## Publications

170 results found

Armstrong J, Brigo D, Hanzon B, 2024, Optimal projection filters with information geometry, *Information Geometry*, Vol: 7, Pages: 525-540, ISSN: 2511-2481

We review the introduction of several types of projection filters. Projection structures coming from information geometry are used to obtain a finite dimensional filter in the form of a stochastic differential equation (SDE), starting from the exact infinite-dimensional stochasticpartial differential equation (SPDE) for the optimal filter. We start with the Stratonovich projection filters based on the Hellinger distance as introduced and developed in Brigo, Hanzon and Le Gland (1998, 1999) [19, 20], where the SPDE is put in Stratonovich form before projection, hence the term “Stratonovich projection”. The correction step of the filtering algorithm can be made exact by choosing a suitable exponential family as manifold, there is equivalence with assumed density filters and numerical examples have been studied. Other authors furtherdeveloped these projection filters and we present a brief literature review. A second type of Stratonovich projection filters was introduced in Armstrong and Brigo (2016) [6] where a direct L2 metric is used for projection. Projecting on mixtures of densities as a manifold coincides with Galerkin methods. All the above projection filters lack optimality, as the single vector fields of the Stratonovich SPDE are projected optimally but the SPDE solution as a whole is not approximated optimally by the projected SDE solution according to a clear criterion. Thisled to the optimal projection filters in Armstrong, Brigo and Rossi Ferrucci (2019, 2018) [10, 9], based on the Ito vector and Ito jet projections, where several types of mean square distances between the optimal filter SPDE solution and the sought finite dimensional SDE approximations are minimized, with numerical examples. After reviewing the above developments, we conclude with the remaining challenges.

Armstrong J, Brigo D, Rossi Ferrucci E, 2024, Projections of SDEs onto submanifolds, *Information Geometry*, Vol: 7, Pages: 397-427, ISSN: 2511-249X

In Armstrong et al. (Proc Lond Math Soc (3) 119(1):176–213, 2019) the authors define three projections of Rd-valued stochastic differential equations (SDEs) onto submanifolds: the Stratonovich, Itô-vector and Itô-jet projections. In this paper, after a brief survey of SDEs on manifolds, we begin by giving these projections a natural, coordinate-free description, each in terms of a specific representation of manifold-valued SDEs. We proceed by deriving formulae for the three projections in ambient Rd-coordinates. We use these to show that the Itô-vector and Itô-jet projections satisfy respectively a weak and mean-square optimality criterion “for small t”: this is achieved by solving constrained optimisation problems. These results confirm, but do not rely on the approach taken in Armstrong et al. (Proc Lond Math Soc (3) 119(1):176–213, 2019), which is formulated in terms of weak and strong Itô–Taylor expansions. In the final section we exhibit examples showing how the three projections can differ, and explore alternative notions of optimality.

Brigo D, Graceffa F, Kalinin A, 2023, Mild to classical solutions for XVA equations under stochastic volatility, *SIAM Journal of Financial Mathematics*, ISSN: 1945-497X

We extend the valuation of contingent claims in presence of default, collateral andfunding to a random functional setting and characterise pre-default value processesby martingales. Pre-default value semimartingales can also be described by BSDEswith random path-dependent coefficients and martingales as drivers. En route, werelax conditions on the available market information and construct a broad class ofdefault times. Moreover, under stochastic volatility, we characterise pre-default valueprocesses via mild solutions to parabolic semilinear PDEs and give sufficient conditionsfor mild solutions to exist uniquely and to be classical.

Armstrong J, Brigo D, Tse A, 2023, The importance of dynamic risk constraints for limited liability operators, *Annals of Operations Research*, Pages: 1-38, ISSN: 0254-5330

Previous literature shows that prevalent risk measures such as value at risk or expected shortfall are ineffective to curb excessive risk-taking by a tail-risk-seeking trader with S-shaped utility function in the context of portfolio optimisation. However, these conclusions hold only when the constraints are static in the sense that the risk measure is just applied to the terminal portfolio value. In this paper, we consider a portfolio optimisation problem featuring S-shaped utility and a dynamic risk constraint which is imposed throughout the entire trading horizon. Provided that the risk control policy is sufficiently strict relative to the Sharpe ratio of the asset, the trader’s portfolio strategies and the resulting maximal expected utility can be effectively constrained by a dynamic risk measure. Finally, we argue that dynamic risk constraints might still be ineffective if the trader has access to a derivatives market.

Bellani C, Brigo D, Pakkanen M,
et al., 2023, Price impact without averaging, *Applied Mathematical Finance*, Vol: 30, Pages: 175-206, ISSN: 1350-486X

We present a method to estimate price impact in order-driven markets that does not require averaging over executions or scenarios. Given order book data associated with one single execution of a sell metaorder, we estimate its contribution to price decrease during the trade. We do so by modelling the limit order book using a state-dependent Hawkes process, and by defining the price impact profile of the execution as a function of the compensator of the state-dependent Hawkes process. We apply our method to a dataset from NASDAQ, and we conclude that the scheduling of sell child orders has a bigger impact on price than their sizes.

Armstrong J, Brigo D, Cass T,
et al., 2022, Non-geometric rough paths on manifolds, *Journal of the London Mathematical Society*, Vol: 106, Pages: 756-817, ISSN: 0024-6107

We provide a theory of manifold-valued rough paths of bounded 3 >p-variation, which we do not assume to be geometric. Rough paths are defined in charts, relying on the vector space-valued theory of [FH14FH14], and coordinate-free (but connection-dependent) definitions of the rough integral of cotangent bundle-valued controlled paths, and of rough differential equations driven by a rough path valued in another manifold, are given. When the path is the realisation of semimartingale we recover the theory of Itô integration and stochastic differential equations on manifolds [É89É89]. We proceed to present the extrinsic counterparts to our local formulae, and show how these extend the work in [CDL15CDL15] to the setting of non-geometric rough paths and controlled integrands more general than 1-forms. In thelast section we turn to parallel transport and Cartan development: the lack of geometricity leads us to make the choice of a connection on the tangent bundle of the manifold T M, which figures in an Itô correction term in the parallelism rough differential equation; such connection, which is not needed inthe geometric/Stratonovich setting, is required to satisfy properties which guarantee well-definedness, linearity, and optionally isometricity of parallel transport. We conclude by providing a few examples that explore the additional subtleties introduced by our change in perspective.

Armstrong J, Brigo D, 2022, Coherent risk measures alone are ineffective in constraining portfolio losses, *Journal of Banking & Finance*, Vol: 140, ISSN: 0378-4266

We show that coherent risk measures alone are ineffective in curbing the behaviour of investors withlimited liability or excessive tail-risk seeking behaviour if the market admits statistical arbitrage opportunities which we term ρ-arbitrage for a risk measure ρ. We show how to determine analytically whethersuch ρ-arbitrage portfolios exist in complete markets and in the Markowitz model. We also consider realistic numerical examples of incomplete markets and determine whether Expected-Shortfall arbitrageexists in these markets. We find that the answer depends heavily upon the probability model selected bythe risk manager but that it is certainly possible for expected shortfall constraints to be ineffective in realistic markets. Since value at risk constraints are weaker than expected shortfall constraints, our resultscan be applied to value at risk.

Brigo D, Buescu C, Francischello M,
et al., 2022, Nonlinear valuation with XVAs: two converging approaches, *Mathematics*, Vol: 10, ISSN: 2227-7390

When pricing OTC contracts in the presence of additional risk factors and costs, such as credit risk and funding and collateral costs, the starting “clean price” is modified additively by valuation adjustments (XVAs) that account for each factor or cost in isolation, while seemingly ignoring the combined effects. Instead, risk factors and costs can be jointly accounted for ab initio in the pricing mechanism at the level of cash flows, and this “adjusted cash flow" approach leads to a nonlinear valuation formula. While for practitioners this made more sense because it showed which discount factor is used for which cash flow (recall the multi-curve environment post-crisis), for academics, the focus was on checking that the resulting nonlinear valuation formula is consistent with the theoretical arbitrage-free “replication approach” that we also analyse in the paper. We formulate specific reasonable assumptions, which ensure that the valuation formulae obtained by the two approaches coincide, thus reinforcing both academics’ and practitioners’ confidence in adopting such nonlinear valuation formulae in a multi-curve setup.

Brigo D, Graceffa F, Neumann E, 2021, Price impact on term structure, *Quantitative Finance*, Vol: 22, Pages: 171-195, ISSN: 1469-7688

We introduce a rst theory of price impact in presence of an interest-rates termstructure. We explain how one can formulate instantaneous and transient price impacton zero-coupon bonds with di erent maturities, including a cross price impact that isendogenous to the term structure. We connect the introduced impact to classic noarbitrage theory for interest rate markets, showing that impact can be embedded in thepricing measure and that no-arbitrage can be preserved. We extend the price impactsetup to coupon-bearing bonds and further show how to implement price impact in aHJM framework. We present pricing examples in presence of price impact and numericalexamples of how impact changes the shape of the term structure. Finally, we show thatour approach is applicable by solving an optimal execution problem in interest ratemarkets with the type of price impact we developed in the paper.

Armstrong J, Bellani C, Brigo D, et al., 2021, Option pricing models without probability: A rough paths approach, Publisher: WILEY

Bellani C, Brigo D, Done A,
et al., 2021, Optimal trading: the importance of being adaptive, *International Journal of Financial Engineering*, Vol: 8, ISSN: 2424-7863

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

Barucci E, Brigo D, Francischello M,
et al., 2021, On the design of sovereign bond-backed securities, *International Journal of Financial Engineering*, Vol: 9, Pages: 1-1, ISSN: 2424-7863

In this paper, we analyze Sovereign Bond-Backed Securities in the Euro area, concentrating our attention on the return of the different tranches and on their riskiness. We show that as the correlation level among States increases, the yield rate of senior tranches increases while the yield rate of junior tranches decreases. A similar effect is observed when introducing a block dependence structure with high correlation among States belonging to the same block. Introducing a nonzero recovery rate, as opposed to a null recovery rate, decreases the yield rate of senior tranches and increases the yield rate of junior tranches. We compute the loss distribution and the Value at Risk (VaR) associated with the market risk of retaining the different tranches of the bond. We also analyze the possibility of reaching a safe asset through pooling tranches of government bonds of different States. In summary, we show that the issue in reaching a comprehensive and safe offering through the securitization of government bonds is not the safety of senior tranches but the risk of the junior ones.

Armstrong J, Bellani C, Brigo D,
et al., 2021, Option pricing models without probability: a rough paths approach, *Mathematical Finance*, ISSN: 0960-1627

We describe the pricing and hedging of financial options without the use of probability using rough paths. By encoding the volatility of assets in an enhancement of the price trajectory, we give a pathwise presentation of the replication of European options. The continuity properties of rough‐paths allow us to generalize the so‐called fundamental theorem of derivative trading, showing that a small misspecification of the model will yield only a small excess profit or loss of the replication strategy. Our hedging strategy is an enhanced version of classical delta hedging where we use volatility swaps to hedge the second‐order terms arising in rough‐path integrals, resulting in improved robustness.

Brigo D, Pisani C, Rapisarda F, 2021, The multivariate mixture dynamics model: shifted dynamics and correlation skew, *Annals of Operations Research*, Vol: 299, Pages: 1411-1435, ISSN: 0254-5330

The multi variate mixture dynamics model is a tractable, dynamical, arbitrage-free multivariate model characterized by transparency on the dependence structure, since closed form formulae for terminal correlations, average correlations and copula function are available. It also allows for complete decorrelation between assets and instantaneous variances. Each single asset is modelled according to a lognormal mixture dynamics model, and this univariate version is widely used in the industry due to its flexibility and accuracy. The same property holds for the multivariate process of all assets, whose density is a mixture of multivariate basic densities. This allows for consistency of single asset and index/portfolio smile. In this paper, we generalize the MVMD model by introducing shifted dynamics and we propose a definition of implied correlation under this model. We investigate whether the model is able to consistently reproduce the implied volatility of FX cross rates once the single components are calibrated to univariate shifted lognormal mixture dynamics models. We consider in particular the case of the Chinese Renminbi FX rate, showing that the shifted MVMD model correctly recovers the CNY/EUR smile given the EUR/USD smile and the USD/CNY smile, thus highlighting that the model can also work as an arbitrage free volatility smile extrapolation tool for cross currencies that may not be liquid or fully observable. We compare the performance of the shifted MVMD model in terms of implied correlation with those of the shifted simply correlated mixture dynamics model where the dynamics of the single assets are connected naively by introducing correlation among their Brownian motions. Finally, we introduce a model with uncertain volatilities and correlation. The Markovian projection of this model is a generalization of the shifted MVMD model.

Bellotti A, Brigo D, Gambetti P,
et al., 2021, Forecasting recovery rates on non-performing loans with machine learning, *International Journal of Forecasting*, Vol: 37, Pages: 428-444, ISSN: 0169-2070

We compare the performances of a wide set of regression techniques and machinelearning algorithms for predicting recovery rates on non-performing loans, using a private database from a European debt collection agency. We find that rule-based algorithmssuch as Cubist, boosted trees and random forests perform significantly better than otherapproaches. In addition to loan contract specificities, the predictors referring to the bankrecovery process – prior to the portfolio’s sale to the debt collector – are also proven tostrongly enhance forecasting performances. These variables, derived from the time-series ofcontacts to defaulted clients and clients’ reimbursements to the bank, help all algorithms tobetter identify debtors with different repayment ability and/or commitment, and in generalwith different recovery potential.

Graceffa F, Brigo D, Pallavicini A, 2020, On the consistency of jump-diffusion dynamics for FX rates under inversion, *International Journal of Financial Engineering*, Vol: 7, Pages: 1-1, ISSN: 2424-7863

We investigate the consistency under inversion of jump diffusion processes in the foreign exchange market. That is, if the EUR/USD exchange rate follows a given type of dynamics, under which conditions will USD/EUR follow the same type of dynamics? After giving a numerical description of this property, we establish a suitable local volatility structure ensuring consistency. We subsequently introduce jumps and analyze both constant and random jump size. While in the first scenario consistency is automatically satisfied, the second case is more involved. A fairly general class of admissible densities for the jump size in the domestic measure is determined.

Graceffa F, Brigo D, Pallavicini A, 2020, On the consistency of jump-diffusion dynamics for FX rates under inversion, Publisher: WORLD SCIENTIFIC PUBL CO PTE LTD

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Bellani C, Brigo D, 2020, Mechanics of good trade execution in the framework of linear temporary market impact, *Quantitative Finance*, Vol: 21, Pages: 143-163, ISSN: 1469-7688

We define the concept of good trade execution and we construct explicit adapted good trade execution strategies in the framework of linear temporary market impact. Good trade execution strategies are dynamic, in the sense that they react to the actual realisation of the traded asset price path over the trading period; this is paramount in volatile regimes, where price trajectories can considerably deviate from their expected value. Remarkably, however, the implementation of our strategies does not require the full specification of an SDE evolution for the traded asset price, making them robust across different models. Moreover, rather than minimising the expected trading cost, good trade execution strategies minimise trading costs in a pathwise sense, a point of view not yet considered in the literature. The mathematical apparatus for such a pathwise minimisation hinges on certain random Young differential equations that correspond to the Euler–Lagrange equations of the classical Calculus of Variations. These Young differential equations characterise our good trade execution strategies in terms of an initial value problem that allows for easy implementations.

Brigo D, Jeanblanc M, Vrins F, 2020, SDEs with uniform distributions: Peacocks, conic martingales and mean reverting uniform diffusions, *Stochastic Processes and their Applications*, Vol: 130, Pages: 3895-3919, ISSN: 0304-4149

Peacocks are increasing processes for the convex order. To any peacock, one can associate martingales with the same marginal laws. We are interested in finding the diffusion associated to the uniform peacock, i.e., the peacock with uniform law at all times on a time-varying support . Following an idea from Dupire (1994), Madan and Yor (2002) propose a construction to find a diffusion martingale associated to a Peacock, under the assumption of existence of a solution to a particular stochastic differential equation (SDE). In this paper we study the SDE associated to the uniform Peacock and give sufficient conditions on the (conic) boundary to have a unique strong or weak solution and analyse the local time at the boundary. Eventually, we focus on the constant support case. Given that the only uniform martingale with time-independent support seems to be a constant, we consider more general (mean-reverting) diffusions. We prove existence of a solution to the related SDE and derive the moments of transition densities. Limit-laws and ergodic results show that the transition law tends to a uniform distribution.

Armstrong J, Brigo D, Rossi Ferrucci E, 2019, Optimal approximation of SDEs on submanifolds: the Ito-vector and Ito-jet projections, *Proceedings of the London Mathematical Society*, Vol: 119, Pages: 176-213, ISSN: 1460-244X

We define two new notions of projection of a stochastic differential equation (SDE) onto a submanifold: the Itô‐vector and Itô‐jet projections. This allows one to systematically develop low‐dimensional approximations to high‐dimensional SDEs using differential geometric techniques. The approach generalizes the notion of projecting a vector field onto a submanifold in order to derive approximations to ordinary differential equations, and improves the previous Stratonovich projection method by adding optimality analysis and results. Indeed, just as in the case of ordinary projection, our definitions of projection are based on optimality arguments and give in a well‐defined sense ‘optimal’ approximations to the original SDE in the mean‐square sense over small times. We also explain how the Stratonovich projection satisfies an optimality criterion that is more ad hoc and less appealing than the criteria satisfied by the Itô projections we introduce.As an application, we consider approximating the solution of the non‐linear filtering problem with a Gaussian distribution. We show how the newly introduced Itô projections lead to optimal approximations in the Gaussian family and briefly discuss the optimal approximation for more general families of distributions. We perform a numerical comparison of our optimally approximated filter with the classical Extended Kalman Filter to demonstrate the efficacy of the approach.

Brigo D, Pede N, Petrelli A, 2019, Multi-currency credit default swaps, *International Journal of Theoretical and Applied Finance*, Vol: 22, ISSN: 0219-0249

Credit default swaps (CDS) on a reference entity may be traded in multiple currencies, in that, protection upon default may be offered either in the currency where the entity resides, or in a more liquid and global foreign currency. In this situation, currency fluctuations clearly introduce a source of risk on CDS spreads. For emerging markets, but in some cases even in well-developed markets, the risk of dramatic foreign exchange (FX)-rate devaluation in conjunction with default events is relevant. We address this issue by proposing and implementing a model that considers the risk of foreign currency devaluation that is synchronous with default of the reference entity. As a fundamental case, we consider the sovereign CDSs on Italy, quoted both in EUR and USD. Preliminary results indicate that perceived risks of devaluation can induce a significant basis across domestic and foreign CDS quotes. For the Republic of Italy, a USD CDS spread quote of 440 bps can translate into an EUR quote of 350bps in the middle of the Euro-debt crisis in the first week of May 2012. More recently, from June 2013, the basis spreads between the EUR quotes and the USD quotes are in the range around 40bps. We explain in detail the sources for such discrepancies. Our modeling approach is based on the reduced form framework for credit risk, where the default time is modeled in a Cox process setting with explicit diffusion dynamics for default intensity/hazard rate and exponential jump to default. For the FX part, we include an explicit default-driven jump in the FX dynamics. As our results show, such a mechanism provides a further and more effective way to model credit/FX dependency than the instantaneous correlation that can be imposed among the driving Brownian motions of default intensity and FX rates, as it is not possible to explain the observed basis spreads during the Euro-debt crisis by using the latter mechanism alone.

Brigo D, Francischello M, Pallavicini A, 2019, Nonlinear valuation under credit, funding, and margins: existence, uniqueness, invariance, and disentanglement, *European Journal of Operational Research*, Vol: 274, Pages: 788-805, ISSN: 0377-2217

Since the 2008 global financial crisis, the banking industry has been using valuation adjustments to account for default risk and funding costs. These adjustments are computed separately and added together by practitioners as if the valuation equations were linear. This assumption is too strong and does not allow to model market features such as different borrowing and lending rates and replacement default closeout. Hence we argue that the full valuation equations are nonlinear, and this paper is devoted to studying the nonlinear valuation equations introduced in Pallavicini et al (2011).We illustrate all the cash flows exchanged by the parties involved in a derivative contract, in presence of default risk, collateralisation with re-hypothecation and funding costs. Then we show how to obtain semi-linear PDEs or Forward Backward Stochastic Differential Equations (FBSDEs) from present-valuing said cash flows in an arbitrage-free setup, and we study the well-posedness of these PDEs and FBSDEs in a viscosity and classical sense.Moreover, from a financial perspective, we discuss cases where classical valuation adjustments (XVA) can be disentangled. We show how funding costs are offset by treasury valuation adjustments when one takes a whole-bank perspective in the valuation, while the same costs are not offset by such adjustments when taking a shareholder perspective. We show that although we use a risk-neutral valuation framework based on a locally risk-free bank account, our final valuation equations do not depend on the risk-free rate. Finally, we show how to consistently derive a netting set valuation from a portfolio level one.

Brigo D, 2019, 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, 2021),where using rough paths theory it is shown that implied volatility isassociated with a purely pathwise lift of the stock dynamics involving noprobability and no semimartingale theory in particular, leading to optionmodels without probability. Finally, an intermediate result by Bender et al.(2008) is recalled. Using semimartingale theory, Bender et al. showed that onecould obtain option prices based only on the semimartingale quadratic variationof the 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.

Armstrong J, Brigo D, 2019, Risk managing tail-risk seekers: VaR and expected shortfall vs S-shaped utility, *Journal of Banking & Finance*, Vol: 101, Pages: 122-135, ISSN: 0378-4266

We consider market players with tail-risk-seeking behaviour modelled by S-shaped utility, as introduced by Kahneman and Tversky. We argue that risk measures such as value at risk (VaR) and expected shortfall (ES) are ineffective in constraining such players, as such measures cannot reduce the traders expected S-shaped utilities. Indeed, when designing payoffs aiming to maximize utility under a VaR or ES risk limit, the players will attain the same supremum of expected utility with or without VaR or ES limits. By contrast, we show that risk management constraints based on a second more conventional concave utility function can reduce the maximum S-shaped utility that can be achieved by the investor. Indeed, product designs leading to progressively larger S-shaped utilities will lead to progressively lower expected constraining conventional utilities, violating the related risk limit. These results hold in a variety of market models, including the Black Scholes options model, and are particularly relevant for risk managers given the historical role of VaR and the endorsement of ES by the Basel committee in 2012–2013.

Armstrong J, Brigo D, 2019, The ineffectiveness of coherent risk measures

We show that coherent risk measures are ineffective in curbing the behaviourof investors with limited liability or excessive tail-risk seeking behaviour ifthe market admits statistical arbitrage opportunities which we term$\rho$-arbitrage for a risk measure $\rho$. We show how to determineanalytically whether such $\rho$-arbitrage 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. By contrast, we showthat reasonable expected utility constraints are effective in anyarbitrage-free market.

Bellani C, Brigo D, Done A, et al., 2018, 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.

Brigo D, Pede N, Petrelli A, 2018, Examples of wrong–way risk in CVA induced by devaluations on default, Innovations in Insurance, Risk- and Asset Management, Publisher: World Scientific Press, Pages: 95-115

When calculatingCredit Valuation Adjustment(CVA), theinteraction between the portfolio’s exposure and the counter-party’s credit worthiness is referred to asWrong–Way Risk(WWR). Making the assumption that the Brownian mo-tions driving both the market (exposure) and the (counter-party) credit risk–factors dynamics are correlated representsthe simplest way of modelling the dependence structure be-tween these two components. For many practical applica-tions, however, such an approach may fail to account for theright amount of WWR, thus resulting in misestimates of theportfolio’s CVA. We present a modelling framework wherea further — and indeed stronger — source of market/creditdependence is introduced through devaluation jumps on themarket risk–factors’ dynamics. Such jumps happen upon thecounterparty’s default and are a particularly realistic featureto include in case of sovereign or systemically important coun-terparties. Moreover, we show that, in the special case wherethe focus is on FX/credit WWR, devaluation jumps provide an effective way of incorporating market information comingfrom quanto Credit Default Swap (CDS) basis spreads and wederive the corresponding CVA pricing equations as a systemof coupled PDEs.

Brigo D, Mai, Jan, et al., 2018, Consistent iterated simulation of multivariate defaults: Markov indicators, lack of memory, extreme-value copulas, and the Marshall–Olkin distribution, Innovations in Insurance, Risk- and Asset Management, Publisher: World Scientific Publishing Co., Pages: 47-93

A current market-practice to incorporate multivariate defaults in global riskfactorsimulations is the iteration of (multiplicative) i.i.d. survival indicator incrementsalong a given time-grid, where the indicator distribution is based on acopula ansatz. The underlying assumption is that the behavior of the resultingiterated default distribution is similar to the one-shot distribution. It is shownthat in most cases this assumption is not fulfilled and furthermore numericalanalysis is presented that shows sizeable differences in probabilities assignedto both “survival-of-all” and “mixed default/survival” events. Moreover, theclasses of distributions for which probabilities from the “terminal one-shot”and “terminal iterated” distribution coincide are derived for problems considering“survival-of-all” events as well as “mixed default/survival” events. Forthe former problem, distributions must fulfill a lack-of-memory type property,which is, e.g., fulfilled by min-stable multivariate exponential distributions.These correspond in a copula-framework to exponential margins coupled viaextreme-value copulas. For the latter problem, while looping default inspiredmultivariate Freund distributions and more generally multivariate phase-type distributions could be a solution, under practically relevant and reasonableadditional assumptions on portfolio rebalancing and nested distributions, theunique solution is the Marshall–Olkin class.

Brigo D, Vrins F, 2018, Disentangling wrong-way risk: pricing credit valuation adjustment via change of measures, *European Journal of Operational Research*, Vol: 269, Pages: 1154-1164, ISSN: 0377-2217

In many financial contracts (and in particular when trading OTC derivatives), participantsare exposed to counterparty risk. The latter is typically rewarded by adjusting the “risk-freeprice” of derivatives; an adjustment known ascredit value adjustment(CVA). A key driverof CVA is the dependency between exposure and counterparty risk, known aswrong-way risk(WWR). In practice however, correctly addressing WWR is very challenging and calls forheavy numerical techniques. This might explain why WWR is not explicitly handled in theBasel III regulatory framework in spite of its acknowledged importance. In this paper wepropose a sound and tractable method to deal efficiently with WWR. Our approach consistsof embedding the WWR effect in the drift of the exposure dynamics. Even though thiscalls for infinite changes of measures, we end up with an appealing compromise betweentractability and mathematical rigor, preserving the level of accuracy typically required forCVA figures. The good performances of the method are discussed in a stochastic-intensitydefault setup based on extensive comparisons of Expected Positive Exposure (EPE) profilesand CVA figures produced (i) by a full bivariate Monte Carlo implementation of the initialmodel with (ii) our drift-adjustment technique.

Brigo D, Piat C, 2018, Static vs adapted optimal execution strategies in two benchmark trading models, Innovations in Insurance, Risk- and Asset Management, Publisher: World Scientific Publishing Co., Pages: 239-273

We consider the optimal solutions to the trade execution problem in the two different classes of i) fully adapted or adaptive and ii) deterministic or static strategies, comparing them. We do this in two different benchmark models. The first model is a discrete time framework with an information flow process, dealing with both permanent and temporary impact, minimizing the expected cost of the trade. The second model is a continuous time framework where the objective function is the sum of the expected cost and a value at risk (or expected shortfall) type risk criterion. Optimal adapted solutions are known in both frameworks from the original works of Bertsimas and Lo (1998) and Gatheral and Schied (2011). In this paper we derive the optimal static strategies for both benchmark models and we study quantitatively the improvement in optimality when moving from static strategies to fully adapted ones. We conclude that, in the benchmark models we study, the difference is not relevant, except for extreme unrealistic cases for the model or impact parameters. This indirectly confirms that in the similar framework of Almgren and Chriss (2000) one is fine deriving a static optimal solution, as done by those authors, as opposed to a fully adapted one, since the static solution happens to be tractable and known in closed form.

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