Credit risk (QRM Chapter 10)

okay so good afternoon so let me tell you what we will do for the remainder of the summer school starting this afternoon we're going to devote it to credit risk and we have in that in the book that you all have there a fort there's a lot of credit risk but it certainly is one of the most important risk categories if not the most important risk category well that may depend on what kind of institution you work for but certainly in banks the most important risk category that so there are four chapters and we will present material from 10 and 11 so 10 this afternoon in 10 we begin with section 10.1 there's not much in there credit risky instruments we look at all the different things that are subject to credit risk but after that the focus of 10 is on the credit risk of a single firm or a single obligor so the way the materials are ranged it begins with 1 obligor how do we model the credit risk in Chapter 11 we put them together in a portfolio so chapter 11 is put his portfolio credit risk chapter 12 which we won't do has some material about dynamic models and CEOs and chapter 17 which we will get on to in the final session tomorrow has some material about counterparty credit risk okay sir - so - go right to the beginning we have a definition of credit risk this is the one that we put forward in the book risk of loss arising from the failure of a counterparty to honor its contractual obligations so this is this is quite general subsumes various things default risk so the the risk of loss is due to the default of a borrower or of a trading party partner or of or of a counterparty in a derivative transaction and also downgrade risk the risk of losses if there's a deterioration in credit quality of a counterparty counterparties and there are you would want them to maintain their credit quality they have a kind of obligation to you to maintain their credit quality if their credit quality deteriorates that may cause you losses some words that we will use this word obligor which i I hadn't heard of in the English language until I started to study credit risk but it comes the obligor comes from obligation so credit risk arises from unfulfilled obligations so the obligor a counterparty who has a financial obligation some examples a debtor who simply owes us money a bond issuer who's promised us interests and promised to redeem the principal at a later date should they fail to either pay us interest or return the principal and a counterparty in a derivative transaction where we have I've agreed with this counterparty - perhaps - to make some kind of swap of payment streams and shoot my counterparty fail to fulfill that obligation that contractual obligation that will be a problem yeah in a certainly in a yeah in a swap as far as I'm concerned obligor as far as I'm concerned and I'm an obligor as far as you are concerned so default is simply the failed the failure to fulfill that obligation failure to repay failure to pay interest etc and of course their various wee reasons why that default may take place a liquidity issue which means the cash isn't available at the right time to make the repayment a lack of liquidity may be symptomatic of a deeper rooted problem the company may be insolvent that's more worrying because the end result of course may be bankruptcy but a lack of liquidity a default caused by lack of liquidity is is a sign that all is not well in the risk management of this company so as I've suggested it's a pretty important risk category hence the amount of space that it gets in the book and it's it arises in in all kinds of context in financial services obviously if you're banking you have a portfolio of loans that's clear if you're an investor with a portfolio of corporate bonds so I'm so not Treasuries from very highly rated and essentially not default about countries but corporate bonds these are obviously affected by credit risk and as you will see and this this will be more tomorrow in an over-the-counter derivatives transaction such as for example an interest rate swap we will take that specific example so tomorrow we'll look at an interest rate swap and we'll analyze it and you'll see where the counterparty credit risk comes from so if one of the parties doesn't fulfill their contractual obligations that is a default that's a loss here in this category in the second category the derivatives we're talking about these may be derivatives which are actually designed to manage other kinds of risk if it's an interest rate swap it's being designed to manage interest rate risk so primarily an interest rate swap is about interest rate risk primarily primarily for an exchange swap is about for an exchange risk but because it's a contract between two parties it carries credit risk counterparty credit risk now there is a third category credit derivatives and these are specifically designed to manage credit risk and so here you have things like credit default swaps in the book we look at securitizations as well under this heading but for today's and tomorrow's purposes we will just look at the the credit default swap as an example of a derivative that is specifically designed to manage credit risk in the first instance clearly banks have a subject to a lot of credit risk but it's very very relevant to insurance companies on the asset side of the balance sheet insurers are exposed to substantial credit risk in their portfolios of bonds moreover there is counterparty credit risk in reinsurance treaties so I so in the insurance industry credit risk is equally a big theme to finish this introduction and I should probably keep my eye on time because we're having to apply a bit of a rearrangement algorithm to the structure of the workshop to fit it all in but I think that will be possible so credit risk management what does it mean the question which is central in the book is the question of capital determining capital for credit risk where they're looking at the regulatory form the regulatory rules are looking at economic capital that a prudent institution would want to have of its own volition and then there are if you have portfolios of credit risky assets then of course there are considerations of diversification of the portfolio of optimizing the return subject to risk considerations or vice versa for the credit derivatives the the things like the CDs there are issues to do with valuation with hedging and then perhaps managing collateral for these trades and so there in the second edition of the textbook a lot of material has been added about this and counterparty credit risk this since the financial crisis has been a big issue for people in banks they need to control their counterparty credit risk and so that's why we we have a separate session on that to close ten point one I'll keep it really brief in in the book we we sort of list credit risky instruments beginning with simple things like loans and bonds and then going on to derivatives but for the purposes of the examples we will be essentially thinking of default herbal bonds and credit default swaps and when we come to value things a little bit later this afternoon we will need the credit default swap so a little bit about those and related credit derivatives so as I've said they're primarily used to manage credit risk that's the point of the credit derivative as opposed to some other kind of derivative which is subject to counterparty credit risk and the payoff will be related to the default of one or more one or more obligor Saur one or more firms and in this chapter ten it will be related to the default of one firm because chapter 10 is about the single firm in this market that lots of participants obviously banks and investment banks are major participants but also insurance companies will use credit default swaps to hedge credit exposures and so it's an important thing to know about in the banking world you will find the retail banks are typically net buyers of protection we're going to see that the credit default swap embodies the notion of one party selling protection and the other party buying protection and so retail banks are generally protecting themselves against credit events such as the default of large obligor who owe them money whereas investment banks and hedge funds they will often enter on the other side because they see opportunities to make money the CD s is the is the it says here the workhorse of the of the credit derivative market and there is a huge volume of these contracts written and before the financial crisis the market grew and through the financial crisis and it's it's still a huge if not an even bigger market since then so the structure of a CD s we will look at this picture but it will come back I think when we when we later on there are three parties listed here but only two of them are actually engaging with each other directly the third party here the reference entity is not involved directly but this is this is the this is the firm whose default we are worried about or one of these parties is worried party a is worried about the default of party parties see so I don't know this let's pretend that C is an oil company or a supermarket group or we were in Vevey yesterday or Nestle a company that issues our bond issues bonds or large on a large scale and a here is a bank or an insurance company that has a very large holding in the bond of C and wants to protect itself against the the default of seen be the protection seller comes in and offers that protection and so what's going to happen is that if there is a default of C B is going to make good the loss that a would suffer so if there is a default so here's a here's a rhetorical question does C default if yes B is making you default payments if no then obviously no payments but the other side of this swap is the premium payments that a makes to be to pay for the protection a little bit of notation there will be a maturity same notation as we've used all week there will be a maturity to this contract tau C is the default time of entity C it may never default but we are worried about the event where tau C is less than team the premiums are paid until maturity and of course and they cease should default take place and the default payment is only made in the event that tau C is less than T so we'll just the yeah we will take it to be a fixed cream there may be a recovery so default this will be sort of legally regulated by the contract it will be it will either happen or not but in the event of default admit it certainly may not be a total loss there may be a recovery and so a recovery element will probably come in and a proportion of A's exposure to C may come in something like that something like a five years CTS those are the kinds of of time horizons we should think of so I think what I said is summarized here but let's meet just make sure the so if the reference entity that was net that was see that was Nestle here experiences a default before tea the protection sailor makes and default payments of protection buyer it mimics the loss or it makes good the loss that the protection buyer actually experiences and this CD s it's said to have two legs and this particular leg is the default payment leg the compensation the premium is paid by the protection buyer to the protection seller it's a regular payments treatment might be could typically quarterly payments are made and after the default premium payments would stop in this swap there is no initial payment it has to be decided at the outset what is a fare premium so the fare premium or the fair swap spread is decided at the outset usually they're quoted as a percentage of the amount of the protection so you might buy ten thousands Swiss francs worth or fifty thousand a hundred thousands with Swiss francs worth of protection against this reference entity and the spread premium will be a percentage and we will use X star as the notation for that when it when it reappears the corporate bond I won't explain that's the other contract that the other instrument that that we will value so just one more slide on the CD s why on earth do these things exist well to begin with it was about protection it was about hedging exposure to large credit risk so if you think of the invent the company that invests so an investor in bonds with a very large exposure to a particular company may genuinely warrior the credit risk so what they could of course do is reduce the bond holding they could sell out of that bond completely they could reduce the holding but the bond market is not particularly liquid and to sell a very large amount of the company's bonds would it would probably lead to a loss through through liquidity risk whereas it is easier to to do a CD s the CDs market is more liquid so rather than reducing the original bond position you could do a CD s because they are more liquid so this was the original reason for CD s it really was about protection but then the market very quickly became speculative the other side the protection sellers are doing it because it offers speculative opportunities and in fact many of these companies appearing credit default swaps on both sides so they appear as buyers they appear as sellers and they're doing this to speculate on changes in the spread and tone changes in the spread and to make money in that way so companies that do it purely speculative lis usually they are actually taking naked CDs positions so a protection buyer if they don't own the bond that's called a naked position you don't have to own the bond that's actually the final point here and that's perhaps surprising at first you don't actually have to own the bond to buy protection so if you don't own the bond and you're buying protection you're speculating basically on the widening of the credit spread you're speculating on the widening because at a later point if it widens you could then sell protection for the same amount and you could lock in a profit and similarly if you sell protection you are you could be speculating on the narrowing of the credit spread so there are both genuine hedging motives and there are speculative motives and 990 it's probably these days 99% yeah there's a yeah yeah I mean the market is huge and those who use it is genuine protection I think of a small fraction I wouldn't like to say you might know I think it would be a lot less liquid if you were forced to have insurable interest so it sounds like insurance and that's what makes it not insurance there's no requirement for protection buyer to actually have the bond no requirement protect insurable interest so it is like insuring someone else's house in effect which of course is you can't do any insurance okay so from now on we think about how we model credit risk how we model default risk and I just want to identify too we come back to the products later and I just want to identify two philosophies one is the the rating or scoring philosophy that is you collect empirical data about the payment records of obligor and you you score them based on their past behavior now whether it's rating or scoring largely depends on whether we're talking about whether we're talking about big companies with debt traded on markets or small companies whose debt is not traded on markets so obviously the rating agencies fulfill the role of rating debt on large companies but in retail credit risk if it's you or I who is borrowing the money we are typically scored using a scorecard so in retail banking there is a whole huge industry in in credit scoring in building score cards that will assign me a credit score and of course you can look up your own credit score based on my payment history and other characteristics of myself so this is very much a statistical thing so that this is very statistical it using empirical data to measure credit quality expressing credit quality on usually often on a on a categorical scale triple a double a a but in retail banking a credit score is usually a numerical score now the other sort of philosophy for determining credit risk is an interesting one and this is using markets to infer credit risk or to try and gain information about credit risk so for obligor whose equity is traded on financial markets prices are often used to infer the markets view of the credit risk of a company and this is where the firm value models come in this is where the structural models come in and this is the thing that I want to explain before the break and so I think we will think we will move fairly rapidly through this section rating and scoring there are a few slides about this I think probably the this is relative relatively straightforward you're well aware of the the role of the rating agencies and you may know something about the use of scorecards and retail credit risk we ratings are often used as a primary measure of credit risk and many people model ratings and rating migrations and in the book we have a section on models for rating migrations Markov chain models both in discrete time and continuous time and they're also there's also one or two nights our scripts in the repository fitting mark is either yet fitting Markov models in discrete time in continuous time to rating migration data so I think that I will pass through this these few slides relatively quickly you've probably seen such things as rating migration matrices before so you rate this as an annual rating migration matrix that I got from Moody's which gives you basically you could effectively take them as probabilities of moving from one particular rating this is the moody system to another particular rating in the course of a year 87 percent of companies Triple A companies remain in Triple A others move into other categories some have their rating withdrawn and so this isn't we have to to use this kind of data we have to deal with them withdrawn ratings as well usually we would correct these figures to account for rating withdrawals but this kind of information can be used to build a model of how companies move among ratings well the rating agencies claim that they rate through the credit cycle so that they so that they take a long view right through the cycle and so a rating should be viewed as a somewhat relative measure rather than an absolute measure so to come so a company that is a rated now and a company that's a rated during the crisis you know that that should be the same company it's not just the fact that it's not just the fact that the rating expresses has a direct relationship to default probability at a particular point in time yes yes I would say but I but the evidence in the data is they have difficulty doing that so they should be through the cycle measures it shouldn't just be the case that all the companies are downgraded when we move into a recession and they're upgraded when we come out of a recession but you know when you analyze the data statistically there's some evidence that they have difficulty doing that this is drawn from statistics so Moody's have a big database which they've collected over the years so every year they will have a cohort of companies of different ratings and they will simply count the defaults and they'll count the the rating transitions so these are empirical migration rates in fact they they have quite rich data and if we'd had a little bit longer I could have shown you the form of a typical data set where you see for each company it maybe starts in one rating and in a particular date its rear ated it's upgraded its downgraded then it has a new rating for a while and then maybe it's rear ated again and then perhaps it defaults and so detailed information on the time that companies spend in ratings from that you can estimate you can estimate matrices of transition probabilities usually Markov assumption the Markov assumption is made so it's assumed that whatever rating you have now that determines your migration and default risk and the history of how you got to that rating whether you've just newly been downgraded or whether you've held the rating a for the last ten years that shouldn't matter if in a Markov model of course it should only be the current state which determines your default risk again the evidence is that that's not the case that there is momentum that the company that was downgraded last week to a is more likely to be downgraded again than the company that was a has been a for the last ten years yeah yeah so rate so okay so for insurance companies are investing yeah yeah I think yes rating companies is perhaps not so problematic but still there is a huge amount of judgment in a rating ultimately it's a committee in a room that decides do we downgrade this month or do we wait another month and see what happens so so for that reason I mean ratings are not universally popular but in the insurance industry we invest in investment grade grade bonds the ratings are often the primary measure you would use of credit quality and in economic scenario generators quite often what is modeled is transitions of bonds around a rating system the one I mentioned yesterday the the Moody's again their economic scenario generator is based on generating transitions of bond portfolios around a rating system the statistics of using historical data to get transition matrices I will admit but as I've said there's a really nice script which shows how you fit a continuous time Markov chain in the repository and instead I'll go on to ten point three which I I which is say it is essential in the world of credit risk the structural or the firm value model there are there are there are a couple of modeling philosophies so I mentioned a couple of measurement philosophies using ratings and using information from prices there are a couple of modeling philosophies there there are the structural models otherwise known as firm volume models and there are the reduced form models so 10.3 is about structural models and the remaining chapter the remaining sections of the chapter will be about reduced form models and you'll see why they're reduced form when I hand over at that point to route again structural models attempt to yeah we're going into the market view we're going we're going into ways of measuring default risk using market information and to do that we need we need models and the structural models are our first attempt and in fact we will only do the model of Merton which is the progenitor the first of these kinds of models and the easiest to understand so the firm finances itself very simply through equity in debt so we have a firm that issues shares to finance itself and it issues debt but the data is assumed to have a really simple form so the debt consists of a single zero coupon bond issue with a fixed face value and a fixed maturity so just one bond issue and you could say what a single zero coupon bond issue and the the nominal of that issue is B and the maturity is time capital T now at any time small T up to the maturity of the debt we will write the value of the equity as s if you like s for stock and we will write the value of the debt as B B for bond and so the value of the firm will simply consists of the value of its equity plus the value of its of its debt s T plus B T and so for merchants firm we can begin to think about what would default look like what would default look like for this firm well the the obligor are the bondholders here the obligation of the bondholders they are owed a repayment at time capital T they must get their their principal back at time capital T so we look at what happens at time capital T there are two ciccone Li two possible scenarios at time capital T the total assets of the company which consists of the value of the of the the the share component the value of the equity component and the value of the debt components if they exceed the amount that has to be repaid the normal of the debt then everything is fine the debt holders are repaid you could think of the company simply being wound up at this point it's a one period model but then of course it may be rolled on for a further period so if VT exceeds B the debt holders get me back and the shareholders are also happy they get they get something they get the residual value and the bigger this is the happier they are so there's no default but if B T is less and B there's a problem the firm can't meet its financial obligations the shareholders well they're wiped out and they walk away they exercise limited liability the bondholders get any reason for any residual value in the firm and here are the shareholders who get nothing and so when Merton wrote down this simple model obviously he saw immediately the option the the contingent claim analogy he saw that what the equity holders get at maturity is the positive part of V T minus B and what the bondholders get is B - the positive part of B minus V T so he saw that these look like contingent claims he saw that this one looked like the value of a call option except it's a little bit odd because the it's a claim on the total assets of the firm V T is the value of the total assets of the firm and the strike is the face value of the debt so the value of equity it's like the payoff of European call option on the assets of the firm with exercised equal to B and this is like the value of the debt where you have the nominal value and you have a short put option a short European put option and so because Merton was involved in the black Scholes Merton model he began to think about how he could value the equity in the debt in this context he probably also looked at it and saw that it made some kind of economic sense in that it explains certain conflicts of interest so between shareholders and bondholders shareholders are a bit more gung-ho they're quite interested in the firm taking on risky projects because that potentially increases the value of their option so they like volatility whereas the bondholders don't like volatility they have a short position and they would like to see volatility reduced so he probably looked at the this analogy from various sites and saw that it made some kind of sense to him and then he applied the model that he'd been working on for valuing contingent claims on on stocks and that is geometric Brownian motion so he decided that he would model the assets of the firm because he's going to he's going to volure contingent claims and the assets of the firm he decided to model them through geometric Brownian motion and I've written down the diffusion model here with two parameters the drift of the assets of the firm they may drift up they may drift down and the volatility of the assets of the firm and then I've also written down the the solution to the stochastic differential equation so at time capital T the value of the assets of the firm in this model would take this form here so we'd have a term here which was an increment of a Brownian motion a Ravena process so this WT here would be a normal variable mean 0 variance team and this formula would give the value of the firm so in the geometric Brownian motion the value of the firm at the time at which it is wound up at which the bondholders are repaid has a log normal distribution this is a log normal distribution the exponential of a normal everything here is just parameter everything else is just parameters v-0 is some starting value and we're going to look at this with our incidentally in a second or if you like the log assets are normal and in this model the default probability is really easy if this is normal the default probability must be a normal probability in some sense it's the probability that the bondholders that the assets do not exceed the nominal of the bond or that the log assets do not exceed the log nominal of the bond this is normal so this is a normal probability and once again I'm imagining that I've never spoken to this point he stood back and he looked at it and he began he asked himself does this behave in the way in a way that would make sense and he saw that it does because the default probability increases in the debt level the more debt the larger the default probability it increases in the volatility of the assets that would lead that so it's increasing the in the asset volatility here but it's decreasing in your initial assets and it's decreasing in the in the drift so it okay we assuming this assuming that these drift up so Mew V is positive right let's let's actually make them let's actually make these pictures because they are instructive and talk about them so okay so if you want to run this script it's called Merton model Merton model dot R and you need two packages you need qrm tools because Kyurem tools has a quick function for computing option values and the black Scholes model and you need SD e so SD is a package that was written by I just forget who wrote it but it accompanies a textbook we could find it in the documentation it accompanies a textbook simulation and inference for s des with our examples 2008 and we're going to use that to generate mertens model and just learn a little bit about how it works so let's assume that these are the parameters I set up so I'll talk through them one by one the initial value of the assets of the firm is one so let's say 1 million 100,000 something like this the process which describes the real-world evolution of the assets so under a real-world probability measure the drift is point zero three per annum so is a three predictor sort of three percent growth in log assets you could say three percent annual growth in log assets the volatility is twenty five percent of the assets will fix interest rates to be two percent and the value of the debt that has to be repaid as 0.85 850,000 whatever and it has to be paid in one year it's Burton's model and we will will use an Euler scheme and just generate some paths of the of this process will generate 50 paths and I'll just sort of run through them so in this library ste there's a GBM here GBM geometric Brownian motion and I've chosen 50 paths so I generate 50 paths and then we look at them you can see you can see a matrix here which has 50 columns and six lines it always starts at 1 million and then you see the the Euler discretization the evolution of the SDE sometimes the assets go up sometimes the assets go down etc and so that's one path and there are many paths ok so starting at 1 million there are many paths and this is one year later so this is the evolution of the firm under under mertens model and certain of those paths are default paths thank you fun ok we started we started yeah certain of those past the default pass any path which doesn't exceed 85 or 0.85 remember that was the face value of the debt any path that that doesn't exceed that is a default path any path that does is a non default path so the probability of default which we have an expression for is the probability of being of a path which ends up down here obviously and one nice thing we could perhaps do we could just lay the density on top and then we'll just so to see how that looks so everything can be computed turn your heads through 90 degrees turn your heads through 90 degrees and you have a log normal distribution sitting on this axis here you have a log normal distribution imagine the total area under this is one and so the the default probability is this area here emergence model emergence model absolutely doesn't matter so a path that goes like this and then this is not a default path other people came along after Merton and they did first passage time models and they said a path which goes below some threshold maybe not this threshold but maybe an even lower one would be a default path right now let's take two paths one which is a default path so example of a default path as you see here we're gonna generate one example of a default path and then one example of a non default path and then talk about what we can learn from them so I I pick a seed that I know will work and here is here is a path it ends up below naught point a five let me add it let me add a green line to start with and let's go back to the slides and talk about just jump over the past at the moment we'll talk about pricing then come back to the two pictures so merton is going to use the machinery of the black Scholes model to value the two contingent claims on the assets of the firm and so in order to apply the black Scholes model he has to he has to invoke the black Scholes assumptions so there is a risk terminus tick and fixed risk-free rate and some less realistic things here that the asset value process is independent of the debt level that's kind of exogenously imposed and moreover the assets are traded on a market with no frictions three is sort of clearly untrue the total value of the assets of firms is not traded on markets the total value of the assets of a firm turns over only very rarely when a firm is bought and sold which happens very rarely you get one observation of what the total value the assets might be what is traded continually on a on a market for large firms like say Nestle or something is the stock so the s T is traded on a market the stock and perhaps a part of the debt but a part of the debt is traded so these are assumptions that are made in order to make progress and simply apply the black Scholes formula equity is a call option on the assets of the firm therefore the value of equity at some time T should be given by the black-scholes call formula so we'll just we'll just call this CBS call black-scholes it will depend on time it will depend on the underlying and it will depend on the various parameters volatility of the assets interest rate time to repayment of the debt in our case 1 and this this nominal value of the debt in our case 85% of v-0 so you get the black Scholes formula it has certain arguments so that should be the value of the debt and then sorry the value of the the equity the value of the debt will be given in the following way so we need some that we need to recall some notation from yesterday's and bond pricing notation so if you have a default free zero coupon bond with maturity T yesterday we said we would write PT t I will write here p 0 2 for default free so p 0 will be for default free and so this is under some continuous model of interest rates we have here the the value of a zero coupon bond now you saw before that the debt payoff and perhaps we just have to go back to that really briefly you saw it consisted of two parts there is the nominal and there is the short put option so those two parts have to be valued and that's those big upon that's those two terms here so that's how we value the nominal we just discounted with this zero coupon bond price it simply discounted and then we have the short put option so PBS for the black Scholes put option and we can work on that formula a little bit and we get this for the value of the debt so in the pictures we could add this we could add the value of the debt and we will add this as a red curve and the green curve that I just put in the green curve is the value of default free debt so just the first term here if the if the debt were non default able so if the debt were non default able it would be valued simply in this way but it is worth less so we take something away because it's default Abul debt it is valued at a lower level and so beg your pardon so that is the value of non default all debts the Green Line it goes up to not 0.85 like this it's it's going up it's an exponential curve and then we add the value of the default Obul debt so so what happens here so red is the value of default of all debt this is the total assets of the firm so the difference between the black line and the red line is the value of equity so this part is the book from the red line to the base down here because that's not 0 from the red line to the base is the value of the debt and from the red lines of the black line is the value of the equity so things begin ok the value of the debt is less than the value at the value of default free debt but then they take a turn for the worse mertens firm heads into difficult territory the value of the debt reduces because at this point we're already starting to think this could be a default path and you see what happens here the equity value just gets smaller it looks like there's a recovery here and so the value of the equity increases a little bit but then the value of the assets dive down ultimately the the equity is wiped out ultimately the value of the equity goes to zero the red line becomes the black line so the bondholders get the residual assets that's a default path and then of course the the opposite kind of scenario okay this is a this is a non this is a nice non default path here's the value of default free debt and here is the value of default herbal debt so you see what happens in this one things go okay for a while it looks like the previous one then things go really badly and the value of the debt reduces and the value of the equity is very small here yeah you perhaps wouldn't invest at this point buy shares the value is very small but then a recovery takes place up here and by the time we get to about here because geometric Brownian motions have continuous paths and we're starting to think it's very unlikely there's going to end up down here it can't jump to defaults then things things are looking quite good both for the bond holders and the equity holders and the value of the equity this difference is significant and the volumes the default herbal debt climbs up to be equal to the value of the default free debt so that's how mertens model works with two paths sorry that so is that not so maybe you have to say they get Frank Soviet a legend would be nice but this is all black as always the assets read as or survived the debt and the difference is always the vive equity withdrawn okay I keep going the wrong way right so we've talked about those two formulas pricing debt a couple of a couple of things that we can look at that are implications of what we've seen we have generated paths under the real world measure of course in the black Scholes theory under the risk-neutral measure the the value process follows this SDE we have this des de here where instead of movie we have the interest rate R we can also write down a risk neutral default probability so this is the same expression as before the save expressions before except that there's an R instead of a movie and we'll call this one Q the risk-neutral we'll call the previous one P the actual physical probability of default and in this model they have a relationship Q and P are related in this way they would be the same thing where it not for this term where it not for this term which is a sort of intuitive ratio it's the growth rate of the log assets minus the risk-free rate divided by the volatility of the of the assets and it's a sort of Sharpe ratio except for the for the firm's assets rather than for the firm's share price so in a sense it's a sort of measure of the risk premium earned by the assets of the firm and we put this one in because it's a rule of it's a sort of little rule of thumb or simple formula that is sometimes used by practitioners to go from the world of P to the world of Q so if you want or vise worth vice versa probably typically from peat you if you can get an estimate of the Sharpe ratio of the firm's assets now given that it's only the equity that's traded you might have to settle for an estimate of the Sharpe ratio Q yeah probably actually yeah so anyway this bit you would have to get from equity data how do you go from P to Q to P as ridiger has just said Q measures you can get from prices and so going in this direction is about inferring ya inferring real-world probabilities of default from information that can be extracted from prices so if we can in a pricing model if we can infer what Q is for a pricing model this will allow us to go to P if we fill in a number here that's the idea and then the other thing is we can look at the credit spreads of the bonds in Mertens models so that the bondholders they have a default able bond so their bond offers a spread over a default free bond so that's related to the difference between the the green line and the red line you had the Green Line was default free debt the red line was default Abell debt and that difference that is related to the extra compensation that mertens bondholders require for buying default herbal debt so that's related to the spread and we can get a formula for the spread so it measures the difference between the yield of the the default free zero coupon bond the guilt or the Treasury or whatever and the default amol bond issued by the firm so whereas we call the other one p0 we shall call this one p1 and what would it be well the total bond issue has nominal B the total bond issue is nominal B so p1 TT we would have to scale the value of the debt by B we'd simply have to divide by B and this would give us the price of the the bond that pays one so as ridiger explained yesterday in this notation this is the bond that pays one so we skip B and we can get an actual formula for the spread it's a difference in yield so one yield - another yield so on the left hand side the yield I'll just keep yeah and you as a formula and what it's and how to look at it to check whether it behaved in a sensible way it actually depends on only three things if we fix the times if I don't only two things if we fix the times it depends on Sigma V that's a key parameter and it depends on a simple ratio be discounted so we discount the nominal debt and we divide by the total value of the assets this is a kind of measure of leverage or indebtedness and so this makes sense if Sigma V increases the spread increases more compensation and if the levirate if this indebtedness or leverage increases the spread increases so more compensation is required and this seemed to make sense but when you actually plot the spreads in Mertens model they are not realistic they're not realistic so for example here's the spread against time to maturity one year two years three years four years the parameters are as we have them and then for short times to maturity quarter year third of a year this spread is essentially zero which is unrealistic in reality market spreads would be larger and this relates to the fact that by the time you get near to the boundary you pretty much know whether you're going to end up in default or not you can't jump to default and you can only be in default if you're under the line at the far end so there are many deficiencies of the model it's clearly not particularly realistic and a whole industry began after Merton adding more realistic features pudding jumps into the stochastic process allowing defaulted into at times etc and you'll find some references in the book but what we have here is a way of going well if I just go back to one formula to conclude and then read again needs to start one formula to conclude on the left hand side we have equity on the right hand side we have the bow of the assets if we want to get a default probability let's go back to the default probability we would need to evaluate this if we wanted a real-world default probability we would need to evaluate this and we would need to know certain things we did for an actual firm we need to know what is Sigma V what is mu V and we need to know what it what are the what is the value of the assets these things Sigma V and b0 are the most are important but difficult to determine movie is probably not so important so to evaluate this we would need to get information somewhere and if we could use stock prices and this formula here to sort of reverse-engineer the value of the assets we could get a measure of default risk lied from stock prices and and this is what happened next people took mertens model mr. Vassa check mr. Crawford was mean they took this model is in stock prices they had a method for reverse engine information they needed to compute default probabilities we go into that in the book we won't go into this afternoon instead we'll go onto the models that are more commonly used for bond pricing so you probably wouldn't use Burton's model to really price of bonds instead you probably use a reduced form model and I'm bout to hand over the baton to ridiger who's gonna do the the remainder of this make a stop before coffee I think in there which one is it no okay so as Alex said in the introduction to creditors there are basically two types of modeling philosophies or approaches for credit risk on the one hand the firm value structure models where Mertens model is really the simplest example where you start from the economic thinking that default has something to do with insolvency it's not exactly the same but closely related so you start modeling the assets of the firm and some default barrier and model default in that way now you can say that is a bit too complicated in some sense you're modeling an awful lot given that for many pricing applications all you really need is something like the distribution of the default time and on the other hand it's also so that the story that default and insolvency is exactly the same so insolvency means really that the value of the assets is lower than the value of the liabilities but there are firms that are insolvent and not default because somehow they are allowed to continue in their firms that are default even if they are solvent because they are in the short term short term cash problems and don't find anybody to lend to them and so there is this alternative idea of using the simpler reduced form models where you model directly the default time and where its distribution of course can depend on economic characteristics of the firm or also of course of the sovereign you want are you interested in so we will always talk here about the default table from being the default identity being affirmed but all of what we are saying in particular the reduced form approach applies just as well to countries I mean in the old days if you were not just investing in Argentinian or Mexican bonds or so you would consider the stuff essentially default free and so many things this has changed so the whole reduced from methodology really does apply to sovereign bonds just as well - which pond what is that yeah 9 over GDP is per GDP bonds exactly they are where the payment depends on the GDP yeah I like it Aisling okay then but when I guess for a GDP bond of course you would probably more use an approach which is closer to the start value approach because there you have to model the well for a GDP bond you would probably have to model the stochastic evolution of GDP if you wanted to price it so there you would be more inclined to having a firm value type model where you have the GDP and then you would perhaps assume that tax revenue of the sovereign is somehow related to this GDP and if this is too low then you have a default something like that so you would be more in the structure modeling philosophy for these things but okay before we get into special here let's stick to these these models now hazard rate models these are the simplest models where we assume that the hazard rate is deterministic they should be familiar to all of you because basically with a slight change of terminology you could call them also like life insurance products more or less I mean tau is here default time of a company could essentially also be mathematically the death of a person or something like that good Heather great models are the simple stood used for models they are the building block for more complicated ones the distribution that's the specific point of the default time is directly specified by so called hazard function we do not bother the mechanism by which default occurs we work on some probability space and we have considered a random default time tau so to start off I will introduce a few mathematical notations around this hazard times before we apply them so a random time is just random variables positive values the distribution function is f the survival function is going to be f bar bar all the survival functions which is 1 minus f in the context of these models of a of more natural to birth with survival functions because it saves you a couple of 1 minus terms and all formulas so the probability that tau is bigger than T we assume that the initial date the firm is still alive and we assume here in these models that there is no a priori given upper limits to the lifetime of the company that can be included then it would simply mean that this probability function will for every t be smaller than 0.5 and sort of half we are mathematically you could agree these models it makes really sense to have to consider it really as a probability distribution on 0 infinity included and tau equals infinity simply means that it's never happening so then it's not a technical problem yeah all you do that okay then for some of the dynamic modeling it's useful to have a piece of notation the so called default indicate it comes here good the so called and the night I should know by now yeah yeah an algorithm is somehow slowly getting gay good I just want some picture the people indicate the process this is time if you assume that the default time happens here here's one then it's a process which is zero here and jumps to one directly at the default time that is the indicator that now is smaller equal to T there's also survivor indicator which is 1 minus that this is 1 here and that Z or from you onward [Music] this will look you sometimes you know formulas good just as a piece of notation and then so this is what I need the cumulative hazard it has a function well normally I guess all of you know this from from life insurance modelling these distribution functions are typically not written down directly but we typically parameterize them in terms of hazard rates or hazard functions and I'm going to introduce these objects now gamma of T which is minus the log of the survival function is the cumulative hazard function and if X is absolutely continuous and has a density which is the case for all the examples considered in this context the function gamma which is the ratio of the density and the survivor function or equivalently minus the derivative of the log of the survival function is called the hazard function and the hazard or hazard rate function and that is the typical object of interest in which we parameterize these models typically and the reason is on these two polar points first of all you see that you can write the survivor function as X of minus integral from 0 to t gamma of s TS so in the simplest case of an exponential distribution this hazard rate would be just standing to get X plus minus lambda T and but you can of course do something more sophisticated and you can't describe basically every function here or distribution function for such a lifetime in terms so in terms of its hazard rate and you have a nice economic interpretation you can show that well let's look at this expression here first without the limits just the probability that default happens in the next instance of we're standing now at time T we're looking for the probability that default happens in the time T cheapest H given that we're still alive at time T that's the probability which is written here of course if H is small this probability will turn to 0 but if you divided by the length of the tyrant retirement above we blow it up again and so that's why this is sort of the instantaneous chance of default what is written here and it's clear basically from the definition well this probability here can using elementary condition expectation be written as f of t plus h minus f of t and then they divided by h and we get of course and divided by the power of t sorry and the whole expression 's equal to the hazard rates of the hazard rate is sort of the instantaneous chance of default given that we've survived so far I already mentioned for the exponential distribution the hazard rate is constant and equal to the parameter of the exponential distribution if you take a true parameter example like the bible distribution vary distribution functions by minus X both minus lambda T to the power alpha lambda alpha bigger than 0 of course for alpha equals 1 you have the exponential distribution as a special case and if you just compute this is the density here and then the hazard rate is lambda alpha T to the power of alpha minus 1 now it's interesting to see if this hazard rate is decreasing or increasing in time so that means that if you've been waiting the interesting bit maybe about the exponential distribution is that it's constant or the standard joke of Freddie their bond used to be that's the mother-in-law distribution to be slightly politically incorrect a constant hazard rate means in the context of the mother-in-law problem no matter if your mother-in-law has been talking to you for two hours the probability that the code should we continue to do so for the next two hours is unchanged and evenly I never met his mother-in-law maybe Paula hands have this is one extremely case and the other one is that you have increasing hazard rate so if you've been waiting longer the chance that something happens increases this is overall the case it's not throughout I guess for human lifetimes for instance at least beyond there are certain small bits I think in the curse words not long it is the typically the chance for an 80 year old person to die the next year is higher than the chance for a 20 year old person and may the opposite thing is that it the intensity is in fact decreasing well this may be for instance the train where the case when you're waiting for a train in whatever yeah Germany they have had enough strikes also say for a train in Germany if you're waiting a little bit you hope that it may come if you've been waiting for two hours and still not coming then maybe it is canceled and it will sort of never come that would be a typical case of a decreasing hazard rate here with this Bible the distribution can model increasing or decreasing you can have more sophisticated examples where you can have as both things good that's just to have this notation clear we will do I will be brief here's more foundation well we will do coffee break now I think and we will continue with this afterwards good why come back the good news is definitely is it's at most an hour to go so there is an upper limit for ones and a few words about the next slide filtrations of course we want to do some pricing of bonds or CBS and in principle if you want to consider pricing models and an enterprise in using risk-neutral pricing hedging and so on we need to an emic model it's very really model how things evolve over time and if we'd like the black shorts like this asset value path Alex show to you before the break and if you do that then in principle we also have to model the fact that the information set of investors and the things they can use and four main trading strategies are making decisions gets larger as time goes on and mathematically this is done by the concept of filtration say model information available to investors over time ramadhir filtration is an increasing family of sub-sigma pass so if G is smaller than s then ft is smaller than SS because in these models people don't forget there is the amount of information that is available is always only increasing and the fact that the set is measurable ft measurable means that at say I said a that at time T an observer of the system can decide if that event has happened or not that's that's the idea so in this theory Sigma fields and filtrations are not just the technical necessity you need to be able to do integration and define expected values or probabilities in a consistent way they are really important tools to model information and it is quite frequent as several informations around at the same time Paul mentioned I think insider trading models in this context there can be also other other such things now I don't want to push this too far this is not mathematical finance workshop here but a little bit we need if only to make the notation of the slides understandable in these very simple models reduced for models where there is sort of not much economics going on there is not so much economic quantities being modeled and so we assume the only observable quantities to default associated with respect to tau the default indicator so basically that means at time T we know only what we know if default has happened or not and if it has happened we know when it has happened that sort of the simplest thing so we know the history of the default indicator up to the time and recall the corresponding filtration HT later on today when we consider models for stochastic hazard rates there will be more economic information and there might be things like stochastic interest rates and some economic state where I will set model maybe GDP or other economic quantities around and then we need bigger filtrations for the moment now next result is useful because it tells us how to compute condition expectations with respect to the simple filtration what this means is that if you want to compute the conditional expectation to suppose that the firmest are alive and we want to compute the conditional expectation of some event with respect to the default given the default information before history up to the time point then okay this is 0 if default has already happened and otherwise what we have to do is we have to take sort of the ordinary exper tation of the product of the default indicator and the thing you want applies normalized by the survival probability and this result can be used for conditional survival probabilities for instance the probability that our bigger than T given the default history up to time T is well if default has happened already this probability 0 that's why we have this indicator here that's why we have this indicator here and the probabilities indicator at is bigger than T times this integral from small T to capital T over the hazard rate this áformer will sometimes use the next slide about the market property of the jump indicator process we can skip this is not really needed for for us here ok so that's sort of the mathematical tools to understand the hazard rate models and now we come to some financial tools we will have a very quick look at risk-neutral pricing to refresh your memories and because it is important to distinguish it from sort of actuarial valuation principles and good the risk-neutral pricing means that in order to compute the price of a quantity you take the expected discounted value of the payoff but you're not doing this with respect to the physical measure you're doing this with respect to some artificial measure q the risk-neutral pricing or equivalent martingale measure and this has the property that it turns the discounted prices of all traded securities into fair bets or martin gates and for those of you who have not seen this before it most of you have a very simple example here it's the example of a corporate font or default herbal bond and this bond has a current price of 0.9 for one and there are two possibilities say in one year either the bond the underlying firm is not defaulting then we get one in return all the firm is defaulting and then we assume in this case there is a recovery payment of 0.6 our recovery rate of 60% the default probability the historical default probability we assume we know that is 1% and the interest rate is 5% so the payoff would be 1 minus P 1 with the probability of 99% and 0.6 with a probability of 1 percent and the first thing we could - I'm not you find this in chapter 2 is to compute the expected discounted payoff the bond this would be this expression here but with the real default probability and if you do that you end up at something that is slightly higher than the assumed price of the bond now what is in this context an equivalent martingale measure well this is a very fancy name for one period model with two states in the world so all we need here is a risk neutral default probability a probability of the default which is a fictitious thing it's just a number a probability and if you use this artificial probability we can explain the current price namely the current price of the bond is the expected discounted value of its future payoff with respect to this artificial probability this leads us to this equation here the current price the man 0.94 one should be equal to the discounted value this is an old example interest rate is 5% so we do discount with 1.05 and we take the expected payoff which is 2 times 0.6 plus 1 minus 2 times 1 and if we solve this equation this is done in the book but I think you believe me then you get a risk neutral default probability of Q equal to 3% of course the example has been re-engineered and I assumed this different probability and then I got this number but nevermind now what are the two important things to take away sort of if you have problems keeping these two concepts apart the first one it is different from the historical default probability that is typical and the typical situation is also as indicated in this example the risk-neutral default probability tends to be higher than historical default probability why this sort of embodies risk aversion on the part of bond investors because if it was equal if cuba equal to P then this would mean that the expected payoff on a default upon another default free bond were the same and then most investors I guess also you would say well if in both investments I make the same expected return then why should I take the additional default risk I'm not a gambler I would just buy the default free pump so investors require some sort of a premium for taking on the default risk so a slightly higher expected return and that's why typically the risk neutral default probability tends to be higher good the other thing while the in fact there isn't free message is to take away and the other thing is how did we find this queue we didn't do any statistics we just forced an equation we looked at current market prices so risk neutral probabilities are those probabilities that somehow explained current market prices under some structural conditions of course on the underlying model so there is no statistics here to determine the risk the historical default probability we might have to do some statistics look at firms with similar characteristics and check their default rate in the past or something like that here we do something in fact simpler and so that's P versus Q and also we had I think in one of the first lectures in this workshop a little bit of a discussion what is estimation and what is Kelly brash I think there's your questionnaire calibration so the terminology is I use it in as I think it is very often being used as a little choice if you do something like that where you try to reverse-engineer model parameters from observed prices we talk of calibration and if you do real statistics you will talk about estimation I think that's sort of the standard terminology but these things are well these both words are not always very carefully used of calibration yet historical volatilities estimation exactly so actuaries know this also they will talk about market consistent valuation as a special case of this good another question is of course why should we do this what is sort of a theoretical justification for using such a pricing principle and also okay here just computed the price of the the computed the risk-neutral default probability but I might then go on and say well I would like to comprise also other derivatives which are non traded for instance I could consider very stylized default swap in this very simple model so the very stylized default swap would be like it if a swap is an insurance against default so that could be something that gives me one here and zero here and I could pry such a thing and that would what I would do and that risk no surprising approaches I would sort of just take this probability and I would price the default swap as expected value of the discounted payoff with respect to this Q that is risk-neutral pricing but why I mean you can come up with many pricing formulas but there should be some justification for that there are two justifications the first one is absence of arbitrage it is well known and Paul's colleague Freddie Durbin has devoted half of his scientific life to figuring this out in ever more mathematical detail that whenever you have an arbitrage free model so whenever you have a model where there are no risk-free profit opportunities then there should be a risk-neutral measure or an equivalent martingale measure up to some technical conditions and the other way around if there is a martingale if there is such a measure then there's no arbitrage opportunity that's easy and so whenever we make a model we would want to make a model that is arbitrage free because a model that even if you believe that the real world out there maybe has some arbitrage pricing model with arbitrage that is a little bit like a hammer was a broken handle if you use it it will flow around your ears at some point so we don't want that and that's why we will want models where we have such a reciprocal probability and of course if you price other products this should be consistent so if you include these other products into the set of traded securities the model should be still arbitrage free and that's why we should use risk-neutral pricing the other may be more convincing argument has the advantage that it's more convincing in the door back that it doesn't always work is based on on hedging you in many situations here this would be a case in point you can reproduce the prices of traded securities by a clever portfolio static or dynamic and then you can say if I can reproduce a payoff of a security with clever trading then the amount of money I need to do this clever trading should be the fair price and it's easily seen that the amount of money you need is exactly the expected value expected discounted value and a such a risk neutral measure however in many real models there is such replicating strategies do not always existed s and also more than one measure and we are in the area of incomplete markets but that is maybe too far for the structure so that's a little bit as justification for risk-neutral pricing it is quite important and it is also important that you really keep the difference between P and Q clear in your mind because in risk management you will very often have the problem that you need to work with both measures at the same time we do an example for that tomorrow LX has a very nice our script when it comes to doing computing or doing risk management for an interest rate swap and there we win the sort of need if you want to understand sort of the risk in an interest rate swap the risk factor is going to be the interest rate and then we need to model the P dynamics of the interest rate to get a distribution of the value of this thing tomorrow in ten days but on the other hand to have the mapping between the value of the interest rate say in ten days and the value of the swap in ten days we need to do some sort of risk-neutral pricing and for that we need the queue dynamics of the interest rate we need to come up with a zero pump curve or years so we need both of these things simultaneously and that's why it's important to keep them apart last thing maybe here what you already seen we made this comment when we talked about the link between risk neutral and historical default probabilities in the mad model typically this is this calibration is somehow easier you have to solve an equation maybe a complicated one with some optimization software service is easier than doing statistics data gathering and in a sense may be also less subject to well I wouldn't say subject to error but it's less ambiguous but what the right approaches you can handle this - a machine exactly and then you can sort of it's it's less arbitrary which method to use and and of course if your evaluation depends on that and you might have an inclination of using another measure where you build balance sheet another statistical approach which makes your balance sheet look nicer and things like that so in that sense this is less arbitrary and that's why there's always a temptation of trying to come up with with neutral default probabilities and then find a magic mapping to historical default probabilities but unfortunate rare so that's a little bit the invariance of difficulties in life and good Oh Alex a job good if you're talking too long but I think this is yeah that's the month doesn't work good but I think this is always important now Martin gave modeling coming back to the slides when we want to set up a pricing model for corporate bonds or for default free ponds with stochastic interest rates it's quite convenient to do a shortcut which martingale modeling and that is to set up the dynamics of things directly under the risk-neutral measure and not as when you do black shorts for instance you start by modeling the stock price dynamics and the p and you have to drift new and then you have to do whatever those of you do to cause and financial mathematics may recall then you discover that the option price doesn't depend on this drift and then you do some change of measure and meet complicated Kisan of serum and stuff and then at the end you arrive under the stock price dynamics under the martingale measure and then you continue to work and from then on the meu never appears in derivative pricing at least and so you could ask yourself why do I need this to begin with I could have started straight aside away under queue and saved myself a lot of hassle and not only that if you price the stock this is not so much of an issue but you have one stock and so the difficulty model in such a way that there's arbitrage when you wanna do interest rate modeling or also default when you have many default Obul securities related to the firm same firm so then constructing models in an arbitrage free way is more complicated most obviously this is when you do some structure modeling because in theory at least you want to model the dynamics of bond prices of all maturities in practice there are only finitely many maturities but this can be many pawn prices and it's maybe not reasonable to have so many risk factors in your model Alec showed us that we need maybe three in his principal component analysis and so there's really an issue of writing down a model in such a way that it's arbitrage free and if you start directly and the martingale measure this problem is solved we have sort of assumed it away things are consistent and for these reasons martingale modeling has become convenient and this is even easier if you know the value of the payoffs at maturity as for zero coupon bonds or default of the coupon bonds so what one does is that you directly use this risk-neutral pricing formula as as the definition you define the price of of an asset by this formula of course again when you use this approach things don't come for free what happens is if you use this approach to set up for instance a model for default double bond prices then you have one problem your model gives you a bond price but of course you also observe one in the market and the overall philosophy of derivative pricing is obviously to do it relative to the market so market price is king and you want to set up your model so that fits to the market price you observe and this is exactly the issue of calibration you need then always to determine the parameters of the model may be of the interest rate then a mixer of the default progress the wisdom to default probability so that the value that comes out of such a pricing formula today coincides with surprise you observe on the market which is called calibration and we want to now apply this approach to default coupons and credit default swaps in the context of this simple reduced for models for some cons of martingale modeling maybe very very briefly only in the interest of time the pros I already mentioned it can be convenient you get by definition models which are arbitrage free the cons this calibration can be very complicated and from a more methodical methodological point of view if you have only relatively little price information only very few securities that are traded that are somehow related to the things you wanna price then this market completeness issue plays a role then it's probably not possible to determine the cue uniquely and then it would be more natural to do something which which you do in insurance anyhow way if you want to apply something like an insurance contract you look at the expected loss plus risk premium but as long as you're in a quite liquid market this is a useful thing to do good bond pricing we consider zero coupon bonds martingale modeling we assume that under the pricing that are used the default time tau is a random time with deterministic hazard paid function gamma Q the information available to investors is the default history interest rates and recovery rates are deterministic not necessarily constant they may depend on time but they are deterministic percentage loss given default is Delta so in case default happens certain percentage of the notional is sort of lost and it's this Delta R of T is the continuously compounded interest rate and the price of a default free zero coupon bond with maturity T is X of minus integral from small T to capital T hour of SDS and if R is constant that's of course equal to X bar minus R times capital T minus mu T that's the setup now let's look at a typical corporate bond then there you can decompose it into two building blocks the one sort of the most important one if you want this is so-called sabbatical we assume the face value is one and if the underlying firm does not default then at the maturity date we will have a payment of one on the other hand that's the first time survivor camera say we get one if the firm survives and we get 0 if the firm defaults before the maturity of the bond are in reality of course when an underlying 74 typically not all is lost but you get some recovery payment and this we need to model as well let's start with a recursive Arthur came the payoff is of course 1 the indicator at tau is bigger than T and so we have the risk-neutral survivor probability the probability tau is bigger than T capital T HT is indicator that the firm is alive at the valuation date small T times X of minus integral from small T to capital T gamma Q of SDS that's the same formula as before and now it will be convenient to introduce an artificial interest rate so RS the real interest the interest rate that stands at Peyton if you sublet till we get some sort of artificial interest rate capital R and if you use this piece of notation the price of the survival claim at time T is okay that's the expected value and a cue of the payoff discount that's discounting with before physio coupon bond this first expression here's just the price of the default physio coupon bond written out times the survival probability and if you use this formula for the survival probability here you see that the whole expression is indicated that the firm is alive that small T times the exist this discounting expression here X both minus integral from small T to capital T our of SDS now if you look at that that looks like the price of a default field bonds only with another interest rate so we see your principal which we will see also later under the more sophisticated models in many models it is possible to reduce the pricing at least of survivor claims of the default with zero coupon bonds to the pricing of default three zero coupon bonds of course the default above zero coupon bond should be worth less because you cannot be sure that you get your money back and this is modeled by discounting with a higher interest rate and this higher interest rate is just the sum of the default free interest rate at the hazard rate so that's what we get here to move on we need to that's what's written here to move on we need to also model the recovery payments and a couple of potential recovery models in the literature what really happens at recoveries is difficult to say but we will consider basically two models which are quite popular and one of the two we also need me for the CDSs the first one is the so-called recovery of Treasury that was proposed by Darren Turnbull and mostly for convenience DD the idea is the following if default happens at tau say we have a 1-year bond and default happens after six months then at the default time you get maybe the recovery rate is 40 percent so you get zero point four units of a now default free pond which has the maturity date one year from now so time to maturity would then be my example six month the beauty of this assumption is that in this case the payoff you get at the maturity of the bond is independent of the timing of the default you get one if there was no default and one minus three the loss given default so the recovery rate in case there is default the recovery of face value which is maybe the most realistic and the most frequently used assumption it's slightly different there it is assumed that you get a recovery payment of size Delta but not at the maturity of the original pompe directly at default in practice you don't get it directly at default but there is some negotiation and so on going on but it's stupid you know not postponed until maturity you get it then maybe half a year later or something and this is more more realistic it has one side tropic the value of this thing depends on the time of the default because there's a discounting going on does make a difference if the default happens very soon after the bond has been issued and then you get your money very early or if the default happens short before maturity and you get your money very late in the setup the value is recovery of face value the the value is the following expression of the recovery payment you get payment of size 1 minus Delta you get it directly a towel that's why we have a discounting here up to time towel and you get it only if ok the bond has to be alive at the time when you do it evaluation and default should happen before maturity otherwise you not get any recovery payment but of course instead to get just the normal payment of the bomb and we have to take the condition expectation with respect to this default history known as a useful formula for that using the fact that this is of course the expectation of a function of tau tau is here and tower is here and because we know the conditioners have a function of tau we know that conditional distribution and its conditional density and we can compute this expectation here and you can show this is 1 minus Delta indicated at hours bigger than capital T times the integral from small T to capital T gamma Q times minus X plus minus integral from small T to s hour of you do yes so that's here the discount factor in here and this density of the default time that is a so-called payment at default claim which we will also use or encounter in the pricing of CDSs good that's the next thing serious pricing so Alex explained to you earlier today what a CD s actually is so let's quickly recall the payment flow Tallis the default times he was just reference entity I think that was Nestle in your case and premium payments are do it deterministic times typically four times per year each premium at each of the premium payment dates a premium of size spread percentage of the notional which is typically quoted in annualized terms that's why this is so complicated so if the test at the city s pet is 2% it means that four times per year you get point five percent that the analyzation is being paid it's a little bit like a life insurance contract so as soon as intrude person is there are no premium payments anymore that's the same thing here and if now happens before the maturity then there is a default payment of size Delta and reality the size of the default payment is linked to the recovery rate of bonds but we assume for simplicity that it's constant if some contract cells also called accrued payments but we ignore them because they are not so important good we cannot price these two things the premium leg is nothing else but a portfolio of survival claims each premium payment has occurred at the deterministic date and it has a deterministic side the size the only thing that is random is that we have to make the premium payment only if the reference entity is still alive at that time so it's a sequence of survival times and each of these of other times we can price and that gives us the value of the premium leg of the premium payments but hopes of legs of such virtual transaction good that's the value of this premium payment leg and this is easy to compute because we know the conditioner you're doing it at equal zeroes I know the survival probability that's the value of this premium payment legs suppose that at the valuation date the firm has not yet defaulted that's the spread and then I'm have to sum over all premium payment dates in the future and I have to disk the sort of discount with this artificial interest rate to take into account that the premium payment is only do if there's no default the default leg is a payment directly at the at our provided at how smaller than T that is exactly what we have just computed to do the recovery payments of the bond on the recovery of face value and we get this expression here it's the same thing as before good and if you want to now you have to remember that these contracts are typically done in such a way that they have they are an exchange of payments and the contracts are designed in such a way that at the moment when the contract is being done its value 0 so if I want to do the CVS contract with Paul and we want to ensure each other against Alex's default because if he is no longer here we have to write our our code ourselves and we need to hire somebody could be true I have to do that and sort of Paul wants to buy protection against the fact that Alex whatever the falls and it's not and I'm willing to sell that yeah then an excessive sperm Alex's firm defaults how's it called QR analytics or Samsung and I wanna sell it and we have to agree maybe Nestle would have been better then we have to both agree on a spread of course if Paul wants to buy protection even service insurance premium as low as possible I'm selling it I want to have it as high as possible we have to simply agree in the premium so they're both willing to do it and typically there's no cash exchanging hands when we do the contract at least not in Switzerland and that's the fair first pair of these things mathematically if you want to compute the fair spread given the pricing model we have to solve an equation M if you have to find the value of x star such that the value of the premium payment give next star and the default intensity gamma Q is equal to the value of the default payment which is independent of the spread now if we go back here the value of the premium payment forgiven spread X is just linear and X x times something and so to solve this equation we simply have to divide the value of the default payment by something which gives this formula here for the default free for the fare spread of the contact good that's the first part that's surprising in reality the market for this credit default swaps is quite liquid the one for swaps onestly is in case of alex that's a little bit more complicated and so what we do it the other way around you will typically say we want to determine this risk-neutral default intensity gamma Q so that the spread we get from our model coincides with the spread on the market this would be Caleb the calibration in the simplest case whenever we have only it depends for some other firms there's only a five year CD s outstanding for some very big firms to find out the CD SS of other maturities like three years or seven years and so on and in the simplest case where there's only one maturity there it's also makes sense to use those a very simple model for the risk-neutral default distribution namely an exponential distribution which has a constant parameter and so we would have to find constant gamma Q Bar the risk-neutral default intensity such that if you use this gamma Q bar the spread that comes out of our model equals the spread we see on the market so gamma Q has to solve this equation here this is the value of the premium payments for the given risk neutral space that's the value of the default payments for the given gamma Q now if you think about it you can also show it that economically it's sort of clear the value of the default payments that increases the default the intensity or the hazard rate goes up because if the hazard rate goes up it's more likely the default is going to happen before the maturity and if it happens it happens sort of early which means that this countless the early of the premium payment is of course decreasing in the default intensity if the default intensity is small the value of the premium payment is quite high because it's likely that I'm going to get all of them if the default intensity is very high the value of the premium payments for fixed spread is low because then it's quite likely that I will not get the later payments and so there's a unique solution if the situation is more complicated and you have more than one city as outstanding you have to either use tor multi parameter distribution for gamma Q for the hazard rate or what practitioners very often do is to use piecewise constant hazard rates and then you can do it with her like what calibration with a backward of its forward induction we first calibrate the shortest and the next and so on good that's how these things work if you look at the numbers you see that the implied hazard rate that fits this spread gamma Q Bar is roughly equal to the ratio of the observed spread and the loss given default so sort of spreads that come out of this reduced for models for shorter I since they are typically like that and they are typically positive that is one of the things that is different in this context then in this simple structure models like Merton Alex talked about before about ok the last bits pricing with stochastic hazard rates now these models we've considered so far are very simple they're enough to value a CVS or to determine risk neutral default intensities from a CVS but in these models the only risk factor is default and there is no real randomness as far as a CVS spread is concerned as long as the underlying is not defaulted the CVS Pat evolves deterministically this is not enough for many things for instance if you use such a model you cannot do risk management for bonds because bond prices would evolve in a deterministic manner that's not what you want you want a model where credit spreads as a Matt and Emily make and have distributions and compute very at-risk or something like that you couldn't price options and bonds or options and CDSs because for that you need also dynamic models so to do many interesting things we need more complicated models where this hazard rate is itself stochastic and the idea is to drive hazard rates by second stochastic process so the default mechanism has two sources of randomness on the one hand we have the default on the other hand we have two randomness in the hazard rates and that's why these models are sometimes also called models with doubly stochastic two types of stochastic random times they're also known as box models and I think in the context of survival analysis they were introduced by David Cox good so now I mean you have to the ideas that I would give you just the Flair of the results you can obtain there for two reasons the one is that we're running out of time the other one is that if you want to do everything in detail then this would become more of a course in continuous-time finance and that's not what you've booked here so you haven't paid for that yeah and in this world the world of finance if you haven't paid for it you don't get it there is no free lunch except for tomorrow of course - yo Joe goes there it's not free you get it on if you listen to this so is this if they pay for it you can probably yeah that's interesting maybe tomorrow you can then auction off the remaining vouchers to students here yeah that's why I have to do tomorrow in the morning let's see how the price is gonna decrease okay um no experimental economics here good so here what we have in abstract terms we have a slightly more complicated filtration we need now to have some additional economic quantities like silastic interest rates like GDP or other variables and they will drive the the hazard rate so the typical thing is you modeled some process and the hazard rate will be a function of that process so that there is some variability in bond prices and credit spreads and in abstract terms you simply say this process generates a filtration F G and the F G contains all information about this economic background events except default and then default is modeled as before plentiful time we have the default indicator we have to default history and then we put those things together and the two things together this filtration is GT and that's just the information about the background process GDP and interest rates and so on and deeper history of the firm and that's information they're having good the table still has a random time the idea is simple it's a two-step process I mean you know people in probability like to draw from urns if they want to visualize the model so to generate a table stochastic random time we have to sort of draw two times in the first draw we draw a realization of the economic environment of the trajectory of the background process and given that process we then consider this s given we considered a model where this hazard rate is now deterministic and equal to this given trajectory and the second random assistant really when does the firm but this given hazard rate with a default and formally you let to this definition here a model has a random time tau default time is called W statistic if there is a background process gamma T the hazard rate process such that the probability that towers bigger than T given s infinity just means given the evolution of the hazard rate from the beginning to of the word to the end is equal to X plus minus integral from 0 to t gamma s TS of course the gamma needs to be positive for that and this integration needs to be finite so given the economic environment we are having a moral worth deterministic has a trade as we studied so far but the economic environment can be also random and so we have sort of a two-step procedure conditional the economic environment you can apply the formula as we had so far but then we have to average over it ok good maybe the best way to understand these times is to look at how you generate them on your computer and I go on directly to the picture because from the picture you probably understand it best you do a two-step thing on the one hand first you generate the trajectory of the hazard rate process how to do that depends on which specific model you are using a typical model people uses the so called coxinha' source model which I will write down then if you have a trajectory of the hazard try to compute the integrated hazard rate gamma which is just the integral from 0 to small T comma SDS and we compute tau as the third and we generates independent of that a standard exponential random variable and then redefined how to be the first time where this integrated has a bad process this is this guy here is bigger than the exponential random variable so that's the e we have this hazard by process and the first time that the process gamma is bigger than e that's you 6 point something that's our town that's how you can sample them and this lemma is just telling us that this way of something them is really the right thing to do okay the next thing is how to test pricing work here and the key message is I want to convey to you is that this idea we've seen before namely that the pricing of the Affordable bonds can be viewed as a pricing problem for default free bonds if we adjust the interest rate that that can be carried over there was quite useful I mean this this observation is now 30 years old or something its original due to david Lando but at that time it was useful because people had sort of a lot of experience already with default free interest rate models and how to compute bond prices in those and discovering that you can use the same type of mathematical techniques to deal also with affordable ones was quite useful you have reduced the problem to something which you already solved maybe some of you know this job you have a mathematical physicist and a mathematician and both of them are given two pots of water and the camping stove and they have a task of boiling two pots of water so the physicist switches on the stove takes the first pot puts it on let's boil put to the side takes the second one puts it on let it boil and stun it the mathematician comes switches on the stove takes the right pot put on the stove let it boil and it takes the left pot puts it to the right side and says now I have reduced the problem to one which I've already solved and in a similar similar way this was here also people who are of course quite heavy happy that they had reduced the surprising problem to something which was already solved so to explain these results we consider an arbitrage free market model Omega s filtration ft and the pricing measure Q and the prices of all the four three securities in particular of all bonds that are do not default are assumed to be ft adapted and we have four default P savings account towers this default time of some company and we assume that the talus adhabi stochastic random time with background filtration FG and hazard rate gamma t and the price of all securities is given by this expression now pricing formulas we cannot price everything but as we've already observed when we looked at our bonds and credit default swaps we need to understand essentially a few building blocks and then we can at least many portfolios we can reconstruct from those so the pricing of bonds and create default swaps can be reduced to the pricing of two building blocks to survive the claim that pays us something if default does not happen for instance one in the case of default to the bond and zero otherwise there are also other interesting survivor claims for instance when it comes to counterparty risk then it could be a derivative security where we get the payoff if the issue of the option has not defaulted in otherwise we might get nothing but here we who mostly take X equal to 1 today and the payment a default claim that pays us something directly at the default time provided that d4 happens before some mature maturity at a default of a bond a sister Steen is a combination of the survival game that models the payments of default has not occurred and payment at default which gives the recovery pain good the key resolve is this one the choice that the pricing of these building blocks can be reduced to a pricing problem for default free claims with adjusted interest rate so define the adjusted interest rate RT as small RT + gamma T and then under the above assumptions so particular palace table stochastic the pricing of the survival cam that surprise of the survival claim to have to discount we get X if tau is bigger than T and we compute the condition expectation with respect to filtration everywhere which contains or everything before these three and background information and that's provided the default has not yet happened we have to compute the expectation of the discounted payoff X but now in the background filtration in particular X is one you just have a default free one price but with the adjusted interest rate and the similar pricing principle also holds for payment at default claims okay maybe you bear with me two or three minutes then I'm done for today applications you can apply this to corporate bonds and we can compute these prices what is may be interesting here is that we can also compute credit spreads and what is interesting is that if you look at credit spreads over short time horizons or here at instantaneous credit spreads where you really look at at the spread of upon the chest maybe a day to go but it's an approximation for a two or three months aspect you see that this spread is equal to loss given default times whose total hazard rate as we had that before for our see the s this is typically bigger than zero so in these models you avoid the problem of the structure models and expanded out that short term credit spreads are unrealistically zero and this model is typically not the case at least if you use a reasonable model for gamma t you can't price credit default swaps and that set up what you get is here an expression for the fare spread and this expression is very similar to the one we had before for the models create a monistic hazard rates only that they're sort of always an additional expectation there where we have to average over this R and the gamma here and the same thing down here but it's a straightforward generalization of the models with a deterministic hazard right good what is left and then maybe I have to take Steve five or ten minutes tomorrow morning's from Alex is of course are there reasonable examples specific parameters Asians where expressions of this form here can be computed nice and easily the answer is yes they are in particular in the field of so-called a fine models like the coxinha service model and I will spend if Alex permits be the first 10 minutes or so tomorrow morning to weekly show these formulas and then we move on to portfolio products so thank you for your good so what I want to do with you in their very last session of this week is to conclude the story from yesterday afternoon about the models worst elastic hazard rates and to talk about counterparty risk and concluding the story maybe means to begin worse to come back to why we're interested in this at all doesn't matter Frank asked me this this morning of a breakfast so you see we're working all the time and I thought maybe I come back to this a little bit because what he mentioned and of course he had a good point there is that he could see that what I was doing yesterday afternoon we were doing this context of credit risk models of introductory chapter had a lot to do with valuation and not so much myth with risk management and what I want to explain to you is that it is an important part which you need to do if you wanna do risk management for bond portfolio so this goes back to sort of what we did in market risk in general so you're an insurance company and you're standing here time to zero and maybe then you have a time t1 and you're interested in the lost distribution of your portfolio and for simplicity your grant portfolio consists of one single default of a bond so we want to find the lost distribution now the value of this bond if you go back to time t1 then the value will be equal to sum' thank you for function G that's our general mapping notation of risk factors and now what would be natural risk factor sort of here to consider it would be probably some interest rate R T so that's interest rate which is changing interest and it will depend on the credit spread of the bond or on the markets estimate of the credit worthiness of that firm this is affected both by the fact if this service may be really become a little bit riskier between t0 and t1 and it has to do with risk aversion of the market now these two things cannot really be separated that easily but it will depend on some endeavors or be typically called they call it outside T and that is the default intensity and that is a measure for the and that measures of the credit quality that can be the markets perceptions and perhaps if this time horizon is long it will also be affected of course but effect if default has happened or not so maybe I put this outside so this is true only if tau is bigger than t1 and so what we need if you want to do that we we need to find this function G and we need a modular came out of a random interest rate we can write down a process and we need a model for this that for this default rate and we need a price pricing model that allows us to turn sort of given this realization of these numbers at time t1 into a price and that is basically what I did yesterday so think about simulation you would then draw if you for instance to begin with say interest rates I may be constant with draw various trajectories of this default intensity so these are all paths of SCI and for each of these pipe would then get a price of this bond this could be like a random mortality rates yeah only that may be with mortality if you go from one day if you make considered over horizon of 10 days you can probably consider it deterministic but if you would want to do this over a longer time horizon you would say maybe you have a random mortality rate it's exactly the same and so that's why I said yesterday we need these models first elastic hazard rates to be able to come up with a bond price in a world where the default intensity or if you want you can think about the spirals as a short term credit spread is fluctuating randomly and then we can draw various of this path for each of this we get some value of the bond and then we can compute value at risk or expected shortfall or think about hedging or whatever and that's why having these valuation models is important that's why it is important to evaluation models where the default intensity hazard rate or short-term credits but turns out to be more or less the same thing is random there is an additional small twist here which is maybe also important to mention once namely when you do that you also have both measures around but there's an old joke about probability probably using all always different probability measures in the alcohol than P in financial mathematics it's a little bit better at least we call them P and Q but that means that you have to bury which ones to take I'm making this point it's not just academic rigor or something it is really important to people in fact in practice for instance when you those of you who still use the economic scenario generator of parent trip at house do you know how it's now called it it's owned by Moody's okay it's still called an ESP they vary a lot about getting exactly this point right maybe when you generate these scenarios here you have to look at the P dynamics of tie however when we then take a realization of the default intensity and compute upon price than we need so when we're standing here and then we sort of continue to look into the future to price the bond then we would need the two dynamics of topside ideally these two would be consistent okay that's why we're doing that essentially why we want this models with random hazard rates and why such things are needed to do risk management for forcing a bond there of course also for pond portfolios maybe at some point we should really work out an example but okay and that's one it doesn't work okay that was the part in operation Ares good so now back to this okay good yeah well you have two sides in here good a little bit my lighter to your boss okay good so that's why da Bistro has two Grandin times and well I recall quickly the definition of the construction Tao established elastic a random time if there is some FG adapted process comity the hazard rate and we will see it's closely related to short term credit spreads such that given that I know what the hazard rate is now and in the future the probability that tau is bigger than T is just X bar minus integral from 0 to t gamma s TS so as I explained yesterday this is really a two-step thinking that's why also table stochastic in the first step I generate my hazard rate and given the hazard right I'm when considering consider this as deterministic and and and sort of back to this survival analysis models with deterministic hazard rate and that underlies also the construction of the simulation for these things the algorithm is based on the following results Paul mentioned suggested I should we call it maybe for all of you so what we do is we have the following result we generate such hazards let such a doubly stochastic random time we need two things we need a simple exponential random variable and then trajectory of the hassle path and these two have to be generated independently and given that we have these two ingredients we can consider the integrated hazard so the integral from 0 to T comma s TS and then we simply say tau is the first time that the integral from 0 to t gamma SD s is bigger or equal to e so it would be just the generalized inverse of the integrated hazard process evaluated at if you want to be fancy but we take the hazard rate to be strictly positive so it's just the inverse so if you let capital gamma T equal to the integral from here to T gamma s then gamma to the minus 1 that's how you would simulate them you first generate your exponentially random variable you generate the trajectory of the hazard rate and you return the tau is given here in point three and that's also clear why this establish stochastic you have these two mechanisms you have to sort of if you want I just pratik generation of the exponential random variable and you have the generation of the trajectory of the hazard rate process and that's that's the picture we talked about yesterday the just may be something which may come surprising to you the hazard rate itself over model it for instance by a cop sitting in service process so that's the diffusion that looks looks like that but of course if you integrate over it you get something relatively smooth and for this plotted here is to integrate it hazard rate process in the picture to hazard a process that may jump in the jumps of the hazard rate are the kings in the integral good and then as I said before what we want to do is we want to compute bond prices and we derive pricing formulas so what we need to compute our expressions as they're given in 61-year expected value under martingale measure of X of margin triggered from small t to capital gr SDS given the payoff now in the simplest case we consider survivor claim that pays us 1 if the firm is alive at the horizon that capital T in 0 otherwise so we would have to compute the expected discounted value of the survival and in this situation here it's sort of important that we keep the terms of the interest rate inside the expectation because this fear is of course able to accommodate two hostage interest rates which is important for bond portfolio risk management because at least over longer time horizons interest rates are random and what we've learned and now we just consider only the first term is that the pricing of such a term here is essentially that surprise we have to compute of such a survivor claim with respect to the information about default and about the background processes and mathematically it's the same as computing the price of something in the default Freebird that's this expectation here but with a higher interest rate and of course we have to do this only if the firm is still alive there is a similar relation for payment at default claims which we need for instance to compute the value of the default leg of a credit default swap I talked about this yesterday's I think today I suggest that we focus mostly on the survival claim because this is notation a little bit easier ok and then we could study various applications corporate bonds and credit default swaps and okay nice thing was for instance for the credit default swap really sold at the fair swap spread we get is just very similar as the one we had in the model switch the deterministic hazard rates only that we always have to average over the path of this R and gamma here and the same thing here so there's an additional layer of randomness what is interesting is that if you take swaps with a very short time to maturity then the fair swap spread converges to Delta times Komachi Delta's a loss keeping default and that's why I said if you want you can interpret the risk neutral default intensity as something like a short-term credit spread so the spread of a bond which is only a few weeks to go or something like that taking into account of course the skipping default and another interesting thing is that in this type of models this quantity is going to be positive varies in the bad model and in many other of these found early models this quantity is going to be 0 for capital T close to large T which is why they're not so good for managing the risk of bonds with a short time to maturity okay so what remains is okay how can i compute expressions of this time at least in some special model because this meant so far we've just said we can reduce it to some simpler situation with deterministic interest rates and I want to just show you quickly one class of models where this works so typically when you set up a model that you will say okay I have this I have an economic state variable and default intensity is a function of said state where a bra is equal to that state variable and so particular the a drastic interest rate is a function of that state variable hazard threat and hazard rate are than the sum of the two are also we want to compute X expressions of this form and we want to be able to do this easily and we could consider in particular the case with G is just equal to one because that's the case of a bond I small so that small RT is the default for interest rate and that can be random in that context it doesn't have to be but it can be and of course if you wanted to say risk management for bond portfolios makes a lot of sense to treat it as a random variable because you have interest rate in there as well and it should depend on the same set of factors in the theory it could be also a multi-dimensional but in the easiest situation it would be one dimensional factor be good and the coccyx is a row so see I Ahmad was urgently model for the short rate of interest also known as square root diffusion for the obvious reasons because there's a square in here what you do is you say okay the stacked up sigh deep xiety has two terms it as a drift in there and there's a diffusion components of this Brownian motion let's look a little bit at the form of these dynamics I will also generate a few paths for you later on the first thing you see is that this this is sort of the systematic term in the dynamics of it and this is the noise term so the systematic term is the following thing the Kappa is typically positive and the theta bar is some long term level of interest rates may be whatever 3% whenever are the lungs and long term level have been its interest rate model whenever the size bigger the theta power you have a tendency to go down whenever the size is smaller you have a tendency to go up so that's why this is called mean reverting and trajectories will fluctuate around the Teta par but of course they will also always be driven away because there's some noise the noise is modeled by the second part here and there it's some volatility and I'm squared upside to the WT the square root is in here well you could have a normal site abut then the model of you know without the square would be not electable anymore you can omit it completely then you would have to pass a check model but then you would at the problem they didn't have a model which can become negative which is in the current environment sort of okay for interest rates but which is sort of total nonsense for hazard rate so it makes no mathematical sense but here this stays positive at least if the parameters are reasonable otherwise it can reach zero but it will not become negative if Kappa theta power is bigger than 1/2 Sigma square then the model will never reach C R good and then these models you have an explicit formula for the price of the default it will bond at debts of the form exponential of alpha times time to maturity plus beta times of time to maturity times the current risk factor interest rate and alpha better follow some complicated expressions just they are there and they are known explicitly so you can compute this and because sort of the the log of this expression is a fine one also talks about affine models that's a yeah well ok at the moment it's a mathematically result if you have it would be a Q integral and pricing application if an expectation of this form then takes this this value but it would be a Q expectation so the message is there are models like oxygen service where these pricing formulas which I gave you can be evaluated so where this G which we had down here is sort of known explicitly and the arguments would then be upside to one in here good you can make things more complicated the primary jump the sigh you can do it for payment 84 claims I jump over jumping side but I'm going to show you one last picture and that gives us credit spreads for default with zero coupon bonds the current value besides ears here about 5% sort of interest rate is 6% it's an old example loss given default is 0.5 and he be a bear's recovery assumptions that is the recovery of face value that's the recovery of a recovery of treasury that's a recovery of face value but small times to maturity they all give you the same value that zero recovery and then the credit spread is the largest because we lose most in defaults for the other models the credit spread is always the same for short times to maturity but if time to maturity gets very long they differ aramis what we have to discuss it was wrong in the picture I had it's like I could remove the label but not add new ones in a new book it's correct but only Sam has that figure so that is the recovery of treasury and that is the recovery of face value that's are the two that I mentioned it's interesting that he credit spreads can in this with these parameters and very long time to maturity it become negatives the idea is the following with the recovery of face value you get money immediately default and whenever loss given default is quite small and interest rates are high it can be better to get fifty percent of your money after two years and all of it after twenty years which is why this may happen so many a very long times to maturity and high interest rates this model might not be so good but at the moment it's probably not a problem okay so now that's the bit which Alex

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Published: Sun Jan 21 2018