impossible. So all the best you can do is do it at some frequent times. So maybe every hour, maybe once a day, maybe once a week, maybe once a month, pending on what the underlying object is and depending on how your risk tolerances are and depending on how much you've charged the clients, basically. If you're making a very thin spread, then you better hedge frequently in order for you to match it exactly. If you're making a large spread, then you don't have to match it so exactly. It's kind of like liquidity issues. If something is very liquid, you better be darn right sure that you're getting it well. If it's illiquid, you can just charge a huge premium for the fact that it's not traded often and therefore make money off of that. So this discrete time here does not mean, it does not mean a discrete model. It means that you're using the continuous time, but you're dynamically hedging it at discrete times. Okay? So let's see what we can do. Let's take the example of a call option. I'm gonna write down the problem that we're solving in a second, but take the example of a call option. We know from general theory from discrete time models taking continuous time limits that the call option price has to look like this, right? It's got to look like that. We hope that the block shows PDE, in fact, gives us the same thing. And I'll tell you, it does. Asymptotically, this is gonna approach S minus K e to negative R, 15 minus little t, the number. That's our asymptotic feature. According to our dynamic hedging strategy that we just arrived, we have to hold this many units of the asset to hedge. Correct? In finance, although I derived everything and I call this alpha in the strategy, this here is called the options delta. Now, why are there two different notations here? It's because if I have a portfolio, my position in the underlying asset can be alpha, but the options, the various options that you have can have separate deltas. And the total delta is some sort of linear combination of those things. So I just used the alpha, beta notation also because it carried over from our discrete time models. But this is what is called delta in option pricing parlance. Okay, if you look at that formula and then you look at that graph and you just ask yourself, well, wait a second. This delta came about by doing a dynamic hedging argument by instantaneously removing risk. Yes, I understand that. But if I was a first year student and I asked you, interpret the right hand side, what would you tell me? You tell me that's the slope of this curve. Wouldn't you? At the point in time t and at the point st. In fact, that is exactly the kind of interpretation that I would like to use. So the slope of the tangent to the curve at the point st is what the delta is. And in fact, you can take that interpretation a little bit further. You can say, well, really, all this option pricing local removal of risk means is if I blow up this region and let's look inside that region, all it says is that I'm locally approximating this curve, which is the price of my option. This is my G. Okay. I'm just locally approximated by a straight line. That's the calculus interpretation. Now, what does that mean financially? Straight line with slope equal to delta. So financially, what does it mean to have a straight line with slope delta? It means you're holding delta units of the stock, right? A straight line on that graph. Remember, on this axis here, this is s. Sorry, yeah, that's s. And this is the value, right? Value of the claim, the value of whatever. This is the value of claim and maturity at an earlier time. If I put a straight line on there, going through the origin, if I put a straight line, that's exactly one unit of the, and slope one, that's one unit of the asset. Slope delta, it's delta units of the asset. Slope delta shifted, that's delta units of the asset plus or minus some bond, right? To move me up or down, because if I have a straight line like this, going through the origin, if I move that up or down, all I'm doing is I'm adding the same quantity at every asset price, which corresponds to a bond or the money market account. So in fact, the straight line corresponds to a portfolio consisting of money market and delta units of the stock. So that's what this dynamic hedging consequences. That dynamic hedging argument of locally removing the risk corresponds to locally make your portfolio tangential to the actual price, okay? So that's an important interpretation and we're gonna come back to this picture again later. So let's work on that interpretation and see how can I then create a strategy that's going to somehow help me hedge the option in this discrete time. So what do I do? So let's say we suppose we sold the call option and we're gonna try to hedge it. So let's write down all these different steps. We're gonna hedge it at these equal times that every day, let's say, t equals zero, one, two, three, or t equals zero, delta t, two, delta t, three, delta t, delta t is some time frame. So t equals zero, since I've sold this option, I now have in my hand, how much money? G zero, right? The value of the claim, whatever it is I've sold it off to. So let's write that as I get G zero by selling the claim. Okay, now, how am I gonna go about hedging this? Well, according to the strategy, I need to hold delta zero of the asset, correct? So that means I need to purchase delta zero G of S, cost. How much is that gonna cost me? Delta zero G is zero. So how much money do I have left after this transaction? G zero minus, sorry, yes, minus this, right? That's how much I have left. This may be positive, this may be negative. Where does it come from? Money market account. If it's positive, then you put it into the money market account. If it's negative, you means you had to borrow at the risk-free rate. I suppose this turns out to be three, then that means you got $3 from this transaction. You sold the option for 10, your delta position's costing you $7, your net transaction is $3. Take the $3, put it in the money market. If on the other hand, you sold the option for 10, the delta position cost you $12, you need $2 to be able to buy that delta. I borrow the $2 from the money market account and I buy the asset, okay? That's my initial state. Now we go to the next day. So I'm just gonna index this by integers, okay? You know what, no, maybe let's put it in there. So what happens by the next day? Well, between now, between time zero and maturity, the claim G, you don't do anything with it. That's your obligation. You owe somebody the payoff of the option G. So you don't touch that. But what you can do and what the dynamic strategy tells you to do is keep changing the delta position. All right, that's what it tells you to do. So once you get to time T equals delta T, first of all, what is the value of this? What's the, what happens to the money market account? M zero grows, right? It has grown. It has now become this. It may be larger if it's, sorry, it will always be larger but it may be a larger positive position or a larger negative position depending on whether you borrowed or actually invested. What happens to the underlying asset position? You don't change it, right? You don't touch it over that instant. You just bought the asset at T equals zero and you hold it till now and that doesn't change. But what's the value of that position? Next position, that's one, right? F of delta T, sorry. Because you've hold, suppose again, delta is a 0.3. You've held 0.3 units from now till the next day. The asset has changed to, and say the original price was $100. It's now $101. The value of your position is now three times 101 or 0.3 times 101. It's grown in some way. So this is my new value but I need to change my delta position. I don't want the position 0.3 anymore. So I need to rebalance because according to the strategy, I need to change this, right? At time T equals zero, my delta was 0.3. At times T equals one, when I calculate this partial derivative, the asset price has moved and time has evolved. So the partial derivative will be different. Let's go back to this curve here for a second. What that means is some little time later S has moved, suppose in a day S moved to there, I now need to match the slope at that point. And the slope at that point is certainly different than the slope at my original point. And this blow up, in this blow up diagram, I move from there to there, for example. And I need to match at this point. And those slopes are clearly different, right? So my delta has to change, time has evolved and the asset price has moved, okay? In fact, this curve, this red curve would have come down a bit when time evolves. Remember, that's what happens, right? Option prices converge to the payoff as time evolves. So the curve would have actually come down a little bit and the asset price moves. So it's a combination of both. I've only shown you one motion here. So I need to rebalance like this. Where do I get that money from? You don't take it from a new source, you borrow it from the money market account. Okay, you have, this is my net, this is the net value, one way to think of it is sell all of the asset and then buy more. Or, in other words, buy this many more units of S. Delta G at time delta T minus delta G, your new position minus your old position, trivial. How much does that cost me? What's this? Where does that money come from? The money market account. I may, maybe I have a positive amount in the money market account in which case I can just pay for it. And if I have enough, or if I don't have enough or if I already owe money in the money market account, I borrow more. So my new money market account position at time delta T has to be equal to whatever it was before, right? This is before the transaction, before rebalancing. That's how much I had in the money market. And I have to pay for this new hedge. So whatever my money market account was, it's now worth this. So now we have got a new, now we basically have a renewal process. We started back exactly in the same position that we did before. I now have this much in the money market account. I have delta T units of the asset. And I just push that forward one more step. So let's just write T equals two delta T down and then I'll show you the simulation. Okay, so T equals delta T, two delta T. What do I need to do? I need to change this position, rebalance that. I'm not gonna write down every step that I had above there, just the rebalance part. I need to rebalance. So my new position in the money market account equals my old position, grown the risk rate, because it's one step later, subtract the cost of the rebalance. And this depends on the current asset price. And I just iterate. So in general, at any time step, n delta T, I can figure out what's in the money market account, old amount of the risk-free rate, subtract the cost of my new delta position, of changing my old delta into my new delta. And I have to pay for that at the new asset price. Okay? So this is the general rebalancing equation, because all I need to do is keep track of the money market account, because I know at any point in time, after rebalancing, I have this much of the asset. Maybe I'll just say here, after rebalancing. And I'll highlight both. That's my general setting. It's a simple recursion relation. You cannot solve this analytically. There is no, I mean, well, you could write down an analytic expression, but it's messy. It doesn't help, doesn't give you any insight. Okay, but what I want you to take away from that strategy is, all you're doing is that every time you're rebalanced, is you're matching the slope of the option, every time. That's how this, and this is a self-financing strategy, because once you set it up at the start, you don't take money from anywhere else. It's all part of the system. You either boring or lending from the money market account, and sloshing it back and forth between there and the asset. And that's it, nowhere else. Okay, so shall we take a look at what this does for you in terms of a hedging strategy? Okay. Oh, sorry. Before we take a look at the simulation, one other little thing. What does the delta of a call actually look like? That's what I need to tell you as well. So you know what the price of a call is from the block-shoulds, from the original block-shoulds formula in discrete time, and I'm gonna tell you, or in discrete time, continuous time limit, I'm gonna tell you that it is also a solution to the partial differential equation. It in fact is, and it equals this, or it is recognized. Okay. And it turns out if you calculate the delta of a call, it has a very simple expression. And in previous years, I used to go through this calculation to show this derivation. It's messy algebra. I'm not even gonna bother. There's no insight into the computation, but it just equals this. And we can plot it, and it has a very nice simple form. So this is what happens in the limit as the little t approaches capital T. You get the blue curve. So delta looks exactly like the slope, well, as it should. You know that that's my payoff. If you just take the derivative at maturity, the payoff, it's zero whenever you're less than k, and it's one whenever you're bigger than k, and this formula actually replicates that. And if you go in some earlier time, what tends to happen is it's gonna be smoothed out here and asymptotically approaches and it touches zero here. Okay, so as you increase maturity, you get more and more smooth and kind of looks like I can't draw this very well. Something like that. I'm gonna keep drawing it until I get, till I'm happy with the shape, something like this. Either a little t decreasing or a big t increasing. It's the same thing. Or you can view it the other way. In fact, that might be better for you. Think of a little t increasing. As little t increases, you go like that and you go like that. And that, again, comes from the intuition of this curve, this payoff function. If you just compute those derivatives, you end up with shapes that look like this. Okay, I'm gonna walk through the code and make mappings back and forth between the code and this strategy that we've just described. So forget that, forget transactions, time steps. So this is gonna do something where I'm using a one year maturity option. I'm gonna hedge every trading day. It's 252 trading days a year. These are my model parameters. There's a few of them here. There's the drift, the real volatility. Forget vol implied for a second. Just think of it as just volatility. It's sigma, both of these are the same. This is my initial stock price. This is the strike and that's a risk-free rate. What this bit of code does is it gives me a set of sample paths for the asset. So if I just run that and I put a break point there, what you'll see, so as I move along this direction along the rows here, this is one sample path. So if I do plot, it's called spot, sorry. Okay, that's one sample path of the asset price. And it's using its real return and the volatility. And what we're gonna do is that each day, each point on this path, we're gonna invoke this strategy that I just described to you. Okay, this just kicks the collection of the time remaining to maturity. Okay, so this here is just gonna compute the delta along every path. So it goes through, you know, we have the sample path here. It's just gonna compute the delta and you see there, there, there, there, there, there all along this path. And it does it for every single sample path. Like I said, the spot, there's a whole bunch of them. There's a whole bunch of sample paths here. There's a thousand of them. Okay, and gotta remind myself not to press F10. Okay, and forget about this thing called delta time-based versus delta move-based. Look only at the thing that says delta time-based. I'll explain the other one afterwards. This is the value of the put along the path. Doesn't really matter. Here, this is the number of asset, number of units of the underlying asset that we're holding. So it's kind of like the delta, right? That's the delta position that we're holding. This is telling me how much money I've saved in the bank account. And for now in all scenarios, there's zero. Here's where I set it up initially. So when I go through that, if I look at this, sorry, delta time-based is what I want you to focus on. So if I open this, see that they're all the same. Every scenario, it's the same because at the start, at the start, I simply, I sold the option for G zero. I need delta zero units of the asset and that costs S zero. So no matter what the scenario is, those are all the same. The asset starts at $100, so it's always fixed. Okay, ignore that line. Ignore that line. Ignore that, ignore that. Ignore that, ignore that. Here we go. No, ignore that, ignore that. I've already pre-computed the, I've already pre-computed this, the delta along the strategy I told you already, right? In this thing. Here, so notice it starts off negative. This is for a put. I'll show you the deltas for a put in a second. And they all start at the same level. If you follow a horizontal line, that's the value that the delta takes as you move along that path. And what we're doing is, ignore that line. This is the only line you need to pay attention to. In fact, those there. That, that I've highlighted, is precisely this recursion relation. The new bank account equals the old bank account grown at the risk-free rate minus the cost of changing my delta position. So if you look here, this is my old bank account at the previous time step. Grown at the risk-free rate. This is my, yeah, sorry. I was confused because the ordering is, it looks wrong. It looks like I have my old position minus my new position. But that's because my coefficient out front is plus. Notice in this recursion, I have it as a minus, right? It's by the change in the position. So I just flip the sign. And times the current value of the spot price, okay? So all I do is I run that through. So let's just do one step here. So, so here in scenario one, we went from owning from $51, we've had to reduce it. So what does that tell you? What happened to my delta? Did it increase or did it decrease? This is the money market account that I'm showing you here. That row, did my delta increase or decrease? It had to increase because it cost me money to buy the asset, right? I have less money now than I did before. Therefore for almost surely, I had to buy more of the asset. In this scenario, there was an increase in the money market account. So what happened? I had to sell some of the asset. My delta actually decreased, okay? So you can, and we can check by looking at this column here. And we can see that, yes, in the scenario one, my delta went from negative 0.44 to negative 0.40. So I had to buy more, right? And that's why the bank account decreased. So from a large negative number to a smaller negative number, I buy, right? To buy. In this scenario, I went from a negative number, negative 0.44 to negative 0.44 to two. So I've had to sell. I'm creating more of a short position. So I'm selling more. So that means I actually gained money in my money market. And you can go through all the scenarios and see that. Okay, so what I do is I run that for all of the scenarios on all of the steps. So this does all of the steps all the way to the very, very, very, very end. Now these terminal values here, each one of them represents the value in my money market account after doing this whole dynamic hedging thing that we described here, all the way to the very end. It's my terminal value. Once I get there, I still have one more task. Remember, I've sold an option. So I owe whatever the options payoff is to the client. So all of this recursion relationship here holds up every time you also have to, in order for you to calculate your actual profits and loss from this methodology, my P and L is equal to whatever it is that I have at the last stage, the value of my portfolio times the value of the underlying asset. But I have to set, I've sold the option. So I owe whatever my payoff function is. That's owed. So this is my P and L at maturity. And you can discount it back to zero if you like or not. It's up to you to view either as a P and L at maturity or as a P and L at times zero. So that's what this last line here does. It takes the spot price, multiplies it by the amount of stock that you're holding added to the bank account, subtract what you owe for the put. In this case, for the put, okay? And let me plot, let me run that. And then what I do, since we have this for many, many, many scenarios, okay, we now have this in scenario one after moving through that entire, after moving through this entire sample path and for the delta, corresponding delta rebalancing and so on, my net value is I, in fact, lost on this transaction. I lost a dollar. So it didn't work out well. In this scenario, I actually gained 40 cents. In this scenario, I gained 30 cents, 35 cents, 31 cents, lost 10 cents, et cetera, okay? So these are my P and L. So what's a natural thing to do to this P and L? Well, look at the distribution for one, right? The average should be, you hope, positive or zero, right? Okay, so let's play this and see what you get. Don't worry about those things. I'll describe them, hopefully still describe them today. Okay, so what's shown here is the red line is the put payoff. The little blue dots, these are the blue circles. These represent the value that the asset, so this is the terminal value. The asset ended up at this price, 200 and say, I guess that's about 200 and what, $230? And the P and L that I received in that scenario was close to zero. And while in here, let's zoom in, okay, there we go. So you can see that they're not all just right on top. While in here, there are some scenarios in which I end up at, say, $98 here, and my P and L is above what I owe to the client. Sorry, not the P and L. This is the value of the replicating. It's above what I owe to the client. My P and L is a difference in these two things. So if I'm above the red line, I have a positive P and L. From below the red line, I have a negative P and L, okay? Again, in this scenario, so in this scenario I lost, in this scenario I won, in this scenario I won, I won, I lost, I lost, I lost, okay? All of these below I lost. If you put this into a histogram as we were just talking about, here's what that histogram looks like. It's clearly not zero and it's kind of, looks like it may be a little bit skewed, maybe a little bit skewed. Now, if I increase, this is hedging once a day. If I, in the continuous time limit, we know that we should end up with zero because we should exactly replicate the option, right? There should just be a big spike there. So keep this in mind, let's go to the simulation and change from doing 250 steps to doing 1,000 steps. So that means trading four times a day. Okay, so here's again our little P and L, our little replication and here's our P and L. Do you remember what it looked like before? All right, you can, this is clearly tighter, right? Smaller variance. So you should be convinced that yes, definitely as I increase the frequency I'm decreasing this variance in my P and L. Now, what have I missed out in this strategy? Something that I didn't do. In the real world, if you were to do this, would this actually be your, and suppose the model was correct, would this actually be your P and L? What am I missing? Yeah? What kind of costs? Transaction costs. You can't just trade, call up your broker and say, okay, hey, I wanna buy five more units of IBM. And they say, okay, sure, here's a market price, that's a market order, you pay this price, that's it, done. No, it's gonna hit you with a transaction cost. Now, if you're a large player, your transaction cost could be essentially zero. For example, I know there are people in the banks downtown that are working on derivatives desks that can just call up their broker and they're more or less the same floor or just a couple floors up and say, I need to execute this, go ahead and do it and they'll do it for nothing. Not, if you don't have that, then you do have transaction costs. So let's turn them on. That's already incorporated into this model. How would I do that? Analytically, what do I need to do? What I need to do is, in this equation, I need to add in another term that's minus transaction costs. Typically, proportional transaction costs are used here. Basically, you pay times delta, let's say. This is proportion to the size of the transactions. Sometimes you wouldn't have the S there, you just pay per transaction. So that's all I'm doing. It doesn't matter whether it's a buy or a sell. So that's why there's an absolute value sign there. And in this code, you see here at the end, there was one more line in there and that one extra line, this is exactly this absolute value of the change in the asset position times a spot times a transaction, which is a spread. And here I've just, whoops. Here I'm assuming the transaction costs are one basis point. Or 10 basis points, right. That's 10 over 100 over 100. Now it's 10 basis points. So what do you expect to happen to this P and L? In fact, sorry, let's go back to the case of zero transactions for a sec and only the 250 time steps. I wanna run that again. Just so that we have in mind, we've got this in mind. What do you expect is gonna happen when I add in that transaction costs to this distribution? Should definitely move off to the left, right. Because you're paying every time you transact. So you're getting hit and you should slide off to the left. And in terms of this, what do you think you should see? You should see this basically the blue, the parts that are blue above the red curve are decreased. There's more blue below than there will be above. More situations with the replication costs more than what you owe. Or it doesn't produce enough. Okay, so let's run that and see. And there we go. They clearly negatively skewed and negative me. And here, it's a little harder to see but I think you can probably convince yourself that this area here is a little more meaty than this area here, right? Okay, okay. So that's a problem with this methodology. Once you start to introduce transactions, you can't transact too often. Otherwise, you get hit by transaction costs. And if we reduce this to say transact once a week, you reduce your transaction cost, yes, but you've increased your risk as well, right? You notice that it looks much wider than before but the mean is kind of closer to zero. Okay? So anyone have an idea of what you might be able to do to maybe somehow take advantage of both? Okay, I'm gonna guide you to the answer by doing this. Okay, I'm only gonna simulate three sample paths but I'm gonna put up a few more plots, something that you haven't seen yet, this one. Yeah, okay. So what's shown here on the top panel, this is a sample path of the spot price once a week. Okay, so we only do once a week. These are three sample paths. They all start at the same point. What's in the bottom panel? Look at the bottom panel. That is the delta position in the underlying asset. So this is how many units of the underlying asset that you're gonna hold at any one point in time. Now you'll notice that the green path in this particular scenario is a path which ends out of the money. This is a put option. So you're ending above $100 and the strike is $100. So that's out of the money. And what happened to the delta position as you approached maturity? Ended up being zero. While the red path and the green and the blue path, they end up in the money. They end up close to $80, say, right? At the end of this transaction. And they end up having a position of negative one in the underlying asset. Right? Now why negative one? I've shown you the delta for a call. Can anyone tell me what the delta for a put has to be? It's an easy argument that tells you what the delta for a put has to be. What's a relationship between put and call? Put, call, parity, right? So you know that a call minus a put is equal to s minus k discount. If you take a partial derivative with respect to s, this gives me delta for call minus delta for put equals one. Therefore, the delta for the put equals the delta for a call minus one. So the put looks identical to the call except shifted down by one in terms of its delta. So it has that kind of shape. And so we can see when you end up above the strike, your position should be liquidated and you should have nothing. When you end up below, you should end up with minus one. And that makes sense because, again, the slope of the put in the money is negative one. The slope of the put out in the money is zero. At maturity. So this explains why these two graphs kind of approach zero or minus one. Or why these scenarios, sorry, approach zero or minus one. Okay, so let's focus a little bit on the red path here. In the current way of doing things, we're rebalancing at every week, right? We're doing it 50 times in a year, should really be 51 trading weeks in a year, but 50 times a year and we're doing that every week. And every time we do it, we get hit by transaction cost, right? In that red path, is it really necessary, for example, to transact between times 20 and 40, or 20 and 30, let's say? About 30 transactions there, which adults is moving a bit, but it's not moving terribly much, is it? So maybe rebalancing every once a week is might not be the best thing to do. Why not instead rebalance only if the delta has moved sufficiently far from what it should be? That's called move-based hedging. And what I described above is called time-based. Now the move-based hedging can be based either on deltas or it can be based on the asset price itself. It really is sort of up to the individual to decide how they want to do it. And most typically it's based on deltas. So the way that this works, one way, one approach, if you put bands equally spaced in delta, and you can start the band at your current level of delta, let's say that's our current level of delta, and then you just put equal spaced bands. And you track the delta of the actual, of the delta of the option, and only if that delta hits the two bands surrounding where you started, that's where I rebalance. And then you flow forward and now I won't rebalance again until I hit there. Now if you're doing this by a time-based, I would be hedging say every, this would be time-based. It's kind of the difference between reman integral and libation integral, right? You know reman integral, you break up equally on the horizontal axis, so the vague is on your vertical, one way of interpreting it. I don't want to clutter the diagram with that, but just to remind you, okay? And then you rebalance once you hit again, stay there. So this is your move-based hedging idea. So the same, the only difference with the formula here is rather than using the delta of the option there, delta G minus delta G, you only take this, you only add in this difference every step if the option delta has moved enough. And that's pretty much it. So in the code, don't worry, in the exam you're not going to have to replicate formulas and code and so on with respect to this. I'm doing this to try to give you intuition as to what people do in practice. What I will at most ask you about these type of strategies is a true false question, okay? That's about at most what I will ask about this stuff. So in here, I have, you notice I kept track of something called number move-based, bank move-based. Well this is exactly that, this is the strategy, I'm going to use the same sample path, but use a move-based strategy and I have a band, the size of the band is called the D delta. So here I've used 5% bands, plus or minus 0.05, okay? That's the width of this band. And these variables simply tell me whether I should change the delta or not, okay? Don't bother to really understand the code, that's all it does. And when I run this, this is what you find for the time-based and let's go to move-based. So time-based, move-based, time-based, move-based. And you can see that what this is doing in terms of the focus on the delta here, you notice that for the move-based, the delta is much more stable, right? There are fewer times in which you've actually changed your position. And what's going to happen is you're going to get hit less with transaction costs. Now, remember that if you don't rebalance frequently enough, you're also going to get hit with the variance. So there's a delicate balance between how big that band should be and the transaction cost. And you've got to know your market well to know how big your band should be. So let's take a look at running this with many scenarios again and compare the P and Ls. Where are simulations, number of simulations, there we go. Let's run a thousand simulations. Okay, this histogram is something that you haven't seen. This is telling me the histogram of the number of transactions that I executed based on the move-based strategy. Remember, for the time-based, I would have had 50, always. Every scenario, I'm rebalancing 50 times through the sample path. Here you can see it's very unlikely. Maybe there might be one event there. I don't know, I think there isn't any. But most of the events are below 40 trades and many of them are quite small, maybe 10 or 20 trades at most. So you're getting hit by much fewer transactions. Let's take a look at the P and Ls or the, there we go, these two things. A little bit hard to see on this replication. What's better here? It's a little bit hard to tell. But there's one place where you can clearly say the move-based is worse. One set of scenarios. There's time, there's move. Time, move. Time, move. Can you tell me? Time, move. Large asset prices. Look at this region out here. Move-based, time-based, move-based, time-based. So it's not doing so well in that region, but let's take a look at, let's see. It could be whether my bans are just not, yeah. In fact, with these parameters, which strategy would you use if you had to hedge on a weekly basis? You would use time, right? With this particular band. Let's see, I don't know if I can make it, where's d delta? I should have worked out which parameters were better in which region here. Now they're kind of close. Move, time, move, time. They're about similar. And let's see. Perhaps I need to bump up my transactions a little bit in order for me to see something significant. Okay, okay. Now we're kind of getting to a critical region where perhaps you might prefer to move-based, slightly. Here's time. It's got a very heavy left tail, right? Here's moved. There's more events off in the right tail. Not significant. Not hugely significant. It's hard to tell. I wanna try some other, one more parameter. Let's see. Oh, that's way really bad. It looks really bad. So I've got to go smaller. I thought I might be able to. Okay. Yeah. Okay. This potentially looks better. The move-based potentially. One thing we can do, let's do a quick little extra coding here. Let's take the difference between the time-based and the move-based for each scenario. Okay. So, uh-oh. Where's my cursor? Yeah. Where it really makes a difference is when we start to do what's called delta gamma hedging. But I was hoping we would see some here. PNL time-based, PNL move-based. Okay. So this is the, this what I've done here, if you can kind of read that code, I've taken the PNL from the move-based under each scenario and looked at the difference between that and the time-based. So you can see that the mean, maybe the mean is positive. Let's see. Mean. Yeah, yeah. Yeah, it's positive. It's not very hugely positive. And standard deviation is 0.15. But, yeah, I don't know. As a trader, I don't know if I would like this actually. I don't know. What do you think? Who would bet on this strategy move-based versus time here? It's kind of iffy, right? It's a little bit iffy. Yeah. So the parameters are really making, can make it make a big difference here. Let's, let's see. Okay. I think that's enough toying with that. Just know that there are, there are certainly cases in which the move-based is better than time-based, but it's not always the case, right? And there's cases when the time-based is better. I also have a bit of code which does something called delta gamma hedging. So I want to describe that for you. And very quickly, it's not hard. You can do it in five minutes. Okay. So the idea behind delta gamma hedging, it's very, very simple. The idea is just a very, it's just a generalization of this delta hedging stuff by recalling when you look locally, the delta hedging is corresponding to saying let's make the price locally be equal to the tangent at the point. The delta gamma hedging is saying let's locally approximate it by a quadratic function. And it follows from this idea that if you look at the increment in the g function after one time step, the value, then using standard calculus ideas, just take an increment with respect to s, not with respect to time. And this increment is approximately the delta of the option times a partial derivative, oh, sorry, times s plus one half two derivatives of the option times s square. This is just a Taylor expansion, right? Times delta s square. This should be delta s. That's my mistake here. That's why I was confused. What the heck is that? Delta s, delta s square. This term here, this is called the gamma. So the gamma is just the derivative of the delta. That's all it is. It's a rate of change of delta. So you know that I think this is something you already know, right? If you just locally approximate by a quadratic, then you have the first derivative times the increment plus a half, the second derivative times the increment square, straight from Taylor's expansion. And then there are higher order terms here which are just ignored because you're just doing an approximation anyway. So how do you modify the stuff that I've talked about before? The idea is that you should hold, instead of using alpha units of s and beta units of the money market, you also introduce gamma units of a second option. Call that h. The idea, the reason is that why you need a second option is because the gamma of an asset is zero. Why is the gamma of an asset zero? Because what is the delta of an asset? Delta of an asset is one, isn't it? Partially the derivative of s with respect to s is one. So the gamma is zero. And because of that, there is no way for you to match the gamma of the option you've sold by trading in the asset. You need another option. So if you write down the value of this portfolio, we'll have alpha times s plus beta. Let's just write it without the sub indices here. Plus beta times gamma times h. And the idea is that you want this, you want the gamma and delta of v to match your claim that you've sold g. So what's the delta of this portfolio? It's alpha, right? Times the delta of an asset, which is one. Plus what's the delta of the money market account? It's zero, isn't it? It's nothing. It doesn't depend on the asset price, so it's zero. Plus gamma times the delta for h. And you want that to be the delta for g. So if you like, you could put your little time in this here, it doesn't really matter. And to match gamma, the gamma of the option of the asset is zero. The gamma of the money market is zero. And you get the gamma for h. So this gives you a very, very simple result. The position that you have in the option, in the extra option should be where the ratio of the two gammas. And the position that you have in the asset, excuse me, is equal to the delta of the option you've traded minus the ratios of the gamma times the delta of the option you're hedging with. So this is another reason why I use alpha and beta and gamma for my positions or small gamma. It's that they're not always exactly equal to the delta of the option. They're sometimes corrected, as this is an example. You can do other kinds of corrections as well. You can do what's called delta gamma vega hedging, where you look at sensitivities with respect to volatility. There's also delta gamma, vana. There's volgas. There's vanavolga. There's a whole bunch of different weird things where you take mixed partial derivatives. But this is the one of the other key type of hedging strategies that I use is delta gamma hedging. And the dynamic strategy that we talked about before, more or less moves along the same basic idea. You just, if you started off as before, except you also have to buy the position in the option. And at any time step, once you want to rebalance, your new money market account is equal to what you had before grown at the risk free rate. You pay for your new position in the asset. So this is my new position in the asset times my new asset price. But then I also have a new position in the other option. So I've got to change that as well. So this becomes my new rebalancing strategy. I think it should be easy for you to convince yourself that's all you have to modify is just this extra line. And at each point you calculate the gamma, the little gamma and the alpha by that equation. Okay, so in the last few minutes, let's take a look at what happens when you do this delta gamma hedging. So this is the delta gamma hedging every day based on the time-based approach. And as you can see, the P&L is much, much tighter than what we had for the delta hedge, right? Significantly tighter. And here's a replication. This is with zero transaction costs. Here is what you get with the move-based hedge. Now you might think, yeah, this definitely has larger variants, but there are a lot of scenarios in which you have a fairly big positive P&L. So as a trader, it's not so obvious whether I want that or I want that, right? What would you prefer? Remember, this is your net P&L. Do you want to almost surely make nothing and with little probability make five cents or 10 cents or almost surely or with a high probability make nothing and with fairly sizable probability make a dollar. But you might lose 25 cents. You know, it depends on your risk tolerance. Most traders would not do this, otherwise it wouldn't be a trader, right? They would probably do that. That's with zero transaction costs. Now let's turn them on. Okay, both for options, both for the underlying asset and options. And options have a slightly higher, higher transaction costs than underliers. Okay, that's your time-based P&L. You can see here from the replication, there are a lot of scenarios in which you end up losing. And in particular, if you look around here, you're almost always losing at the money, which is kind of the interesting area because typically these options are written, you know, not so deep in the money. There are deep in the money options, but typically they're close to at the money. While the move-based, here's what our strategy is looking like. And yes, it looks like we might be losing some here, but we're nowhere near as bad as we were in the time-based. And if we look at our P&L, you're comparing that with that. So with transaction and doing a delta gamma hedge, there's a significant advantage to do the move-based approach versus a time-based approach. So if I didn't convince you with just delta hedging, I hope for sure you're convinced with delta gamma hedging that there are advantages. And here's a number of trades, significantly smaller than 250 trades every day, every year. Okay, so with that, let's call it, call it, oh, question? Yep. Okay, sorry, can we be quiet? There's still a question, yeah? Yes. You can also do that. So the move, so the one that I implemented here is simply the, I checked to see whether my replicating portfolio has moved sufficiently far from my targeted option. So I don't just look at alpha, for example. It's not just, I don't just track whether this error, whether alpha has moved outside of a band. I checked whether the total delta, which is given by this equation here, I checked whether the total delta has moved sufficiently away from the options delta, because that's what's really relevant. But you're absolutely correct. What I could do is I could introduce two stopping times. Sorry, these are mathematically called stopping times when you do the trade. But I can introduce two times at which I can trade at. One, if the delta has moved far away enough and, or whatever comes first, the gamma has moved far away enough. You can do both. And it may or may not be better. It all depends, again, on the specifics of the transaction costs and the market. Any further questions about this? Or anything else covered today? Okay, so I know you might have noticed you did not have a quiz today. I'm not about to give you one now. But that dynamic hedging methodology, an argument, like I said, know it well. Okay? All right, so I'll see you guys next week. We've got two more classes left.