## Fitting market data

The exponent p and correlation p affect the volatility smile in similar ways. They both cause a downward sloping skew in aB(K, f ) as the strike K varies. From a single market snapshot of ob(K, f ) as a function of K at a given f, it is difficult to distinguish between the two parameters. This is demonstrated by Figure 8. There we fit the SABR parameters a, p, v with p 0 and then re-fit the parameters a, p, v with p 1. Note that there is no substantial difference in the quality of the fits,...

## Structural breaks

Granger and Hyung (1999) take a different approach to the analysis of long term serial autocorrelation effects. Their starting point is the standard I(d) representation of an fractionally integrated process yt of the form where d is the fractional integration parameter and, from its Maclaurin expansion d j 1 d (1 L) > jtjLJ, Jtj - -Jtj i, 7T0 1 The researchers examine the evidence for structural change in the series of absolute returns for the SP500 Index by applying the sequential break...

## Info

Credit Risk Appraisal From the Firm Structural Approach to Modern Probabilistic Methodologies The purpose of the present chapter is to review the evolution of credit risk modelling developments in the recent past and to present some main research lines proposed in 2003. Particular attention is devoted to the increasing need to study the implications of the credit risk modelling framework on the general cash-flow cycle of the firm, represented by the ability of the firm to create value and...

## Psychology in Financial Markets

During the 1980s financial economists, confronted with phenomena in financial markets that were difficult to explain within the rational expectations and expected utility framework, started to consider the possibility that some market participants behave less than rationally, and to study whether this might affect markets as a whole. Initially, they made no explicit use of insights from psychology. Although the literature by psychologist Daniel Kahneman and his co-author Amos Tversky on...

## The model unwillingness as an option

The argument for modelling the unwillingness to pay as an American style call option goes as follows. Based on the generally accepted concept of national sovereignty, a government has Contact address Middlesex University, The Burroughs, London NW4 4BT, UK E-mail e.clark countrymetrics.com Telephone (44) 0181 411 5130 www.countrymetrics.com an ongoing de facto right to repudiate or default on its foreign debt, if this is deemed in the national interest. There is, however, no obligation on the...

## Change of numeraire for forwardstart options

Where T is the expiry, T0 lt T is the strike set date, and K is the percentage strike. Another common version of this contract, which primarily serves as a building block for cliquet options, has the payoff Forward-start option is one of the simplest exotics, with the terminal payoff Contact address TD Securities, Triton Court, 14-18 Finsbury Square, London, EC2A 1DB, UK. E-mail vladimir.lucic tdsecurities.com Valuing these options in the Black-Scholes framework is standard see Wilmott 1998 ....

## Bootstrapping method 2 constant forward rates CFR

The second method, Constant Forward Rates CFR , constrains the problem by enforcing that all one year forward rates, effective at 5, 6, 7, 8 and 9 years, be equal. Let F be this rate. This implies Df6Y Df5Y 1 F , Df7Y Df5Y 1 F 2 and so on. In our example 4.6 1 F -i 104.6 1 F -5 Df5YM 100 - 4.6 DfiY i 1 i 1 and it is straightforward to solve for F. The results are labeled CFR in Graph 1. These two bootstrapping methods are fairly standard and we will go out on a limb and say they are market...

## References

Andersen, L. 1999 A simple approach to the pricing of Bermudan swaptions in the multifactor LIBOR market model. The Journal of Computational Finance, 3 2 5-32. Andersen, L. and Andreasen, J. 2000 Volatility skews and extensions of the libor market model. Applied Mathematical Finance, 7 1 . Bachelier 1900 Theorie de la Speculation. PhD thesis, Universite de Paris. Breeden, D. T. and Litzenberger, R. H. 1978 Prices of state-contingent claims implicit in option prices. Journal of Business, 51 4...

## Who wins and who loses

Trading an option is not a two-person zero-sum game, because both the issuer and the holder can trade the underlying asset with other investors. If the issuer maintains a risk-free portfolio by trading the underlying asset while the holder leaves her position naked, the issuer's balance will be non-negative in the end while the holder's depends upon whether or not she guessed market direction correctly. If the holder exercises her option at her own optimal time, there is also a possibility that...

## The implied volatility of a country

V, E bt - at bt 1 - at 1 R-1 bn - an R- n-t Xt - Mt Ct Vt 1 - Vt rVt Ct We recognize the left hand side LHS of equation 2 as net domestic product where net investment in any year is equal to Vt 1 Vt. The right hand side RHS is profits rVt plus cost consumption Ct. There are many ways to get an estimate of Vt. One technique that I have used is to define a process for b a . For reasons I won't go into here but which are linked to the balance of payments identity, a mean reverting process seems...

## Credit spread and the fixedincome logic

The quantitative measure of credit quality has traditionally been the credit spread. Risky bonds are priced by the market at a discount to sovereign debt, and the price difference, when expressed in terms of the excess in the implied yield, is the credit spread. Bonds maturing at different dates can imply different credit spreads, hence creating credit spread term structure. When the bonds are zero coupons, the term structure of credit spread is equivalent to giving the whole array of risky...

## Local volatility models

An apparent solution to these problems is provided by the local volatility model of Dupire 1994 , which is also attributed to Derman and Kani 1994, 1998 . In an insightful work, Dupire essentially argued that Black was too bold in setting the coefficient C t, to obF. Instead one should only assume that C is Markovian C C t, F . Re-writing C t, F as oloc t, F F then yields the local volatility model, where the forward price of the asset is dF Oloc t, F F dW, F 0 f 5a in the forward measure....

## Appendix B Analysis of the dynamic SABR model

We use effective medium theory Clouet, 1998 to extend the preceding analysis to the dynamic SABR model. As before, we take the volatility y t a and volvol v t to be small, writing Y t gt eY t , and v t gt ev t , and analyze in the limit e 1. We obtain the prices of European options, and from these prices we obtain the implied volatility of these options. After obtaining the results, we replace eY t gt Y t and ev t gt v t to get the answer in terms of the original variables. Suppose the economy...

## Stable distributions

In spite of wide-spread awareness that most risk factor distributions are heavy-tailed, to date, risk management systems have essentially relied either on historical, or on univariate and multi-variate Normal or Gaussian distributions for Monte Carlo scenario generation. Unfortunately, historical scenarios only capture conditions actually observed in the past, and in effect use empirical probabilities that are zero outside the range of the observed data, a clearly undesirable feature. On the...

## The SABR model

The failure of the local volatility model means that we cannot use a Markovian model based on a single Brownian motion to manage our smile risk. Instead of making the model non-Markovian, or basing it on non-Brownian motion, we choose to develop a two factor model. To select the second factor, we note that most markets experience both relatively quiescent and relatively chaotic periods. This suggests that volatility is not constant, but is itself a random function of time. Respecting the...

## Managing exotics

First let us briefly discuss how we get in these jams. During the normal course of business, the pricing and management of fixed income derivatives depend on two key markets. First is the swap market delta market , which is encapsulated by the yield curve. Swap desks maintain current yield curves by continually stripping and re-stripping a set of liquid swaps, futures, and deposit rates throughout the day. This curve determines all current swap rates, FRA rates, forward swap rates, etc. The...

## The total expected loss from default as the value of a hypothetical insurance policy

Modelling the total expected loss from default as the value of a hypothetical insurance policy that pays off any and all losses due to de facto or de jure default addresses the reality that there is no recognized legal framework for sorting out sovereign defaults and very little scope for creditors to seize assets as a means of recovering their loans. Thus, default is usually not a one-off affair, as in corporate default. It is more likely to be a series of events where losses result from a...

## Managing smile risk

The complexity of the above formula for aB K, f obscures the qualitative behavior of the SABR model. To make the model's phenomenology and dynamics more transparent, note that formula 17a-17c can be approximated as b K, f Al - - I - p - pX log K f 12 1 2 3p2 A2 l0g2 K f - provided that the strike K is not too far from the current forward f. Here the ratio v i 0 measures the strength v of the volatility of volatility the volvol compared to the local volatility a f1-0 at the current forward....

## Ephraim Clark

Contemporary credit risk modelling is dominated by two types of models, the structural models and the reduced-form models. The structural models are based on Merton 1974, 1977 and view bonds as contingent claims on the borrowers' assets. The credit event is modelled as timing risk when the assets of the borrower reach a threshold. In Merton 1974, 1977 , Black and Cox 1976 , Ho and Singer 1982 , Chance 1990 and Kim, Ramaswamy and Sundaresan 1993 , default is modelled as occurring at debt...

## Wilmott magazine September 2002

European options are often priced and hedged using Black's model, or, equivalently, the Black-Scholes model. In Black's model there is a one-to-one relation between the price of a European option and the volatility parameter aB. Consequently, option prices are often quoted by stating the implied volatility aB, the unique value of the volatility which yields the option's dollar price when used in Black's model. In theory, the volatility aB in Black's model is a constant. In practice, options...

## Volatility modeling and Stable vs Normal VaR

It is well known that risk factors returns exhibit volatility clustering, and that even after adjusting for such clustering the returns will still be non-normal and contain extreme values. There may also be some serial dependency effects to account for. In order to adequately model these collective behaviors we recommend using ARIMA models with an ARCH GARCH time-varying volatility input, where the latter has Stable non-Gaussian innovations. This approach is more flexible and accurate than the...

## Portfolio optimization and ETL vs VaR

To the surprise of many, portfolio optimization with ETL turns out to be a smooth, convex problem with a unique solution Rockafellar and Uryasev, 2000 . These properties are in sharp contrast to the non-convex, rough VaR optimization problem. The contrast between VAR and ETL portfolio optimization surfaces is illustrated in Figure 12 a and b for a simple two-asset portfolio. The horizontal axes show one of the portfolio weights from 0 to 100 and the vertical axes display portfolio VAR and ETL...

## Stable ETL leads to higher risk adjusted returns

ETLOP Expected Tail Loss Optimal Portfolio techniques, combined with multivariate Stable distribution modeling, can lead to significant improvements in risk adjusted return as compared to not only Normal VAROP methods but also compared to Normal ETL optimization. In practice, a VAR Optimal Portfolio VAROP is difficult to compute accurately with more than two or three assets. Figures 13 and 14 are supplied to illustrate the claim that Stable ETLOP produces consistently better risk-adjusted...

## Phialpha optimal portfolios

A Phi-Alpha gt a optimal portfolio is one that minimizes portfolio expected tail loss ETL subject to a constraint of achieving expected portfolio returns at least as large as an investor defined level, where both quantities are evaluated in 4 gt a. Alternatively, a 4 gt a optimal portfolio solves the dual problem of maximizing portfolio expected return subject to a constraint that portfolio expected tail loss ETL is not greater than an investor defined level, where again both quantities are...

## Fabrizio Lillo Rosario N Mantegna Jean Philippe Bouchaudt and Marc Potters

Esterday the S amp P500 went up by 3 . Is this number telling all the story if half the stocks went up 5 and half went down 1 Surely one can do a little better and give two figures, the average and the dispersion around this average, that two of us have recently christened the variety Lillo and Mantegna, 2000 . Call ri t the return of asset i on day t. The variety V t is simply the root mean square of the stock returns on a given day where N is the number of stocks and rm 1 N J2i ri is the...

## Appendix A Analysis of the SABR model

Here we use singular perturbation techniques to price European options under the SABR model. Our analysis is based on a small volatility expansion, where we take both the volatility a and the volvol v to be small. To carry out this analysis in a systematic fashion, we re-write a gt ea, and v gt ev, and analyze in the limit e 1. This is the distinguished limit Cole, 1968 Kevorkian and Cole, 1985 in the language of singular perturbation theory. After obtaining the results we replace ecu gt a, and...

## Generation of copuladependent random numbers

Despite the importance of an accurate model for the dependency structure of the returns of the assets in a portfolio, an obstacle for practical implementation of any copula-based model was the absence of an efficient method for generating copula-dependent random variates. These dependent random variates are essential for the simulation of the portfolio's risk return profile, and also for the development and testing of estimation methods for the parameters of these distributions. The most...

## R Douglas Martint Svetlozar Zari Rachevt and Frederic Siboulet

When anyone asks me how I can describe my experience of nearly forty years at sea, I merely say uneventful. Of course there have been winter gales and storms and fog and the like, but in all my experience, I have never been in an accident of any sort worth speaking about. I have seen but one vessel in distress in all my years at sea I never saw a wreck and have never been wrecked, nor was I ever in any predicament that threatened to end in disaster of any sort. E. J. Smith, Captain, 1907, RMS...

## The future

The monuments of ancient Egypt are the most spectacular buildings of the ancient world. They are covered with intricate hieroglyphic writing that seems to offer insight into the wisdom of the ancients. However for a millennium and a half, all knowledge of how to read the hieroglyphics was lost. In 1799, a French engineer discovered the Rosetta stone, which led to decipherment of the ancient texts. Imagine the excitement of seeing the solution to such a problem, knowing that soon the long-dead...

## Copula functions and Laplace transforms

Whenever several dependent dimensions of uncertainty have to be modelled, the standard and often also the only approach is to somehow transform the problem in such a way that a multivariate normal distribution can be used to model the uncertainty. In the modelling of equities, exchange rates and interest-rates, multivariate lognormal distributions are used i.e. exponentials of normals , squared Gaussian and related models are also popular, and in portfolio credit risk modelling the Credit...

## Chooser range note

The vanilla range note has cashflows linked to the number of days that the reference rate typically a LIBOR rate lies within a specified band. In the Chooser Range Note CRN , the band is not pre-specified in the contract but is chosen by the contract holder at the start of each period. In the example in the term sheet shown in Figure 2 there are four decisions to be made, one at the start of each period. And that decision is not of the simple binary type Do I exercise or not, Do I pay the...

## Philipp J Schonbucher

N his influential papers, Vasicek 1987, 1997 showed that in a simplified multi-obligor version of the Merton 1974 credit risk model, the distribution of the losses of a large loan portfolio can be described by the inverse Gaussian distribution function. In his setup, the probability that the fraction L of defaults in the portfolio is less than a given level q is given by where p is the default probability of any individual obligor in the portfolio, and q is the asset value correlation between...

## ETL and Stable vs Normal distributions

Expected Tail Loss ETL is simply the average or expected value loss for losses larger than VaR. ETL is also known as Conditional Value-at-Risk CVaR , or Expected Shortfall ES . Usual 1 to 5 Normal ETL is close to Normal VaR See VaR by Jorion, 2001 p. 98 For CI 5 , VaR 1.645 and ETL 2.062. For CI 1 , VaR 2.336 and ETL 2.667. By failing to capture kurtosis, Normal distributions underestimate ETL. The ubiquitous Normal assumption makes ETL difficult to interpret, in spite of ETL's remarkable...

## Hyperbolic absolute risk aversion HARA

Merton 1990 provides a complete description of this family of utility functions. The hyperbolic absolute risk aversion means W u' y m a for a positive constant y . This utility applies in the case when the wealth of the investor is bounded below m a gt 0. Thus the richer the investor is, the less risk averse. Up to a constant shift where 3 gt 0. The parameter a is assumed positive as the option payoff could be zero. Simple algebra reduces the inequalities in 16 and 17 to quadratic inequalities....

## Appendix

Figure 1 Comparative logic of the inhomogeneous and homogeneous equity-to-credit models. The inhomogeneous model consists of one default regime and one non-default regime. The implied volatility surface and the credit smile surface are explained by a local volatility surface and a local hazard rate surface. In the homogeneous model, volatility and hazard rate are stochastic and switch between the three non-default regimes Stock Component of the Hazard Rate Function Stock Component of the Hazard...

## Selmi and Jean Philippe Bouchaud

It is well known that the perfect Black-Scholes hedge only works in the ideal case of a continuous time, log-Brownian evolution of the price of the underlying. Unfortunately, this model is rather remote from reality the distribution of price changes has fat tails, which persist even for rather long time lags see, e.g. Granger and Ding, 1997 Guillaume et al, 1997 Gopikrishnan et al, 1998 Plerou et al, 1999 Bouchaud and Potters, 2000 . This makes the whole idea of zero-risk strategies and perfect...

## Interpretation of the TF model in our framework

We argued earlier that T amp F do not provide a justification of their mathematical model in terms of what happens in effect to the convertible bond and to its components in case of default. Their splitting is just a heuristic splitting which tries to fulfil at best the desiderata that we have listed above, to the effect that the bond component should capture the cash-flows, fixed and contingent, that the holder is owed, and the equity component should capture his right to convert, etc., only...

## Optimal trading strategy The classical formulation

First, we explain the classical valuation of American options. For a rigorous derivation see Myneni's article 1992 , which contains a survey of the literature on the subject. In what follows, we assume that the underlying is tradable and its price satisfies the following stochastic differential equation SDE dS t M t S t dt aS t dW t 1 where W is a standard Wiener process, a is volatility, and M is an adapted process. The only motivation for assuming a constant volatility is brevity. In fact,...

## Example 2 Maximization of expected CARA utility

Figure 11 is the issuer's expected profit as a function of the absolute risk aversion. The option holder's exercise time maximizes the expected CARA utility, while the physical drift coincides with the rate 8 . Thus, the outcome is the marginal effect of the absolute risk aversion to the issuer's expected profit. Again, the option is an at-the-money American put, and parameters are set as before. If the absolute risk aversion vanishes and the physical drift coincides with the rate, then the...

## The Hull White model

In this model the credit index x of a company follows a standard Brownian motion. Default can only occur at discrete times Tn's, n 1, ,N where TN is the maturity of the contract. At each time Tn there's a wall and defaults occur when the credit index hits the wall. The height B Tn of the wall is such that the probability for the credit index for not hitting any of the walls up to time Tn is equal to S Tn , the survival probability of this company at that time. In their paper Hull and White 2000...

## The copula method

The Gaussian copula method see Li, 2000 . Suppose there are a assets in the basket. For each simulation we do the following Simulate a correlated N 0,1 random variables fa, i 1 - a. Find xi's such that CND fa xi for i 1 - a. Find the default times ti 's by Si ti xi for i 1 - a. Find the first default time t mini ti. In the above, CND is the cumulative normal distribution function and Si t 's are the marginal survival curves. Repeating this procedure allows us to construct the first to default...

## Hyungsok Ahn and Paul Wilmott

The price of an American option is dictated by the concept of optimal, exercise. But optimal is defined from the perspective of the option writer, who is assumed to be able to delta hedge. This theory for when to exercise the option is well known. However, buyers of American options may, and do, exercise early or late for a variety of reasons. Suppose you are long an in-the-money American call, and you are concerned that the market may collapse. What can you do You may close the position by...

## Conditional expectation

People often use the correlation coefficient to form a conditional expectation. For example, if scores on the math and verbal SAT tests have a correlation of 0.75, and we know that a student scored two standard deviations above the mean on the math test, we predict a score of 0.75 x 2 1.5 standard deviations above the mean on the verbal test. When people make this argument, they are thinking of a graph like Figure 1. The points are randomly scattered around a line with slope of 0.75. If you...

## Credit spread and the convertible bond

The relation between the convertible bond and the credit spread seemed at first to arise only from the bond character of the convertible. The embedded equity option would be priced in the Black-Scholes framework alright, where discounting takes place at the risk-free interest rate, but the presence of a fixed-income part of course implied that something had to be discounted under a risky curve, if only to be consistent with the fixed-income analysis of the issuer's debt. The difficulty,...

## Summary and conclusions

Behavioural finance has made two valuable innovative contributions to finance theory and to empirical research. In the first place, it shows that market participants evaluate financial outcomes in accordance with prospect theory, rather than expected utility theory. Many anomalies in preferences result from rules of thumb that are applied when editing prospects to facilitate decision making. Moreover, a greater sensitivity to losses than to gains implies that decisions differ according to how a...

## Introduction

In September 2002 a small, keen group working for a small but perfectly formed website, serving a very niche financial market, joined forces with a book publisher to create a new magazine, Wilmott, aimed at mathematicians and scientists working in investment banks. The cunning plan was to bring together cutting-edge content, incisive articles and fab design to combine the logic of the research papers for the left-brainers with an easy-on-the-eye look for the right-brainers. Can't be done, you...

## Antony Penaud and James Selfe

Default dependence is one of the biggest issues in quantitative finance at the moment. One of the most popular products in which default dependence occurs is the first to default swap. In this short article we are going to review different pricing methodologies for this product. The product. In a first to default swap FTDS the buyer has to pay a fixed rate at each payment date until the default of any of the reference names defined in the swap or until the maturity date of the swap - whichever...

## Next Generation Models for Convertible Bonds with Credit Risk

Vetzal Convertible bonds are hybrid securities which offer equity-like returns when the share of the issuing firm is strong, yet behave like conservative fixed-income investments when the stock market is either stagnant or negative. Indeed the convertible bond is essentially a bond that can be converted into shares, a feature which allows the equilibrium of interests between the three parties involved, the issuing company, the equity investor and the...

## Structural and reduced form modelling of sovereign debt

There has been no reduced-form modelling specific to sovereign debt. Work on sovereign debt with respect to reduced-form models has concentrated on testing existing models and especially the Duffie and Singleton 1999 model. For example, this is the case for Andritzsky 2003 and Zhang 2003 for Argentine bonds, Merrick 2001 for Argentine and Russian bonds and Keswani 1999 and Pages 2000 on Latin-American Brady bonds. Dullman and Windfuhr 2000 test two affine diffusion models, the Vasicek 1977 and...

## Rewind

Dan Tudball reviews the major events of the year just gone. Aaron Brown, Vice President of risk architecture at Citigroup, provides perspective. Thematic resonance, that's what the literati call it the consistent and thereby satisfying emergence and re-emergence of the same refrain delivering the moral and political caution of the work. Explicit, implicit - it matters not their presence differentiates art from mere reportage. At this end of the scriptorial spectrum, however, there just isn't...

## Quantitative Finance Review 2003

The first Wilmott Quantitative Finance review gathered together some of the industry's leading lights. On 11 November 2003 the first conference designed for quants by quants took place in London. Rather than taking the approach of a three-ring circus, which seems to be the norm these days, the QFR 2003 was designed for a relatively small number of delegates over the course of one day. The structure of the Review, held at the headquarters of 7city Learning in the city of London, allowed speakers...

## Education in Quantitative Finance

Quantitative Finance as a branch of modern banking is one of the fastest growing areas within the corporate arena. This, together with the sophistication of modern and complex financial products, has acted as the motivation for new mathematical models and the subsequent development of mathematical and computational techniques. Investment decisions for predicting risk and return are being increasingly based on principles taken from the Quantitative Finance arena, providing a challenge for both...

## Lucic Volatility

I Education in Quantitative Finance 1 III Quantitative Finance Review 2003 7 Chapter 3 A Perspective on Quantitative Finance Models for Beating Chapter 4 Psychology in Financial Markets 39 Chapter 5 Credit Risk Appraisal From the Firm Structural Approach to Modern Probabilistic Methodologies 59 Chapter 6 Modelling and Measuring Sovereign Credit Risk 69 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 The Equity-to-credit Problem or the Story of Calibration,...