CAHIERS - Institut Louis Bachelier...By Julien Guyon Modeling and Managing Financial Risks E ankov Quantitative trading TION IN THE MARKETS By Charles-Albert Lehalle No-arbitrage bounds

A BETTER UNDERSTANDING OF FINANCIAL RESEARCH

Special Edition

With

Aurélien AlfonsiNicole El KarouiEmmanuel GobetJulien GuyonCharles-Albert LehalleNizar TouziPeter Tankov

DE L’ILBCAHIERSL E S

N°3The Institut Louis Bachelier Research Review

JULY 2011

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Optimal order execution AND PRICE MANIPULATION

By Aurélien Alfonsi

Fast formulas for option valuation AND REAL-TIME CALIBRATION

By Emmanuel Gobet, Eric Benhamou and Mohammed Miri

The smile IN STOCHASTIC VOLATILITY MODELS By Julien Guyon

Modeling and Managing Financial Risks A FINANCIAL RISK CHAIR CONFERENCEBy Peter Tankov

Quantitative tradingRATIONALIZATION OF NEGOTIATION IN THE MARKETS By Charles-Albert Lehalle

No-arbitrage bounds FOR LOOKBACK OPTIONS

By Pierre Henry-Labordère and Nizar Touzi

“Probabilities and Finance” Master’sCELEBRATES ITS 20TH ANNIVERSARY By Nicole El Karoui

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editorialFinancial risk: a challenge for mathematicians For some 40 years now mathematicians have been working in partnership with the finance industry on models of stock price fluctuations, in order to develop methods of valuing assets andcontrolling risks. However, the recent subprime crisis has shown thatwe have a long way to go to fully understand the functioning of marketsand associated risks. The models used still fall well short of reality,particularly in regard to liquidity risk and interactions between agents.

The “Risques financiers” Chair of La Fondation du Risque was createdshortly before the crisis to promote better use of mathematics in financial institutions. Currently held by Nicole El Karoui, an internatio-nally recognized expert in financial markets and the founder of thehighly reputed Master’s "Probability and Finance", this Chair bringstogether research teams in financial mathematics from the Ecole Polytechnique and the Ecole des Ponts et Chaussées with the SociétéGénérale quantitative research team, headed by Lorenzo Bergomi.These teams are at the forefront of financial research in the Paris region and in some fields are world leaders. The objective of the Chair is on the one hand to “move up a gear” in terms of research, inparticular by further developing international collaborations to makethe Paris market more attractive for researchers, and on the otherhand to promote collaborations with practitioners with a view to identifying relevant research topics and aligning academic researchmore closely with the concerns of the financial industry.

Research topics on which we work include:

• Effective numerical methods for the problems of valuation and riskmanagement.

• Modeling liquidity risk, i.e. the risk associated with the impossibilityof quickly finding a buyer or a seller for a large amount of assets.

• Nonlinear problems, including questions related to the interactionbetween agents.

• Modeling market microstructure and order books; processing extremely high frequency financial data.

• Mathematical methods for solving optimization problems in finance, in particular through the use of “backward” stochastic differential equations.

The “Modeling and Managing Financial Risks” conference was anopportunity for us to make a preliminary assessment of the first threeyears of the Chair, an assessment which proved overwhelmingly positive. Research teams have grown, because the existence of theChair on the one hand attracts support from institutions and on theother hand results in top quality recruitment. We have been able toinvite a large number of leading foreign researchers and to initiateongoing collaborations with some of them. Collaborations betweenacademics and Société Générale researchers are also under way.Much progress has been made in the Chair’s research fields. Thiscompilation presents the “Modeling and Managing Financial Risks”conference, focusing in particular on the contributions of membersof the Chair, and provides an overview of what has been achieved.

Nicole El Karoui and Nizar TouziScientific Directors of the Financial Risks Chair

Publication of Institut Louis BachelierPalais Brongniart28 place de la Bourse75002 ParisTél. : 01 49 27 56 40www.institutlouisbachelier.orgwww.e-fern.org

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CorrespondentPeter Tankov [email protected]

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Financial Risk Chair's Partners


BIOGRAPHY

Aurélien Alfonsi

Aurélien Alfonsi is a researcher in

probability and finance at the Ecole

Nationale des Ponts et Chaussées,

at CERMICS (Centre for Teaching and

Research in Mathematics and Scientific

Computing). His research focuses on

modeling financial markets and

numerical [email protected]

Introduction

These orders, known as limit orders, canonly be placed on a price grid determi-ned by the market. When a participantwishes to acquire the asset, he may either place a limit order to buy and waituntil it finds buyers or buy at the bestoffer price by acquiring the cheapest sell orders. The first solution has the advantage of lower cost, but the imple-mentation and the time of the transactionare uncertain. The second solution ismore expensive but immediate. In ourstudy [AS], for greater simplicity weconsider only the latter possibility.

When an agent buys a small quantity ofassets on the market, he will acquireonly the least expensive offers. The costof the transaction will then approximateto the product “quantity” × “quoted price”,

and consequently the information givenby the order book is superfluous. The situation is different if the agent buys alarge quantity of assets. More precisely,we here mean that the quantity purcha-sed is of an order of magnitude compa-rable to the number of sell limit orderspending in the order book. In this case,the agent will acquire limit orders to sellat a much higher price than the quotedprice. Knowledge of the order book isthen necessary to determine the cost of the transaction, which will anyway behigher than the product '”quantity” ×“quoted price”. In fact in this case, theagent has no incentive to buy all the assets in a single transaction. It is betterfor him to divide up his purchase, so thatnew, less expensive limit sell orders appear between his purchase orders.This brings us to the optimal execution

Optimal order execution AND PRICE MANIPULATION


KEY POINTS

■ In their study [AS], Aurélien Alfonsi and Alexander Schied offer a simple and intuitive model of the dynamics of order books in financialmarkets.

■ This allows them to obtain an explicit optimal strategy for placing apurchase order.

In general, the value of an asset quoted on a market is indicated by aprice. This price is the average between the highest bid and lowest selloffer pending on that asset. It does not contain all the information givenby the market. Such information is r epresented by the order book thatcontains all pending purchase and sell orders on the asset.

By Aurélien Alfonsi Ecole des Ponts ParisTech - Université Paris-Est Marne-la-Vallée


The problem of optimal execution of orders is closely related to the question of market viability.

A p p l i c a t i o n sfor the actors concerned

■ Solving the optimal execution problemis not only interesting for practitionersseeking to reduce their transactioncosts but may also allo w regula tors to identify ho w listing mechanisms influence optimal stra tegies and thebehaviour of market agents.


problem we have studied: given a timehorizon (a few hours or days), how doesone optimally divide up the purchaseorder so as to minimize the expectedtotal cost of the transaction?

Methodology

To answer this question, we have proposedin [AS] a very simple model of the orderbook and its dynamics. The order bookis represented by a shape function thatdescribes the number of sell orders pen-ding at a given price. When the agentpurchases assets, he acquires the leastexpensive offers pending in the orderbook. This has the effect of increasingthe price of the best sale offer. When theagent is inactive, new sale offers appearin the order book at a rate that we assume is proportional either to the volume or to the price movement causedby the agent. In both cases, we obtain asimilar structure for the optimal strategywhen transactions take place on a homogeneous time grid. To minimizecost, the agent should begin by makingan initial purchase of significant size.Then, at intermediate times, he shouldmake small purchases that exactlyconsume all new orders that have appeared in the order book. At the finaldate, he should buy the outstandingquantity needed to meet his target. Theheuristics of this finding is very clear. Thegreater the size of the first purchaseorder, the more it will attract new limit orders into the order book. Conversely,the greater the size of orders, the higher

the marginal cost of the asset. Thus theoptimal strategy is a compromise between reducing the cost of orders andattracting new sales offers into the orderbook. In addition, our study gives an explicit expression of the optimal stra-tegy. This reveals from a quantitativestandpoint the impact of the shape ofthe order book and the resilience of themarket on the optimal strategy and its cost. We find in particular that the optimal strategy is relatively insensitive tothe shape of the order book, which is agood indicator of its robustness.

The problem of optimal execution of orders is closely related to the questionof market viability. According to Huber-man and Stanzl [HS], price manipulationis a strategy in which one has the samenumber of risky assets at the final timeas at the initial time, but the expectedcost of these is negative. A market is viable if there is no such strategy. Theidea is that if such a strategy existed,one could certainly make money by repeating it indefinitely, which wouldconstitute a standard arbitrage. Fromour standpoint, price manipulation is aspecial case of the optimal executionproblem in which one wants to buy zeroassets. If it were possible to purchase noassets with an average negative cost,there would be price manipulation. In ourorder book model, we have been able toshow that this is not feasible, althoughthe market that it represents is viable.

“”

Further reading

■ [AS] Alfonsi, A., Schied, A. (2010), OptimalTrade Execution and Absence of PriceManipulations in Limit Order Book Models. SIAM J. Finan. Math., Vol. 1, pp.490-522.

■ [HS] Huberman, G., Stanzl, W. (2004).Price manipulation and quasi-arbitra ge,Econometrica, Vol. 72, No. 4, pp. 1247-1275.


BIOGRAPHY

Emmanuel Gobet

Emmanuel Gobet is professor of appliedmathematics at the Ecole Polytechnique.After studying at X, he defended his thesisin 1998 and obtained his accreditation todirect research in 2003. He has publishedsome forty scientific papers in internationaljournals. His work mainly focuses on probabilistic numerical methods (MonteCarlo simulations, stochastic approxima-tion), processes statistics and financialmathematics (calibration methods, options valuation and hedging tools). He is also involved in many professionalcollaborations with financial institutions,insurance companies and energy providers.

[email protected]

Eric Benhamou

Eric Benhamou is CEO of Pricing Partners.He has extensive experience in the tradingroom, with stints at Goldman Sachs andNatixis before founding Pricing Partners, an independent company that provides thefinancial community with development andrisk management software based on latestgeneration mathematical models, thus allowing greater transparency in [email protected]

The financial market needs r eal-timecalculations

This expansion has been accompaniedand made possible by many advancesin computer technology and the simul-taneous development of the tools of financial mathematics, especially in ma-thematical analysis (stochastic calculus,Malliavin calculus, etc.) and numericalcalculus (Monte Carlo methods, solutionmethods for partial differential equations,etc.)

On the operational side, the industriali-zation of market finance has led to muchhigher requirement levels in terms of response time to the development of

products, calculation of their hedging orthe calculation of various risk indicators.These must be computable in real time,or at least be very rapidly computable,so that professionals are able to adjusttheir banking risk management in responseto fast-changing financial markets. Fromthis standpoint, calculation speed is avery powerful operational constraint. If arelevant model it is not computable orcapable of being simulated in a matter ofa second or two (or a minute or so, formore global banking management orportfolio calculation problems), it has tobe rejected, even if it is highly suited torisk modelling. One consequence of this

Fast formulas for option valuation AND REAL-TIME CALIBRATION

By Emmanuel Gobet, Ecole Polytechnique ; Eric Benhamou, PricingPartners and Mohammed Miri, Pricing Partners


KEY POINTS

■ The authors’ presentations (E. Gobet’s plenary lecture, M. Miri’s minisymposium contribution) at the “Modelling and Managing FinancialRisks” conference last January provide new and effective solutions to certain real-time calculation problems in regard to option prices and their hedging.

■ In fact, in many cases, an approximate formula of the price can bequite sufficient for objectives of calibration (determination of the model’s parameters) or large portfolio valuation.

■ If, in addition, the formula is analytical and thus very rapid to estimate(as the famous Black-Scholes formula can be), then the resulting process is carried out in real time and can respond effectively to large-scale calculation problems involving thousands of pricing options.

Since the opening of the options market in Chicago in 1973 and theground-breaking work of Black-Scholes and Merton, financial engineeringhas undergone dramatic growth, not only in terms of the types of markets(equities, for eign exchange, inter est rates, cr edit, commodities, life insurance, etc.) and the types of financial products available (so-calledvanilla options, exotic options, swing options, etc.) but also in terms ofproblematics (product valuation and hedging, portfolio management,risk identification and management, etc.).


Further reading

■ E. Gobet, A. Suleiman. New Approximationsin Local Volatility Models. To appear in MarekMusiela Festschrift. Springer Verlag, 2011.

■ E. Benhamou, E. Gobet, M. Miri: Analyticalformulas for local vola tility model with stochastic rates. Quantitative Finance, 2011.

■ E. Gobet, P. Etore. Stochastic expansion forthe pricing of call options with discrete dividends. Preprint, 2010.

■ E. Benhamou, E. Gobet, M. Miri: Time dependent Heston model. SIAM Journal onFinancial Mathematics, Vol.1, pp.289-325,2010.

■ E. Benhamou, E. Gobet, M. Miri: Expansionformulas for European options in a localvolatility model. Interna tional Journal ofTheoretical and Applied Finance, Vol.13(4),pp.602-634, 2010.

■ E. Benhamou, E. Gobet, M. Miri: Smart ex-pansion and fast calibra tion for jump dif-fusion. Finance and Stochastics, Vol.13(4),pp.563-589, 2009.



■ The tools developed by E. Gobet and his co-authors allow prices and hedge indices of options to be calcula ted in real time in relatively sophistica ted models. F inancial establishments will thus be able to take intoaccount risk factors tha t were previouslyinaccessible because of computa tional timeconstraints.

■ Furthermore, with their better grasp of non-linear effects in options portfolios, thesenew algorithms improve the calcula tion of indicators for estima ting regula tory capital or adjustments for counterparty risk andcontribute to a c learer view of risk a t banklevel.

real-time computing constraint is that,in practice, simple models which can provide risk indicators in real time are preferable to more comprehensive models requiring more computing time.It is a choice made necessary by theconstraints of high market volatility andvery fast risk management, which maybe at the expense of more fine-tunedrisk management but with a delay effectthat would obviously be harmful. Obviouslythis practice may have adverse conse-quences.

Although it might be assumed that tech-nological progress (through the celebratedMoore's law) now allows financial pro-ducts and their associated risks to beassessed ever more quickly, we are stillfar from the real-time calculations wantedby all trading room operatives. There aremany reasons for this. On the one hand,models tend to be constantly improvedto better capture the physical phenomenaof financial markets and thus to betterreproduce the data or information avai-lable. This refinement of models inevitablyresults in an inflation of computing time.On the other hand, risk management is approached more comprehensively, particularly in the context of regulatorycapital calculations through Value atRisk, or that of counterparty risk knownas Credit Valuation Adjustment. This callsfor making calculations on very largeportfolios and aggregating infinitesimalrisks of each option or financial productat the level successively of the trader’sportfolio, the trading floor, and the bank.It is of course essential to have a consistentapproach at all levels, and one as discer-ning as possible, taking into account theinterdependencies among risk factors.

Methodology

The idea presented at the conference isthat of choosing an approximate or so-called proxy model or financial contract,in which explicit calculations are possible.This proxy is then used to approximatethe quantities concerned, possibly byproposing correction terms to this ap-proximation. The choice of proxy is on acase by case basis. There follow varioussituations where it has been possible toobtain fast and accurate approximateevaluation formulas.

■ In Heston’s stochastic volatility model(with possibly time-dependent para-meters), one possible proxy modelmay be the Black-Scholes model bytaking volatility to be 0.

■ In local volatility models, the Black-Scholes model can be a proxy by setting the function of the local volatilityeither at spot or strike prices. One canalso incorporate Hull and White-typeGaussian interest rates to obtain aBlack-type proxy.

■ When Gaussian jumps are added tolocal volatility, the same type of approxi-mation leads to the Merton model.

In these three examples, one can approxi-mately calculate vanilla option prices asthe sum of the price in the proxy modelplus correction terms equal to a combi-nation of sensitivities (Greeks) calculatedin the proxy model. The numerical testscarried out generally show an impressivegain of about a factor of 100 in the com-puting time over the best previously available numerical method (FFT in theHeston model, resolution of dual EDP for the local volatility model with jumps).A complementary procedure can also deduce changes in implied volatilities, further facilitating calibration.The case of exotic options is also possible.Studies currently being finalized processaverage (Asian) or basket options. Incor-porating linear dividends can obtain newapproximations that are very accurateand fast. This study with affine dividendshas similarities with the valuation of gua-ranteed life insurance policies, commonlyknown as variable annuities, in particularin the case of GMABs.

Although approximation formulas aresimple and precise, their mathematicalderivation is quite complex and requiresa detour via stochastic analysis (Malliavincalculus) and stochastic disturbances,combined with ingenious parameteriza-tions. In addition to the formulas, fine errorestimates are established, clearly showingthe role of the parameters of the model or of the payoff in the accuracy of the formulas, and thus defining the frame-work for reasonably applying these approximations in real time.

Far from being closed, the subject is expanding rapidly and is thus promisesmajor advances.

The models tend to be constantly improved.“ ”

Mohammed Miri

Mohammed Miri is Research and Development

team leader at Pricing Partners. After studying

at Ensimag, in 2009 he defended his doctoral

thesis on approximation methods for obtaining

real-time calculations.

[email protected]


BIOGRAPHY

Julien Guyon

Julien Guyon works in Paris in the

Société Générale quantitative research

team. He has a doctorate in applied

mathematics from the Ecole des

Ponts (Paris), with particular expertise

in probability and statistics. He holds

degrees from the Ecole Polytechnique

(Paris), Université Paris 6 and ENPC.

He is also a professor at the Ecole

des [email protected]

Introduction

Although the Black-Scholes model canexplain the prices observed in the mar-ket, all the implied volatilities σBS(T,K)are equal, irrespective of the maturityand the strike price of the option. Butthis is not what occurs. Typically, we find that implied volatilities depend onmaturity T, and for a fixed maturity, there is a volatility “smile” i.e. the impliedvolatility depends on the strike price K.The term “smile” is used because thegraph K → σBS(T,K) often looks like a smile, i.e. it is often convex, first decreasing, then increasing. Figure 1shows such a volatility smile.

In the case of equity markets, the smilesignificantly decreases in the low strikeprice zone. In this zone, traders correctprices produced by the Black-Scholesmodel by increasing implied volatility,

mainly for two reasons: first, to take account of the possibility of a crash or amajor downward price movement,events that in the Black-Scholes modelare underweighted (for example, we observe that the historical volatility of ashare is typically greater when its priceis lower); and second, to take accountof the lack of liquidity options whosestrike price diverges considerably from

The smile IN STOCHASTIC VOLATILITY MODELS

By Julien GuyonSociété Générale, Global Markets Quantitative Research


KEY POINTS

■ Julien Guyon and Lorenzo Bergomi are concerned with “stochasticvolatility” models. In a completely general way, they calculate an exact limited development of the implied volatility smile at order 2 in volatility of volatility.

Louis Bachelier in 1905, followed by Francis Black and Myron Scholes in1973, put forward the first two theories of derivative products, also calledoptions. In these two models, uncertainty is fully quantified by a singlenumber, σ, the volatility of asset prices. For example, in the Black-Scholesmodel, which is the market benchmark, the price of call options is givenby the well-known Black-Scholes formula, and at each observed price inthe market, for a given maturity T and strike price K, there is one and onlyone constant volatility σBS(T,K), called implied volatility , obtained by reversing the formula.

Figure 1: A smile observed in the 5-10 year swaptions market




■ The results have a twofold value, both quan-titative and qualitative. On the one hand, theyallow the structural constraints on implied volatility to be specified.

■ On the other, they can be used for parametercalibration purposes and can significantly reduce the model risk associa ted with the development of an exotic option.

the current value of the asset. This second reason explains why we as oftensee the smile “accentuate” for very highstrike prices.

Methodology

To account for these observations, it is therefore necessary to modify theBlack-Scholes model in order to make itproduce a smile. One solution oftenused is to replace constant volatility σ bya process that itself has random fluctua-tions. It is these “stochastic volatility”models that we, together with LorenzoBergomi, head of quantitative researchat Société Générale, are interested in.Using perturbation expansion techniques,we have shown that in fact there is acompletely general formula for the smile generated by stochastic volatilitymodels. We presented our findings atthe "Modelling and Managing FinancialRisks" conference, organized by theRisques Financiers Chair in Paris from10 to 13 January 2011.Our main finding is as follows. Just asprice fluctuations are measured by vola-tility σ, volatility fluctuations can be mea-sured by the “volatility of volatility” ω. Inan entirely general way, in stochastic volatility models, we can calculate anexact limited smile expansion at order 2 in volatility of volatility ω. At order 2,the smile is described by a simple func-tional form:

where FT is the initial value of the forwardand the coefficients , and , whichrepresent respectively the implied spotprobability (i.e. for the value of the strikeprice equal to the price of the underlying),its slope and its curvature, are expressedsimply using the model’s parameters.Whichever stochastic volatility model is used, we can calculate these threenumbers and deduce the smile fromthem. The expression of the smile andthe expressions of the coefficients ,

and show that the smile of sto-chastic volatility models is structurallyconstrained.

For example:

■ The spot slope is generated by thecovariance between the price of theunderlying asset and its volatility.

■ The spot smile is convex if the price ofthe underlying asset and its volatilityare uncorrelated, but this is not neces-sarily true in the general case.

■ We can show, for short maturities,how the slope and curvature of thesmile depend on the implied spot volatility.

In particular, our work allow us to find thelimited development of the smile at order2 in volatility of volatility carried out byAlan Lewis [5] in the case of the Hestonmodel – more precisely, in the case of a parameterized family of Heston-typemodels. Our result is of course muchmore powerful, since we do not assumeany particular form for the stochastic volatility model. We get the developmentat order 2 in volatility of volatility of theBergomi model [1]. The exact calculationof the smile by Monte Carlo allowed usto verify that the development at order 2 is very accurate for a wide spectrumof volatilities of volatility ω and for a largerange of maturities and strike prices.

Figure 2 shows smiles obtained for typical values of volatility of volatility (ω = 200%) and of correlations betweenspot and volatilities for three maturities(1, 3 and 8 years):• first, the exact smile obtained using a

Monte-Carlo simulation (MC),• second, the smile obtained from our

new (order 2) formula.

The approximation is highly satisfactory.

Further reading

■ [1] Bergomi L., Smile Dynamics 2, Risk Magazine, pages 67-73, October 2005

■ [2] Hagan, P., Kumar, D., Lesniewski, A. et Woodward, D., Managing smile risk, Wilmott Magazine,pages 84-108, September 2002.

■ [3] Heston, S., A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bondand Currency Options, Review of Financial Studies, Oxford University Press for Society for FinancialStudies, 6(2):327-343, 1993.

■ [4] Hull, J. White, A., The pricing of options on assets with stochastic volatilities, Journal of Finance,42:281-300, 1987.

■ [5] Lewis A., Option valuation under stochastic volatility, Finance Press, 2000.

■ [6] Renault, E. and Touzi, N., Option pricing and implied vola tilities in a stochastic vola tility model,Mathematical Finance, 6:279-302, 1996.

Figure 2: Exact smile (MC) and our (order2) smile formula in the Bergomi model

In conclusion, although our results canbe used for parameter calibration purposes, we do not think it is alwaysadvisable to calibrate the parameters ofa stochastic volatility model on the smileit produces. In particular, if the model isused to evaluate exotic options whoserisks are orthogonal to those of conven-tional purchase options, such a calibrationwould only offer unfounded psychologicalwell-being and could have undesirableconsequences (mispricing). However, itfrequently happens that some well-chosenpurchase options can significantly reducethe risk of an exotic option. In this case,our formula may be used to choose themodel’s parameters so that it correctlyassesses the price of these hedging instruments. Finally let us insist on thefact that our work not only allows thesmile produced by stochastic volatilitymodels to be precisely quantified, butalso and especially allows the structuralconstraints of that smile to be specified.It thus has a twofold value, both quanti-tative and qualitative.



The “Modeling and Managing Financial

Risks” inter national confer ence has

been the main scientific event of the

first five years of the Financial Risk Chair. This

four-day event was organized in January 2011

by the CMAP financial mathematics team. The

conference presentations on various aspects

of financial risk took place in the amphitheatres

of the Campus de Cordeliers. Located on the

site of the old C ordeliers monastery, these

historical buildings are now owned by the Uni-

versity of Paris VI and are used for organizing

prestigious events.

The plenary lectures were given by distingui-

shed speakers invited by the scientific com-

mittee, and pr esentations from the parallel

sessions were selected from the many contri-

butions received by the organizing commit-

tee. The call for papers was circulated widely

in France and abroad and a special effort was

made to attract contributions fr om young

researchers.

Some figures:66 presentations in total, including 12 by practitio-ners, 25 by speakers from abroad, 9 by members ofthe Financial Risk Chair, and 21 in the plenary session.219 participants registered, including 53 practitionersand 87 from abroad.

Among the plenary session speakers duringthe first three days:

■ Ioannis Karatzas talked about models for large-scale equity markets. He is interested in particular inthe relationship between the size of a listed company,i.e. the proportion of its stock in the overall volume,and its performance, and the implications of this relationship on long-term portfolio optimization.

■ Emmanuel Gobet presented a method for calcu-lating option prices in the presence of dividends. Itdevelops highly accurate approximations for real timeevaluation.

■ Nicole El Karoui gave a three-session short courseon different aspects of the historical interaction between mathematics and financial markets. The twomain topics were on the one hand portfolio managementunder the constraint of drawdown and its link with thetheory of martingales and the other hand backwardstochastic differential equations and their applicationsfor risk measurement.

MODELING AND MANAGING FINANCIAL RISKS a Financial Risk Chair Conference

Peter Tankov CMAP-Ecole Polytechnique



■ Xunyu Zhou spoke about optimal timing for buyingor selling an asset, in the case where the buyer/seller is not completely rational but underestimates or overes-timates certain probabilities.

■ Paul Embrechts presented his thinking on the role offinancial engineering in financial crises and their prevention.Emphasis was placed in particular on the social role offinancial mathematics in managing pensions.

■ Jean-Pierre Fouque presented new approximationsfor stochastic volatility models, i.e. models in which thevariability of the assets may itself change randomly fromone day to another.

■ Walter Schachermayer introduced innovative workon multidimensional measures of risk. The aim is to beable aggregate various types of risk into a single measure.

■ Nizar Touzi presentation focussed on price boundsin the presence of calibration constraints. This involvescalculating all possible values of an asset without ma-king any assumptions about its behaviour, but usingonly the prices of other assets available on the market.

The last day was devoted specifically to the interactionbetween academics and practitioners, with presentationsby Bruno Dupire, Julien Guyon, Lor enzo Bergomi,Salah Amraoui, Alex Lipton, Jean-Philippe Bouchaud,Charles-Albert Lehalle, Rama Cont and FrédéricAbergel. There was also extensive participation bypractitioners throughout the four days of the conference.

The afternoon of the second day of the conference wasan opportunity to celebrate the 20th anniversary of the"Probability and Finance" Master’s. Many former Master’sstudents attended this event with presentations by Helyette Geman, Nicole El Karoui, Shige Peng, GillesPagès, Jean Jacod and Bruno Dupire. It was followedby cocktails.

Overall, the conference was a great success, both interms of the quality of presentations and in terms of theinteraction between academic communities and prac-titioners. The conference programme together with thepresentations is available on the conference website atwww.cmap.polytechnique.fr/f inancialr isks/conference2011/

BIOGRAPHY

Peter Tankov is a lecturer and resear-cher in financial mathematics at the EcolePolytechnique. The author of “FinancialModeling with Jump Processes”, his research focuses on modelling jump riskin finance and on stochastic processes

with jumps. Together with C. Hillairetand J. Guyon, he was one of the orga-nizers of the “Modeling and ManagingFinancial Risks” conference.


BIOGRAPHY

Charles-Albert Lehalle

Charles-Albert Lehalle heads the

Credit Agricole Cheuvreux quantitative

research team responsible for optimization

of negotiation in equity markets. He

has published numerous articles on the

optimal control of trading, estimation

of market impact and measurement

of the effectiveness of a negotiation

process. He teaches the Quantitative

Trading course in the Masters in

Probability and Finance at the

University of Paris VI and [email protected]

In order to optimize this balance – tra-ding slowly so as to avoid too muchmarket impact, trading quickly so as toavoid excessive exposure to market risk–, it is first necessary to model the natureof market impact, and hence to studythe microstructure of markets. It is alsoimportant to correctly measure the riskinvolved in waiting (the role of random-ness in the price formation process) andtherefore to make use of the findings of processes statistics. Once these twostages are complete, one can clearly articulate an optimization problem corresponding to the negotiation proble-matic and then solve it.

The microstructure of markets The microstructure of markets is com-monly understood as the set of rules and behaviour that structure the biddingprocedures established for price forma-tion. In terms of markets driven by orders, there are two main types of bidding procedure: call auctions, whichusually begin and close trading days,and continuous auctions, which applyduring the rest of the trading day.

Call auctions are very similar to regularauctions of works of art, except that thesellers participate as well as the buyers.This is made possible through the fungi-bility of the shares of a listed company.

Quantitative tradingRATIONALIZATION OF NEGOTIATION IN THE MARKETS

By Charles-Albert Lehalle, Crédit Agricole Cheuvreux


KEY POINTS

■ Optimal trading is a recent field which establishes a link between an investment strategy in the broad sense and the state of supply and demand in an electronic market. In this way it provides a betterunderstanding of one type of hitherto little explored risk by combiningthe findings of applied mathematics and quantitative finance with cexpertise in modelling discrete processes on large samples.

■ C.-A. Lehalle and his colleagues are developing powerful digital toolsfor optimal trading and for geographical optimization in placing orders.

Quantitative trading brings together techniques allowing the implementa-tion of market investment decisions to be streamlined. These techniqueswere created around the purchase and sale of large quantities of sharesin electronic markets. If one buys or sells too quickly , the price will beaffected (this effect is known as "market impact"), whereas if one buysor sells very slowly to avoid disrupting the price formation process, oneruns the risk that the price changes during the implementation of thedecision, thus making it obsolete.




■ The optimization of interactions with the orderbooks of electronic markets, even though infinitesimal in terms of implementa tion time (deciding to inter vene in seconds), is a genuine source of economies for much ofthe business of financial market actors suchas market makers and fund managers.

During the “auctioning” stage, buyersand sellers declare their interests simul-taneously in the form of quantities andprices. The sum of buy and sell offersform two cumulated order books. At thetime of the “call” an equilibrium price iscalculated which corresponds to the intersection of the two order books.

Continuous auctions are conductedaccording to a different principle: the cumulated order books do not overlapand the matching of buy and sell ordersare implemented continuously. As soonas a buy order is declared at a higherprice than a sell order already present in the order book (or a seller declares a price lower than that of an already present buyer), a trade is generatedwhich “cleans” the book in real time. For continuous auctions, it is useful todistinguish between liquidity providers,who are present in the order book, and liquidity consumers, whose ordersgenerate immediate trades by meetingthe orders of liquidity providers. Liquidityproviders are patient, willing to wait before engaging in a transaction; liquidityconsumers are impatient and are willingto “pay for” the supply-demand spreadin return for the certainty of a transaction.

The price formation process thereforecomes from the concatenation of calland continuous auctions, and arisesfrom the dynamics of order books. Therecent increase (uninterrupted in Europesince 2007) in the frequency of orderchanges by the various participants is currently leading to most types of negotiation being conducted in electronichigh frequency trading markets, incontrast to the situation a few years agowhen orders were submitted by phone.In 2011, a CAC 40-listed tradable assetgenerates about 50,000 transactions a day, conveyed by some 600,000 movements in the ten top prices in theorder book.

The modeling of the arrival processes oforders according to their price, quantity,state of order books and the recent pasthas been the subject of many studies,though without any definitive conclusionemerging. Because the rules of the various order books available and thebehaviour of the different actors areconstantly changing, it is difficult to drawconclusions as to a “best” model of theprice formation process. A number ofpromising proposals have nonethelessemerged, including point processes(Bacry et al. 2011), models of price mo-vements immediately after a transactionfollowed by an order book relaxation pe-riod (Bouchaud, Mezard and Potters,2002 and Gatheral, 2010), and a jointmodelling of order volumes and quotedprices (Cont and De Larrard 2011).

The negotiation problematicOnce the problem is properly stated in terms of bidding procedures, the problematic of the purchase or sale of alarge number of shares clearly emerges.

The first part of the rationalization of trading involves controlling risk. Severalequivalent formal frameworks were putforward in the late 1990s by various academics (Almgren, Chriss 2000). Theprinciple is that of specifying a criterionto be minimized: this was initially amean-variance equilibrium that can reflect the desire to act slowly (in orderto minimize the market impact cost,which is on the side of the mean) and the desire to act quickly (in order to minimize the variance of prices duringthe trading period). Today more sophis-ticated techniques are commonly used(Bouchard, Dang, Lehalle 2011). Theseenable one to determine, though the numerical optimization of “optimal tradingcurves” (see Figure 1) that govern thenumber of shares, whether it is wise tobuy or sell in order not to be too heavilyexposed to various risks.

It is then a matter of actually buying or selling shares by participating in the auction procedure. This is done by meansof trading robots, also called “liquidityseekers”. In this context one should notneglect the importance of optimizing thechoice of order book with which the trading robot will interact. Indeed for thesame listed share, auction procedurestake place in parallel in traditional markets(NYSE or Euronext for French shares)and alternative exchanges such as BATS,Chi-X or Turquoise, as well as in “DarkPools” (anonymous order books). Geo-graphical optimization of the placing oforders is supported by what is generallycalled a “Smart Order Router” (SOR).Proposals to formalize these problematicshave recently been made by Pagès, Laruelle, Lehalle (2009).

Further reading

■ (Bacry et al. 2011) Modeling microstruc-ture noise with mutually exciting point processes, by E. Bacry, S. Delattre, M. Hoff-mann, J. F. Muzy (18 Jan 2011), pre-publi-cation.

■ (Bouchaud, Mezard, Potters, 2002)Statistical properties of stock order books:empirical results and models, by J. P . Bouchaud, M. Mezard, M. Potters (18 Jun2002); pre-publication.

■ (Gatheral, 2010) No-Dynamic-Arbitrage andMarket Impact, by Jim Ga theral; SocialScience Research Network Working PaperSeries (31 October 2008).

■ (Cont, Larrard 2011). Price Dynamics in aMarkovian Limit Order Book Market, byRama Cont, Adrien De Larrard; SocialScience Research Network Working PaperSeries (7 January 2011).

■ (Bouchard, Dang, Lehalle 2011) Optimalcontrol of trading algorithms: a general impulse control a pproach, by Bruno Bouchard, Ngoc-Minh Dang, Charles-AlbertLehalle, SIAM J. F inancial Ma thematics(forthcoming 2011).

■ (Pagès, Laruelle, Lehalle 2009) Optimalsplit of orders across liquidity pools: a stochastic algorithm a pproach, by GillesPagès, Sophie Laruelle, Charles-Albert Lehalle (2009); pre-publication.

■ (Almgren, Chriss 2000) Optimal executionof portfolio transactions, by R. F. Almgren,N. Chriss in Journal of Risk, Vol. 3, No. 2.(2000), pp. 5-39.

Figure 1: A real example of the implementation of a trading algorithm. At the top, market prices (values of the

“best bid” and “best ask” in green and blue, coloured dotsfor transactions obtained by the algorithm) on which aresuperimposed volume weighted average prices (VWAP)

of the market (black) and the algorithm (grey). Below, low volumes traded 100% rebased in late trading: black

for the market, grey for the algorithm, and colour for the different limits calculated by a trading optimizer.


BIOGRAPHY

Nizar Touzi

Nizar Touzi is professor of applied

mathematics at the Ecole

Polytechnique. His research interests

include financial mathematics,

stochastic control and Monte Carlo

[email protected]

Pierre Henry-Labordère

Pierre Henry-Labordère is a researcher

in the Société Générale quantitative

research department. [email protected]

Methodology

We assume that the underlying asset isavailable for exchange in continuoustime with no constraints on trading volumes. We also assume a zero interestrate context. It is well known that the absence of arbitrage implies that theprice process of the underlying asset isa martingale under a certain measure ofprobability, commonly called risk neutralprobability.

The investor also has access to a European options market and can tradeEuropean options with maturity T for allpositive prices during the fiscal year. Theno-arbitrage assumption together with

the linearity of the functional assessment

then allows us to identify the marginal

distribution of the price of the underlying

asset at time T under risk neutral proba-

bility. This possibility was noted some

time ago by Breeden and Litzenberger.

With the two preceding assumptions –

the linearity of the evaluation function

and the absence of arbitrage in the

market for the underlying asset and in

the European call options market – we

thus have two conditions applying to

the underlying asset: under risk neutral

probability, it is a marginal distribution

martingale at a given T. The question ad-

dressed in our study is to find the bounds

No-arbitrage bounds FOR LOOKBACK OPTIONS

By Pierre Henry-Labordère, Société Générale, and Nizar Touzi, Ecole Polytechnique, CMAP


KEY POINTS

■ We provide a general method for seeking bounds on derivativesprices, given the market information on a specific date.

■ These bounds should be robust in the sense that they do not dependon a particular modelling choice, but are based solely on the singlebasic law of market finance, namely the concept of no-arbitrage.

■ However, we make an additional assumption of linearity of the functionalevaluation and of the continuity of the price process of the underlyingasset.

We will focus on the evaluation at the date of origin of a derivative writtenon a single underlying of a pay-off, payable at maturity T, depending onthe overall trajectory of the underlying asset.



R e c o m m a n d a t i o n sfor the actors concerned

■ These bounds are ver y useful for actors operating in financial markets since theyallow them to directly test whether a givenprice is consistent with all a vailable prices in the market at a given date.

■ The price inter val provided by the methodalso allows banks to quantify the model riskassociated with an exotic product.

on the price of an exotic option for thisunderlying asset without any other as-sumptions in the model.

We express these bounds as robusthedging problems, i.e. the hedge shouldensure risk protection, whatever the underlying model. We show that thesehedging problems allow a dual formula-tion which is more suited to numerical or analytical calculation methods andwhich treats our problem both.

■ as an optimal transportation problem,of a different kind to the problemsstudiedin the traditional literature because of themartingale condition imposed on us bythe absence of arbitrage,

■ and as the well-known Skorohod embedding problem, which has generatedextensive literature in the probabilitycommunity.

David Hobson had noticed the relationshipwith Skorohod’s embedding problemback in 1998. In particular, Hobson andKlimmek (2011) have shown that for aclass of Lookback options, the so-calledAzéma and Yor solution to Skorohod’sembedding problem gives the optimalupper bound. Out approach also allowsthis explicit result to be obtained.

A natural extension of our problem is toassume that the investor has access tothe marginal law of the underlying assetprice on several dates before maturity. Inparticular, the case of two marginals andone Lookback option depending only onthe maximum was fully solved by Brown,Hobson and Rogers (2001). For caseswith several marginals, Madan and Yor(2002) gave an explicit solution under a monotonicity condition for related barycentric functions. With our approach,we are able to obtain a solution to theproblem in the general case, avoidingMadan and Yor’s condition.

Conclusion

Our approach offers a general methodo-logy for the formulation of the problem of no-arbitrage bounds. These boundsare very useful for the practice of financial markets as they can directly test whether a given price is consistentwith all the prices available on the marketat a given date. It is clear that in generalthese bounds give a sufficiently wide acceptable price interval. However, theinterval size decreases with the amountof information made available to the investor.

Several extensions to our work may beenvisaged. First, we should not expectto obtain explicit formulas for generalexamples of structured products. Theimportant thing is to come up with effective numerical methods for approxi-mating the solution of our problem. Thistask is partially addressed in Xiaolu Tan’sthesis work at the Ecole Polytechnique.

Finally, not all explicit calculation proce-dures have been exploited, even in the context of very specific Lookbackoptions. What happens for a generalpayoff that does not conform to Hobsonand Klimek’s conditions? What happensif all the marginals are known? This lattercondition could lead to a simplifiedcontext thanks to the power of differentialcalculus for continuous variables.

The question addressed in our study is to findthe bounds on the price of an exotic option for thisunderlying asset without any other assumptions inthe model.

“”

Further reading

■ [1] Galichon, A., Henr y-Labordère P. andTouzi, N. (2011), A stochastic controlapproach to no-arbitrage bounds givenmarginals, a pplication to Lookback options. Preprint.

■ [2] Henry-Labordère P. and Touzi, N. (2011),No arbitra ge bounds for Lookback options given multiple marginals. Preprint.

A n e x a m p l e . We used our analytic results to determinethe optimal upper bound of an option call written on the Eurostock maximumbetween two dates t1=1 year and t2=2years for different strikes. By way of com-parison, we evaluated this product inMonte Carlo with a local volatility modelintroduced by Dupire (1993).

StrikeBorneHaute

Vol.Locale

80 41,24 33,15

90 32,10 24,78

100 23,49 17,17

110 15,77 10,73

120 9,40 5,86

130 4,83 2,75



Nicole El Karoui is a researcher

at the Ecole Polytechnique’s

Center of Applied Mathematics

(CMAP) and a mathematics

professor at the University of

Paris VII’s (Univ.Pierre & Marie

Curie) Laboratory of Probabilities

and Random Models (PMA).

She is in charge of the latter

institution’s Master’s 2 program,

“Probabilities and Finance”,

a joint degree with the Ecole

Polytechnique. She is scientific

director of the Institut Louis

Bachelier, director of the financial

risk research chairs for the

Fondation du Risque (FdR) as

well as of the future derivatives

research chairs co-sponsored by

the Federation of French Banking.

BIOGRAPHY

The teaching of financial mathematics is at a very high levelin France and young graduates in this discipline ar e highlysought after in the job market. With its 900 graduates andtwenty years of existence, the “Pr obability and Finance"Master’s (UPMC/EP) has played a decisive r ole in this development, as attested by an article in the W all StreetJournal in 2006.

Key figures

The Master’s has so far trained900 graduates, most of themscience students and halffrom the Grandes Ecoles. Ahigh proportion of those taking

the Master’s are foreign students.“Quants” (quantitative engineers)make up the leading occupationafter training, followed some consi-derable way behind by trading, risk-management and hedge funds.At least a third of our graduates currently work in the four majorFrench banks, and another third in foreign banks in London or Japan.For the past five years, more thanhalf of our graduates’ first jobs havebeen in London.

Key dates In the late 1980s, the first financial futures markets opened in Paris(MATIF in 1986 and MONEP in 1987).Antoine Paye set up the options products business at the Société Générale. The NYSE’s Black Monday(October 1987) was on everyone’smind.

In 1990, Nicole El Karoui (Professor at Paris VI) and Helyette Geman (Professor at ESSEC) created theProbabilities and Finance option within the Probabilities DEA at Paris VIUniversity (Director Jean Jacod). The DEA was co-accredited with the Ecole Polytechnique, lENPC andESSEC. It was the first training of itskind in a science environment.

20 years of training,900 students, 2 major crises

THE

MAS

TER

PROBABILITIES AND FINANCE MASTER’

S

CELEBRATES ITS 20TH ANNIVERSARY



The aim was to train quants, who understood and used the methodo-logies developed in the United Statesover the previous fifteen years, particularly in the area of interestrates, for which there was a strongdemand. The training scheme thatwas set up is still relevant today, as isexplained below.

1990-2000

From modest beginnings – six gra-duates the first year – the Master’sgrew rapidly in response to conside-rable demand. There were two specificchallenges. One was to draw attentionto the course outside of trading rooms,especially with Human Resourcesdepartments. The other difficulty wasthat the Anglo-Saxon banks in Londondid not usually take trainees on along-term basis, partly because ofthe issue of professional secrecy.Here it was a matter of waiting until ahigher proportion of managers wereFrench.

It was a very exciting time for both ourstudents and the teams in the banks,because everything was happeningat incredible speed – broadly spea-king, that of the calculating power ofcomputers. Tools that seemed im-possible to use were fully integrated

within three years. There was no pro-blem finding jobs, wages were highbut still reasonable, and quant bo-nuses were far from exorbitant.

At the same time, academic researchwas actively being developed, oftenin association with banks. Marc Yor,Professor at Paris VI and a memberof the Academy of Sciences, playeda key role, along with other teacherson the Master’s. A new research discipline, “financial mathematics”,was needed in Europe in particular.Mathematicians did not always viewit in a favourable light.

A number of crisis – LTCM, the Asiancrisis, Enron and Barings, amongothers – were occasionally remindersthat these markets can lead to veryheavy losses and that there is alwaysa temptation to operate with huge leverage. 1998 saw the regulator require that a daily Value at Risk beproduced on market risks, i.e. on theaggregate activity of a trading floor.Even though computers had becomemore powerful and decreased incost, this was a real challenge. It wasa major task for financial institutions,as well as for the educational esta-blishments that produced the quantswho would have to implement thisnew measure. Academic exchangeswith the market were intensifying, especially around the appropriatenessof VaR as a measure of risk.

Students from the Grandes Ecoleswere becoming more numerous inour training programme, as well as in the programmes created at Paris VII,Dauphine, Paris I and then Marne laVallée and Evry and in the past fewyears at the Grandes Ecoles them-selves.

Between 1995 and 2000, Nicole ElKaroui left the Ecole Polytechnique

and set up a Financial Mathematicsresearch team, though remaining veryinvolved in the training under theEcole Polytechnique; Helyette Gemantook over as head of Paris Dauphine’sDEA 203 and left the training pro-gramme; and Gilles Pagès arrived atParis VI, where he developed nume-rical probabilities and participated in the management of the Master’s,together with Marc Yor.

2000-2008

The coming of the millennium sawthe bursting of the dot.com bubble,which disrupted our student intakefor a year and a half (2001-02) butnot the demand for training. Mergersbetween banks were becoming morefrequent. The first credit derivativeswere being introduced, and manycourses started focussing on modellingthem. The academic community hadits reservations about the models putforward.

The “El Karoui DEA” proved popularand our students began to acquire international exposure, culminating in2006 with the investigative article inthe Wall Street Journal on the 30% of French quants in the financial markets, followed a little later by apiece in Le Monde. We had createdanother luxury product for export,with bonuses having reached astro-nomical levels, especially for traders.The French approach to training “engineers”, with a strong backgroundin mathematics and a great capacityfor adaptation, supplemented by high-level academic training, was particu-larly well suited to professions of thiskind.



2008-2010

The bankruptcy of Lehman Brothersin September 2008 badly affectedthe Master’s. Financial mathematicswas stigmatized and held responsiblefor all sorts of evils. Even though themethodological criticisms were oftenvery superficial, the more interestingquestion of the role and responsibilityof scientists in the real world wasperspicacious, particularly as financeis far from having the supposed “neutrality” of other more traditionaldisciplines.

The subprime crisis had little affect onthe derivatives business apart fromcredit. Applications for the Master’s(for the 2009 academic year) reacheda very high level, and despite ourstringent selection we accepted alarge number of candidates. 2009 wasa difficult year for finding traineeships,and was hardly easier for first jobs.This proved particularly hard for foreignstudents, for whom the issue of “papers” was problematic yet crucial.

The crisis gave a very negative imageof financial markets, which led toquestioning by a number of students.Moreover, the crisis in Greece showedthat considerable uncertainty still remained. Under regulation injunc-tions, control measures in institutionswere being strengthened, and quan-titative engineers were hired for thispurpose. The situation in London wasvery different, since many teams hadbeen laid off as a result of the crisis.As the economy recovered, teamswere re-formed and our studentswere in great demand. Over the pasttwo years, the major London banks

(both Anglo-Saxon and others) haverecruited many more of our studentsthan French banks.

For several years, banks and hedgefunds have been developing high-frequency trading on their own accountor on behalf of others. Given the scaleof demand, we have oriented an optional part of our training in this direction, linking it to a wider debatearound risk. Although this area repre-sents only a part of our curriculum, it is something we devote muchthought to.

Students

The number of students coming fromGrandes Ecoles has greatly increasedin recent years, largely through mediacoverage of the Master’s. At theEcole Polytechnique, since the arrivalof Nicole El Karoui in 1998, trainingin financial mathematics has beenreorganized, and the research teamhas been strengthened and becomehighly visible. As a result, Polytech-nique students (including foreigners)are strongly represented in our training programme – somewhat at

the expense of University students,who have been slightly put off by theirstrong presence. The number ofwomen students is extremely low;and their focus is most often oncontrol organizations. There are alarge number of foreign students,more than 50% of the total, eventhough the teaching is always inFrench. Some of these students follow the joint Master’s organized by Fudan University and the EcolePolytechnique. On their return to China,and recruited by U.S. banks, they arebeginning to build a future financial industry, since all the signs are thatthe market is shifting, faster than expected, to Asia and particularlyChina. The hope is that French bankswill also benefit from this.

The curriculum

The core curriculum of the originalMaster’s has changed little in termsof its principles, which have remainedbasically the same for the last twentyyears, except for increased emphasison the teaching of regulation and risk management. Firstly, there is anintroduction to finance for all stu-dents, essential for understanding theworld of economics, business, and financial markets. Above all, however,we offer specialized finance marketteaching, in the field of financial deri-vatives. This sector has a strong applied mathematics component,both in terms of probability and in calculation methods: the problemsconcern modelling with a view to calculating the price of the productsold and hedging the risks it generates,resulting in real-time calculations, but



calling for little by way of financialanalysis. Following the crisis, we areheightening students’ awareness ofrisk issues within a broader frameworkthrough courses on risks given jointlyby academics and professionals, aswell as through a specific course onregulation.

One of the keys to the success of thismaster is undoubtedly the spirit inwhich it was conceived by its founders– Nicole El Karoui and HelyetteGeman – who had previously justspent a year in a bank dissecting andexplaining to practitioners the firststochastic models of interest rates.This experience clearly showed thatthese could be not simply a collectionof current models – Black-Scholes,Vasicek, HJM, etc. – in the style ofPrévert , but very much a fundamentalknowledge base to provide studentswith the wherewithal for an autono-mous future, essential for their evolutionin a highly dynamic industry constantlyseeking innovations and new territo-ries. This is also indispensable forwithstanding the pressure of busi-ness ("the market is always right")when necessary.

Most of the theoretical teaching isprovided in a relatively traditional way(hypotheses, theorems, proofs, etc.),and the core of the training consistsof courses in stochastic calculus, numerical simulations and statisticsand their use in relation to derivatives.The optional courses are very flexible

(e.g. credit calibration, U.S. options in commodity markets and climatederivatives, insurance products for highfrequency management, etc.), andenable students to choose options tofit their personal requirements. Here,academics and professionals are involved on a roughly 50-50 basis,since we aim always to keep abreastof the market issues emerging fromcurrent models and future topics, something that is crucial for trainingsuch as ours.

In the second stage of the pro-gramme, it is important that each student works in a bank research unit(or comparable unit at a specializedsoftware publisher, energy company,etc.) in order to apply the spectrumof tools acquired in the first semesterin an operational framework – “in acombat situation”, to use the martialmetaphor of an equity research manager of a large bank. This is theonly way to measure the real-timecomplexity of markets, their interactions,liquidity, operational and regulatoryconstraints, and many other realities ;and is a prerequisite for becoming agood quant capable of developingmodels that lead to appropriate strategies for hedging derivativesmarket risks.

The dynamism of the market is reflec-ted in the ever higher level of trainingrequirements, across an increasinglywide spectrum of knowledge. Wealso offer the opportunity for students

wishing to spend a term bringing together all this knowledge by conduc-ting case studies jointly supervised byan academic and a professional, priorto their traineeship in a bank.

The issue of risk is central to the training, since it is inherently the corebusiness of the quant engineer, andalso because it is not possible to treatrisks separately. Training technicallycompetent students, who are awareof the structural risks in financial markets as well as of their “social”responsibility, is the challenge thatour Master’s programme aims tomeet.

The issue of risk is central to the training, since it is inherently the core business of the quant engineer.

“”

MASTER’S SUPERVISORYPERSONNEL

For Pierre et Marie Curie University

■ Nicole El Karoui

■ Gilles Pagès

■ Marc Yor

For Ecole Polytechnique

■ Emmanuel Gobet

■ Nizar Touzi


4th EUROPEAN SUMMER SCHOOL IN FINANCIAL MATHEMATICS

ETH Zürich & École PolytechniqueZürich, 5-9 September 2011

An EMS Applied Mathematics school

The European Summer School in Financial Mathematics aims at br inging together themost talented young researchers in the field, looking at the v ery young who only juststarted their PhD studies . The Scientific Committee consists of European leaders andrepresentatives of financial mathematics. We warmly thank them for their encouragementand for accepting to be part of this committee. Their first and main task is to identify themost promising candidates. The successful candidates will be sponsored for their traveland living expenses during the summer school.

The fourth European Summer School is held at ETH Zurich for the first time. We gratefullyacknowledge the suppor t of the F rench Federation of Banks (Fédér ation Bancaire Française) and the ETH Foundation.

The Summer School is centred around one or tw o advanced courses taught by widely recognised experts. We are par ticularly interested in a w ell-balanced combination of theoretical and practical input. We welcome suggestions for topics to cover in the courses.There will also be some students' seminars where the y oung par ticipants can get teaching experience and discuss their research.

We hope that the Summer School leads to an activ e cooperation between the various European institutions. We also hope that the Summer School provides an opportunity toopen doors for European students amongst different research centres. Our ambition isalso to put the foundations for a more visible European network at the doctoral level inthe field of financial mathematics. We very much count on the members of the ScientificCommittee to help us make this wish a reality.

This school belongs to the series of the EMS Applied Mathematics schools.

Contact: [email protected]

CMAP UMR 7641 École Polytechnique CNRS, Route de Saclay, 91128 Palaiseau Cedex France Tel : +33 1 69 33 46 00 Fax : +33 1 69 33 46 46

ETH Zürich, Rämistrasse 101, 8092 Zürich Switzerland Tel : +41 44 632 64 65 Fax : +41 44 632 14 74


Documents

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