Financial Decision Making

The aim of this course is to present the foundations of financial decision making and some of its applications, especially in financial markets but not limited to these. The concepts of price, financial risk and asset allocation are at the heart of this course. The first part is devoted to these questions whilst the second shows how these notions are used for investment decisions. This course gives in depth analyses and contains theoretical and technical developments that ask for some prerequisites: knowledge of probability theory and stochastic processes and basic results on derivative securities and fixed income.

This course gives an overview of the methods and techniques used in financial decision-making. It begins with the foundations of finance and general principles. Then it applies these concepts to investment decisions, assets valuation and financial risk management. The emphasis of this course is more on the methods than the formulas. It outlines the validity domain of the underlying theories, pointing out their importance and drawbacks. It justifies rigorously the use of tools and results necessary in other master courses in finance and economics. The students following this course learn about arbitrage opportunities, complete markets and equilibriums, and investigate the foundations of the CAPM, of option pricing and of yield curve modeling.

Life Insurance Theory

This is a first class in life insurance theory. As such, it covers only part of the syllabus of SOA exam MLC. Quoting excerpts from this syllabus, the student attending this class will be able to:  “Explain and interpret the effects of transitioning between states, the survival models and their interactions. Calculate and interpret standard probability functions including survival and mortality probabilities, force of mortality, and complete and curtate expectation of life.  For models dealing with multiple lives and/or multiple states, explain the random variables associated with the model; calculate and interpret marginal and conditional probabilities, and moments. Using the factors mentioned above, construct and interpret survival models for cohorts consisting of non-homogeneous populations, for example, smokers and non-smokers or ultimate-and-select groups. Describe the behavior of continuous-time and discrete-time Markov chain models, identify possible transitions between states, and calculate and interpret the probability of being in a particular state and transitioning between states. Apply to calculations involving these models appropriate approximation methods such as uniform distribution of deaths, constant force, Woolhouse, and Euler. “

Also in accordance with the syllabus of SOA exam MLC, the class enables the student to: “Calculate and interpret probabilities, means, percentiles and higher moments of present values random variables. Calculate and interpret probabilities, means, percentiles and higher moments of random variables associated with these premiums, including loss-at-issue random variables. Using any of the models mentioned above, calculate and interpret the effect of changes in policy design and underlying assumptions such as changes in mortality, benefits, expenses, interest and dividends.”

Which certification is best for you ?

Not yet sure on which journey to embark? Here are a few suggestions.

If you like mathematics and are sure that you want to become an actuary and work for an insurance company, then you already know that you have to take or continue taking actuarial exams. From which society? If you are interested in life insurance, risk management, the frontiers of finance and insurance, pension funds, health insurance, then you should take the exams of the Society of Actuaries (SOA). If it is perfectly clear that you want to work in the Property/Casualty (P/C) field, then you should take the exams of the Casualty Actuarial Society (CAS). The difficult situation is when you are interested in the P/C field, but you also have broader interests. Then, perhaps it is better to contemplate taking the fellowship exams of the general insurance track of the SOA. See or

If you are not a big fan of mathematics and you have a minimal background in calculus and if you want to work in corporate finance or to apply non-quantitative or semi-quantitative skills to the financial markets by working for a bank or a fund? Then, you should take the three Chartered Financial Analyst (CFA) exams. The advantage of this certification is that it gives you a very solid and broad knowledge about financial markets, economics, accounting, and other related fields. So many people are taking this certification now that taking the exams seems inevitable. This certification will give you a common culture on which you will be able to build your career. See

But, if you like mathematics and you want to work in risk-management for a bank, then, the Financial Risk Manager (FRM) or Professional Risk Manager (PRM) certifications are for you. They are now virtual prerequisites for working in risk management. I will give more details about the FRM certification, because it is the one I took. The FRM exams are very well designed and you can reasonably expect to pass them if you have attended solid quantitative finance classes, in a business school for instance. The FRM exams allow you to build a vision beyond these classes. The curriculum is very up to date. See or

The above discussion offers a first-order approximation of what is needed for your career. There are many situations that are blurred or mixed. For example, if you want to work for a pension fund, it may make sense to both take the SOA exams and to become a CFA Charterholder.

In what order should you take the Associateship exams and modules of the SOA ?

If you have not already done so, take a look at the following SOA’s webpage, which is pretty instructive:

My first advice is to start the Fundamentals of Actuarial Practice (FAP) e-learning course as soon as possible. The eight modules of the FAP course come with a lot of activities to perform: lots of things to read, lots of spreadsheets to play with, six end-of-module reports to write, an interim assessment, and a final assessment. It can take quite a while (one year or more) before you complete it plus it is very time consuming while you are doing it, so I really recommend that you start with it as soon as possible.

My second recommendation is to start preparing for the probability exam P at the same time you enroll in the FAP course. What you learn when preparing for exam P will be useful for all subsequent exams, so prepare for it well. Actuarial science, and finance for a large part, is built on probability theory. The next exam to take after exam P is the Financial Mathematics exam FM. Preparing for this exam will allow you to master the passage of money backward and forward in time. This discounting / compounding process is also critical to actuarial science and a key tool to master before considering taking the MLC/LTAM, C/STAM, and MFE/IFM exams.

About one year has passed and you have typically completed – or are nearing completion of – the FAP course and exams P and FM. Your second year of distance learning should be devoted to starting the preparation for the MLC/LTAM, C/STAM, MFE/IFM plus SRM and PA exams.

In what order should you take these exams? There is no unequivocal answer to this question. All exams are hard and cover mostly distinct topics (except SRM and PA, which go together and make use of the R software). The best idea is perhaps to start with the exam that best matches your current fields of competence. Indeed, there is a huge gap in difficulty between exams P and FM on the one hand and the next exams on the other hand. Therefore, it may be wiser to gradually advance by degree of difficulty. What is the common feature between these exams? As already hinted at, they are extremely hard. You should prepare them one at a time and devote a substantial amount of time and energy to prepare for each of them.

A few words about VEEs. In case your university credits do not give you the three VEEs, you should consider taking additional exams or certifications to achieve them. If you are interested in finance, the best idea is probably to take and pass the first two levels of the CFA program, which allow you to achieve the three VEEs at the same time.

Once you have achieved the three VEEs, passed all the FAP requirements, and have obtained a mark superior or equal to six for each of the ASA exams; congratulations. Now you only need to complete the Associateship Professionalism Course. This is a (little bit less than) one-day seminar in which case studies and ethics considerations are put forward. On completion of this seminar, you are now an Associate of the Society of Actuaries.

In what order should you take the Fellowship (QFI track) exams and modules of the SOA ?

I will only discuss the case of the Quantitative Finance and Investment (QFI) track, which is the track I chose to follow. However, most of what is written here is also applicable to the other tracks.

As in the case of associateship activities, I recommend that you take the three modules (except the Decision Making And Communication or DMAC module discussed later on) before you take the exams. Why? First, completing the modules gives you a preliminary vision of what the SOA wants from its future fellows. It gives you hints of what you will find in exams, and you can start reading additional material that will be useful later on. Then, if you thought the ASA exams were hard, you will discover that the FSA exams are at another level. The fellowship exams are extremely hard. Each of them requires reading about 2,000 pages of articles and books. Although there are good companion texts made available by Actex, the material tested is so broad that it is not possible to find a unique and synthetic book for each of these exams, which implies you will need to study for an uncountable number of hours. So, completing the modules first will give you a sense of accomplishment before you cross a very long and dry desert.

More specifically, what order should you take the exams of the QFI track in? I suggest that you take the ERM exam first – in case you are interested in becoming a Chartered Enterprise Risk Analyst (CERA) – because it is shorter and slightly “easier” (or if you prefer very very hard instead of crazily hard) than the QFI Core and QFI Advanced exams. So, taking the ERM exam before the QFI exams prepares you for what you will be facing when taking these latter exams. If instead of taking the ERM exam you prefer to take the Investment Risk Management exam, it is even clearer that you should take the IRM exam first, because this is a two hour exam only, while the ERM exam is a four hour exam.

All right, now you have passed all three fundamental modules plus exam ERM or exam IRM. Now, what should you do? What is for sure is that you must absolutely take the QFI Core exam before the QFI Advanced exam. This is because there are lots of things about stochastic analysis that you will learn when you prepare for the QFI Core exam that are reused when you take the QFI Advanced exam. For instance, you cannot master the contents related to advanced interest rate models that are tested in the QFI Advanced exam without first mastering the fundamental recipes about Brownian motion that you will learn when you prepare for the QFI Core exam.

Let us say that two years have elapsed and you have passed the three fundamental fellowship modules and two out of three fellowship exams. Thus, you just have one more exam to take and pass, presumably the QFI Advanced exam. It is about time to register for and complete the Decision Making And Communication (DMAC) module. When your DMAC report is validated, you have two years to validate your remaining exams and modules. If you are working sufficiently hard, it is reasonable to expect validating the final fellowship exam in less than two years, given that you can take an exam every six months. Put differently, if you could pass each of the ERM (assuming you chose the CERA subtrack) and QFI Core exams in much less than four trials, then it is reasonable to expect the same thing for the remaining QFI Advanced exam. So, do it, register for the DMAC exam and enjoy: you will learn lots of non-technical skills that will be extremely useful for the rest of your career. Better advice is to go even further: buy all the books recommended in the DMAC e-learning system, read them and reread them!

That’s it, you have now completed all the activities in your track: the three fundamental modules, the three exams and the DMAC module. You can now attend the Fellowship Admissions Course. What can you expect from this course? Lots of case studies and lots of discussion and thinking about ethics, entertaining keynote presentations, and a lot of networking ; all of this in a top-notch hotel. It is not difficult to pass (at last!), you just need to be polite and always on time. However, aren’t these conditions also required in everyday professional life? You are now a Fellow of the Society of Actuaries: this is the beginning, not the end, of your career.


This class teaches how to price and hedge vanilla and exotic derivatives based on the fundamental principles of arbitrage. Some of the key concepts of the class are: pricing, hedging, replication, simulation, optimal stopping, smile, volatility surface, Greeks, Gaussian and non-Gaussian models.

Here are the main topics covered: description of main derivatives products and strategies, Black Scholes formula, American options with binomial trees, path-dependent options with Monte-Carlo simulations, arbitrage-free relations, futures and forwards, options on indices, currencies and futures, limits of the Black and Scholes theory, the smile and the volatility surface, Heston’s model, applications to corporate finance.

This class is technical in essence. It alternates between formal developments (mathematical proof of the Black and Scholes formula,…), exercises, and a lot of programming in VBA. Homework consists in designing a pricer similar to those found on the desks of investment banks.

Students will learn useful tricks concerning derivatives for the Series 7 license. The knowledge gained in this class is also useful for SOA exam MLC and the FRM and CFA exams. Students should read the book by John Hull: Options, Futures and Other Derivatives, Pearson, at home in parallel with the class.

Lebesgue Integration Theory

This is a classic class in integration theory that allows students to understand what is behind probability theory. The class starts by defining and studying measurable spaces, by concentrating on sigma algebras – including Borel sigma algebras. Then, measurable functions and Borel functions are introduced. Within the study of measures, sigma additivity and sigma subadditivity are examined together with the Lebesgue measure and the Carathéodory condition.

All these ingredients being prepared, we arrive at the construction of Lebesgue integrals – after providing reminders on the Riemann integral viewed as a limit of Darboux sums. Fundamental theorems are obtained and illustrated: the Beppo-Levi monotone convergence theorem, the Lebesgue dominated convergence theorem – Fatou’s lemma being examined in passing. Then, we derive theorems for integrals that depend on a parameter and for multiple integrals (Fubini-Tonelli’s theorem in the latter case). Convolution is also studied.

Finance for Managers

The aim of this course is to introduce the student, in its role as a future firm manager, to the basic notions of finance and to the main criteria of an investment decision. The course provides the tools any manager should use to correctly quantify the impact of a decision in order to correctly evaluate its value creation.

This course illustrates the basic concepts of corporate finance and modern finance theory. Concepts such as the valuation of risk and the weighted average cost of capital of a firm are covered. At the end of the course, students should be able to make simple capital budgeting decisions and to evaluate financial securities.

Other main takeaways are: the estimation of cash flows, understanding the difference between earnings and cash flows, the net present value of cash flows and other methods of cash flow valuation, capital markets and the price of risk, capital asset pricing model (CAPM).

To sum it up, this course allows the student to: learn the main methods of project valuation, know the principles underlying the pricing of risk in financial markets, know the determinants of the cost of capital for corporations, compute cash flows and their net present value and internal rate of return, and compute the cost of capital for a corporation.

Derivatives and Financial Modeling

This class offers a broad investigation of derivatives and quantitative methods. It starts with reminders on stochastic processes and Brownian motion. It also covers stochastic integrals, martingales, Ito’s lemma and Girsanov’s theorem. Then, it constructs the Black Scholes framework and discusses arbitrage relationships.

This class relies a lot on numerical methods and explains the mechanics of trees, PDE schemes and Monte Carlo simulations. Other advanced concepts examined are stochastic volatility (with the Heston model derived from one end to the other) and the volatility smile.

Basic interest rate models, such as the Vasicek and CIR models are studied before looking at the Heath-Jarrow-Morton and Brace-Gatarek-Musiela frameworks. Credit intensity and structural models are also studied in this class.

Management of Extreme Risks

The contents of this class are based on my book “Extreme Financial Risks and Assets Allocation” published by Imperial College Press and coauthored with Christian Walter. The first part of the class offers a broad introduction to infinitely divisible distributions and to Lévy processes. The main distributions and processes are surveyed and their main properties (activity, variation, moments…) are examined. A strong focus is made on Fourier transform methods.

The second part of the class is dedicated to applications. It shows how static and dynamic portfolio choice problems can be solved in the presence of extreme risks (modeled via Lévy processes in the dynamic case). The course also examined the management of risks and the computation of risk indicators such as Value at Risk or Tail Conditional Expectation when extreme risks are considered.


This class is an introduction to time series and to their broad use in management science. After some reminders on random walks and white noises, the first part of the class introduces autoregressive processes and then moves on to moving averages. It also covers more complex dynamics such as ARCH and GARCH processes.

The second part of the class consists in student presentations. These presentations can be related to Vasicek processes for instance and involve a strong implementation dimension. Students are graded via these presentations.

Model Implementation

All sessions start with the description of a numerical method to be implemented and are followed by an actual implementation in Matlab by the students. Among the methods that are studied, we can cite: Monte-Carlo simulations, numerical PDE solving, design of binomial trees, and the implementation of Fourier transforms.

Therefore, the students learn how to price various types of options (path-dependent and American) assuming various types of models (diffusive, with jumps, with stochastic volatility,…). When implementing Monte Carlo simulations, a special focus is made on the computation and interpretation of confidence intervals.

Probability Theory and Stochastic Processes

The objectives of this class are two-fold. Half of the sessions are dedicated to classic probability (and correspond to about two thirds of the syllabus of SOA exam P) and half of the sessions provide an introduction to stochastic processes. The student who attends this class gains knowledge about all the necessary tools that are prerequisite to the study of option pricing and hedging. This class is also useful to those who contemplate furthering their studies in risk management or actuarial science. 

The contents are as follows. Quick reminders on set theory. Independent and mutually exclusive events. Bayes theorem and law of total probability. Moments, including high-order moments, of probability distributions. Main probability distributions. Brownian motion. Martingales and Markov processes. Stochastic differential equations. Ito’s lemma. Girsanov’s theorem. Change of numéraire.

Financial Mathematics

This course presents the fundamental principles of financial mathematics, as applied in the fixed income departments of banks and in the life insurance and pensions businesses.  The syllabus exactly follows the first part of the 2015 syllabus of the financial mathematics exam of the Society of Actuaries (the second part of this latter exam is about derivatives and is not covered). See This syllabus is as follows.

“Definitions of the following terms: interest rate (rate of interest), simple interest, compound interest, accumulation function, future value, current value, present value, net present value, discount factor, discount rate (rate of discount), convertible m-thly, nominal rate, effective rate, inflation and real rate of interest, force of interest, equation of value. Given any three of interest rate, period of time, present value and future value, calculate the remaining item using simple or compound interest. Solve time value of money equations involving variable force of interest. Given the effective discount rate, the nominal discount rate convertible m-thly, or the force of interest, calculate any of the other items. Write the equation of value given a set of cash flows and an interest rate.

Definitions of the following terms: annuity-immediate, annuity due, perpetuity, payable m-thly or payable continuously, level payment annuity, arithmetic increasing/decreasing annuity, geometric increasing/decreasing annuity, term of annuity. For each of the following types of annuity/cash flows, given sufficient information of immediate or due, present value, future value, current value, interest rate, payment amount, and term of annuity, the candidate will be able to calculate any remaining item. Arithmetic progression, finite term. Level annuity, finite term. Arithmetic progression, perpetuity. Level perpetuity. Geometric progression, finite term. Non-level annuities/cash flows. Geometric progression, perpetuity. Other non-level annuities/cash flows.

Definitions of the following terms: principal, interest, term of loan, outstanding balance, final payment (drop payment, balloon payment), amortization, sinking fund. Given any four of term of loan, interest rate, payment amount, payment period, principal, calculate the remaining item. Calculate the outstanding balance at any point in time. Calculate the amount of interest and principal repayment in a given payment. Given the quantities, except one, in a sinking fund arrangement calculate the missing quantity.”


Treasury Management

La trésorerie est aujourd’hui le cœur opérationnel de la moyenne ou grande entreprise. Pas de solvabilité et donc pas pérennité de l’entreprise sans trésorier. Au-delà du rôle fondamental de garant de la survie de l’entreprise, le trésorier assure aussi de plus en plus le rôle de stratège : par le recours à des outils financiers adaptés, il intervient dans la gestion des risques de taux et de change.

Ce cours couvre les sujets suivants : budgets de trésorerie, échelles d’intérêts et dates de valeurs, modes de financement, arbitrages de financement, cash management, gestion du risque de change (théorie), gestion du risque de change (pratique), gestion du risque de taux (théorie), gestion du risque de taux (pratique), économie financière.

Fixed Income

This course offers a broad viewpoint on fixed income markets and investments. It starts with a description of bond and mortgage international markets. Then, it quickly reviews fundamental elements of financial mathematics allowing evaluators to shift money forward and backward in time. Duration, convexity, effective duration, effective convexity, option adjusted spread, Z-spread, DVO1, and other similar risk management indicators are then examined.

The class also examines complex products, like exotic floating rate notes, and lets students learn by practicing, by manipulating realistic pricers with trees implemented in XL spreadsheets. The pricing, risk management, and challenges associated with mortgage backed securities are also studied. The class also discusses callable and putable bonds and structural models such as that of Merton (1974).

Probability Theory and Stochastic Processes

The objectives of this class are two-fold. Half of the sessions are dedicated to classic probability (and correspond to about two thirds of the syllabus of SOA exam P) and half of the sessions provide an introduction to stochastic processes. The student who attends this class gains knowledge about all the necessary tools that are prerequisite to the study of option pricing and hedging. This class is also useful to those who contemplate furthering their studies in risk management or actuarial science. 

The contents are as follows. Quick reminders on set theory. Independent and mutually exclusive events. Bayes theorem and law of total probability. Moments, including high-order moments, of probability distributions. Main probability distributions. Brownian motion. Martingales and Markov processes. Stochastic differential equations. Ito’s lemma. Girsanov’s theorem. Change of numéraire.

Stochastic Processes and their Applications

This class is a general theoretical introduction to stochastic processes and to some of their applications. The class starts with the classic elements of the general theory of semimartingales, such as stopping times and Brownian motion. It constructs the Wiener integral and then the Ito integral. Key results such as Ito’s lemma and Girsanov’s theorem are then introduced and some illustrations are provided.

This class offers an introduction to stable distributions and stable processes, and more generally to fractals and self-similarity. It also provides a description of infinitely divisible distributions and of Lévy processes – from simple Poisson processes to the more complex and contemporaneous CGMY processes. More generally, this class develops a Fourier vision of the theory of stochastic processes.

VBA for Finance

This class teaches Visual Basic for Applications via the construction of an option pricer. The use of the main statements (“if” condition, “for” loop …) of this programming language is mastered at the end of the class. Students also learn how to properly indent and comment their codes, and to structure them within VBA modules.

The VBA case study that is jointly built with students is an American put option pricer. Students learn how to plot trees in XL that are controlled from VBA modules. Interfaces are also implemented that allow the students to construct a realistic option pricer. The code computes option prices but also option Greeks. 

Introduction to the Risk Management of Financial Institutions

The course is devoted to the regulation and risk management of banks and insurance companies. The goal of its first part is to learn about the nature and management of the main risks that impact financial institutions. Market, credit and operational risks are its main objects. An introduction to the risks associated with derivatives is also offered. Securitization is studied in a special session. This class offers key conceptual tools to tackle the preparation of the FRM. Useful complements in terms of ERM, multivariate risks… are given in the consecutive block of classes. In the second part, the general principles of insurance are introduced and the notion of risk for insurers is developed. This part ends with the presentation of Solvency 2, which is the equivalent of the Basle agreements in the insurance sector. Part I of this class is made of: introduction to market risks, basic statistical tools, Value-at-Risk and Expected Shortfall, introduction to credit risk, rating transitions, structural models, introduction to operational risk, introduction to options, Greeks, smile, securitization. The details of part II of this class are not given here, as it was taught by a colleague. 

Health and Portfolio Choices : a Diffidence Approach

The effect of health status on portfolio decisions has been extensively studied from an empirical viewpoint. In this paper, we propose a theoretical model of individuals’ choice of financial assets under bivariate utility functions depending on wealth and health. Our model relies on the diffidence theorem, which pertains to the class of hyperplane separation theorems. We establish the conditions under which the share of wealth held in risky assets falls as: 1) individuals’ health status deteriorates and; 2) individuals’ health status becomes risky. These conditions are shown to be related to the behaviour of the intensities of correlation aversion and of cross prudence as wealth increases.

Inside the Solvency 2 Black Box: Net Asset Values and Solvency Capital Requirements with the least-squares Monte Carlo method

The calculation of Net Asset Values and Solvency Capital Requirements in a Solvency 2 context – and the derivation of sensitivity analyses with respect to the main financial and actuarial risk drivers – is a complex procedure at the level of a real company, where it is illusory to be able to rely on closed-form formulas. The most general approach to perform- ing these computations is that of nested simulations. However, this method is also hardly realistic because of its huge computation resources demand. The least-squares Monte Carlo method has recently been suggested as a way to overcome these difficulties. The present paper confirms that using this method is indeed relevant for Solvency 2 computations at the level of a company.

Portfolio Optimisation with Jumps: Illustration with a Pension Accumulation Scheme

In this paper, we address portfolio optimisation when stock prices follow general Lévy processes in the context of a pension accumulation scheme. The optimal portfolio weights are obtained in quasi-closed form and the optimal consumption in closed form. To solve the optimisation problem, we show how to switch back and forth between the stochastic differential and standard exponentials of the Lévy processes. We apply this procedure to both the Variance Gamma process and a Lévy process whose arrival rate of jumps exponentially decreases with size. We show through a numerical example that when jumps, and therefore asymmetry and leptokurtosis, are suitably taken into account, then the
optimal portfolio share of the risky asset is around half that obtained in the Gaussian framework.

Imprudent Risk-Lovers are Inconsistent

Research characterizes most risk averters as prudent and temperate but devotes little attention to the study of risk lovers. The risk lovers who prefer to combine good with good are prudent and intemperate. This paper shows how the assumption of “combining good with good » can be relaxed, and how similar results can come from the consistency hypothesis for decision makers. Namely, the risk lovers who are consistent are prudent and intemperate with positive derivatives of their utility function for all orders. However, empirically risk lovers do exist who are both imprudent and intemperate. These risk lovers, being imprudent, are inconsistent.


This article extends credibility theory by making quadratic adjustments that take into account the squared values of past observations. This approach amounts to introducing non-linearities in the framework, or to considering higher order cross moments in the com- putations. We first describe the full parametric approach and, for illustration, we examine the Poisson-gamma and Poisson-single Pareto cases. Then, we look at the non-parametric approach where premiums must be estimated based on data only, without postulating any type of distribution. Finally, we examine the semi-parametric approach where the con- ditional distribution is Poisson but the unconditional distribution is unknown. The goal of this paper is not to claim that q-credibility always brings better results than standard credibility, but it is to provide several building blocks for understanding how credibility changes when quadratic corrections are added.

Credit Risk and Solvency Capital Requirements

Credit risk permeates the assets of most insurance companies. This article develops a framework for computing credit capital requirements under the constant position paradigm and taking into account recovery rates. Although this framework was originally derived under the Solvency 2 regulation, it also provides concepts that can be useful under other international regula- tions. After a brief survey of the existing technology on rating transitions and default probabilities, the paper provides new results on risk premium adjust- ment factors. Then, three different procedures for reconstructing constant po- sition market-consistent histories of credit portfolios from quoted Merrill Lynch indices are given. The reconstructed historical credit values are modeled via mixed empirical-Generalized Pareto Distribution (GPD) dynamics and a de- tailed parameter estimation is performed. Several validations of the estimation are also provided. Finally, credit Solvency Capital Requirements are computed and an analysis of the results per rating class is given.

Some Further Results on the Tempered Multistable Approach

Typical approaches incorporating jumps in financial dynamics, such as the Variance Gamma and CGMY models, can be made to depart from the i.i.d. hypothesis by using a stochastic clock. In such a context, the introduction of a dispersion of the clock is equiva- lent to the introduction of a dispersion of the volatility itself. A distinct route that yields
comparable features is that of adding a jump component to a stochastic volatility process, or of considering, in discrete time, leptokurtic innovations within a GARCH process. In this article, we take a third route and we provide a study on tempered multistable pro- cesses, which convey both jumps and autocorrelation from their very construction, and on some of their applications in finance. We obtain the multivariate characteristic function of the asymmetrical field-based tempered multistable process and we study the autocorrela- tions that stem from the use of this model. We concentrate on three types of applications in finance: we study the term structures of Value-at-Risk that can be obtained with this model, we perform a calibration on stock index data, and we also conduct a calibration on derivatives prices.

An Intensity Model for Credit Risk with Switching Lévy Processes

We develop a switching regime version of the intensity model for credit risk pricing. The default event is specified by a Poisson process whose intensity is modeled by a switching Lévy process. This model presents several interesting features. Firstly, as Lévy processes encompass numerous jump processes, our model can duplicate sudden jumps observed in credit spreads. Also, due to the presence of jumps, probabilities do not vanish at very short maturities, contrary to models based on Brownian dynamics. Furthermore, as parameters of the Lévy process are modulated by a hidden Markov process, our approach is well suited to model changes of volatility trends in credit spreads, related to modifications of unobservable economic factors.

Decreasing Downside Risk Aversion and Background Risk

In this paper, we show that risk vulnerability can be associated with the concept of down- side risk aversion (DRA) and an assumption about its behavior, namely that it is decreas- ing in wealth. Specifically, decreasing downside risk aversion in the Arrow-Pratt and Ross senses are respectively necessary and sufficient for a zero-mean background risk to raise the aversion to other independent risks.

The Computation of Risk Budgets under the Lévy Process Assumption

This paper revisits the computation of Value-at-Risk and other risk indicators based on the use of Lévy processes. We first provide a new presentation of Variance Gamma Pro- cesses with Drift: we reconstruct them in an original way, starting from the exponential distribution. Then, we derive general Fourier formulas that allows to compute VaR quickly and efficiently, but also other typical indicators like Tail Conditional Expectation (TCE), TailVaR or Expected Shortfall. Based on this formula, we conduct a study of the term structure of VaR, and provide a discussion of the Basle 2 and Solvency II agreements.

On Surrender and Default Risks

This article examines the impact that surrender risk can have on the default of insurance 4
companies. The companies that we study issue contacts similar to the ones studied earlier in the literature by Briys and de Varenne (2001), Grosen and Jorgensen (2000), or Bernard, Le Courtois and Quittard-Pinon (2005). They are subject to interest rate and default risk; they offer a guaranteed amount plus a bonus indexed on the performance of the underlying portfolio. In this article, we assume in addition that policyholders have the option to surrender. Surrender risk has been extensively studied in an arbitrated market by Bacinello (2001), using trees for the valuations. We deal with surrender risk in another way, assuming policyholders have sets of information and preferences that differ from the ones of financial market agents. In particular, policyholders are supposed to be only partially rational (at least in the financial sense).

On the Bankruptcy Risk of Insurance Companies

The fall of AIG is another confirmation that the insurance business is not immune to bankruptcy. Contrary to the actuarial literature which postulates that insurance firms can survive forever, we believe that this is not the case, and that a realistic and business- oriented risk management approach needs to be designed in order to prevent the actual, finite-time, bankruptcy of insurance companies. In this article we model the surplus process of an insurance firm firstly by a stable Lévy process, secondly by a double exponential compound Poisson process. We compute finite-time survival and bankruptcy probabilities under such hypotheses. To achieve this, we make use of the Wiener-Hopf factorization and compute bankruptcy formulas written in terms of inverse Laplace transforms. The Abate and Whitt, and Gaver-Stehfest algorithms are used to obtain numerical estimations.

Concentration des portefeuilles boursiers et asymétrie des distributions de rentabilités d’actifs

This article develops on the link between the asymmetry of asset return distributions and the concentration of portfolios. We start by recalling the rationale behind the theory of diversification, in order to let appear that this theory relies on a reduction of risk viewed only at order 2 and on the related application of the theory of errors, as developed during the XVIIIth century. We also expose the controversy opened by E. Fama in 1965 on this theory of errors, in order to let appear that a change in the type of underlying randomness can lead to the concentration and not the diversification of portfolios. Then, we examine how the inclusion in the optimization program of the asymmetry between gains and losses can lead to a propensity to concentrate.We recall the main aspects of the Mitton and Vorkink (2007) model, and then we propose a new approach in the spirit of this model. We end up with an illustration of the latter framework on American data, letting appear important differences between the performance obtained with a classically diversified port- folio, a portfolio concentrated along existing models, and a portfolio concentrated along the model that we propose.

Asset Risk Management of Participating Contracts

In this paper we study the asset-liability management of an insurance company selling “participating contracts”. Participating contracts are typical insurance policies sold world- wide. The payoff of a participating policy is linked to the portfolio or the surplus of the insurance company. We examine the impact of the choice of assets’ investment strategy on the company value, its solvency, and how well the company may meet the commitments associated with its liabilities. Four strategies are investigated and compared: a simple buy-and-hold strategy, a dynamic CPPI (Constant Proportion Portfolio Insurance), an investment in Equity Default Swaps (EDS), and a protection by ways of forward-starting puts. For example it is shown that an active protection strategy by CPPI may significantly reduce the company’s default risk but is very costly to policyholders. Our study illustrates how to compare asset management strategies and how to choose the parameters of a suit- able allocation such that the policyholders’ market value is preserved and the default risk is reduced.

Performance Regularity: a New Class of Executive Compensation Packages

The ability of standard executive stock options to incite managers to adequately select the assets of their firm has been extensively questioned by academics and practitioners. However, very few alternatives exist or have been proposed to better control the investment strategies of top managers. The present article studies the evaluation and sensitivity of a new class of executive stock options well designed for the control of managers. Such packages are aimed at giving incentives to CEOs to maintain a regular performance over time and a stable volatility level. The importance and implications of the choice of the different parameters as well as their robustness with respect to standard financial criteria are examined. In brief, this article studies in a utility-based framework a new type of executive stock options that can be useful to protect and enhance the economic performance of corporations.

Protection of a Company Issuing a Certain Class of Participating Policies in a Complete Market Framework

In this article, we examine to what extent policyholders buying a certain class of partici- pating contracts (in which they are entitled to receive dividends from the insurer) can be described as standard bondholders. Our analysis extends the ideas of Bühlmann [2004], and sequences the fundamental advances of Merton [1974], Longstaff and Schwartz [1995], and Briys and de Varenne [1994, 2001]. In particular, we develop a setup where these
participating policies are comparable to hybrid bonds but not to standard risky bonds (as done in most papers dealing with the pricing of participating contracts). In this mixed framework, policyholders are only partly protected against default consequences. Continu- ous and discrete protections are also studied in an early default Black and Cox [1976] type setting. A comparative analysis of the impact of various protection schemes on ruin prob- abilities and severities of a Life Insurance company which only sells this class of contracts concludes this work.

Fair Valuation of Participating Life Insurance Contracts with Jump Risk

The purpose of this article is to value participating life insurance contracts when the linked portfolio is modeled by a jump-diffusion. More precisely, this process has a Brownian com- ponent and a compound Poisson one, where the jump size is driven by a double exponential distribution. Specifically here, the bankruptcy risk of the insurance company is considered. Thus, market and credit risks are taken into account. A quasi-closed-form formula is ob- tained in fair value for the price of the considered life insurance contract. This allows us to investigate the impact of strategic parameters as well as structural ones, as is shown in the numerical section of this paper. In particular, we study the impact on the contract of the volatility, jump intensity, jump asymmetry, company leverage, guaranteed rate, participa- tion rate and level of the default barrier, and comment on how they are likely to increase the probability of early default of the issuer.

Pricing Derivatives with Barriers in a Stochastic Interest Rate Environment

This paper develops a general valuation approach to price barrier options when the term structure of interest rates is stochastic. These products’ barriers may be constant or stochastic, in particular we examine the case of discounted barriers (at the instantaneous interest rate). So, in practice, we extend Rubinstein and Reiner (1991), who give closed- form formulas for pricing barrier options in a Black and Scholes context, to the case of a Vasicek modeling of interest rates. We are therefore in the situation of pricing barrier options semi-explicitly or explicitly (depending on the shape of the barrier) with stochastic Vasicek interest rates. The model is illustrated with a specific contract, an up and out call with rebate, hence a typical barrier option. This example is merely here to show how any standard barrier option can be priced and its Greeks be obtained in such a context. The validity of the approximation is analyzed and the sensitivity to the barrier level and to discretization schemes are also derived.

The Optimal Capital Structure of the Firm with Stable Lévy Assets Returns

This article builds a new structural default model under the assumption that a firm’s assets return follows a dynamics displaying jumps of both signs. In essence, we expand the work of Hilberink and Rogers (itself an extension of the Leland and Toft framework), which deals only with negative jumps. In contrast, we make use of stable Lévy processes, and we compute the values of the firm, debt and equity under this assumption. Theoretical credit spreads can also be obtained in our framework. They prove to be consistent with the empirical credit spreads observed in financial markets.

Risk-Neutral and Actual Default Probabilities with an Endogenous Bankruptcy Jump- Diffusion Model

This paper focuses on historical and risk-neutral default probabilities in a structural model, when the firm assets dynamics are modeled by a double exponential jump diffusion process. Relying on the Leland [1994a, 1994b] or Leland and Toft [1996] endogenous structural approaches, as formalized by Hilberink and Rogers [2002], this article gives a coherent construction of historical default probabilities. The risk-neutral world where evolve the firm assets, modeled by a geometric Kou process, is constructed based on the Esscher measure, yielding useful and new analytical relations between historical and risk-neutral probabilities. We do a complete numerical analysis of the predictions of our framework, and compare these predictions with actual data. In particular, this new framework displays a greater predictive power than current Gaussian endogenous structural models.

Development and Pricing of a New Participating Contract

This article designs and prices a new type of participating life insurance contract. Partic- ipating contracts are popular in the United States and European countries; they present many different covenants and depend on national regulations. In the present article, we design a new type of participating contract very similar to the one considered in other studies, but with the guaranteed rate matching the return of a government bond. We prove that this new type of contract can be valued in closed form when interest rates are stochastic and when the company can default.

Le Point Sur… Les Options Parisiennes et leurs Applications

This survey paper is dedicated to some options that are not extremely well-known, but that bring along very promising applications: Parisian options. We start by describing these products, then we detail their use in the contexts of bank deposit insurance, the theory of real options, and finally the structural theory of default. The reader will consider these examples to be illustrative, though not exhaustive.

A New Procedure for Pricing Parisian Options

Parisian options extend barrier options in that their covenant depends on the time spent by the underlying beyond a given threshold. Due to their very nature, they are hard to price and hedge, though some quite involved material has been made available in that direction. Valuation of Parisian Options can be done by using four different main methods: Monte- Carlo simulations, lattices, partial differential equations, inverse Laplace transform. In this article, we develop a new inverse Laplace transform method, quick and well-suited to the problem under study. This method could also be used to treat other financial problems where inversion of a Laplace transform is required.

Market Value of Life Insurance Contracts under Stochastic Interest Rates and Default Risk

The purpose of this article is to value some life insurance contracts in a stochastic interest rate environment taking into account the default risk of the underlying insurance company. The participating life insurance contracts considered here can be expressed as portfolios of barrier options as shown by Grosen and Jørgensen [1997]. In order to price these options, the Longstaff and Schwartz [1995] methodology is used with the Collin-Dufresne and Goldstein [2001] correction.

A Study of Mutual Insurance for Bank Deposits

This article displays a study on the mutual insurance of bank deposits. A system where deposits are first insured by a consortium then by the Government is envisaged. We wish to compute the fair premia due to both the consortium and the Government. Various types of covenants aiming at making banks reduce their risks are detailed. These provisions can be, as is the case in Chapter 11, of a Parisian type. This means that surveillance is based on the path followed by the assets or the leverage. We compare these various types of covenants and conclude on the proposal for new regulatory provisions.

Changes of Probability Measures in Finance and Insurance: A Synthesis

Changes of probability measures now pave a very efficient way towards the study of many problems in Finance and Insurance. To quote only a few, this approach has proven to be useful in portfolio selection, option pricing and premium valuation in insurance. Many techniques have been developed more or less independently: changes of numéraire, pricing kernels (equivalents of probability measure changes) , Esscher transforms. The aim of this paper is to give a survey on these approaches and illustrate them with the Black and Scholes model taken as a benchmark; and then to see to what extent these changes of measure can be adapted to a more general context than the standard diffusion processes extensively assumed in the financial literature for modelling price processes. More specifically, we explore these techniques when price movements are modelled as Lévy processes. In sections two, three, four, five of this article we concentrate on diffusions – our aim being to exhibit the fundamental ideas behind the commonly used tools. From section six, we move on to the case of jump-diffusions, and in particular to the case of assets following geometric Lévy motions.

Modelling Stock Returns with Lévy Processes

Several approaches to model stock returns with Lévy Processes have been developed in the past years. Firstly, this article will review existing approaches and compare the latest ones through an analysis of the Lévy density. Secondly, this article will provide a simple but general parameterization for the Lévy density which yields a class of Lévy processes that can be used in a financial context. These processes – titled α-β Lévy motions – will allow for excessive arrival rates of average size jumps, in correspondence to humped return distributions at short time scales.