Imperial College London


Faculty of Natural SciencesDepartment of Mathematics

Chair in Mathematical Finance







805Weeks BuildingSouth Kensington Campus





Publication Type

139 results found

Brigo D, Pistone G, Projection based dimensionality reduction for measure valued evolution equations in statistical manifolds

We propose a dimensionality reduction method for infinite-dimensionalmeasure-valued evolution equations such as the Fokker-Planck partialdifferential equation or the Kushner-Stratonovich resp. Duncan-Mortensen-Zakaistochastic partial differential equations of nonlinear filtering, withpotential applications to signal processing, quantitative finance, heat flowsand quantum theory among many other areas. Our method is based on theprojection coming from a duality argument built in the exponential statisticalmanifold structure developed by G. Pistone and co-authors. The choice of thefinite dimensional manifold on which one should project the infinitedimensional equation is crucial, and we propose finite dimensional exponentialand mixture families. This same problem had been studied, especially in thecontext of nonlinear filtering, by D. Brigo and co-authors but the $L^2$structure on the space of square roots of densities or of densities themselveswas used, without taking an infinite dimensional manifold environment space forthe equation to be projected. Here we re-examine such works from theexponential statistical manifold point of view, which allows for a deepergeometric understanding of the manifold structures at play. We also show thatthe projection in the exponential manifold structure is consistent with theFisher Rao metric and, in case of finite dimensional exponential families, withthe assumed density approximation. Further, we show that if the sufficientstatistics of the finite dimensional exponential family are chosen among theeigenfunctions of the backward diffusion operator then the statistical-manifoldor Fisher-Rao projection provides the maximum likelihood estimator for theFokker Planck equation solution. We finally try to clarify how the finitedimensional and infinite dimensional terminology for exponential and mixturespaces are related.

Journal article

Brigo D, Pede N, Petrelli A, Multi Currency Credit Default Swaps Quanto effects and FX devaluation jumps

Credit Default Swaps (CDS) on a reference entity may be traded in multiplecurrencies, in that protection upon default may be offered either in thedomestic currency where the entity resides, or in a more liquid and globalforeign currency. In this situation currency fluctuations clearly introduce asource of risk on CDS spreads. For emerging markets, but in some cases even inwell developed markets, the risk of dramatic Foreign Exchange (FX) ratedevaluation in conjunction with default events is relevant. We address thisissue by proposing and implementing a model that considers the risk of foreigncurrency devaluation that is synchronous with default of the reference entity. Preliminary results indicate that perceived risks of devaluation can induce asignificant basis across domestic and foreign CDS quotes. For the Republic ofItaly, a USD CDS spread quote of 440 bps can translate into a EUR quote of 350bps in the middle of the Euro-debt crisis in the first week of May 2012. Morerecently, from June 2013, the basis spreads between the EUR quotes and the USDquotes are in the range around 40 bps. We explain in detail the sources for such discrepancies. Our modelingapproach is based on the reduced form framework for credit risk, where thedefault time is modeled in a Cox process setting with explicit diffusiondynamics for default intensity/hazard rate and exponential jump to default. Forthe FX part, we include an explicit default-driven jump in the FX dynamics. Asour results show, such a mechanism provides a further and more effective way tomodel credit / FX dependency than the instantaneous correlation that can beimposed among the driving Brownian motions of default intensity and FX rates,as it is not possible to explain the observed basis spreads during theEuro-debt crisis by using the latter mechanism alone.

Journal article

Brigo D, Durand C, An initial approach to Risk Management of Funding Costs

In this note we sketch an initial tentative approach to funding costsanalysis and management for contracts with bilateral counterparty risk in asimplified setting. We depart from the existing literature by analyzing theissue of funding costs and benefits under the assumption that the associatedrisks cannot be hedged properly. We also model the treasury funding spread bymeans of a stochastic Weighted Cost of Funding Spread (WCFS) which helpsdescribing more realistic financing policies of a financial institution. Weelaborate on some limitations in replication-based Funding / Credit ValuationAdjustments we worked on ourselves in the past, namely CVA, DVA, FVA andrelated quantities as generally discussed in the industry. We advocate as adifferent possibility, when replication is not possible, the analysis of thefunding profit and loss distribution and explain how long term funding spreads,wrong way risk and systemic risk are generally overlooked in most of thecurrent literature on risk measurement of funding costs. As a matter of initialillustration, we discuss in detail the funding management of interest rateswaps with bilateral counterparty risk in the simplified setup of our frameworkthrough numerical examples and via a few simplified assumptions.

Journal article

Brigo D, Rapisarda F, Sridi A, The arbitrage-free Multivariate Mixture Dynamics Model: Consistent single-assets and index volatility smiles

We introduce a multivariate diffusion model that is able to price derivativesecurities featuring multiple underlying assets. Each asset volatility smile ismodeled according to a density-mixture dynamical model while the same propertyholds for the multivariate process of all assets, whose density is a mixture ofmultivariate basic densities. This allows to reconcile single name andindex/basket volatility smiles in a consistent framework. Our approach could bedubbed a multidimensional local volatility approach with vector-state dependentdiffusion matrix. The model is quite tractable, leading to a complete marketand not requiring Fourier techniques for calibration and dependence measures,contrary to multivariate stochastic volatility models such as Wishart. We proveexistence and uniqueness of solutions for the model stochastic differentialequations, provide formulas for a number of basket options, and analyze thedependence structure of the model in detail by deriving a number of results oncovariances, its copula function and rank correlation measures andvolatilities-assets correlations. A comparison with sampling simply-correlatedsuitably discretized one-dimensional mixture dynamical paths is made, both interms of option pricing and of dependence, and first order expansionrelationships between the two models' local covariances are derived. We alsoshow existence of a multivariate uncertain volatility model of which ourmultivariate local volatilities model is a Markovian projection, highlightingthat the projected model is smoother and avoids a number of drawbacks of theuncertain volatility version. We also show a consistency result where theMarkovian projection of a geometric basket in the multivariate model is aunivariate mixture dynamics model. A few numerical examples on basket andspread options pricing conclude the paper.

Journal article

Sarais G, Brigo D, Inflation securities valuation with macroeconomic-based no-arbitrage dynamics

We develop a model to price inflation and interest rates derivatives usingcontinuous-time dynamics that have some links with macroeconomic monetary DSGEmodels equipped with a Taylor rule: in particular, the reaction function of thecentral bank, the bond market liquidity, inflation and growth expectations playan important role. The model can explain the effects of non-standard monetarypolicies (like quantitative easing or its tapering) and shed light on howcentral bank policy can affect the value of inflation and interest ratesderivatives. The model is built under standard no-arbitrage assumptions. Interestingly,the model yields short rate dynamics that are consistent with a time-varyingHull-White model, therefore making the calibration to the nominal interestcurve and options straightforward. Further, we obtain closed forms for bothzero-coupon and year-on-year inflation swap and options. The calibrationstrategy we propose is fully separable, which means that the calibration can becarried out in subsequent simple steps that do not require heavy computation. Amarket calibration example is provided. The advantages of such structural inflation modelling become apparent whenone starts doing risk analysis on an inflation derivatives book: because themodel explicitly takes into account economic variables, a trader can easilyassess the impact of a change in central bank policy on a complex book of fixedincome instruments, which is normally not straightforward if one is usingstandard inflation pricing models.

Journal article

Brigo D, Graziano GD, Optimal execution comparison across risks and dynamics, with solutions for displaced diffusions

We solve a version of the optimal trade execution problem when the mid assetprice follows a displaced diffusion. Optimal strategies in the adapted classunder various risk criteria, namely value-at-risk, expected shortfall and a newcriterion called "squared asset expectation" (SAE), related to a version of thecost variance measure, are derived and compared. It is well known thatdisplaced diffusions (DD) exhibit dynamics which are in-between arithmeticBrownian motions (ABM) and geometric Brownian motions (GBM) depending of thechoice of the shift parameter. Furthermore, DD allows for changes in thesupport of the mid asset price distribution, allowing one to include a minimumpermitted value for the mid price, either positive or negative. We study thedependence of the optimal solution on the choice of the risk aversioncriterion. Optimal solutions across criteria and asset dynamics are comparablealthough differences are not negligible for high levels of risk aversion andlow market impact assets. This is illustrated with numerical examples.

Journal article

Brigo D, Mai J-F, Scherer M, Consistent iterated simulation of multi-variate default times: a Markovian indicators characterization

We investigate under which conditions a single simulation of joint defaulttimes at a final time horizon can be decomposed into a set of simulations ofjoint defaults on subsequent adjacent sub-periods leading to that finalhorizon. Besides the theoretical interest, this is also a practical problem aspart of the industry has been working under the misleading assumption that thetwo approaches are equivalent for practical purposes. As a reasonable trade-offbetween realistic stylized facts, practical demands, and mathematicaltractability, we propose models leading to a Markovian multi-variatesurvival--indicator process, and we investigate two instances of static modelsfor the vector of default times from the statistical literature that fall intothis class. On the one hand, the "looping default" case is known to be equippedwith this property, and we point out that it coincides with the classical"Freund distribution" in the bivariate case. On the other hand, if allsub-vectors of the survival indicator process are Markovian, this constitutes anew characterization of the Marshall--Olkin distribution, and hence ofmulti-variate lack-of-memory. A paramount property of the resulting model isstability of the type of multi-variate distribution with respect to eliminationor insertion of a new marginal component with marginal distribution from thesame family. The practical implications of this "nested margining" property areenormous. To implement this distribution we present an efficient and unbiasedsimulation algorithm based on the L\'evy-frailty construction. We highlightdifferent pitfalls in the simulation of dependent default times and examine,within a numerical case study, the effect of inadequate simulation practices.

Journal article

Brigo D, Garcia J, Pede N, CoCo Bonds Valuation with Equity- and Credit-Calibrated First Passage Structural Models

After the beginning of the credit and liquidity crisis, financialinstitutions have been considering creating a convertible-bond type contractfocusing on Capital. Under the terms of this contract, a bond is converted intoequity if the authorities deem the institution to be under-capitalized. Thispaper discusses this Contingent Capital (or Coco) bond instrument and presentsa pricing methodology based on firm value models. The model is calibrated toreadily available market data. A stress test of model parameters is illustratedto account for potential model risk. Finally, a brief overview of how theinstrument performs is presented.

Journal article

Brigo D, The direct L2 geometric structure on a manifold of probability densities with applications to Filtering

In this paper we introduce a projection method for the space of probabilitydistributions based on the differential geometric approach to statistics. Thismethod is based on a direct L2 metric as opposed to the usual Hellingerdistance and the related Fisher Information metric. We explain how thisapparatus can be used for the nonlinear filtering problem, in relationship alsoto earlier projection methods based on the Fisher metric. Past projectionfilters focused on the Fisher metric and the exponential families that made thefilter correction step exact. In this work we introduce the mixture projectionfilter, namely the projection filter based on the direct L2 metric and based ona manifold given by a mixture of pre-assigned densities. The resultingprediction step in the filtering problem is described by a linear differentialequation, while the correction step can be made exact. We analyze therelationship of a specific class of L2 filters with the Galerkin basednonlinear filters, and highlight the differences with our approach, concerningparticularly the continuous--time observations filtering problems.

Journal article

Brigo D, Chourdakis K, Consistent single- and multi-step sampling of multivariate arrival times: A characterization of self-chaining copulas

This paper deals with dependence across marginally exponentially distributedarrival times, such as default times in financial modeling or inter-failuretimes in reliability theory. We explore the relationship between dependence andthe possibility to sample final multivariate survival in a long time-intervalas a sequence of iterations of local multivariate survivals along a partitionof the total time interval. We find that this is possible under a form ofmultivariate lack of memory that is linked to a property of the survival timescopula. This property defines a "self-chaining-copula", and we show that thiscoincides with the extreme value copulas characterization. The self-chainingcondition is satisfied by the Gumbel-Hougaard copula, a full characterizationof self chaining copulas in the Archimedean family, and by the Marshall-Olkincopula. The result has important practical implications for consistentsingle-step and multi-step simulation of multivariate arrival times in a waythat does not destroy dependency through iterations, as happens wheninconsistently iterating a Gaussian copula.

Journal article

Brigo D, Pistone G, Projecting the Fokker-Planck Equation onto a finite dimensional exponential family

In the present paper we discuss problems concerning evolutions of densitiesrelated to Ito diffusions in the framework of the statistical exponentialmanifold. We develop a rigorous approach to the problem, and we particularizeit to the orthogonal projection of the evolution of the density of a diffusionprocess onto a finite dimensional exponential manifold. It has been shown by D.Brigo (1996) that the projected evolution can always be interpreted as theevolution of the density of a different diffusion process. We give also acompactness result when the dimension of the exponential family increases, as afirst step towards a convergence result to be investigated in the future. Theinfinite dimensional exponential manifold structure introduced by G. Pistoneand C. Sempi is used and some examples are given.

Journal article

Brigo D, Francischello M, Pallavicini A, An indifference approach to the cost of capital constraints: KVA and beyond

The strengthening of capital requirements has induced banks and traders toconsider charging a so called capital valuation adjustment (KVA) to the clientsin OTC transactions. This roughly corresponds to charge the clients ex-ante theprofit requirement that is asked to the trading desk. In the following we tryto delineate a possible way to assess the impact of capital constraints in thevaluation of a deal. We resort to an optimisation stemming from an indifferencepricing approach, and we study both the linear problem from the point of viewof the whole bank and the non-linear problem given by the viewpoint ofshareholders. We also consider the case where one optimises the median ratherthan the mean statistics of the profit and loss distribution.

Journal article

Armstrong J, Brigo D, Optimizing S-shaped utility and implications for risk management

We consider market players with tail-risk-seeking behaviour as exemplified bythe S-shaped utility introduced by Kahneman and Tversky. We argue that riskmeasures such as value at risk (VaR) and expected shortfall (ES) areineffective in constraining such players. We show that, in many standard marketmodels, product design aimed at utility maximization is not constrained at allby VaR or ES bounds: the maximized utility corresponding to the optimal payoffis the same with or without ES constraints. By contrast we show that, inreasonable markets, risk management constraints based on a second moreconventional concave utility function can reduce the maximum S-shaped utilitythat can be achieved by the investor, even if the constraining utility functionis only rather modestly concave. It follows that product designs leading tounbounded S-shaped utilities will lead to unbounded negative expectedconstraining utilities when measured with such conventional utility functions.To prove these latter results we solve a general problem of optimizing aninvestor expected utility under risk management constraints where both investorand risk manager have conventional concave utility functions, but the investorhas limited liability. We illustrate our results throughout with the example ofthe Black--Scholes option market. These results are particularly importantgiven the historical role of VaR and that ES was endorsed by the Baselcommittee in 2012--2013.

Journal article

Armstrong J, Bellani C, Brigo D, Cass Tet al., Option pricing models without probability

We describe the pricing and hedging practices refraining from the use ofprobability. We encode volatility in an enhancement of the price trajectory andwe give pathwise presentations of the fundamental equations of MathematicalFinance. In particular this allows us to assess model misspecification,generalising the so-called fundamental theorem of derivative trading (seeEllersgaard et al. 2017). Our pathwise integrals and equations exhibit the roleof Greeks beyond the leading-order Delta, and makes explicit the role of Gammasensitivities.

Journal article

Armstrong J, Brigo D, Ferrucci ER, Projections of SDEs onto Submanifolds

In [AB16] the authors define three projections of $\mathbb R^d$-valuedstochastic differential equations (SDEs) onto submanifolds: the Stratonovich,It\^o-vector and It\^o-jet projections. In this paper, after a brief survey ofSDEs on manifolds, we begin by giving these projections a natural,coordinate-free description, each in terms of a specific representation ofmanifold-valued SDEs. We proceed by deriving formulae for the three projectionsin ambient $\mathbb R^d$-coordinates. We use these to show that theIt\^o-vector and It\^o-jet projections satisfy respectively a weak andmean-square optimality criterion "for small t": this is achieved by solvingconstrained optimisation problems. These results confirm, but do not rely onthe approach taken in [AB16], which is formulated in terms of weak and strongIt\^o-Taylor expansions. In the final section we exhibit examples showing howthe three projections can differ, and explore alternative notions ofoptimality.

Journal article

Bellani C, Brigo D, Done A, Neuman Eet al., Static vs Adaptive Strategies for Optimal Execution with Signals

We compare optimal static and dynamic solutions in trade execution. Anoptimal trade execution problem is considered where a trader is looking at ashort-term price predictive signal while trading. When the trader creates aninstantaneous market impact, it is shown that transaction costs of optimaladaptive strategies are substantially lower than the corresponding costs of theoptimal static strategy. In the same spirit, in the case of transient impact itis shown that strategies that observe the signal a finite number of times candramatically reduce the transaction costs and improve the performance of theoptimal static strategy.

Working paper

Armstrong J, Brigo D, Statistical arbitrage of coherent risk measures

We show that coherent risk measures are ineffective in curbing the behaviourof investors with limited liability if the market admits statistical arbitrageopportunities which we term $\rho$-arbitrage for a risk measure $\rho$. We showhow to determine analytically whether such portfolios exist in complete marketsand in the Markowitz model. We also consider realistic numerical examples ofincomplete markets and determine whether expected shortfall constraints areineffective in these markets. We find that the answer depends heavily upon theprobability model selected by the risk manager but that it is certainlypossible for expected shortfall constraints to be ineffective in realisticmarkets. Since value at risk constraints are weaker than expected shortfallconstraints, our results can be applied to value at risk.

Working paper

Brigo D, Probability-free models in option pricing: statistically indistinguishable dynamics and historical vs implied volatility

We investigate whether it is possible to formulate option pricing and hedgingmodels without using probability. We present a model that is consistent withtwo notions of volatility: a historical volatility consistent with statisticalanalysis, and an implied volatility consistent with options priced with themodel. The latter will be also the quadratic variation of the model, a pathwiseproperty. This first result, originally presented in Brigo and Mercurio (1998,2000), is then connected with the recent work of Armstrong et al (2018), whereusing rough paths theory it is shown that implied volatility is associated witha purely pathwise lift of the stock dynamics involving no probability and nosemimartingale theory in particular, leading to option models withoutprobability. Finally, an intermediate result by Bender et al. (2008) isrecalled. Using semimartingale theory, Bender et al. showed that one couldobtain option prices based only on the semimartingale quadratic variation ofthe model, a pathwise property, and highlighted the difference betweenhistorical and implied volatility. All three works confirm the idea that whilehistorical volatility is a statistical quantity, implied volatility is apathwise one. This leads to a 20 years mini-anniversary of pathwise pricingthrough 1998, 2008 and 2018, which is rather fitting for a talk presented atthe conference for the 45 years of the Black, Scholes and Merton option pricingparadigm.

Working paper

Graceffa F, Brigo D, Pallavicini A, On the consistency of jump-diffusion dynamics for FX rates under inversion

In this note we investigate the consistency under inversion of jump diffusionprocesses in the Foreign Exchange (FX) market. In other terms, if the EUR/USDFX rate follows a given type of dynamics, under which conditions will USD/EURfollow the same type of dynamics? In order to give a numerical description ofthis property, we first calibrate a Heston model and a SABR model to marketdata, plotting their smiles together with the smiles of the reciprocalprocesses. Secondly, we determine a suitable local volatility structureensuring consistency. We subsequently introduce jumps and analyze both constantjump size (Poisson process) and random jump size (compound Poisson process). Inthe first scenario, we find that consistency is automatically satisfied, forthe jump size of the inverted process is a constant as well. The second case ismore delicate, since we need to make sure that the distribution of jumps in thedomestic measure is the same as the distribution of jumps in the foreignmeasure. We determine a fairly general class of admissible densities for thejump size in the domestic measure satisfying the condition.

Working paper

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