 What I'd like to do is give you a quick review of where we were last date because that's one of our main key results and I guess actually this is not dark enough, is it? Yeah, it's better. Okay, so last time we were focusing on this Black-Scholes partial differential equation and I demonstrated how you can go ahead and derive it. And I want to again remind you of the key steps involved in this. So first of all we're making an assumption on the underlying dynamics of the stock price and so this is we're assuming stocks are geometric Brownian motions. So ds over s is mu dt plus sigma dw, mu and sigma are constants here and the money market account grows at a constant rate r, so dm over m is equal to r dt. And then we said, okay, what we're going to do is we're going to set up a strategy which consists of holding certain units of the stock, holding certain units of the money market account and shorting some contingent claim which we're trying to value. So we're trying to figure out what is the equation that g must satisfy in order for us to avoid arbitrage in the market. And to do this we set up this initial portfolio, we make it cost nothing at the start. We then impose, we then look at the dynamics of that value, of that portfolio value by imposing the self-financing constraint. So that means that the value in the portfolio is gained only by changes in the value of the underlying assets and not by the change in the position over that instance in time. So the positions are effectively thought of as being held constant. Alpha sub t, beta sub t are being held as constant over that short interval in time and all the changes is the asset prices. So s and m, that's why we have that equation. Alpha ds plus beta dm and then we have the short position in the claim so there's the minus the dg. Okay, then we made one further assumption as we assumed that that claim actually can be written, the price of that claim can be written as a function of time and the value of the stock at that point in time. Not all options can be written that way but some can. So under this assumption and further, not only must it be a function of time and the stock, it must also be once differentiable in time and twice differentiable in the second argument. Okay, why did we have to make that a differentiability assumption? We made the differentiability assumption so that we can apply ito's lemma on that process g. Once we did that, we realized that there are basically two types of terms in the increments of the value of the portfolio. One is a deterministic or I should say a pre-visible part, that's our drift and the other part is the risk, the coefficients of the dw term. That's the part that we don't actually know. So we don't know what dw is although we know it's coefficient. We do know what the drift is because we know it's coefficient and the key idea, one of the key fundamental ideas is remove the instantaneous risk. Choose the portfolio such that this instantaneous risk is absent and that choice forces us to choose a position in the underlying asset to be the partial derivative of the price with respect to asset price. Once we did that, then we find that the increment in the portfolio value has only a drift term, there is no volatility term. The coefficient of the dw equals zero, which means the portfolio is instantaneously riskless. That means we know whether that portfolio is going to have a positive return or a negative return or a zero return over that instant in time. Now we impose an arbitrage argument. Say that if that coefficient of the dt term there is anything but zero, there will be an arbitrage. Because we know what its drift is, we started with nothing, we would have something that either grows to be a positive value or grows to be a negative value and if it's negative take the short of everything, in which case we would create an arbitrage. So it has to be zero. This is a necessary condition. We don't know if it's sufficient and in this course I'm not going to bother to prove the sufficiency of it. But it's a necessary condition. Given that we now know that the increment in the portfolio value is zero, so that's the statements that are written here, and we started with nothing, the portfolio always has to be worth nothing. This gives us a second criteria to determine the amount of money in the bank account. That gives us our beta coefficient. After plugging that back in, we then end up with the Blochscholz partial differential equation. So it is a number of key steps involved to get here and I really want you to make sure you understand those key steps. And again, what are the ideas? Build a portfolio, start with nothing. Make it self-financing. Instantaneously remove the risk. Invoke a no arbitrage argument about the fact that there is no risk. Conclude that the portfolio must always be zero. Substitute back. Conclude that the price function must satisfy this partial differential equation. And this partial differential equation is our Blochscholz PDE. Okay? So that's the line of reasoning that you have to use. Are there any questions that I can clarify about this set of... this argument? No? Okay. Well, once we have that result, we're then curious about how do you solve that equation? All right? How do we solve... sorry, this equation here. That's what we would like to know how to do. In the last lecture, I simply told you, well, let's guess at the solution. Let's guess based on what we already know from the discrete models when we take their continuous time limit. We guess that the solutions are expectations under some risk neutral measure of the discounted price. And when we made that guess, that's for example here, we're looking at a binary option. It pays one if the asset is large enough and zero otherwise. So we simply guess that it actually satisfies this equation, or sorry, that the price is given by this expectation. And then what we would like to do is check whether it actually does satisfy the Blochscholz PDE. And we started the calculation and it suggested that you sit at home and work through the rest of the algebra, because it's a little bit...it's just annoying algebra. It's nothing...no financial intuition is in there. It's just algebra. So you were able to convince yourself hopefully on your own that that function does indeed satisfy the Blochscholz partial differential equation. We looked at another example and again had our guess that this relationship should still hold. And once again, in this case, we explicitly checked it and PDE does in fact get satisfied. So we were happy that at least in the few examples we checked, solutions of that partial differential equation do have a representation given as a discounted expectation of a payoff. That's what we were able to convince ourselves of, at least with these few examples. And one of the main things that I'm going to do today is convince you that this is not only true for these specific examples, but it's true generically. And it requires using a particular result and more or less deriving the particular result. Okay, so putting that aside for a moment, we then said, let's suppose that we actually have price functions now. We've solved the PDE, we've already done it. And I was addressing what a practitioner would do in the real world of how they would use this dynamic hedging argument to hedge a claim that they have written and given to someone else, the intuition that you should take away from you, that you should take away from the analysis, the intuition of the final result is that locally what this position in the underlying asset is, is locally it's saying you're going to match the option price function locally by a straight line. And the slope of that straight line is, well, it is the partial derivative and that is the position that you put in the underlying asset because that's what a straight line in asset price space represents. And that's pretty much it, is that this straight line is what you're using locally to approximate the price function. And if you do this continuously, after epsilon changes in the prices, then you will of course match that price function exactly. But in reality you cannot do that, so what does a practitioner do? Practitioner will trade at regular time intervals and we talked about two approaches, time-based approaches and the move-based approach. In the time-based approach, what we did is we said let's consider selling this claim, purchasing delta zero, where delta zero represents my position now of the underlying asset. And where do we get that money from? Well, we've sold the claim, we have some money in our hand, we purchase it from the money that we have in our hand, and if we need, we borrow some extra. So we have a money market account as well. We then looked at the evolution of that money market account at each point in time, and this is our key result, is that the money market account after one time step is equal to whatever you started with grown at the risk-free rate and then subtract the cost of changing your position from your old delta to your new delta. And remember, when you go from your old delta to your new delta, there are two main things that have changed. One, the price has moved, the underlying asset price has moved. Two, there's less time remaining until maturity. So the option price curve has... So the option price curve is not just one sixth curve because that's when you draw a price function, say for a call, that curve represents the price with the maturity of one year from today. But tomorrow, the maturity is one day less than a year, so that curve would slightly change a little bit, and then where you are in the horizontal axis has moved because the asset price has moved as well. So those are the two main things that you have to account for when you purchase that new position in the underlying asset. Okay, so this is our strategy, and then we just went ahead and set some scenario generation for this. And well, I guess I don't... Let me... I guess I can pull it back up here. And while it's loading, let me remind you of what the difference between time-based and move-based were. So in the time-based approach, you do this kind of rebalancing, this equation here. You do this rebalancing where you change from your old delta to your new delta at equal intervals, maybe once a week, something like this. In the move-based approach, instead what you're doing is you're going to rebalance only when the delta has changed sufficiently, has changed enough. And there's a little bit of an issue with this, with this move-based approach. Imagine the case when we do delta-gamma hedging. Remember what delta-gamma hedging is? So gamma, when you do a gamma hedge, you are further approximating the curve locally, not by straight line, but by a quadratic function. So when you have a delta-gamma hedge, there are two things that you have to account for. There are two sources of delta, I should say. When you do delta-gamma hedge, you have the underlying asset and you have the option. The delta of the underlying asset is one. The delta of the option is something else. And you're using both of those to hedge your option that you've sold. So delta comes from two sources when you have a delta-gamma hedge. When you try to rebalance in this way, what you need to do is you need to track the total delta relative to the actual delta. So the actual delta is the delta of the option you've sold. That's delta G. The total delta of your replicating portfolio is delta due to the asset and the delta due to the other claim that you're using to hedge with. So there are two sources and you should be tracking basically the difference between the portfolio delta and the delta of your option. So only when that has moved sufficiently enough then you rebalance. That's the slight subtlety with the move-based hedge. But the principle is easily understood as you make a trade whenever delta change is enough. And that's it. That's all I want you to take away from this. Okay, so let's see if I can find... Here it is. Okay, so here was an simulation and we had a bunch of different results here. So we're looking at... P&L for the time-based approach. This is a P&L for the move-based approach under this particular gap size. And in this case they more or less look similar, right? More or less the same thing. So that was for the delta hedge. Now when we went to delta gamma hedging on the other hand that's your time-based and that's your move-based. There's a clear advantage to do move-based hedging although there it's not always positive but the time-based approach is almost always negative here. Why is that? What was the reason that the time-based approach was always negative? Almost always negative. Transaction costs, exactly. In this whole analysis, if the block shows approach was correct I should have a mean of zero. And as I increase the number of transactions, if I rebalance more frequently, we saw that this distribution gets thinner and thinner and becomes concentrated around zero when there are no transaction costs. But if it costs you something to make a trade every time you make a trade you shift your distribution to the left. And so this is why under the time-based approach here in this particular example we saw that there were a lot of hedges going on and you're getting pushed significantly to the left and this is causing your P&L to look horrific. You wouldn't want to do that. But for the move-based approach because, let's see, where is it? Here we go. Oh, actually this was an extreme move-based approach where I've only made two rebalances. One at the start and one at the end. So in this particular case we find that your P&L, yes, it's shifted to the left but there are a lot of scenarios in which you win quite a bit of money because you've only made two transactions. Now, that is perhaps a very big extreme case for the move-based. Let's put a delta band of 0.1. So when I had my delta band equal to 1 all I've done is I chose a position at the start and I did nothing until maturity and then I liquidated. So that's why it's two rebalances. So this is a case where I do some more rebalancing and as you can see we're concentrated around say maybe the average might be about five trades in that time frame and here hopefully we'll see that my P&L looks a lot. It looks better. So this looks better than the other one that we had when we only had two transactions because with two transactions the mode was to the very far left. We had a huge spike out here at negative 2. Now we can see that that spike has moved in closer towards 0. So having a delta band that's of some reasonable size is important. You sometimes do better than time-based heads. Sometimes you don't. And you have to tune that according to the dynamics of the underlying asset and your own transaction cost. Okay. Are there any questions that I can address with respect to these simulation analysis and simulations? Like I said, on an exam I'm not going to ask you to derive anything with respect to move-based approach or time-based approach. You can't simulate on an exam. But I can ask you something about qualitative features that we've observed here such as I don't know what's a qualitative feature that I haven't pointed out. Maybe you can tell me one. And then I can put that on the exam. I don't know. I've already said it, sent the exam so I can't change it. So what's another qualitative feature that you observe here? So I spoke about the mean just now, right? And the mode. What are your other basic statistics? Variance, skewness, kurtosis. What can you say qualitatively about well I think there's one that's obviously different than the other skewness, right? Time-based has a huge negative skewness. The move-based has a significantly positive skewness. You can see the tail is much heavier to the right. While over here the tail is much heavier to the left. And kurtosis would be a little bit hard to tell. You've got to actually compute it here. My bet is that you have more kurtosis in the time-based than the move-based. I'm not 100% sure. Skewness is one that's just obvious. Okay. If there are no questions about what I just went over, I'd like to go on to the new material then. Nope. Okay. Yep. Sorry, the recording. Yeah, it is posted now. I posted it last night. I completely forgot to compile it and post it. So I did that last night. Okay, so one of the main things today before our first break is to show you how to solve the partial differential equation that the block shows partial differential equation that we had, so in general. So this was our partial differential equation. And the solution is going to be using and kind of deriving at least showing you, heuristically, why this is true in the Feynman-Cat's theorem. Okay, so if I were to ask you to write down what you expect the solution to be in complete generality for this equation, what is your guess for this function based on what you know before, not based on knowledge about PDEs, but based on what we've seen already. We know, we should be able to guess that of the equation could be written in this way, where s t equals in distribution s e to the r minus a half sigma squared big t minus little t plus sigma w big t minus w little t. We should be able to guess that it equals this where, let me put a bar on top of these things here to denote that these are not the original Brownian motions that we had before. These are q Brownian motions. So I'm going to put a question mark over that equal sign. This is what we would like the answer to be. Because if it was this answer, then everything corresponds exactly to what we had in discrete time when we took the limit of the steps going to zero. And so it would be great if we had that result, because this result also has a very nice, financial intuition. All you do is you compute the expected random payoff and discount it. The only difference is the measure, the probabilities that you compute the expectation under are not the ones you began with. They're not the p probabilities. It's some other thing, some other q probability. That's what we would like. Okay, so it turns out this is true. So how do we see, how can I convince you that that function actually does solve that PDE? Seems like a difficult thing to do. One is a partial differential equation which has no stochasticity in it. It's a PDE. There's nothing stochastic in a PDE. PDE is a purely deterministic thing. It's just about derivatives of functions. While the second equality, while the G written as an expectation, requires the introduction of a stochastic variable. So it's a little bit odd that it's actually connected in some way. So here's the I'm going to basically go through the derivation showing you that suppose you write G in that way, then what can we do? In order for us to get through the main result I'm going to prove to you one key observation. Suppose if I define a function HT,S I simply define it to be the expectation under the risk-neutral measure of a function of terminal stock price conditional on ST equals little s. Okay? Suppose I just define that object. This object has a very interesting in fact let me not let me, yeah sure that's fine conditional on that, that's fine. Okay, this object has one very important property. It is what's called a martingale. Remember we talked about martingales in discrete time? Long time ago? The basic idea that a martingale is a free game it's a process in which when you compute its expected value at any point in the future it just equals its current value and we've seen it come up again and again relative price of a traded asset to the money market account is a martingale under the risk-neutral measure right? If you take the ratio it equals its expected value of its future ratio so objects of this kind turn out to also be martingales. Let me write out the word for you fully. So I'm going to put in brackets here what do I mean by martingale that means if you take the expected value under Q of the process at any future time given information about the process at an earlier time this equals the process at the earlier time. Now this is not strictly formally correct but it's more or less what you need. So it's telling you that take the and here S is bigger than T and in fact I want it to be less than or equal to the capital T. So it has exactly this idea that it's a fair game in some sense. If I look at its future value and I compute its expectation given what I know about the process today that future expected value is just its value today. Nothing more. So I need to convince you of that. I'm just telling you this is true. So how would you show that? So the idea to show that is take the expression you know what because I'm using S for the asset price maybe S is not such a good idea to be used as a time index here so I'm going to change it to U just so that you're not confused with asset price. So what I need to convince you is I basically just need to convince you of this equality here. So let's just do that. Let's do the calculation. Let's check. And you'll be surprised at how powerful that one result is. That one result is going to lead to very interesting things. So if you take the expectation I'm just simply going to write I'm just going to compute this expectation of H U given H T all I'm going to do is I'm going to insert what is H U? What's the definition of H subscript U? Well that is the expectation of S at capital T given S at capital at little U. That is what H U is and then we have to be given H little T. That is really H U. Just look at its definition. H at any point in time is its expected value of the five function given to the asset price at that point in time. So H at time U is the expected value of the future asset price function of the future asset price given the asset price at time U. And now you're more or less done. This is an iterated expectation, isn't it? You can view this as an iterated expectation. Knowing the future asset price taking an expectation and then conditioning only on H right now and knowing H right now because H is just a simple transformation of that. So I've had to make one assumption here and this is why I said it's not strictly correct there. I have to assume that that H is invertible in the second argument. But it doesn't really change much. Conditioning on H is the same thing as conditioning on S at time little T. So just the law of iterated expectation nothing more than that simply tells me this is the expectation of phi of S at capital T given S at little T, right? I just remove the inner expectation and then now you just make an observation. Well, the definition of H at time little T is in fact the left-hand side here. So this is H little T. That's it. You're done. There's nothing more to do. The whole idea is that you're taking some object which is random in the future taking its expected value given the information you have today and whenever you do that that process turns out to always have the same mean. That's what this is telling us. The expected value of the future process is always equal to its current value. Today is always the best estimate of that future expectation. So you can think maybe I'll write that down here. So today's value is the best estimate of the future expectation. That's what this statement is saying. Okay, now why the heck is that a useful thing to know? It looks like a completely just pure mathematical or probabilistic result. If you're awake you realize that this expectation is in fact part of our guess for the price. It's part of our guess. So what we know at this point in time that part of our guess the part without the discounting has this property that it is a martingale. Okay? So how are we going to use that for that result? So here's the idea. So we have the expected value of H at any time u given H at time t is equal to H at time t. So this implies that the expected value of H at time u minus H at time t given H at time t is equal to zero. All I've done is I've put the H sub t on the right-hand side. I've pulled it onto the left-hand side and since the expectation is a conditional one you're conditioning on knowing H sub t H sub t is in fact a constant and I can put it under the expectation. Okay? Is that clear? So let me maybe I'll put one more line in between these two just to be crystal clear and I'll put a note here since H sub t is a quote-unquote constant under the expectation given H t. It's only because I've conditioned on H t if I didn't do that I wouldn't be able to pull it under the expectation. Okay, anyone have an idea of what I might want to do next? If you know that the increment, so by the way what is the same word that lost the quality says the increment of the value of the process in mean is zero. Okay? And it's true no matter what u is as long as u is bigger than t. So why not? Remember what are we trying to do? What's our ultimate goal? Our ultimate goal is to come up with a partial differential equation. Right? That's our ultimate goal. What objects appear in your PDE? Derivatives, correct? What are derivatives? They have something to do with small changes in whatever it is you're taking the derivative of. Right? Doesn't it make sense to perhaps think of this increment over a small distance? Take this u to be t plus epsilon so that we look at an incremental change in H and if we look at an incremental change in H then we could perhaps say something. We will, in fact, be able to say something about its derivatives. So what's under this expectation then is ht plus epsilon minus ht. That's what we have under the expectation. What serum that you know of can tell you something about this increment? In fact, it's a lemma. It's O's lemma. Right? It's O's lemma tells you that let me just write O's lemma not in the form. I'm going to use it here in the standard form. O's lemma tells me that dh I want to write this in a different color because it's not the form. I'm going to use the integrated form, the actual form. The qualitative version of O's lemma is dh is equal to partial remember h is a function of time in s so there will be a partial of h ds there's going to be partial in time dt and then there's the itto correction. Right? This is our itto's lemma in differential form. Now in order for me to have written this, I actually had to tell you what is the dynamics of s. What sd e does it satisfy? So let's slide back up here for a second. I've simply defined h as this expectation and we've told you that s at capital T in distribution equals that. Well, we could also since we just have an equality in distribution, but we kind of guess that probably once we changed, once we use Brownian motions, we don't have to put equality in distribution because the Brownian motion tells the whole path. So probably this equality is not just in distribution, it's probably for the whole Brownian motion. It's probably for the whole path, sorry. So in order for me to write down what ds is here, so let's just say recall our guess was that st equaled in distribution s little t e to the r minus a half sigma squared was this. So further a guess, we can't simply write s be equal to e to the minus a half, not in distribution, but in fact, path-wise, this guy. And we know from our previous work on when we just applied it as lemma, the statement the last two statements are equivalent to one another. If f solves this stochastic differential equation it can be written in that exponential form and if f is written in that exponential form, it actually solves that differential equation. We saw that, we did that example in class maybe two or three lectures ago. So this tells me so why did I do that? Because now I know what I can put here, right? And as well strictly I also needed to know this in order to know what to put there. In iddo's lemma what should happen is I need to first of all put ds on this side of my equation and what appears in front of the dw term here is always the coefficient of sigma of the, sorry, the coefficient of the dw term squared and whatever appears in ds is well, whatever ds happens to be. That's just straightforward application of iddo's lemma. Nothing complicated. The complicated thing here was to go from the guess about the distributional property of s which we guess from that discrete model to saying that perhaps when we look at this as a process we should use the whole Brownian motion and not just the distribution of the Brownian motion. Okay? Remember those were two distinct things. We clarified their difference last class but if you still have question about it ask me because it's important distinction. Okay, so let's get back on track to where we want to go. So what we're trying is we're trying somehow to make this guess about the expectation show that that guess actually solves this PDE. That's what we want to do. So we're looking at this increment and we're now going to use iddo's lemma but let's use it in integrated form because epsilon is just something positive. We don't really, it's not an infinitesimal thing and we know that the equation that's written doesn't actually make mathematical sense. We have to be a little bit careful about it. So what does this increment mean? This increment, according to iddo's lemma in integrated form means integrate from t to t plus epsilon of the partial derivative of s dsu plus the integral from little t to the little t plus epsilon partial derivative of s du and this will be hu. I forgot the h over here. And one half sigma squared integral little t to little t plus epsilon partial derivative dwu. So all I'm doing is I'm using that second line and just putting integrals around them. Because what's written in black actually is correct. It actually makes proper sense. All of those objects are defined. What's written in green is not defined. Partial t, thank you, sorry. So now what we do is we're going to use that expectation and apply it to this object. And there's a couple of things that are going to happen. What is the expectation of that third term? When I apply the expectation on that third term expectation of this is zero. So remember one of those key results that we showed when we defined stochastic integrals. I showed you that the expected value of a stochastic integral is zero. So that's going to be zero. As well when we insert the expression for ds over here when we take this expression and insert it in for ds there's one term, this dw term which gives us a second stochastic integral. And that expected value will also be zero, right? For the same reason. So let me just write down this equals zero implies the expected value of the integral t to t plus epsilon partial derivative of h with respect to s and the only term, the dsu what's the only term that isn't zero? It's r times s. That's the only term that isn't zero from there. And then the second term is not zero. And the third term is zero. Conditional on ht. So let me put a note here this is all because the integral from little t to little t plus epsilon expected value of gu dw u or not g let's call it sum arbitrary l u dw u is equal to zero. That's the key result that we're using. Now you'll notice that these two terms are actually of the same kind. They're both Riemann integrals. They're both being integrated with respect to the Lebesgue measure here. Do you? So we can in fact combine them. There's no reason to put these as two separate terms. Let's just combine them. Did I? You know what? I've made a really stupid mistake and you're all just sort of confused by my voice and completely forgotten the mistake. The itto correction term, what should it be? What is the itto correction term? It's this, right? It's not dw, it's dt. That's why when I looked at my result I was wondering, wait a second, there's something wrong. The itto correction term is dt, not dw. So we actually have three terms that remain. So in the video I'll try to correct that by voicing it over if I get time. So there's one half, sigma squared, su squared and similarly over here there should be a du. So there are two places to correct. This and this. Okay? Those two. That's all. So then I'm two derivatives with respect to asset price u du is equal to zero. And this integrand is time u, so all the time indices should be u indices under this integral. Okay, good. Now hopefully you're starting to see that PDE kind of coming alive now, right? All of a sudden there are partial derivatives there. There weren't any a moment ago. Just by using the fact that the increment has a zero mean we now have that the expected value of this integral is equal to zero. So now in order to really prove it rigorously, we have to do some we have to use some boundiness relationships. We have to use something called a big dominated convergence, which we're not going to go into. Basic intuition. What do you think I need to do for a basic intuition? To reduce this, what I want is somehow I want to get rid of that integral. I want to get rid of that expectation somehow. That's what I'd like to do. So what would the intuition be? Partition? No. We've got an epsilon there. What do you know about integrals from a to a plus epsilon of some arbitrary function of u, du? If you took one over epsilon of that and you took the limit as epsilon went down to zero what is that? l evaluated to the point a. Imagine this is a Riemann sum. The integral from a to a plus epsilon is really approximated by l a times epsilon for small epsilon. And then you're dividing by epsilon. It's approximately it will become equal to l at that end point. It's almost like the fundamental theorem of calculus. You're taking the derivative of an integral. It's effectively what you're doing. So this is what we can do. But in order to do it properly we can easily divide by epsilon. That doesn't pose a problem. What poses a problem is interchanging the limit and the expectation. And that's where you need to use these dominated convergence. But we'll assume that we can interchange the limit and the expectation. So if we can do that then we would simply have that the expected value of the partial derivative in this way R times s partial derivative of h. So now everything is a time t all of a sudden. All the u's become a little t. This is zero. And then the final thing to notice is that well everything that I've underlined there is actually just some function of h. It's a property about the function h. And you're conditioning on unknowing it. So in fact everything that I've underlined is not random relative to that information. So the expectation is in fact irrelevant. Since we're doing this condition on a specific s we can just change it to being from the process to the state variable again. And this gives me let's interchange the order. We'll write the partial in time first. This gives me partial time ht,s plus rs ht,s plus one half sigma squared s squared two derivatives ht,s equals zero. Does it look familiar? That equation? This is almost the black shoal's partial differential equation. The only difference is that the right hand side here is zero. In the black shoal the right hand side would be r times h. So we've almost got it. Do you see how powerful that martingale result is? All we use is the fact that if we compute the expected value of this phi function, given what you currently know that thing is a martingale apply it to oslama and outcomes that h has to satisfy this PDE. So all we need to do is the last thing we need to do is show that g in fact satisfies the black shoal's PDE. So that just requires two lines of work. g is in fact this times h. So in other words h equals this times g. So simply substitute this expression for h into the box and what will you find? When the partial derivative of time hits the exponential you'll pull on a factor of minus r times in fact at every term you'll have this exponential term coming back. So we'll just pull that out front. When the time derivative acts on the exponential you'll get an extra factor of minus r times g itself. Then you're going to have when the time derivative acts on the function g and then every other term the middle term and the last term are derivatives with respect to s and s does not appear in that exponential factor so those derivatives simply hit the g function. And we're done. We have the exponential divide it, put the minus r times g on the other side. Last thing to check is does g satisfy the boundary condition? Does g at capital T actually equal five s? So let's just check that. Well g at capital T actually equals h at capital T doesn't it? Because the exponential factor is e to the zero so that's one. And what is h at capital T? We'll slide back up. h at capital T is this expected value but if you're at capital T the five s is known. It's not random. So it comes out of the expectation. So h at capital T equals five s. And then we're done. That is the Bloch-Scholes PDE together with the appropriate boundary condition. Okay. So what we've proven what we've basically proven is the Feynman-Katz result in a slightly hand-waving manner but it's reasonably close. So here's the general form of Feynman-Katz. This is a specific one. I'm just going to give you the general one and on your exam it's your very first page of your exam. It's actually given to you. So you don't have to memorize it. You don't have to remember how to derive it. I won't ask you to derive it. I'm only going to ask you to use it. Okay. After all that hard work right now you can just forget what we did. But hopefully you've learned something by seeing how it's derived. So the Feynman-Katz theorem basically says if it satisfies the PDE partial T G plus any function of time and s s partial s G plus any other function of time and s squared times s squared two derivatives G equals any third function of T and s G and you have this boundary condition then G admits a unique solution given by and here's the result given by G T, s equals expectation some probability q of e to the negative the integral from 0 to little t of c of u, s u du times phi of s sorry that should be integral from little t to big t s capital T given s little t where s t equals a t, s t d t plus b t, s t d w q so this is a q Brownian motion it's a bit of a mouthful as you can see which is why I give it to you but I want to point out a couple of things about it and then you'll realize actually it's easy to remember once you realize a couple of things it's not too bad so I'll wait till I see your hand stop scribbling away so here are the key points whatever appears in front of the G on the right so first of all this equation is kind of like a black-shell's equation sort of except instead of r you have this function a and instead of sigma you have this function b and on the right hand side c so it's just a generic thing so whatever appears in front of the coefficient on the right hand side that is what shows up like a discount factor whatever appears in the right hand side is effectively showing up as a discount factor whatever shows up in front of the linear derivative term shows up as a drift in this expectation representation and whatever shows up in front of the second derivative term shows up as the volatility so those are the three things that you just need to identify the right hand side is my discount factor the coefficient of the linear term is my drift the coefficient of the second derivative term is my variance so take it square root and I get the volatility okay that's the key thing to notice and if you're keen you can try to prove it it's not something I will ask on the exam I'll let you know now so you don't all have to be worried oh gee so I have to try to prove that we're often reproducing on the exam no it's not but if you really want to understand this you can basically use the approach that I just showed you and see if you can convince yourself that this result in fact is true okay so what I'd like to do we'll take a little bit of a break and what I'd like to do when we come back from the break is show you a couple of examples of how to use the result okay so let's pause for 10 minutes alrighty so during the break I had a number of people coming and asking me here we go why why am I allowed to make that second equality why is this true okay basically how do I go from here to there and the reason when I said it I didn't write it down but I said it it's because of the law of iterated expectation if you take the expected value of the expected value of a random variable conditional on another random variable given a third this is identical to taking the expectation in this manner okay that's all it is you've probably more you've probably seen it in this form before without the last conditioning event but it's also true when you have a conditional event there and the event y contains z okay so y is a set of events and it contains z so that's what we have in our situation because knowing the future asset price what I should really strictly be writing here is something called conditional on the sigma algebra generated by the asset price which means knowing the path of the asset and if you know it's path then you know if you know it's path in the future you know it's value now so it's a condition it's a contained event okay so all that's all it is this law of iterated expectation alright so as promised I want to try to use this result in some way and see if we can do something that we haven't solved yet so let's go to a new slide and suppose I asked you to solve this equation partial derivative of g plus a constant x plus one half another constant equals c times g and g at capital T the function of x is equal to one let's suppose we've got this actually not equal one let's say equals x why not and I want you to solve it all we do is we just apply fine and tact so the result is going to tell us that g t comma x is going to be equal to the expected value of e to the negative the integral of c c is a constant so I just get that time a random variable or the process I should say given that the process starts at little x and this process has to satisfy this stochastic differential equation now you notice I don't have dx divided by x there why? if you go back to the form of the fine and tacts whatever is in front of the partial derivative of g with respect to the state variable shows up in front of the dt term so when there's an f there I have to divide by s on the left hand side but we don't have that extra factor of s in our case the a function would be the constant a divided by x or just think of it as whatever appears in front of the partial derivative of the g term that entire thing is what shows up in front of in front of the term here so let me let me write this also in this form so that you can and I'll put the circles here here so that's the same thing as what I had before but I'm just identifying to show you that it's not just the function a but it's the a times s that's showing up in front of the dt term and it's not just the function b squared s squared the square root of that that shows up sorry it's not just the function b squared that shows up in the dw term but it's the b squared s squared that shows up so in this example since those things are actually constants there are no x's and so on I simply have an equation of this kind for the fd okay so we've got to compute this expected value and what's our answer well the solution to this stochastic differential equation the way that you do it is you solve the stochastic differential equation and then you compute expectations of that stochastic differential equation so s of capital T has to be equal to x at little t plus a times big t minus little t plus b times the increment of this Brownian motion alright that's clearly a solution of this sdE it's just a Brownian motion constant coefficient and therefore the g function is the expectation under q of e to this negative c times little x plus a big t minus little t plus b times the increment of my Brownian motion and I've already used the conditioning event x x t equals little x is already incorporated by the fact that I have replaced x t by little x so I don't need to condition anymore so what's the answer here the mean of the last term is zero so this is actually a solution to that partial differential equation and those of you who want check to see that this actually is a solution you can check it it's not too difficult okay what if the payoff was x squared instead suppose we have the same PDE but instead of suppose instead we had x squared there what would you do what changes well we'd have everything that we had there squared right that's the only thing that would change the payoff is the only thing that changes and let's just do that case so then this discount factor just comes out front and we get a couple of terms we get x squared plus a squared plus b squared twice that term and then twice it's getting messy at the end of the screen here I think I got all the terms so we can see that we'll get x squared here plus a squared big t minus little t all squared what will this third term be what's the expected value it's a Brownian motion experience is equal to its increment and the only other term that's non-zero is this one and that's zero because it's the increment of the Brownian motion the last term so again you can check does this actually satisfy the partial differential equation and hopefully you'll convince yourself if you do the work that you do get the answer so you should try it one more interesting example for the same equation is what if you had e to the minus x squared here so to pay off the terminal condition is that sorry I don't want the squared in there I just want x the squared is too hard so let's see what's so it would be again this kind of discount like factor and I would simply have e to the minus x capital t which we know is little x plus a plus b increment and you can pull out this constant bit and then you have expectation of this increment and what's the expectation of this increment the exponential of this well this is normal the increment in terms of distributions normal zero variance capital t minus little t so I should be getting e to the and the mean is zero so I should get e to the minus e to the plus one half b squared so that's your result and again if you want go ahead and check and you'll see that this does indeed satisfy the equation the boundary conditions clear right as little t equals big t it just reduces to e to the minus x questions about this quick application okay let's see let's do one one more application trying to think of a good one to do you know what no I think that's fine any equation that I'm going to ask you is going to be either the block shows type it'll reduce something to the block shows type or this kind of linear type so let you know ahead of time you don't have to know anything more than those two those two types of equation so it will not need it will not necessitate some arbitrary functions of time and s that show up here as long as you know how to do the block shows type and the constant type that's it and only one of those two will show I'm not asking you both yeah what's that what do you mean you mean just as a general question if there are hard questions on the test of course there are hard there there must be right otherwise it wouldn't be a test so there there's a mix of questions there is easy question not so easy question moderately difficult questions and a difficult question yeah the example okay which half term what you mean you're worried about this factor of a half year well if you go to the the statement of the theorem even oh shit I see you know what it's my fault there is a half here there's a half there the statement of the theorem should have a half and you see that when we did the derivation in fact there is a half there right so somehow I just when I was writing it down I completely forgot the half my apologies let me make sure yeah I think that's it that's it thank you for pointing that out that's good anything else about this if you look at the last year I have some of the old exams posted right and you can take a look and get a sense of the kinds of questions that I have been asking in the past now of course just like all mutual funds past performance does not represent future same thing here past exams do not necessarily represent your exam but it'll give you a good idea right as opposed to mutual funds it actually does give you a good idea so let me see if I can hook on here and show you an example of the kind of question pertaining to this usually the question pertaining to the Feynman-Hatz result is one of the difficult questions okay is probably the difficult question on the exam just so you know ahead of time as well and I urge you when you are working on the exam to to focus your time first on the easy questions because they're all worth the same amount right they're not it's not like if you worked on the hard question you'll get more points so this is uploaded and you'll see that the questions are more or less standard this here did I ask you Feynman-Hatz no in this exam I did not ask you about Feynman-Hatz in fact at all sorry oh that's term test there we go that's why there's a lot of empty space don't get frightened by the page numbers actually why don't we kind of run through this a little bit just to give you an idea on this exam you had nine questions actually there are quite a few I think I um yeah so I almost always do this I almost always ask you briefly explain two concepts and they're different usually then there's please indicate sure or false type question okay so you know here's an example of a kind of question pertaining to that delta hedging stuff okay and in this case in this case I talked about what happens if you hedge with an incorrect volatility I didn't tell you guys about that this year so you won't expect that but you'd expect something else that I did discuss in class there's always a sketching question about sketching deltas and sketching gammas okay there's always some sort of discrete time model question it depends sometimes it's interest rate sometimes it's equity but it's just a basic discrete time model and in this case that's continuing the question in this year I asked them to check that this thing actually satisfies the block shows PDE but with zero interest rate it's much easier calculation than the one with general interest rate and so right the first part was actually just derive show that the price of this kind of digital call was that and the part B is to confirm that it actually satisfies the block shows PDE and then here you've got correlated Brownian motions and you're doing an integration by parts formula should all be familiar with that mean invariance of this kind of thing so this is using Edo's isometry now we get into serious questions these are the ones that have become more difficult and here so this is something we haven't covered yet is multiple assets how to deal with that I'll tell you about that next class the Edo's lemma for two asset case with correlation we've studied how to correlate Brownian motions we've actually introduced that already but we haven't studied how to model two assets at the same time and then do valuation on that so that's something I would like to cover and I think the last this was a question that they were told was coming so I usually do that I think I've already told you one that was coming and I hope you took note I made it very clear on the time when I post it and the last question this is hard this is a hard question so I've given them the Vasacek model and I'm asking them to show something about the distribution of the integral of that model and I only ever expect just a couple of people to solve in a reasonable way the last question and but typically the the average typically is somewhere around 70 70 something like 71 around there in terms of your final grade so I expected the same to be this year and if we look at last year's exam it's more or less the same format in fact in this year I only had eight questions because I think one of them was longer so this year I actually asked the same briefly explained concept the exact same thing it is not true for your year I will tell you that now but it is okay I'll tell you this one of these will come but not both it's not a contradiction I said it's not you will not get both the same this year and there will always be true or false just like this there will always be a sketch of a delta sketch of a gamma here's a discrete time model once again do something with the discrete time model here's a continuous time model and this question again was very similar to the year before except there was a modification of the payoff and confirm satisfies block-shoulds once again integration by parts formula another it was isometry like question and a multiple offset question broken up into two parts and the last question is a dynamic hedging argument question but I'm asking you to do it with two assets as opposed to one like how we proved in class going to use both assets and then there's a question about Feynman-Tab's result so your exam is as you can see there's somewhat of a similarity between the format of the previous year's exam and yours has a similarity as well but the specific question particularly the specific difficult question is going to be different and the moderate questions will have very very similar form okay yeah I wanted to I wanted to post it at the same time but I didn't find it and then I got busy with other things so I'll try to post it as soon as I can find it is it somewhere on this computer and I just don't know where okay let's see I wanted to do one more sort of one piece of theoretical like work but you know I think you're kind of overloaded with that at this point so what I'm going to do instead is I'm just going to tell you a result and then we'll just use it and I won't even get into the detail and then I'll come back to something practical okay so one of the this result that I want to tell you about is dynamic hedging in incomplete markets so I'm not going to go through the analysis so you remember what the concept of incomplete markets meant in discrete time it meant that the number of traded assets is less than the number of outcomes so if I had two traded assets but three outcomes you cannot exactly replicate a contingent claim so incomplete markets and interest rates was an example of that you always have more states than you have fundamental traded assets such as bond so when you want to do the dynamic hedging in incomplete markets the way what happens is if you assume and let's talk about it particularly in the context of interest rates because that's what is mostly relevant for you guys in your future careers actually so let's take the example of an interest rate model such as the vasochec model the problem with going through the dynamic hedging argument as we had it for the Bloch-Scholes model is that R is not the price of a traded asset and if you recall let's just go back here for a second to the Bloch-Scholes hedging argument what we did before is we took a position in the traded asset S a position in the money market and then short the claim and that allowed us to be able to instantaneously remove the risk because we could trade S now we cannot trade R because R is not the price of a traded asset so what happens in this case is that you can still do a dynamic hedging argument but your portfolio is going to consist of alpha units a different claim call it H cannot trade R let's say to price G so G is the thing you want to price and you cannot trade the interest rate to price G what you do is you introduce yet another contingent claim and you trade that together with the money market account and at the same time short the claim G so this is the idea when you do this it turns out that the partial differential I'm not going to bother to go through the whole dynamic hedging and tell you all the details instead I'm going to tell you that the pricing function for G turns out to satisfy this kind of equation it has that equation and this looks very similar to the Black-Scholes PDE the main difference is that this in Black-Scholes was R itself it was equal to R but for interest rate models it's not for incomplete markets in general the drift and remember that coefficient there tells you about the drift in the expectation representation that drift is not equal to R what is equal to instead is the original drift for the interest rate modified by something called the market price of risk and this market price of risk it's sometimes also known as a sharp ratio if you're keen on understanding how this works you can actually look up last year's notes I'll keep it online and you can see how this derivation works that's our end result is that you have something that looks very similar to Black-Scholes where you get the volatility squared of the underline process you get instead of R you have the old drift minus the market price of risk times the volatility and this then becomes an issue because you might remember when we dealt with interest rate models there was sort of a degree of freedom we were able to choose the probabilities and then make the drift of this little Hoenn Lee model to match market prices and I said that we actually have a lot of degree of freedom we have no arbitrage does not fix the probabilities for us instead we get to pick them the analog of that freedom that we had in the SCREED model is the degree of freedom that we have in choosing the market price of risk now I say we have a degree of freedom but in terms of actual implementation you have to choose that market price of risk to make sure that it matches real world data so although you have a degree of freedom you're not completely free to make it be anything you want it's got to be chosen to make it match the way that the world actually works and so a typical assumption for market prices of risk in interest rates is that this is equal to some linear function of R so the idea is that if B is 0 the assumption is that the market assumes just a fixed constant market price of risk adjustment to interest rates and if B is non-zero then that adjustment depends on the level of interest rates for large interest rates you get more market there's more at risk there's more risk aversion if you want to think of it like that for smaller interest rates it's less now what does this do in terms of the dynamics if you put this equation into this what I've underscored in blue here then that whole coefficient you can rewrite it as in that same form a new kappa and a new theta you can see the coefficient there in the blue parenthesis is just the linear function of R and all I'm doing is I'm subtracting another linear function of R so I've got to end up with a linear function of R and I've just simply written that linear function of R in this form so what happens when you change from the real world to the risk neutral where you're trying to do pricing because what happens is that the level that interest rates mean historically do not have to be the level in which they mean revert to when pricing options because you have different level theta showing up here as well the rate at which the price at which the interest rates converge to that mean reversion level can be higher or lower than the actual rate observed that's quite different from the volatility you can see the volatility stays the same sigma is showing up in the same place sigma doesn't change but the drift does so this is kind of an interesting fact is that when you go from P to Q for interest rate models you can change from the parameters kappa theta to kappa bar theta bar and you can go to you can basically modify the level and rate of mean reversion and it's an important fact that you can do this if you couldn't do that you would find that your models do not match market prices if you assumed that lambda was zero and you just looked historically at what interest rates did and then use that for pricing purposes you would find that you get incorrect answers for bond prices they do not match the market so it's very necessary to incorporate this market price of risk this extra degree of freedom that you have and that you only have it in complete market settings so one question that I'll address right now and then we'll take another short break is suppose we have this how can we price options or how can we price first of all just the bond I'm actually not even going to touch options on bonds because it's a little too complicated for this course so what would how could we apply Feynman cat to this result what would Feynman cat say if we're going to solve that equation then Feynman cat implies that we could write g as a function of t and my short rate now equal to the expected value under q of the discount factor and what must the discount factor be e to the negative the integral of whatever appears on the right-hand side now the right-hand side is not a constant because it's our state variable so we actually have to do the integral of this process times whatever payoff happens to be conditional on our little t equals this and we're not done yet because Feynman cats we still need to specify what is the sd e that are satisfied when we compute this expectation and what is the answer what's the sd e we're supposed to just read off the coefficients the coefficient in front of the partial derivative with respect to our term is our drift so that's kappa bar theta bar minus r the coefficient of the second derivative in our term is our variant modulo the factor of a half so that's just sigma so we've got that so in fact what we need to do for this expectation has exactly the same form as what we started with we started with a p model with a real historical model that is a vasochec type and we end up with a model that is a vasochec type but with different parameters that's what I was saying before so the goal then is how do you compute expectations of this kind and I think I don't really want to spend too much time but let me sketch the idea for you suppose I wanted to compute a bond price in continuous time how would you do that in this model how do you calculate a bond price what's the payoff of a bond suppose interest rates go up to 100 at the time in which the bond matures what's my payoff notional is $100 well at maturity I just get my notional don't I always regardless of what interest rates are if interest rates are 2% I still get $100 at maturity so this five function in fact is just one and so our pricing equation it's very simple and it makes intuitive sense it's just the price of the discounted $1 but the discount factor is stochastic so in order to compute this expectation you need to know something about how this distribution how the integral of R is distributed and I'll tell you now that this is normal with some mean and some variance I'm not even going to bother to tell you if you look at one of your previous exams you'll see that that was part of the exam question is actually coming up with that form so they were given all of this information and they're given also the solution for R it turns out to be normal and hopefully that makes sense to you because you might remember in the vaso-check model R was normally distributed do you remember that in the vaso-check model it was normally distributed so if you integrate R a bunch of points in time it's kind of like summing up a bunch of normals so it's not too surprising that the answer is normal so suppose you in terms of this M and this V what would the bond price function actually look like what's the answer and this is a distribution under the measure Q what is the answer for this in terms of M and V isn't it just exponential negative M plus a half the variance you're computing the expected value of the exponential of a normal random variable so it's just that these are the steps that you would have to do in order for you to compute bond prices and then how do you actually find Kappa bar and Theta bar what you do in practice is one of the things that you could do I should say is you would have yields for bonds in the market that look like that your model from this I can compute a yield your model will tell you that the price is this for one set of parameters and you keep playing with the parameters Kappa bar and Theta bar until you get the best possible match to your data and the best possible match could mean minimize the least squared error for example so it's very different than for equity here you actually have to do a calibration procedure but you might remember in discrete time we actually did exactly that if you think back when we had the discrete time models for interest rates we had these little Thetas that were changing in time and we had that Excel spreadsheet and we were finding them to make the error between the bond yield from the model and the bond yield in the market be equal do you recall doing that it was before your midterm it was a while ago but we basically did that exact same procedure but this is the counterpart in the continuous time setting as you approach it like this okay and there are some ways to do this in an analytical closed form approach but I'm not going to get into the details I just want to sketch the ideas for you okay so why don't we take a quick break say five minutes and we'll come back to the very practical thing tell you a little bit about some insurance embedded options that appear in insurance products since you are actuaries after all okay so the last thing I want to do today is a very very simple down to earth example of an actual option that shows up in an insurance type product and these are called the guaranteed minimum with death benefits withdrawal benefits and I'm not going to go into a huge amount of detail I simply want to give you an idea of what they are so the idea of these kinds of insurance products is an investor so just a big picture investor such as yourself typically it's someone who's almost in retirement is going to give an insurance company a big chunk of money say $50,000 whatever it is some chunk of money and what the insurance company does is they give you a product that looks like this they say here's time here's the initial investment that you've given them let's for concrete purposes let's say one million okay you give them a million dollars you tell them I want you to invest this in a very particular application of stock and bond and what they will normally do is they'll say here's a collection of typical portfolios here's a balanced portfolio it has 50% stock, 50% bond here's an aggressive portfolio it's got 80% stock, 20% bond here's a very conservative portfolio it's got 20% stock, 80% bond one of these and these will be sort of standard things one of those so you have a fund you put your cash into that fund and that fund is going to grow at some rate whatever the market return happens to be and what this product does is it says it could be on an annual basis it could be on a quarterly quarter annual, it could be on a monthly basis you are allowed to withdraw a certain amount of the money let's say it's $10,000 and let's say that happens every quarter just for the sake of illustration here so this is that date you get to withdraw I've drawn it much larger than 10% there but you withdraw however much it is it's $10,000 let's say and then this keeps growing at whatever rate so that sounds like a cool thing doesn't sound like anything terribly interesting so why would an investor want something like this the reason is that if the fund is kind of just doing okay because market conditions are just sort of okay as you withdraw you can see what's happening here as you keep withdrawing there's going to be a point in time in which when you withdraw there's not enough money in the fund you see that there's some point in time in which there's not enough money in the fund so what does the insurance company do the insurance company says okay I am still going to allow you to withdraw the original amount that you were that you were told you could withdraw so each of these are $10,000 and they're going to simply say if you withdraw $10,000 they're going to pay to you so the fund drops to nothing and these cash flows these are all the obligations of the insurance company so in other words what it's doing is it's protecting you for downward trends in the markets basically so if the market does well so this is one particular scenario that could happen and this has some finite maturity it might be you have one, two, three, four, five, six, seven, eight say eight quarters you did this over two years and this is a really extreme example but suppose after eight quarters then the contract ends but it could so happen that in another scenario of the market or another scenario whenever you withdraw I'm going to draw this on a slightly different scale but the idea is the same the market is just constantly growing and it's outpacing or at least on average maintaining your portfolio level so you actually never go bust the fund never goes bust and in this scenario the insurer never pays you anything because it's all coming from the fund the fund is automatically replenishing itself this guarantee minimum withdrawal benefit is precisely this benefit this is the true optionality in there if we get the scenario like in this case then the insurer never didn't have to really do anything they simply had to take the money put it in the fund, invest it done, they never owed any cash flows in scenario number one they owed a sequence of one, two, three, four cash flows and one partial cash flow this partial cash flow here because part of it was in the fund and part of it was not so this kind of option there's actually an optionality in here isn't there? you can view this as an option that the insurer has written to you and that option protects you from this fund going below zero and you might imagine how you could value this kind of option let's ask the question let's say, let's look at and see if we can formulate what that option actually looks like because this is all in words how it's working more or less all in words so let's try to formulate it by looking at one little period and seeing if you can come up with a nice little per period formula so from here to there you know that in this period N if we assume geometric Brownian motion for the fund that would be our obvious simple standard assumption that we could use it's wrong partly because the fund is a balanced fund so it's going to have some equity and some bond and you're not dynamically trading it you just got a fixed position so it's actually not GBM so this equals this guy under the risk neutral measure which is where you might be doing your pricing purposes it would be r minus a half sigma squared times the time between these guys plus sigma times the increment of the Brownian motion that's basically what we've been dealing with for quite a while now so this is the fund value at the end of that period then T sub N minus before you made your withdrawal right? that's actually what you made just before withdrawal so you withdraw you get this minus the new fund amount is equal to this minus whatever that $10,000 is let's call that K agreed? so the new fund amount is equal to this you get protected this is maxed out at zero so if this drops below zero if F is too small the extra portion is coming from somewhere else this is what the fund is trying to do for you so the optionality is effectively looking kind of like a call from one side of the party on the other side it actually looks like a put and insurance companies the difficulty that they have with this kind of option is when this drops below zero is at a random point in time so although this can be written in terms of a strip of put option it's actually a put option on a process which itself is sort of recursively being dropped in this manner and it gets linked to something like an Asian option remember this idea of Asian options I think we talked about it way back Asian option is kind of the average of an asset price over a future over a past set of time this you can rewrite in something like an Asian option and like I said I'm only giving you these I'm just shooting this out there so you kind of have an idea of some of the products that insurance companies are dealing with and to show you a little bit of a tie in with what we've done I'm not trying to spell out every detail here and if you want more information I'm just going to show you a little bit of a different direction so that's basically their challenge it's trying to model this thing model this default this is kind of like a default time it's the default of the fund the fund drops too low and that's the challenge and then even this model of geometric Brownian motion is not valid you can see that the returns and there's no reason to believe that those future volatilities are the same as our past volatilities in fact in recent years you noticed that there's much higher volatility in the market than there were in previous years have you noticed that I hope you've all taken some interest in the financial markets you've seen that volatility is very high come back down again but there were spikes in the volatility this type of model of motion does not capture those features at all what you need to do is you need to have a model in which volatility itself is a stochastic variable and not just stochastic variable but a stochastic process and that process changes with time so to give you a little foresight into some of the things that you can do when you go on and do your masters and your PhD in mathematical finance one kind of model for asset prices is called the Heston model and here what you do is you change the sigma and you make it a process and this process has mean reverting features remember that quality that I said that there's spike in volatility but it comes back down so it's kind of like a mean reverting process similar to Vastichek model but the difference is you're not allowed to let volatility become negative variance in particular cannot ever become negative and to prevent that what's often done is you put a factor of square root of that variance in front of the db term and this will prevent it from becoming negative this particular model it's a very well-known model and it's used a lot there are a lot of flaws in it for some reasons or another but it's still popular in practice it's called Heston model Heston model and like I said this volatility process is basically a mean reverting process and just like the sorry the Vastichek model it gets pulled to this level theta there are spikes and then it comes back down comes back up and it's going to fluctuate around here so the Heston volatility this is giving us a stochastic volatility model and this is important in products this can be important in products like this one why? because the timeframes that are relevant here can be decades not just a couple of months or a year but decades now in for volatility to move around it's an important part of the pricing of these instruments if you don't incorporate them then you're going to misprice them severely and in fact many insurance companies have been hit not just by misprices because they didn't incorporate stochastic volatility but because they didn't appropriately hedge their instruments they were holding static positions they sold these kinds of guarantees but they did not at the same time dynamically replicate the obligation as you've learned through this dynamic hedging strategy you've seen that you do need to do this if you want your risk exposure to be small if you purely want to be exposed to the potential of anything then you don't hedge but you're going to potentially be hit by a lot of losses and that's what's happened to some insurance companies and a number of them are starting up what are called hedging groups which are trying to model and incorporate more sophisticated financial models and the risk management of those models within their groups let me see another way to incorporate volatility, stochastic volatility are something called regime switching models regime switching models are models that are based on Markov chains so the idea here is that you have a number of states the volatility or the variance is just simply a function of some sort of Markov chain so H is a Markov chain and it's a Markov chain and it takes on, you know, it has a number of states so these could represent low vol high vol if you think of just two situations there's only two states in one state of the world or three states of volatility, low volatility so you can have a Markov chain that switches between these three states and that Markov chain evolves in continuous time but it only takes on these three values and those three values determine the volatility in the market you can modify the block show this dynamic hedging argument that we've developed and so on to come up with partial differential equations which models based on Markov chain also satisfy and this will help you again in the pricing and risk management of these kinds of products so they're sort of very important features let me give you one more example of another kind of guarantee that ensures sometimes right and this one is called an equity indexed annuity so in an equity indexed annuity what the firm will do is they'll say, okay you give them some investment they're going to have some upper bound that grows at a fixed rate in fact since this is usually over a month it doesn't look that high there's a lower bound and the company will say if you drop below that lower bound return in that month they will match it but if you go above some upper bound your return will get pulled down to it so what happens is the firm will do it so you continue your guarantee from the same previous level because it was never enforced so this is your previous level here in this particular scenario you notice that the fund underperforms it underperforms the guarantee so what the firm will do is they will bump up and they will add money to your fund so that it hits this lower bound it just goes on again so from this point oh my bad you're always getting the guarantee from where where your fund is at the end of that time frame and if you underperformed you get bumped up to the lower boundary and if you overperformed you get bumped down it's kind of odd but the idea is that that helps to curb the cost if they allowed you to just get the upside and none of the downside because this is effectively protecting you from downside if it allowed you to get all of the upside and none of the downside it'll cost too much and people won't want to buy it so they put an upper bound on how much you can gain and let's draw one more scenario here so this is going to this got bumped up to there and that's my lower bound my upper bound so again in the second scenario it got bumped up to that point and always a challenge drawing these things so in this scenario it overperformed in that time so what they would do is push you down and your fund would be worth this amount so this is one form of inequity indexed annuity it's given you lower and upper bound protection and what's interesting is that it protects you over each period it's not just your overall return and it's not just your overall lower bound return and overall upper bound return but it's the return over each sub period so in every month you're guaranteed at least 3% and at most 12% for example that's completely different than saying over one year I'll guarantee you 3% and at most 12% right it's a very different beast okay is it done? yeah what's the number written on it? last number