Some asymptotic results about American options and volativity
De Marco, Stefano
The valuation of American options (a widespread type of financial contract) requires the numerical solution of an optimal stopping problem. Numerical methods for such problems have been widely investigated. Monte-Carlo methods are based on the implementation of dynamic programming principles coupled with regression techniques. In lower dimension, one can choose to tackle the related free boundary PDE with deterministic schemes. Pricing of American options will therefore be inevitably heavier than the one of European options, which only requires the computation of a (linear) expectation. The calibration (fitting) of a stochastic model to market quotes for American options is therefore an a priori demanding task. Yet, often this cannot be avoided: on exchange markets one is typically provided only with market quotes for American options on single stocks (as opposed to large stock indexes - e.g. S&P500 - for which large amounts of liquid European options are typically available). In this talk, we show how one can derive (approximate, but accurate enough) explicit formulas - therefore replacing other numerical methods, at least in a low-dimensional case - based on asymptotic calculus for diffusions. More precisely: based on a suitable representation of the PDE free boundary, we derive an approximation of this boundary close to final time that refines the expansions known so far in the literature. Via the early premium formula, this allows to derive semi-closed expressions for the price of the American put/call. The final product is a calibration recipe of a Dupire's local volatility to American option data. Based on joint work with Pierre Henry-Labordère.
 Thanks, so I'd like to thank Andrea, Href, Cyril and Jean-François for the invitation and for accepting to move my talk on schedule because I was supposed to speak tomorrow, which is also the reason I apologize with you because you will have a second long talk today. So the talk of today, so my talk, sorry, will be somehow different with respect to many other ones we've had here at CEMACS, which we are focusing on probabilistic numerical methods for high dimensional problems or for their mean field limits. I would rather focus on the problem of finding exact approximate solutions for low dimensional nonlinear problems and precisely the nonlinear PDs that appear in optimal stopping. So it will be somehow orthogonal to the direction that many talks had until now. So my motivation comes from mathematical finance. So this first part will be maybe somehow obvious for some of you, but hopefully not for others. So the motivation is the following. In mathematical finance, we have a stochastic process that modifies financial asset, a stock, let's say a stock, so the stock of Google or Societé Générale, for example, where my co-author works. So what we say is that this is a price expressed in euros of Societé Générale stock and we model it with an SDA. So WT is a marine motion in dimension one. And by the way, what do I mean by low dimension? Once a colleague of mine said that high dimension starts from D equal 2, so by contrast, low dimension will be D equal 1. So Brownian motion dimension one and sigma is a function of time and space with values possibly in some, well, with values we're taking some bounded intervals, sigma bar, low and high, plus some standard leapshield conditions in space uniformly in time. So this is our model. What do we have on the market? On the market we have some financial products that are called European options. What is that? Well, we fix a future time horizon, you can think of T equal one year or T equal one month. We fix a certain constant, K, positive. This is a price, so this is expressed in euros, okay? And we can buy a contract on the market that gives us this amount of money in the future. The maximum of K minus the future value of S, okay? So you can think, for those who have never seen this, you can think of this as an insurance product, right? You are protecting yourself against a drop or against a decrease in the value of S. You are thinking that S in the future will be very low, we go from one other to 50 and if you have both this contract you will get some money, okay? Otherwise if S is high you will get zero, but okay? So you're protecting yourself against drops in the values of the asset. Okay, so this we can rewrite in a simple way as positive part of K minus S T. Okay, good. So the theory for pricing and managing these kind of products tell us this object, okay? I'm buying the right of getting some money in the future, a capital T. I have to pay something now in order to have this right in the future, okay? So the price of this contract now, at time zero, is given by an expectation. Expectation of exponential minus R T, positive part K minus X T, where X is another process. That satisfies this S D, where R, small r is a constant that I'm introducing, positive. That represent the interest rate that you get if you invest money in a bank account, okay? So we have to compute, sorry, and maybe we have an initial condition here that is the initial value S zero that we observe over there, okay? So the problem, so the price is given an expectation, the problem we have is computing an expectation, okay? For which you can use, now the numerical problem we have is computing expectation based on stochastic process, solution of an S D for which you can use your favorite numerical method, right? You can use Monte Carlo. In some situations you can maybe compute this explicitly. This is an integral against the law of X. Or maybe you can solve the linear PDE that is satisfied by this expectation, right? Okay. So you can choose your numerical method. And maybe at this level, I'd like to point out that if we call this function, I'm not managing this blackboards very well. This function we would call it P, TK. There's two parameters, T and K. P, you can think P stands for price. Actually, this is a put option. So it's the P for put option. It will depend on sigma as well, right? On the sigma that is inside my process, okay? I'd like to point out that something that is known is this thing is that this function solves a certain PDE. Solves the following PDE. Okay. Parabolic PDE with an initial condition P0K equal to, well, what you obtain when you inject T equals zero there. Okay. So we have a forward PDE, which is nothing but the Fokker Planck PDE for this process written in terms of the function P, okay? You can think of this, so there's many ways of proving. This PDE, you can think of it as the Fokker Planck forward PDE written in terms of the function P sigma. Okay. Good, so you can tackle the problem from the side that you want and everything here is linear. Linear expectations, linear PDEs. Okay, then we come to another type of product, which are this American option. I'm going to focus on... During the talk, okay? So we still fix a future time and some positive price K, okay? And this American option is a financial contract that still gives you some money in the future, okay? But you don't necessarily look at this difference K minus S at final time. You can look at this at any time tau is more than capital T, okay? So at any time tau, you can stop the game and you say, now I want to get this amount of money, okay? Which means that you exercise your option at tau. So this is what you get if you exercise your right at time tau is more than capital T, okay? Okay, so again, the theory, which I'm not going to recall indeed as, but the theory for pricing and hedging, for pricing this kind of object tell us that now the price I have to pay today in order to have this right in the future, okay? This is the American option. The price of this American option is not an expectation but a supremum of expectation. Okay, it is in the end the value function of an optimal stopping problem, okay? So it's the supremum of these expectations over what? Over all, toes that belong to this base capital tau that I denote in a standard way like that, which is the space of stopping times in the augmented filtration of the running motion W, okay, with values in zero capital T, okay? So we have this, yes? So just to remind you that perhaps you are using a different signal from there to there because you changed that from the motivation that the price increases S, you have the, for I think the C, C, C, S, U, U, U, U, U, U, U, U, U, U, U, U, U, U, U, U, U, U, U, U, which, what do you mean, so why might it not be the same function I� Sorry, maybe the dú´ cos hasn�t it fine. So, the variable is just a variable. It has to be changing from the single to having a problem. They're changing variables. I'm changing variables, yes, but you agree that the price of this option is a number? No, yeah. That I can write in this way? Yes, that's possible, but just that you're changing the process is different. But, okay, but you... Okay, so we can discuss about that later. So you agree with this identity and with this signal? I agree with the future. And I agree with the future. Okay. But the sigma in the X is the same as the sigma in the X. Yes, it's the same, yes. This is what I mean, it's really the same. But the sigma in the P is just like a symbol, is it a symbol of Pk? Yes, yes, this is a symbol. So this is an expectation that depends on a lot of things, on T, on K, on R, which I don't put, and it will depend also on the function sigma that I put there. Yeah. Okay. So, George, there's something I missed in what you said, I believe, but okay, if it's okay to go on like that, I would be happy to. So, yes, so now we have this problem to solve, okay? So, I'm going to take a small remark, en passant, which I'm going to go back to afterwards. So you take in that problem tau equal capital T, okay? And what do we get? We get that this value, sorry again, this price now I will denote it as ATK, A for American, TK, and sigma, the function sigma is still there, okay? So, the point is that A sigma, TK is larger than this expectation computed at T. So this will be larger than what I call P, right? Just this expectation, the price of the European option plus something, plus a certain amount that will be positive, okay? So, this amount, this is what we call the early premium in mathematical finance, okay? Early premium because it's what you have to pay in addition to the previous price of the European option, it's the additional price that you have to pay if you want the additional right of exercising your option at any stopping time before capital T, okay? And I will go back to this formulation in the future, so in my talk, okay? So, now the problem we have is to solve numerically if we want to apply the model to market data, we have to solve numerically this problem over there, okay? Which is more complicated than just computing an expectation, okay? So, this is the moment when we wonder about efficient valuation numerical methods for optimal stopping problems, okay? So, hopefully, quickly, I'd like to remind what are standard classes of methods to solve this optimal stopping problem, okay? Well, and now, from now on, I will start denoting UTX for T before capital T and X positive value. So, what's the value function of this optimal stopping problem? So, the price of the American option if I enter into the game at times small t, okay? So, it will be the supremum of these expectations. So, we have now taken over all stopping times, though, the take values in small t capital T, okay? So, we use this notation, which should by now be clear. Supremum over all stopping times in the filtration of W that take values in the interval small t capital T, okay? So, we have here where we, the standard notation again, X T X S, right? Is this solution of the ST starting at small x at time t, okay? So, this value function is in the, a lot of previous talks, okay? So, this value function satisfies what we call a backward dynamic programming principle, okay? Which is suitable for time discretization in such a way that we can actually simulate the random variables, right? So, there's a backward dynamic programming principle. Which if we discretize time, okay, for which we can discretize time, take n dates between 0 and capital T, regularly spaced, so Tk, K, T over n, which means that I'm taking a fixed time grid, T over n. I can consider, and I know how to simulate, the Euler scheme for the process X, right? So, what I can construct is a discrete time process U starting from the beginning, from the end, sorry, okay? And then iterating backward from time ti plus 1 to time ti with this principle, okay? This will represent my gain at time ti, it is the maximum between, so the value of my option at time ti is the maximum between what I gain if I stop at ti and the conditional expectation of what I get if I wait one more step, and if I wait one more step, my value is U ti plus 1. Conditional expectation with respect to my process at time ti, and I have to take the maximum between these two. Possibly I'm forgetting this exponential term here, sorry, and I iterate from time n to time 0, okay? And I'm using the value U0 that I obtain as an approximation of the solution of my value function at time 0, okay? I forgot the plus, yes, okay, thank you, thank you, yes. Did I forget it? Everywhere, okay, everywhere, yeah, thanks. So this now opens the way to Monte Carlo methods, right? Because the problem I'm left with is to simulate this conditional expectation to which I can apply my favorite regression method, okay? Which means using the IID simulations of the Euler scheme, I can switch to regress, and then choosing a basis of functions on which to project in order to replace this conditional expectation with a finite dimensional projection, okay? And we have heard about this in a lot of presentations during some rags, for example, Plum and Trocadius, right? So this opened the ways to Monte Carlo methods, which are potentially applicable in higher dimension, okay? Which means that you have several processes, X1, X2, up to XD. But as I mentioned in the beginning, what I'm interested in now is the analytical counterpart of this problem in low dimension. And this is a free boundary PDE, okay, which is written in the following way, okay? So let me notice, first of all, that this value function over there is always larger than K minus X positive part, right? Just take tau equal small t, okay? It's always larger than this, which means that it can be larger, which can be strictly greater or equal, okay? So it makes sense to define this object, small x of t, this is a function of time, which is the infimum of point X, at which this function is strictly larger than the value function is strictly larger than this fixed function, okay? And if we do this, the picture looks like that. If we do this, the picture looks like that. This is time, capital time, strike is here, okay? This function X looks, has this behavior, okay? So this is small x of t, and I'm dividing the time-space domain in two regions, one in which my value function is larger than this, and one in which it is equal, okay? So we can call this C and this other one, we can call it D, okay? So for the notation capital X is the process, small x of t is the boundary, okay? If you, I have complaints about this notation, I'd be happy to forward them to my co-author who chose it. So if we do this, then the statement is that U and this function of time X is the unique solution to the following problem. Okay. To the following problem. Okay, so in this region, what this identity is telling us is that it's optimal to stop immediately, right? If we look at this optimal stopping problem, UTX equal K minus X means that tau equals small t is an optimal stopping time, right? So here it's optimal to stop immediately, here it is not. Here it's optimal to wait a little bit before exercising, okay? So it makes sense that in this region, the function, the value function satisfies the standard parabolic valuation PD. Okay, what we call C. While in the complement, we know that we have this identity. Oh, the other way around, sorry. Correct. And this we can call D bar, the closure of D, okay? The value function needs to be larger than the fixed payoff, okay? And then we have the finite condition, the terminal condition, right? And maybe a boundary condition at infinity. This function has to tend to zero at infinity, okay? Okay, so I'm looking at the couple value function and this boundary as the solution of this problem, which is a free bound, which is what we call free boundary PD, okay? We're looking for the solution and this boundary at the same time. Okay, so this small x is, oh, by the way, L by L, you already imagine that, but this is the infinitesimal generator of the process x. So this would be Rx, first derivative, plus one-half, second-order term, okay? So this x, the boundary between these two regions is what we call the exercise boundary of the American option. This x is what we call exercise boundary. Why? Because, well, the stopping time tau star, defined by the infimum of times at which I go below the level of this boundary, is optimal. Is optimal for the stopping problem, defining the value function, okay? So it starts from here, if I start from zero, okay, the process diffuses and this is the time, the first time it goes below this boundary, it's the optimal time for stopping. Okay, so if we wonder about the properties of this function, okay, so a lot of properties are known, because this function has been studied in several works, okay? So in this setting, we know that this function is monotone, it is C1 over zero T, opening T, and its limit at capital T is equal to K, okay? So it looks like the function I've drawn over there, okay? So it's defined by a lot of authors, so by the way, this kind of system goes back to the relation of these to the optimal stopping, goes back to one of the usual aspects of these same racks, okay? It goes back to works of MacKinn 65, and then properties of these exercise boundaries have been studied by a lot of several authors between the 70s and nowadays, okay? Which I'm not going to quote so that I don't forget anyone. So the properties that we know, okay? So what I like to use now is the fact that for this PDE, for this free boundary problem, we have an integral, integral, no, an integral representation. Yes, yes. The boundary needs to be equal to minus one, yes, something I forgot, yeah. Good point, yes. This is something I forgot, so I will write it here. In addition, you want that the derivative in space of this function, we can write it like that. The limit from above is equal to minus one, okay? Okay, so this from below in this region, we will have VTX equal to K minus X positive part, and this is actually equal to K minus X, okay? Because we are below K, so positive part K minus X is positive, okay? So the derivative of K minus X is minus one, and indeed a property that function U satisfies and that we need to ask in this system is that the first derivative in space is continuous while crossing the boundary, okay? Yes, thank you. So we are going to, what I'm calling integral representation, we can derive it in this way. We can derive it in this way. We just wrote for the function U, we wrote it on the set C, the function U satisfies this equation, and that on D, which is an open set for what I've just said on the function X, the same thing will be equal to, okay? On this set D, the function U is equal to K minus X without positive part, because this is positive. So if we apply this operator to K minus X, time derivative, zero. In L, there is a second derivative which hits a linear function, zero. So the only part I'm left with is RX, the X of this minus R, K plus RX, right? So what we get, this is the most complicated computation of the whole talk. So what we get is minus RX plus RX minus RK, okay? On the region D, we have that this is a negative constant, okay? Good. So if we can write in one line the PDE, the equation satisfied by U, this is what? This is equal to zero on C, and on D, this is equal to minus RK. So I can write, I take it to the other side, plus RK, indicator of what? Indicator of X in D, so X is smaller than XT, right? I've written the two conditions in only one line, okay? But so we can see this as a linear PDE for U with a source term here, right? With a source term, which depends on T and X, and which is also a function of my exercise boundary that I will put there, okay? So we can see this equation for the couple UX as a nonlinear equation, okay? Because X, the function X of T is there inside the indicator function. In any case, we have a finite-cast representation for U, right? We have a finite-cast representation for U from this PDE. And this goes like that, expectation of the terminal value, K minus X, plus what? Plus integral of the source term, so RK, here I have expectation of what? Well, here I have expectation of F of XS, right? And so this is, according to my F, indicator function of XS smaller than the boundary, right? And I have to put initial condition here, okay? And so, well, we can rewrite this in a shorter way that I will use, okay? So this is exactly the price of the European option, right? This is what I call P, capital TK, and now I'm changing notation a little bit. I don't put sigma anymore. I put the initial condition TX over there, and then I have plus RK integral of the CDF of the process, okay? And then I have this, which I will write like that, where I'm introducing, okay, notation, somehow standard for the CDF of the process, okay? And my notation with F is the, as I said, the CDF, so probability that XS is smaller than Y if I start from TX, okay? So here we, what we have just written is the, what we have identified is the early premium that I introduced before, right? So we are writing the price of the American option as the price of the European option plus a positive correction, okay? Which is the early premium, and this is why we call this equation early premium formula. Okay, this is the early premium formula, okay? So why is this interesting? Because now from this we can get, we can get rid of U in this equation and get an integral equation, equation for the exercise boundary X. So let's say I want to get an integral, just inject, inject what the fact that along, along this curve, the function U, which is a continuous function on my space-time domain, is equal to K minus XT, okay? And if we have colors, if we do that, well, we will obtain here K minus XT, right? And we will get XT here, XT here, and XS there, okay? So we have this equation plus the terminal condition for, the terminal condition X capital T equal to K. This is giving me a way of numerically computing this exercise boundary, right? I start from the end, XT equal to K, and then I discretize time exactly as we did before. So I'm going to use some time TN to time TN minus one, solving this nonlinear equation, okay, for XTN minus one, okay? XTN minus one, XTN minus one, XTN minus one, okay? And then we have, of course, to, well, discretize, approximate this integral in some way, okay? So this is, okay, so this is something that we can do provided that we know how to compute PNF, okay, this expectation and the CDF, which we know only specific cases, okay, not in general. But so why is this interesting for us, for me, for us? If we compute this exercise boundary, okay, then we can re-inject it into the equation, okay? We go back to the white, suppose we have computed X of T, we have evaluated X of T, we go back to the white equation, UTX, PTX, FTX, XS is there, okay? But if I know, suppose that I know X of S, okay? Now what we have, what is left to do, and I can compute F, what is left to do is to compute this integral, this, compute numerically this one-dimensional integral with respect to time, with my favorite quadratore scheme, okay? This is a very simple task. So we have nailed down the problem of compute solving the optimal stopping problem to computing a one-dimensional integral with respect to time. Provided we know these functions, right? So back to my issue, this expectation and this CDF, we know it explicitly, we know that explicitly only specific situations. For example, when my process X is geometric when I motion, this will be written in terms of the Gaussian CDF, but I want to do it in my more general setting with a generic function sigma X, okay? And so this is the moment when we can call upon asymptotic calculus for diffusion processes, okay? And so, last part, time left, what can we work out for the asymptotics of the problem? Asymptotics in which sense? Asymptotics when time goes to zero, okay? So for this, I will consider a slightly simpler setting for the presentation, which is a function sigma that does not depend on time. Okay? Only sigma of X, which is of course a limitation that we don't want to have in practice. But okay, I'll consider this setting in the presentation, okay? So what we know about this exercise boundary X is that it tends to K, okay? What we want to know is more, we want to have a whole asymptotic expansion of X, T as T goes to capital T, okay? Which means that for this function here that I drew, we want to know how it behaves here, okay? We want to refine what is known as asymptotic behavior, okay? And I will just denote, okay, let me notice that X, T is a homogeneous market process, okay? And then I can denote, I can reverse time, I can reverse time and denote U bar, okay? U bar tau of X as the value function in terms of time to maturity. This will be U of T minus tau X, okay? For any T, for any capital T, okay? The same for the exercise boundary. The same for the exercise boundary, so we'll denote X bar of tau, the exercise boundary as a function of time to maturity, okay? What we know now is that X bar of tau tends to K as tau tends to zero, okay? We are sending, tau now is no more stopping time, of course. We are just sending time to maturity to zero. Okay, so our contribution in this setting with my collaborator, by the way, Pierre-Anne Labarder, whom I haven't mentioned yet, our contribution is done in two steps. So step one is to derive an alternative representation for the exercise boundary, okay? Which is written in terms of what? In terms of the density, the transition density of my process, okay? So I will denote, if you forgive me for introducing notation now, I will denote F of T, Y, X. This should be readable. The transition density of the process from point X to point Y in time T. Homogeneous microprocess, we only have a time lapse, okay? And in symbols, this is this low, right? So the alternative representation that we have is the following. I will write it. Okay, so the density, the transition density of my process from X bar tau to K equal to... Okay, well... I may be forgot some exponentials before in the Feynman-Ketz representation. I believe I forgot this exponential in the integrator, but okay, let's say it's not such a big deal. Integral of the density computed at this point. Okay, so... Okay, so what do we have? This exercise boundary here, here, here, and this derivative, okay? So the picture would be like that. T minus tau. Okay, we have an equation that involves this value, X bar of tau, and we are looking at the transition of the process from X bar of tau to K here, and here we have an integral functional that depends on the density of the process computed along this... transition density along this curve, okay? So you might say earlier we had this early premium formula written in terms of expectations, right? Put price and CDF. We might manipulate it in such a way that I might prove, but I probably won't. And we manipulate it in such a way that we write it as an equivalent, actually, representation that makes the density of the process appear, okay? Good. So what did I gain? Okay, what did I gain? Well, what we gained is the fact that now, to this representation, we can apply step two, which is using what we know under asymptotic behavior of the density of the process as time goes to zero, right? So now we open the way to decades of literature about heat kernel expansions in small time, right? And in particular, what we know in this simple one-dimensional case is that the density of the process looks like that. There is an exponential term, okay, which depends on x, y over tau, okay? For Brownian motion, this d would be x minus y square. D would be x minus y. Sorry, I'm not finished, okay? But just let me point out what d is. This d square is, okay, in a fancy way, it is the square geodesic distance of x to y in the Riemannian metric induced by the function sigma, okay? Which means what? This is a Riemannian distance, so it is the infimum of lengths of curves which are integral from 0 to 1 of phi dot t divided by my sigma phi square. The infimum computed over all the phi that are, while in camera or marketing space, of Brownian motion, h1, okay? Absolutely continuous. We dialed two first derivatives and that connects x to y. Okay? So this is really a distance, right? You look at all the curves that go from x to y and you compute the one that has minimum, you consider the one that has minimum distance according to this metric, okay? But then in a much simpler way, in a much simpler way, in this one dimensional setting, this function d x, y is actually just equal to the integral from x to y of... 1 over sigma, okay? Okay, which is an easier way of looking at it. Okay? So this function, we know it and then there's something in front, okay? A one dimensional diffusion has a time scale that is square root of 2, okay? And then we have some coefficient in front which is precisely this, okay? So a certain coefficient depending on sigma divided by sigma y 3 half, why not? Sorry for that. 1 plus big O of tau, okay? So this is an exact expansion for the density of the process and now what you do is to inject this into my new integral representation, okay? So you do this, you inject this expression and now the problem you are left with is the problem of working out the asymptotic expansion of x as tau goes to zero, the asymptotic expansion of x from the explicit asymptotic expansion of the density f, okay? So you are left with a problem of asymptotic analysis, which is maybe not a piece of cake, I mean in which in particular you will need to use Laplace method to study the asymptotic of this integral, okay? But so you are left with that and conclusion, you obtain an asymptotic expansion of this function, okay? Of this function here, which you can push a priority to arbitrary order, which refines the expansions for the exercise boundary that we are knowing in the literature, okay? So here we will stop in one minute, so you obtain an explicit, so the message that you obtain an explicit approximation for this function, you re-inject this function into the early planning formula that we had before and you have semi-closed approximation as a final result for the price of the American option. So for the solution of this optimal stopping problem, okay? So the numerical, which is of course very accurate, right? And the numerical results of which you can look in the mice lights, which will be on the, which will be online, which I wanted to show you, but I think it's better to stop because it's 11. Okay, thank you very much. Thank you very much, Stefano. Is there any question? Just can you recall where does your last representation come from? Yes, this one, so it really in a nutshell, in a nutshell, you basically take the, you start from the early planning formula for the original early planning formula for this function, okay? So here we have what? Sorry, we have this plus I'm changing time, okay, I'm making some time change, so we have this, okay? So we have this, we have this plus I'm changing time, okay? So we have this plus I'm changing time, okay? So this D will pop up from Dupers PD, right? This is the second order term in Dupers PD, and what you're left with is still the integral term that you have when you differentiate with respect to tau, you see you have to differentiate here, the density comes from the derivative of the CDF, and then you have this derivative that appears, okay? So this type is simple, this type is simple, okay? So what is more involved is, than the second one. Yes, please. So you develop this expansion by more, you have only the big old tau down, but of course you can make a complete series for this expansion, and if you inject this expansion in the integral formula, then could you handle this, maybe taking more terms? So potentially yes, potentially yes, then what we notice is that, what we noticed is that actually the asymptotic of the exercise boundary, you know, here this function behaves as what, k minus sigma k times square root of tau log of gamma over tau, okay? So it has a behavior, almost square root behavior in tau times log of one over tau, okay? This is what pops up, and actually these and the further refinements here only depend on what is here. So they don't depend, because there's square root of tau, the square root of tau, so all the terms that come from here, okay, that comes from this big old tau, let's say heuristically, these terms they don't count in the, so I don't know, I don't need this. Any more questions? You have a kind of a integral equation, for your boundary, do you apply a kind of method to solve this equation, for example, if there's a kind of big alteration, something like this, why? Of course, of course, as you say, we could apply methods for this integral equation for this one or for the original one, okay, for this one, if we know the density. So we could, of course, we could. What our aim was to obtain this explicit formulas for the exercise boundary, in the sense that at the end, you know, for you plug in this formula, in the end globally what you have is an almost explicit formula for an American option. So, the only numerical task left is to compute this integral with respect to time, okay. So, for calibration, what we have in mind is calibration purposes that I haven't mentioned here, but okay, so as usual, right, you want semi-closed formulas to have faster computations, and so our aim was really to work out an explicit formula rather than a numerical solution of this equation. I don't know if this answers to. Yeah, but I mean, is this is only up to this error? Yes. So, if your options are not short, might be original? Of course, yeah, of course. If the options are not short, maturity, this will worsen, the quality of the approximation will worsen. So, the options typically are, have no, maturity is no longer than one and a half or two years, for which we tested our approximation, and of course it works very well. You have discrete time on the. You have discrete time on the, so you mean Bermudan? Okay, yeah, fair enough. Then either we just assume that our Bermudan are American or, well, yes, of course, for discrete time American option, this will change. Okay, let's thank Stefano once again.
 Thanks, so I'd like to thank Andrea, Href, Cyril and Jean-François for the invitation  and for accepting to move my talk on schedule because I was supposed to speak tomorrow, which  is also the reason I apologize with you because you will have a second long talk today.  So the talk of today, so my talk, sorry, will be somehow different with respect to many  other ones we've had here at CEMACS, which we are focusing on probabilistic numerical  methods for high dimensional problems or for their mean field limits.  I would rather focus on the problem of finding exact approximate solutions for low dimensional  nonlinear problems and precisely the nonlinear PDs that appear in optimal stopping.  So it will be somehow orthogonal to the direction that many talks had until now.  So my motivation comes from mathematical finance.  So this first part will be maybe somehow obvious for some of you, but hopefully not for others.  So the motivation is the following.  In mathematical finance, we have a stochastic process that modifies financial asset, a stock,  let's say a stock, so the stock of Google or Societé Générale, for example, where  my co-author works.  So what we say is that this is a price expressed in euros of Societé Générale stock and  we model it with an SDA.  So WT is a marine motion in dimension one.  And by the way, what do I mean by low dimension?  Once a colleague of mine said that high dimension starts from D equal 2, so by contrast, low  dimension will be D equal 1.  So Brownian motion dimension one and sigma is a function of time and space with values  possibly in some, well, with values we're taking some bounded intervals, sigma bar,  low and high, plus some standard leapshield conditions in space uniformly in time.  So this is our model.  What do we have on the market?  On the market we have some financial products that are called European options.  What is that?  Well, we fix a future time horizon, you can think of T equal one year or T equal one month.  We fix a certain constant, K, positive.  This is a price, so this is expressed in euros, okay?  And we can buy a contract on the market that gives us this amount of money in the future.  The maximum of K minus the future value of S, okay?  So you can think, for those who have never seen this, you can think of this as an insurance  product, right?  You are protecting yourself against a drop or against a decrease in the value of S.  You are thinking that S in the future will be very low, we go from one other to 50 and  if you have both this contract you will get some money, okay?  Otherwise if S is high you will get zero, but okay?  So you're protecting yourself against drops in the values of the asset.  Okay, so this we can rewrite in a simple way as positive part of K minus S T. Okay, good.  So the theory for pricing and managing these kind of products tell us this object, okay?  I'm buying the right of getting some money in the future, a capital T.  I have to pay something now in order to have this right in the future, okay?  So the price of this contract now, at time zero, is given by an expectation.  Expectation of exponential minus R T, positive part K minus X T, where X is another process.  That satisfies this S D, where R, small r is a constant that I'm introducing, positive.  That represent the interest rate that you get if you invest money in a bank account,  okay?  So we have to compute, sorry, and maybe we have an initial condition here that is the  initial value S zero that we observe over there, okay?  So the problem, so the price is given an expectation, the problem we have is computing  an expectation, okay?  For which you can use, now the numerical problem we have is computing expectation based on  stochastic process, solution of an S D for which you can use your favorite numerical method,  right?  You can use Monte Carlo.  In some situations you can maybe compute this explicitly.  This is an integral against the law of X.  Or maybe you can solve the linear PDE that is satisfied by this expectation, right?  Okay.  So you can choose your numerical method.  And maybe at this level, I'd like to point out that if we call this function, I'm not  managing this blackboards very well.  This function we would call it P, TK.  There's two parameters, T and K.  P, you can think P stands for price.  Actually, this is a put option.  So it's the P for put option.  It will depend on sigma as well, right?  On the sigma that is inside my process, okay?  I'd like to point out that something that is known is this thing is that this function  solves a certain PDE.  Solves the following PDE.  Okay.  Parabolic PDE with an initial condition P0K equal to, well, what you obtain when you  inject T equals zero there.  Okay.  So we have a forward PDE, which is nothing but the Fokker Planck PDE for this process  written in terms of the function P, okay?  You can think of this, so there's many ways of proving.  This PDE, you can think of it as the Fokker Planck forward PDE written in terms of the  function P sigma.  Okay.  Good, so you can tackle the problem from the side that you want and everything here is  linear.  Linear expectations, linear PDEs.  Okay, then we come to another type of product, which are this American option.  I'm going to focus on...  During the talk, okay?  So we still fix a future time and some positive price K, okay?  And this American option is a financial contract that still gives you some money in the future,  okay?  But you don't necessarily look at this difference K minus S at final time.  You can look at this at any time tau is more than capital T, okay?  So at any time tau, you can stop the game and you say, now I want to get this amount  of money, okay?  Which means that you exercise your option at tau.  So this is what you get if you exercise your right at time tau is more than capital T,  okay?  Okay, so again, the theory, which I'm not going to recall indeed as, but the theory  for pricing and hedging, for pricing this kind of object tell us that now the price I have  to pay today in order to have this right in the future, okay?  This is the American option.  The price of this American option is not an expectation but a supremum of expectation.  Okay, it is in the end the value function of an optimal stopping problem, okay?  So it's the supremum of these expectations over what?  Over all, toes that belong to this base capital tau that I denote in a standard way like that,  which is the space of stopping times in the augmented filtration of the running motion  W, okay, with values in zero capital T, okay?  So we have this, yes?  So just to remind you that perhaps you are using a different signal from there to there  because you changed that from the motivation that the price increases S, you have the,  for I think the C, C, C, S, U, U, U, U, U, U, U, U, U, U, U, U, U, U, U, U, U, U, U, U,  which, what do you mean, so why might it not be the same function I�  Sorry, maybe the dú´ cos hasn�t it fine.  So, the variable is just a variable.  It has to be changing from the single to having a problem.  They're changing variables.  I'm changing variables, yes, but you agree that the price of this option is a number?  No, yeah.  That I can write in this way?  Yes, that's possible, but just that you're changing the process is different.  But, okay, but you...  Okay, so we can discuss about that later.  So you agree with this identity and with this signal?  I agree with the future.  And I agree with the future.  Okay.  But the sigma in the X is the same as the sigma in the X.  Yes, it's the same, yes.  This is what I mean, it's really the same.  But the sigma in the P is just like a symbol, is it a symbol of Pk?  Yes, yes, this is a symbol.  So this is an expectation that depends on a lot of things, on T, on K, on R, which I don't put,  and it will depend also on the function sigma that I put there.  Yeah.  Okay.  So, George, there's something I missed in what you said, I believe, but okay, if it's okay to go on like that,  I would be happy to.  So, yes, so now we have this problem to solve, okay?  So, I'm going to take a small remark, en passant, which I'm going to go back to afterwards.  So you take in that problem tau equal capital T, okay?  And what do we get?  We get that this value, sorry again, this price now I will denote it as ATK, A for American, TK,  and sigma, the function sigma is still there, okay?  So, the point is that A sigma, TK is larger than this expectation computed at T.  So this will be larger than what I call P, right?  Just this expectation, the price of the European option plus something, plus a certain amount that will be positive, okay?  So, this amount, this is what we call the early premium in mathematical finance, okay?  Early premium because it's what you have to pay in addition to the previous price of the European option,  it's the additional price that you have to pay if you want the additional right of exercising your option  at any stopping time before capital T, okay?  And I will go back to this formulation in the future, so in my talk, okay?  So, now the problem we have is to solve numerically if we want to apply the model to market data,  we have to solve numerically this problem over there, okay?  Which is more complicated than just computing an expectation, okay?  So, this is the moment when we wonder about efficient valuation numerical methods for optimal stopping problems, okay?  So, hopefully, quickly, I'd like to remind what are standard classes of methods to solve this optimal stopping problem, okay?  Well, and now, from now on, I will start denoting UTX for T before capital T and X positive value.  So, what's the value function of this optimal stopping problem?  So, the price of the American option if I enter into the game at times small t, okay?  So, it will be the supremum of these expectations.  So, we have now taken over all stopping times, though, the take values in small t capital T, okay?  So, we use this notation, which should by now be clear.  Supremum over all stopping times in the filtration of W that take values in the interval small t capital T, okay?  So, we have here where we, the standard notation again, X T X S, right?  Is this solution of the ST starting at small x at time t, okay?  So, this value function is in the, a lot of previous talks, okay?  So, this value function satisfies what we call a backward dynamic programming principle, okay?  Which is suitable for time discretization in such a way that we can actually simulate the random variables, right?  So, there's a backward dynamic programming principle.  Which if we discretize time, okay, for which we can discretize time, take n dates between 0 and capital T,  regularly spaced, so Tk, K, T over n, which means that I'm taking a fixed time grid, T over n.  I can consider, and I know how to simulate, the Euler scheme for the process X, right?  So, what I can construct is a discrete time process U starting from the beginning, from the end, sorry, okay?  And then iterating backward from time ti plus 1 to time ti with this principle, okay?  This will represent my gain at time ti, it is the maximum between, so the value of my option at time ti is the maximum  between what I gain if I stop at ti and the conditional expectation of what I get if I wait one more step,  and if I wait one more step, my value is U ti plus 1.  Conditional expectation with respect to my process at time ti, and I have to take the maximum between these two.  Possibly I'm forgetting this exponential term here, sorry, and I iterate from time n to time 0, okay?  And I'm using the value U0 that I obtain as an approximation of the solution of my value function at time 0, okay?  I forgot the plus, yes, okay, thank you, thank you, yes.  Did I forget it?  Everywhere, okay, everywhere, yeah, thanks.  So this now opens the way to Monte Carlo methods, right?  Because the problem I'm left with is to simulate this conditional expectation to which I can apply my favorite regression method, okay?  Which means using the IID simulations of the Euler scheme, I can switch to regress,  and then choosing a basis of functions on which to project in order to replace this conditional expectation with a finite dimensional projection, okay?  And we have heard about this in a lot of presentations during some rags, for example, Plum and Trocadius, right?  So this opened the ways to Monte Carlo methods, which are potentially applicable in higher dimension, okay?  Which means that you have several processes, X1, X2, up to XD.  But as I mentioned in the beginning, what I'm interested in now is the analytical counterpart of this problem in low dimension.  And this is a free boundary PDE, okay, which is written in the following way, okay?  So let me notice, first of all, that this value function over there is always larger than K minus X positive part, right?  Just take tau equal small t, okay?  It's always larger than this, which means that it can be larger, which can be strictly greater or equal, okay?  So it makes sense to define this object, small x of t, this is a function of time, which is the infimum of point X,  at which this function is strictly larger than the value function is strictly larger than this fixed function, okay?  And if we do this, the picture looks like that.  If we do this, the picture looks like that.  This is time, capital time, strike is here, okay?  This function X looks, has this behavior, okay?  So this is small x of t, and I'm dividing the time-space domain in two regions, one in which my value function is larger than this,  and one in which it is equal, okay?  So we can call this C and this other one, we can call it D, okay?  So for the notation capital X is the process, small x of t is the boundary, okay?  If you, I have complaints about this notation, I'd be happy to forward them to my co-author who chose it.  So if we do this, then the statement is that U and this function of time X is the unique solution to the following problem.  Okay.  To the following problem.  Okay, so in this region, what this identity is telling us is that it's optimal to stop immediately, right?  If we look at this optimal stopping problem, UTX equal K minus X means that tau equals small t is an optimal stopping time, right?  So here it's optimal to stop immediately, here it is not.  Here it's optimal to wait a little bit before exercising, okay?  So it makes sense that in this region, the function, the value function satisfies the standard parabolic valuation PD.  Okay, what we call C.  While in the complement, we know that we have this identity.  Oh, the other way around, sorry.  Correct.  And this we can call D bar, the closure of D, okay?  The value function needs to be larger than the fixed payoff, okay?  And then we have the finite condition, the terminal condition, right?  And maybe a boundary condition at infinity.  This function has to tend to zero at infinity, okay?  Okay, so I'm looking at the couple value function and this boundary as the solution of this problem, which is a free bound, which is what we call free boundary PD, okay?  We're looking for the solution and this boundary at the same time.  Okay, so this small x is, oh, by the way, L by L, you already imagine that, but this is the infinitesimal generator of the process x.  So this would be Rx, first derivative, plus one-half, second-order term, okay?  So this x, the boundary between these two regions is what we call the exercise boundary of the American option.  This x is what we call exercise boundary.  Why? Because, well, the stopping time tau star, defined by the infimum of times at which I go below the level of this boundary, is optimal.  Is optimal for the stopping problem, defining the value function, okay?  So it starts from here, if I start from zero, okay, the process diffuses and this is the time, the first time it goes below this boundary, it's the optimal time for stopping.  Okay, so if we wonder about the properties of this function, okay, so a lot of properties are known, because this function has been studied in several works, okay?  So in this setting, we know that this function is monotone, it is C1 over zero T, opening T, and its limit at capital T is equal to K, okay?  So it looks like the function I've drawn over there, okay?  So it's defined by a lot of authors, so by the way, this kind of system goes back to the relation of these to the optimal stopping, goes back to one of the usual aspects of these same racks, okay?  It goes back to works of MacKinn 65, and then properties of these exercise boundaries have been studied by a lot of several authors between the 70s and nowadays, okay?  Which I'm not going to quote so that I don't forget anyone.  So the properties that we know, okay?  So what I like to use now is the fact that for this PDE, for this free boundary problem, we have an integral, integral, no, an integral representation.  Yes, yes.  The boundary needs to be equal to minus one, yes, something I forgot, yeah.  Good point, yes.  This is something I forgot, so I will write it here.  In addition, you want that the derivative in space of this function, we can write it like that.  The limit from above is equal to minus one, okay?  Okay, so this from below in this region, we will have VTX equal to K minus X positive part, and this is actually equal to K minus X, okay?  Because we are below K, so positive part K minus X is positive, okay?  So the derivative of K minus X is minus one, and indeed a property that function U satisfies and that we need to ask in this system is that the first derivative in space is continuous while crossing the boundary, okay?  Yes, thank you.  So we are going to, what I'm calling integral representation, we can derive it in this way.  We can derive it in this way.  We just wrote for the function U, we wrote it on the set C, the function U satisfies this equation, and that on D, which is an open set for what I've just said on the function X,  the same thing will be equal to, okay?  On this set D, the function U is equal to K minus X without positive part, because this is positive.  So if we apply this operator to K minus X, time derivative, zero.  In L, there is a second derivative which hits a linear function, zero.  So the only part I'm left with is RX, the X of this minus R, K plus RX, right?  So what we get, this is the most complicated computation of the whole talk.  So what we get is minus RX plus RX minus RK, okay?  On the region D, we have that this is a negative constant, okay?  Good.  So if we can write in one line the PDE, the equation satisfied by U, this is what?  This is equal to zero on C, and on D, this is equal to minus RK.  So I can write, I take it to the other side, plus RK, indicator of what?  Indicator of X in D, so X is smaller than XT, right?  I've written the two conditions in only one line, okay?  But so we can see this as a linear PDE for U with a source term here, right?  With a source term, which depends on T and X, and which is also a function of my exercise boundary that I will put there, okay?  So we can see this equation for the couple UX as a nonlinear equation, okay?  Because X, the function X of T is there inside the indicator function.  In any case, we have a finite-cast representation for U, right?  We have a finite-cast representation for U from this PDE.  And this goes like that, expectation of the terminal value, K minus X, plus what?  Plus integral of the source term, so RK, here I have expectation of what?  Well, here I have expectation of F of XS, right?  And so this is, according to my F, indicator function of XS smaller than the boundary, right?  And I have to put initial condition here, okay?  And so, well, we can rewrite this in a shorter way that I will use, okay?  So this is exactly the price of the European option, right?  This is what I call P, capital TK, and now I'm changing notation a little bit.  I don't put sigma anymore.  I put the initial condition TX over there, and then I have plus RK integral of the CDF of the process, okay?  And then I have this, which I will write like that, where I'm introducing, okay, notation, somehow standard for the CDF of the process, okay?  And my notation with F is the, as I said, the CDF, so probability that XS is smaller than Y if I start from TX, okay?  So here we, what we have just written is the, what we have identified is the early premium that I introduced before, right?  So we are writing the price of the American option as the price of the European option plus a positive correction, okay?  Which is the early premium, and this is why we call this equation early premium formula.  Okay, this is the early premium formula, okay?  So why is this interesting?  Because now from this we can get, we can get rid of U in this equation and get an integral equation, equation for the exercise boundary X.  So let's say I want to get an integral, just inject, inject what the fact that along, along this curve, the function U, which is a continuous function on my space-time domain,  is equal to K minus XT, okay?  And if we have colors, if we do that, well, we will obtain here K minus XT, right?  And we will get XT here, XT here, and XS there, okay?  So we have this equation plus the terminal condition for, the terminal condition X capital T equal to K.  This is giving me a way of numerically computing this exercise boundary, right?  I start from the end, XT equal to K, and then I discretize time exactly as we did before.  So I'm going to use some time TN to time TN minus one, solving this nonlinear equation, okay, for XTN minus one, okay?  XTN minus one, XTN minus one, XTN minus one, okay?  And then we have, of course, to, well, discretize, approximate this integral in some way, okay?  So this is, okay, so this is something that we can do provided that we know how to compute PNF, okay, this expectation and the CDF,  which we know only specific cases, okay, not in general.  But so why is this interesting for us, for me, for us?  If we compute this exercise boundary, okay, then we can re-inject it into the equation, okay?  We go back to the white, suppose we have computed X of T, we have evaluated X of T,  we go back to the white equation, UTX, PTX, FTX, XS is there, okay?  But if I know, suppose that I know X of S, okay?  Now what we have, what is left to do, and I can compute F, what is left to do is to compute this integral, this,  compute numerically this one-dimensional integral with respect to time, with my favorite quadratore scheme, okay?  This is a very simple task.  So we have nailed down the problem of compute solving the optimal stopping problem to computing a one-dimensional integral with respect to time.  Provided we know these functions, right?  So back to my issue, this expectation and this CDF, we know it explicitly, we know that explicitly only specific situations.  For example, when my process X is geometric when I motion, this will be written in terms of the Gaussian CDF,  but I want to do it in my more general setting with a generic function sigma X, okay?  And so this is the moment when we can call upon asymptotic calculus for diffusion processes, okay?  And so, last part, time left, what can we work out for the asymptotics of the problem?  Asymptotics in which sense? Asymptotics when time goes to zero, okay?  So for this, I will consider a slightly simpler setting for the presentation, which is a function sigma that does not depend on time.  Okay? Only sigma of X, which is of course a limitation that we don't want to have in practice.  But okay, I'll consider this setting in the presentation, okay?  So what we know about this exercise boundary X is that it tends to K, okay?  What we want to know is more, we want to have a whole asymptotic expansion of X, T as T goes to capital T, okay?  Which means that for this function here that I drew, we want to know how it behaves here, okay?  We want to refine what is known as asymptotic behavior, okay?  And I will just denote, okay, let me notice that X, T is a homogeneous market process, okay?  And then I can denote, I can reverse time, I can reverse time and denote U bar, okay? U bar tau of X as the value function in terms of time to maturity.  This will be U of T minus tau X, okay? For any T, for any capital T, okay?  The same for the exercise boundary.  The same for the exercise boundary, so we'll denote X bar of tau, the exercise boundary as a function of time to maturity, okay?  What we know now is that X bar of tau tends to K as tau tends to zero, okay?  We are sending, tau now is no more stopping time, of course.  We are just sending time to maturity to zero.  Okay, so our contribution in this setting with my collaborator, by the way, Pierre-Anne Labarder, whom I haven't mentioned yet,  our contribution is done in two steps.  So step one is to derive an alternative representation for the exercise boundary, okay?  Which is written in terms of what? In terms of the density, the transition density of my process, okay?  So I will denote, if you forgive me for introducing notation now, I will denote F of T, Y, X.  This should be readable.  The transition density of the process from point X to point Y in time T.  Homogeneous microprocess, we only have a time lapse, okay?  And in symbols, this is this low, right?  So the alternative representation that we have is the following.  I will write it. Okay, so the density, the transition density of my process  from X bar tau to K equal to...  Okay, well...  I may be forgot some exponentials before in the Feynman-Ketz representation.  I believe I forgot this exponential in the integrator, but okay, let's say it's not such a big deal.  Integral of the density computed at this point.  Okay, so...  Okay, so what do we have? This exercise boundary here, here, here, and this derivative, okay?  So the picture would be like that.  T minus tau.  Okay, we have an equation that involves this value, X bar of tau,  and we are looking at the transition of the process from X bar of tau to K here,  and here we have an integral functional that depends on the density of the process computed along this...  transition density along this curve, okay?  So you might say earlier we had this early premium formula written in terms of expectations, right?  Put price and CDF.  We might manipulate it in such a way that I might prove, but I probably won't.  And we manipulate it in such a way that we write it as an equivalent, actually, representation that makes the density of the process appear, okay?  Good.  So what did I gain? Okay, what did I gain?  Well, what we gained is the fact that now, to this representation,  we can apply step two, which is using what we know under asymptotic behavior  of the density of the process as time goes to zero, right?  So now we open the way to decades of literature about heat kernel expansions in small time, right?  And in particular, what we know in this simple one-dimensional case is that the density of the process looks like that.  There is an exponential term, okay, which depends on x, y over tau, okay?  For Brownian motion, this d would be x minus y square.  D would be x minus y. Sorry, I'm not finished, okay?  But just let me point out what d is.  This d square is, okay, in a fancy way, it is the square geodesic distance of x to y in the Riemannian metric induced by the function sigma, okay?  Which means what?  This is a Riemannian distance, so it is the infimum of lengths of curves which are integral from 0 to 1 of phi dot t divided by my sigma phi square.  The infimum computed over all the phi that are, while in camera or marketing space, of Brownian motion, h1, okay?  Absolutely continuous. We dialed two first derivatives and that connects x to y.  Okay? So this is really a distance, right?  You look at all the curves that go from x to y and you compute the one that has minimum, you consider the one that has minimum distance according to this metric, okay?  But then in a much simpler way, in a much simpler way, in this one dimensional setting, this function d x, y is actually just equal to the integral from x to y of...  1 over sigma, okay? Okay, which is an easier way of looking at it.  Okay? So this function, we know it and then there's something in front, okay?  A one dimensional diffusion has a time scale that is square root of 2, okay?  And then we have some coefficient in front which is precisely this, okay?  So a certain coefficient depending on sigma divided by sigma y 3 half, why not?  Sorry for that.  1 plus big O of tau, okay?  So this is an exact expansion for the density of the process and now what you do is to inject this into my new integral representation, okay?  So you do this, you inject this expression and now the problem you are left with is the problem of working out the asymptotic expansion of x as tau goes to zero, the asymptotic expansion of x from the explicit asymptotic expansion of the density f, okay?  So you are left with a problem of asymptotic analysis, which is maybe not a piece of cake, I mean in which in particular you will need to use Laplace method to study the asymptotic of this integral, okay?  But so you are left with that and conclusion, you obtain an asymptotic expansion of this function, okay?  Of this function here, which you can push a priority to arbitrary order, which refines the expansions for the exercise boundary that we are knowing in the literature, okay?  So here we will stop in one minute, so you obtain an explicit, so the message that you obtain an explicit approximation for this function, you re-inject this function into the early planning formula that we had before and you have semi-closed approximation as a final result for the price of the American option.  So for the solution of this optimal stopping problem, okay?  So the numerical, which is of course very accurate, right? And the numerical results of which you can look in the mice lights, which will be on the, which will be online, which I wanted to show you, but I think it's better to stop because it's 11.  Okay, thank you very much.  Thank you very much, Stefano. Is there any question?  Just can you recall where does your last representation come from?  Yes, this one, so it really in a nutshell, in a nutshell, you basically take the, you start from the early planning formula for the original early planning formula for this function, okay?  So here we have what? Sorry, we have this plus I'm changing time, okay, I'm making some time change, so we have this, okay?  So we have this, we have this plus I'm changing time, okay?  So we have this plus I'm changing time, okay?  So this D will pop up from Dupers PD, right? This is the second order term in Dupers PD, and what you're left with is still the integral term that you have when you differentiate with respect to tau, you see you have to differentiate here, the density comes from the derivative of the CDF, and then you have this derivative that appears, okay?  So this type is simple, this type is simple, okay?  So what is more involved is, than the second one.  Yes, please.  So you develop this expansion by more, you have only the big old tau down, but of course you can make a complete series for this expansion, and if you inject this expansion in the integral formula, then could you handle this, maybe taking more terms?  So potentially yes, potentially yes, then what we notice is that, what we noticed is that actually the asymptotic of the exercise boundary, you know, here this function behaves as what, k minus sigma k times square root of tau log of gamma over tau, okay?  So it has a behavior, almost square root behavior in tau times log of one over tau, okay?  This is what pops up, and actually these and the further refinements here only depend on what is here.  So they don't depend, because there's square root of tau, the square root of tau, so all the terms that come from here, okay, that comes from this big old tau, let's say heuristically, these terms they don't count in the, so I don't know, I don't need this.  Any more questions?  You have a kind of a integral equation, for your boundary, do you apply a kind of method to solve this equation, for example, if there's a kind of big alteration, something like this, why?  Of course, of course, as you say, we could apply methods for this integral equation for this one or for the original one, okay, for this one, if we know the density.  So we could, of course, we could.  What our aim was to obtain this explicit formulas for the exercise boundary, in the sense that at the end, you know, for you plug in this formula, in the end globally what you have is an almost explicit formula for an American option.  So, the only numerical task left is to compute this integral with respect to time, okay.  So, for calibration, what we have in mind is calibration purposes that I haven't mentioned here, but okay, so as usual, right, you want semi-closed formulas to have faster computations, and so our aim was really to work out an explicit formula rather than a numerical solution of this equation.  I don't know if this answers to.  Yeah, but I mean, is this is only up to this error?  Yes.  So, if your options are not short, might be original?  Of course, yeah, of course. If the options are not short, maturity, this will worsen, the quality of the approximation will worsen.  So, the options typically are, have no, maturity is no longer than one and a half or two years, for which we tested our approximation, and of course it works very well.  You have discrete time on the.  You have discrete time on the, so you mean Bermudan?  Okay, yeah, fair enough.  Then either we just assume that our Bermudan are American or, well, yes, of course, for discrete time American option, this will change.  Okay, let's thank Stefano once again.
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Thanks, so I'd like to thank Andrea, Href, Cyril and Jean-François for the invitation and for accepting to move my talk on schedule because I was supposed to speak tomorrow, which is also the reason I apologize with you because you will have a second long talk today. So the talk of today, so my talk, sorry, will be somehow different with respect to many other ones we've had here at CEMACS, which we are focusing on probabilistic numerical methods for high dimensional problems or for their mean field limits. I would rather focus on the problem of finding exact approximate solutions for low dimensional nonlinear problems and precisely the nonlinear PDs that appear in optimal stopping. So it will be somehow orthogonal to the direction that many talks had until now. So my motivation comes from mathematical finance. So this first part will be maybe somehow obvious for some of you, but hopefully not for others. So the motivation is the following. In mathematical finance, we have a stochastic process that modifies financial asset, a stock, let's say a stock, so the stock of Google or Societé Générale, for example, where my co-author works. So what we say is that this is a price expressed in euros of Societé Générale stock and we model it with an SDA. So WT is a marine motion in dimension one. And by the way, what do I mean by low dimension? Once a colleague of mine said that high dimension starts from D equal 2, so by contrast, low dimension will be D equal 1. So Brownian motion dimension one and sigma is a function of time and space with values possibly in some, well, with values we're taking some bounded intervals, sigma bar, low and high, plus some standard leapshield conditions in space uniformly in time. So this is our model. What do we have on the market? On the market we have some financial products that are called European options. What is that? Well, we fix a future time horizon, you can think of T equal one year or T equal one month. We fix a certain constant, K, positive. This is a price, so this is expressed in euros, okay? And we can buy a contract on the market that gives us this amount of money in the future. The maximum of K minus the future value of S, okay? So you can think, for those who have never seen this, you can think of this as an insurance product, right? You are protecting yourself against a drop or against a decrease in the value of S. You are thinking that S in the future will be very low, we go from one other to 50 and if you have both this contract you will get some money, okay? Otherwise if S is high you will get zero, but okay? So you're protecting yourself against drops in the values of the asset. Okay, so this we can rewrite in a simple way as positive part of K minus S T. Okay, good. So the theory for pricing and managing these kind of products tell us this object, okay? I'm buying the right of getting some money in the future, a capital T. I have to pay something now in order to have this right in the future, okay? So the price of this contract now, at time zero, is given by an expectation. Expectation of exponential minus R T, positive part K minus X T, where X is another process. That satisfies this S D, where R, small r is a constant that I'm introducing, positive. That represent the interest rate that you get if you invest money in a bank account, okay? So we have to compute, sorry, and maybe we have an initial condition here that is the initial value S zero that we observe over there, okay? So the problem, so the price is given an expectation, the problem we have is computing an expectation, okay? For which you can use, now the numerical problem we have is computing expectation based on stochastic process, solution of an S D for which you can use your favorite numerical method, right? You can use Monte Carlo. In some situations you can maybe compute this explicitly. This is an integral against the law of X. Or maybe you can solve the linear PDE that is satisfied by this expectation, right? Okay. So you can choose your numerical method. And maybe at this level, I'd like to point out that if we call this function, I'm not managing this blackboards very well. This function we would call it P, TK. There's two parameters, T and K. P, you can think P stands for price. Actually, this is a put option. So it's the P for put option. It will depend on sigma as well, right? On the sigma that is inside my process, okay? I'd like to point out that something that is known is this thing is that this function solves a certain PDE. Solves the following PDE. Okay. Parabolic PDE with an initial condition P0K equal to, well, what you obtain when you inject T equals zero there. Okay. So we have a forward PDE, which is nothing but the Fokker Planck PDE for this process written in terms of the function P, okay? You can think of this, so there's many ways of proving. This PDE, you can think of it as the Fokker Planck forward PDE written in terms of the function P sigma. Okay. Good, so you can tackle the problem from the side that you want and everything here is linear. Linear expectations, linear PDEs. Okay, then we come to another type of product, which are this American option. I'm going to focus on... During the talk, okay? So we still fix a future time and some positive price K, okay? And this American option is a financial contract that still gives you some money in the future, okay? But you don't necessarily look at this difference K minus S at final time. You can look at this at any time tau is more than capital T, okay? So at any time tau, you can stop the game and you say, now I want to get this amount of money, okay? Which means that you exercise your option at tau. So this is what you get if you exercise your right at time tau is more than capital T, okay? Okay, so again, the theory, which I'm not going to recall indeed as, but the theory for pricing and hedging, for pricing this kind of object tell us that now the price I have to pay today in order to have this right in the future, okay? This is the American option. The price of this American option is not an expectation but a supremum of expectation. Okay, it is in the end the value function of an optimal stopping problem, okay? So it's the supremum of these expectations over what? Over all, toes that belong to this base capital tau that I denote in a standard way like that, which is the space of stopping times in the augmented filtration of the running motion W, okay, with values in zero capital T, okay? So we have this, yes? So just to remind you that perhaps you are using a different signal from there to there because you changed that from the motivation that the price increases S, you have the, for I think the C, C, C, S, U, U, U, U, U, U, U, U, U, U, U, U, U, U, U, U, U, U, U, U, which, what do you mean, so why might it not be the same function I� Sorry, maybe the dú´ cos hasn�t it fine. So, the variable is just a variable. It has to be changing from the single to having a problem. They're changing variables. I'm changing variables, yes, but you agree that the price of this option is a number? No, yeah. That I can write in this way? Yes, that's possible, but just that you're changing the process is different. But, okay, but you... Okay, so we can discuss about that later. So you agree with this identity and with this signal? I agree with the future. And I agree with the future. Okay. But the sigma in the X is the same as the sigma in the X. Yes, it's the same, yes. This is what I mean, it's really the same. But the sigma in the P is just like a symbol, is it a symbol of Pk? Yes, yes, this is a symbol. So this is an expectation that depends on a lot of things, on T, on K, on R, which I don't put, and it will depend also on the function sigma that I put there. Yeah. Okay. So, George, there's something I missed in what you said, I believe, but okay, if it's okay to go on like that, I would be happy to. So, yes, so now we have this problem to solve, okay? So, I'm going to take a small remark, en passant, which I'm going to go back to afterwards. So you take in that problem tau equal capital T, okay? And what do we get? We get that this value, sorry again, this price now I will denote it as ATK, A for American, TK, and sigma, the function sigma is still there, okay? So, the point is that A sigma, TK is larger than this expectation computed at T. So this will be larger than what I call P, right? Just this expectation, the price of the European option plus something, plus a certain amount that will be positive, okay? So, this amount, this is what we call the early premium in mathematical finance, okay? Early premium because it's what you have to pay in addition to the previous price of the European option, it's the additional price that you have to pay if you want the additional right of exercising your option at any stopping time before capital T, okay? And I will go back to this formulation in the future, so in my talk, okay? So, now the problem we have is to solve numerically if we want to apply the model to market data, we have to solve numerically this problem over there, okay? Which is more complicated than just computing an expectation, okay? So, this is the moment when we wonder about efficient valuation numerical methods for optimal stopping problems, okay? So, hopefully, quickly, I'd like to remind what are standard classes of methods to solve this optimal stopping problem, okay? Well, and now, from now on, I will start denoting UTX for T before capital T and X positive value. So, what's the value function of this optimal stopping problem? So, the price of the American option if I enter into the game at times small t, okay? So, it will be the supremum of these expectations. So, we have now taken over all stopping times, though, the take values in small t capital T, okay? So, we use this notation, which should by now be clear. Supremum over all stopping times in the filtration of W that take values in the interval small t capital T, okay? So, we have here where we, the standard notation again, X T X S, right? Is this solution of the ST starting at small x at time t, okay? So, this value function is in the, a lot of previous talks, okay? So, this value function satisfies what we call a backward dynamic programming principle, okay? Which is suitable for time discretization in such a way that we can actually simulate the random variables, right? So, there's a backward dynamic programming principle. Which if we discretize time, okay, for which we can discretize time, take n dates between 0 and capital T, regularly spaced, so Tk, K, T over n, which means that I'm taking a fixed time grid, T over n. I can consider, and I know how to simulate, the Euler scheme for the process X, right? So, what I can construct is a discrete time process U starting from the beginning, from the end, sorry, okay? And then iterating backward from time ti plus 1 to time ti with this principle, okay? This will represent my gain at time ti, it is the maximum between, so the value of my option at time ti is the maximum between what I gain if I stop at ti and the conditional expectation of what I get if I wait one more step, and if I wait one more step, my value is U ti plus 1. Conditional expectation with respect to my process at time ti, and I have to take the maximum between these two. Possibly I'm forgetting this exponential term here, sorry, and I iterate from time n to time 0, okay? And I'm using the value U0 that I obtain as an approximation of the solution of my value function at time 0, okay? I forgot the plus, yes, okay, thank you, thank you, yes. Did I forget it? Everywhere, okay, everywhere, yeah, thanks. So this now opens the way to Monte Carlo methods, right? Because the problem I'm left with is to simulate this conditional expectation to which I can apply my favorite regression method, okay? Which means using the IID simulations of the Euler scheme, I can switch to regress, and then choosing a basis of functions on which to project in order to replace this conditional expectation with a finite dimensional projection, okay? And we have heard about this in a lot of presentations during some rags, for example, Plum and Trocadius, right? So this opened the ways to Monte Carlo methods, which are potentially applicable in higher dimension, okay? Which means that you have several processes, X1, X2, up to XD. But as I mentioned in the beginning, what I'm interested in now is the analytical counterpart of this problem in low dimension. And this is a free boundary PDE, okay, which is written in the following way, okay? So let me notice, first of all, that this value function over there is always larger than K minus X positive part, right? Just take tau equal small t, okay? It's always larger than this, which means that it can be larger, which can be strictly greater or equal, okay? So it makes sense to define this object, small x of t, this is a function of time, which is the infimum of point X, at which this function is strictly larger than the value function is strictly larger than this fixed function, okay? And if we do this, the picture looks like that. If we do this, the picture looks like that. This is time, capital time, strike is here, okay? This function X looks, has this behavior, okay? So this is small x of t, and I'm dividing the time-space domain in two regions, one in which my value function is larger than this, and one in which it is equal, okay? So we can call this C and this other one, we can call it D, okay? So for the notation capital X is the process, small x of t is the boundary, okay? If you, I have complaints about this notation, I'd be happy to forward them to my co-author who chose it. So if we do this, then the statement is that U and this function of time X is the unique solution to the following problem. Okay. To the following problem. Okay, so in this region, what this identity is telling us is that it's optimal to stop immediately, right? If we look at this optimal stopping problem, UTX equal K minus X means that tau equals small t is an optimal stopping time, right? So here it's optimal to stop immediately, here it is not. Here it's optimal to wait a little bit before exercising, okay? So it makes sense that in this region, the function, the value function satisfies the standard parabolic valuation PD. Okay, what we call C. While in the complement, we know that we have this identity. Oh, the other way around, sorry. Correct. And this we can call D bar, the closure of D, okay? The value function needs to be larger than the fixed payoff, okay? And then we have the finite condition, the terminal condition, right? And maybe a boundary condition at infinity. This function has to tend to zero at infinity, okay? Okay, so I'm looking at the couple value function and this boundary as the solution of this problem, which is a free bound, which is what we call free boundary PD, okay? We're looking for the solution and this boundary at the same time. Okay, so this small x is, oh, by the way, L by L, you already imagine that, but this is the infinitesimal generator of the process x. So this would be Rx, first derivative, plus one-half, second-order term, okay? So this x, the boundary between these two regions is what we call the exercise boundary of the American option. This x is what we call exercise boundary. Why? Because, well, the stopping time tau star, defined by the infimum of times at which I go below the level of this boundary, is optimal. Is optimal for the stopping problem, defining the value function, okay? So it starts from here, if I start from zero, okay, the process diffuses and this is the time, the first time it goes below this boundary, it's the optimal time for stopping. Okay, so if we wonder about the properties of this function, okay, so a lot of properties are known, because this function has been studied in several works, okay? So in this setting, we know that this function is monotone, it is C1 over zero T, opening T, and its limit at capital T is equal to K, okay? So it looks like the function I've drawn over there, okay? So it's defined by a lot of authors, so by the way, this kind of system goes back to the relation of these to the optimal stopping, goes back to one of the usual aspects of these same racks, okay? It goes back to works of MacKinn 65, and then properties of these exercise boundaries have been studied by a lot of several authors between the 70s and nowadays, okay? Which I'm not going to quote so that I don't forget anyone. So the properties that we know, okay? So what I like to use now is the fact that for this PDE, for this free boundary problem, we have an integral, integral, no, an integral representation. Yes, yes. The boundary needs to be equal to minus one, yes, something I forgot, yeah. Good point, yes. This is something I forgot, so I will write it here. In addition, you want that the derivative in space of this function, we can write it like that. The limit from above is equal to minus one, okay? Okay, so this from below in this region, we will have VTX equal to K minus X positive part, and this is actually equal to K minus X, okay? Because we are below K, so positive part K minus X is positive, okay? So the derivative of K minus X is minus one, and indeed a property that function U satisfies and that we need to ask in this system is that the first derivative in space is continuous while crossing the boundary, okay? Yes, thank you. So we are going to, what I'm calling integral representation, we can derive it in this way. We can derive it in this way. We just wrote for the function U, we wrote it on the set C, the function U satisfies this equation, and that on D, which is an open set for what I've just said on the function X, the same thing will be equal to, okay? On this set D, the function U is equal to K minus X without positive part, because this is positive. So if we apply this operator to K minus X, time derivative, zero. In L, there is a second derivative which hits a linear function, zero. So the only part I'm left with is RX, the X of this minus R, K plus RX, right? So what we get, this is the most complicated computation of the whole talk. So what we get is minus RX plus RX minus RK, okay? On the region D, we have that this is a negative constant, okay? Good. So if we can write in one line the PDE, the equation satisfied by U, this is what? This is equal to zero on C, and on D, this is equal to minus RK. So I can write, I take it to the other side, plus RK, indicator of what? Indicator of X in D, so X is smaller than XT, right? I've written the two conditions in only one line, okay? But so we can see this as a linear PDE for U with a source term here, right? With a source term, which depends on T and X, and which is also a function of my exercise boundary that I will put there, okay? So we can see this equation for the couple UX as a nonlinear equation, okay? Because X, the function X of T is there inside the indicator function. In any case, we have a finite-cast representation for U, right? We have a finite-cast representation for U from this PDE. And this goes like that, expectation of the terminal value, K minus X, plus what? Plus integral of the source term, so RK, here I have expectation of what? Well, here I have expectation of F of XS, right? And so this is, according to my F, indicator function of XS smaller than the boundary, right? And I have to put initial condition here, okay? And so, well, we can rewrite this in a shorter way that I will use, okay? So this is exactly the price of the European option, right? This is what I call P, capital TK, and now I'm changing notation a little bit. I don't put sigma anymore. I put the initial condition TX over there, and then I have plus RK integral of the CDF of the process, okay? And then I have this, which I will write like that, where I'm introducing, okay, notation, somehow standard for the CDF of the process, okay? And my notation with F is the, as I said, the CDF, so probability that XS is smaller than Y if I start from TX, okay? So here we, what we have just written is the, what we have identified is the early premium that I introduced before, right? So we are writing the price of the American option as the price of the European option plus a positive correction, okay? Which is the early premium, and this is why we call this equation early premium formula. Okay, this is the early premium formula, okay? So why is this interesting? Because now from this we can get, we can get rid of U in this equation and get an integral equation, equation for the exercise boundary X. So let's say I want to get an integral, just inject, inject what the fact that along, along this curve, the function U, which is a continuous function on my space-time domain, is equal to K minus XT, okay? And if we have colors, if we do that, well, we will obtain here K minus XT, right? And we will get XT here, XT here, and XS there, okay? So we have this equation plus the terminal condition for, the terminal condition X capital T equal to K. This is giving me a way of numerically computing this exercise boundary, right? I start from the end, XT equal to K, and then I discretize time exactly as we did before. So I'm going to use some time TN to time TN minus one, solving this nonlinear equation, okay, for XTN minus one, okay? XTN minus one, XTN minus one, XTN minus one, okay? And then we have, of course, to, well, discretize, approximate this integral in some way, okay? So this is, okay, so this is something that we can do provided that we know how to compute PNF, okay, this expectation and the CDF, which we know only specific cases, okay, not in general. But so why is this interesting for us, for me, for us? If we compute this exercise boundary, okay, then we can re-inject it into the equation, okay? We go back to the white, suppose we have computed X of T, we have evaluated X of T, we go back to the white equation, UTX, PTX, FTX, XS is there, okay? But if I know, suppose that I know X of S, okay? Now what we have, what is left to do, and I can compute F, what is left to do is to compute this integral, this, compute numerically this one-dimensional integral with respect to time, with my favorite quadratore scheme, okay? This is a very simple task. So we have nailed down the problem of compute solving the optimal stopping problem to computing a one-dimensional integral with respect to time. Provided we know these functions, right? So back to my issue, this expectation and this CDF, we know it explicitly, we know that explicitly only specific situations. For example, when my process X is geometric when I motion, this will be written in terms of the Gaussian CDF, but I want to do it in my more general setting with a generic function sigma X, okay? And so this is the moment when we can call upon asymptotic calculus for diffusion processes, okay? And so, last part, time left, what can we work out for the asymptotics of the problem? Asymptotics in which sense? Asymptotics when time goes to zero, okay? So for this, I will consider a slightly simpler setting for the presentation, which is a function sigma that does not depend on time. Okay? Only sigma of X, which is of course a limitation that we don't want to have in practice. But okay, I'll consider this setting in the presentation, okay? So what we know about this exercise boundary X is that it tends to K, okay? What we want to know is more, we want to have a whole asymptotic expansion of X, T as T goes to capital T, okay? Which means that for this function here that I drew, we want to know how it behaves here, okay? We want to refine what is known as asymptotic behavior, okay? And I will just denote, okay, let me notice that X, T is a homogeneous market process, okay? And then I can denote, I can reverse time, I can reverse time and denote U bar, okay? U bar tau of X as the value function in terms of time to maturity. This will be U of T minus tau X, okay? For any T, for any capital T, okay? The same for the exercise boundary. The same for the exercise boundary, so we'll denote X bar of tau, the exercise boundary as a function of time to maturity, okay? What we know now is that X bar of tau tends to K as tau tends to zero, okay? We are sending, tau now is no more stopping time, of course. We are just sending time to maturity to zero. Okay, so our contribution in this setting with my collaborator, by the way, Pierre-Anne Labarder, whom I haven't mentioned yet, our contribution is done in two steps. So step one is to derive an alternative representation for the exercise boundary, okay? Which is written in terms of what? In terms of the density, the transition density of my process, okay? So I will denote, if you forgive me for introducing notation now, I will denote F of T, Y, X. This should be readable. The transition density of the process from point X to point Y in time T. Homogeneous microprocess, we only have a time lapse, okay? And in symbols, this is this low, right? So the alternative representation that we have is the following. I will write it. Okay, so the density, the transition density of my process from X bar tau to K equal to... Okay, well... I may be forgot some exponentials before in the Feynman-Ketz representation. I believe I forgot this exponential in the integrator, but okay, let's say it's not such a big deal. Integral of the density computed at this point. Okay, so... Okay, so what do we have? This exercise boundary here, here, here, and this derivative, okay? So the picture would be like that. T minus tau. Okay, we have an equation that involves this value, X bar of tau, and we are looking at the transition of the process from X bar of tau to K here, and here we have an integral functional that depends on the density of the process computed along this... transition density along this curve, okay? So you might say earlier we had this early premium formula written in terms of expectations, right? Put price and CDF. We might manipulate it in such a way that I might prove, but I probably won't. And we manipulate it in such a way that we write it as an equivalent, actually, representation that makes the density of the process appear, okay? Good. So what did I gain? Okay, what did I gain? Well, what we gained is the fact that now, to this representation, we can apply step two, which is using what we know under asymptotic behavior of the density of the process as time goes to zero, right? So now we open the way to decades of literature about heat kernel expansions in small time, right? And in particular, what we know in this simple one-dimensional case is that the density of the process looks like that. There is an exponential term, okay, which depends on x, y over tau, okay? For Brownian motion, this d would be x minus y square. D would be x minus y. Sorry, I'm not finished, okay? But just let me point out what d is. This d square is, okay, in a fancy way, it is the square geodesic distance of x to y in the Riemannian metric induced by the function sigma, okay? Which means what? This is a Riemannian distance, so it is the infimum of lengths of curves which are integral from 0 to 1 of phi dot t divided by my sigma phi square. The infimum computed over all the phi that are, while in camera or marketing space, of Brownian motion, h1, okay? Absolutely continuous. We dialed two first derivatives and that connects x to y. Okay? So this is really a distance, right? You look at all the curves that go from x to y and you compute the one that has minimum, you consider the one that has minimum distance according to this metric, okay? But then in a much simpler way, in a much simpler way, in this one dimensional setting, this function d x, y is actually just equal to the integral from x to y of... 1 over sigma, okay? Okay, which is an easier way of looking at it. Okay? So this function, we know it and then there's something in front, okay? A one dimensional diffusion has a time scale that is square root of 2, okay? And then we have some coefficient in front which is precisely this, okay? So a certain coefficient depending on sigma divided by sigma y 3 half, why not? Sorry for that. 1 plus big O of tau, okay? So this is an exact expansion for the density of the process and now what you do is to inject this into my new integral representation, okay? So you do this, you inject this expression and now the problem you are left with is the problem of working out the asymptotic expansion of x as tau goes to zero, the asymptotic expansion of x from the explicit asymptotic expansion of the density f, okay? So you are left with a problem of asymptotic analysis, which is maybe not a piece of cake, I mean in which in particular you will need to use Laplace method to study the asymptotic of this integral, okay? But so you are left with that and conclusion, you obtain an asymptotic expansion of this function, okay? Of this function here, which you can push a priority to arbitrary order, which refines the expansions for the exercise boundary that we are knowing in the literature, okay? So here we will stop in one minute, so you obtain an explicit, so the message that you obtain an explicit approximation for this function, you re-inject this function into the early planning formula that we had before and you have semi-closed approximation as a final result for the price of the American option. So for the solution of this optimal stopping problem, okay? So the numerical, which is of course very accurate, right? And the numerical results of which you can look in the mice lights, which will be on the, which will be online, which I wanted to show you, but I think it's better to stop because it's 11. Okay, thank you very much. Thank you very much, Stefano. Is there any question? Just can you recall where does your last representation come from? Yes, this one, so it really in a nutshell, in a nutshell, you basically take the, you start from the early planning formula for the original early planning formula for this function, okay? So here we have what? Sorry, we have this plus I'm changing time, okay, I'm making some time change, so we have this, okay? So we have this, we have this plus I'm changing time, okay? So we have this plus I'm changing time, okay? So this D will pop up from Dupers PD, right? This is the second order term in Dupers PD, and what you're left with is still the integral term that you have when you differentiate with respect to tau, you see you have to differentiate here, the density comes from the derivative of the CDF, and then you have this derivative that appears, okay? So this type is simple, this type is simple, okay? So what is more involved is, than the second one. Yes, please. So you develop this expansion by more, you have only the big old tau down, but of course you can make a complete series for this expansion, and if you inject this expansion in the integral formula, then could you handle this, maybe taking more terms? So potentially yes, potentially yes, then what we notice is that, what we noticed is that actually the asymptotic of the exercise boundary, you know, here this function behaves as what, k minus sigma k times square root of tau log of gamma over tau, okay? So it has a behavior, almost square root behavior in tau times log of one over tau, okay? This is what pops up, and actually these and the further refinements here only depend on what is here. So they don't depend, because there's square root of tau, the square root of tau, so all the terms that come from here, okay, that comes from this big old tau, let's say heuristically, these terms they don't count in the, so I don't know, I don't need this. Any more questions? You have a kind of a integral equation, for your boundary, do you apply a kind of method to solve this equation, for example, if there's a kind of big alteration, something like this, why? Of course, of course, as you say, we could apply methods for this integral equation for this one or for the original one, okay, for this one, if we know the density. So we could, of course, we could. What our aim was to obtain this explicit formulas for the exercise boundary, in the sense that at the end, you know, for you plug in this formula, in the end globally what you have is an almost explicit formula for an American option. So, the only numerical task left is to compute this integral with respect to time, okay. So, for calibration, what we have in mind is calibration purposes that I haven't mentioned here, but okay, so as usual, right, you want semi-closed formulas to have faster computations, and so our aim was really to work out an explicit formula rather than a numerical solution of this equation. I don't know if this answers to. Yeah, but I mean, is this is only up to this error? Yes. So, if your options are not short, might be original? Of course, yeah, of course. If the options are not short, maturity, this will worsen, the quality of the approximation will worsen. So, the options typically are, have no, maturity is no longer than one and a half or two years, for which we tested our approximation, and of course it works very well. You have discrete time on the. You have discrete time on the, so you mean Bermudan? Okay, yeah, fair enough. Then either we just assume that our Bermudan are American or, well, yes, of course, for discrete time American option, this will change. Okay, let's thank Stefano once again.
numerical methods for high dimensional problems computing expectation based on stochastic right? Okay. So you can choose your numerical Okay. Good, so you can tackle the problem from then we come to another type of product, which this American option is a financial contract that This is the American option. The price of this American option is not an expectation but a the value function of an optimal stopping numerically if we want to apply the model to market data, we have to solve numerically this efficient valuation numerical methods for optimal this optimal stopping problem? So, the price of So, there's a backward dynamic programming approximation of the solution of my value free boundary PDE, okay, which is written in the this boundary, it's the optimal time for complicated computation of the whole talk. So American option as the price of the European upon asymptotic calculus for diffusion processes, done in two steps. So step one is to derive an alternative representation that we have is the this early premium formula written in terms of problem you are left with is the problem of to arbitrary order, which refines the expansions the literature, okay? So here we will stop in one you have semi-closed approximation as a final for the solution of this optimal stopping problem, okay? So the numerical, which is of which will be on the, which will be online, which our aim was to obtain this explicit formulas for only numerical task left is to compute this than a numerical solution of this equation. I the approximation will worsen. So, the options