## Publications

159 results found

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

Armstrong J, Brigo D, Rossi Ferrucci E, 2023, Projections of SDEs onto submanifolds, *Information Geometry*, 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.

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.

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, Publisher: WILEY

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, Huang X, Pallavicini A, et al., 2021, Interpretability in deep learning for finance: a case study for the Heston model

Deep learning is a powerful tool whose applications in quantitative financeare growing every day. Yet, artificial neural networks behave as black boxesand this hinders validation and accountability processes. Being able tointerpret the inner functioning and the input-output relationship of thesenetworks has become key for the acceptance of such tools. In this paper wefocus on the calibration process of a stochastic volatility model, a subjectrecently tackled by deep learning algorithms. We analyze the Heston model inparticular, as this model's properties are well known, resulting in an idealbenchmark case. We investigate the capability of local strategies and globalstrategies coming from cooperative game theory to explain the trained neuralnetworks, and we find that global strategies such as Shapley values can beeffectively used in practice. Our analysis also highlights that Shapley valuesmay help choose the network architecture, as we find that fully-connectedneural networks perform better than convolutional neural networks in predictingand interpreting the Heston model prices to parameters relationship.

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.

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.

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.

Armstrong J, Brigo D, 2018, Rogue traders versus value-at-risk and expected shortfall, *Risk -London- Risk Magazine Limited-*, Pages: 63-63, ISSN: 0952-8776

We show that, in a Black and Scholes market, value at risk and ex-pected shortfall are irrelevant in limiting traders excessive tail-risk seekingbehaviour as modelled via Kahneman and Tversky’s S-shaped utility. Tohave effective constraints one can introduce a risk limit based on a secondbut concave utility function.

Armstrong J, Brigo D, 2018, Intrinsic stochastic differential equations as jets, *Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*, Vol: 474, ISSN: 1364-5021

We explain how Itô stochastic differential equations(SDEs) on manifolds may be defined using 2-jets ofsmooth functions. We show how this relationship canbe interpreted in terms of a convergent numericalscheme. We also show how jets can be used toderive graphical representations of Itô SDEs, and weshow how jets can be used to derive the differentialoperators associated with SDEs in a coordinatefreemanner. We relate jets to vector flows, givinga geometric interpretation of the Itô–Stratonovichtransformation. We show how percentiles can be usedto give an alternative coordinate-free interpretation ofthe coefficients of one-dimensional SDEs. We relatethis to the jet approach. This allows us to interpretthe coefficients of SDEs in terms of ‘fan diagrams’. Inparticular, the median of an SDE solution is associatedwith the drift of the SDE in Stratonovich form for smalltimes.

Brigo D, Hvolby T, Vrins F, 2018, Wrong-way risk adjusted exposure: analytical approximations for options in default intensity models, Innovations in Insurance, Risk- and Asset Management, Publisher: World Scientific Publishing Co.

We examine credit value adjustment (CVA) estimation under wrong-way risk(WWR) by computing the expected positive exposure (EPE) under an equiva-lent measure as suggested in [1], adjusting the drift of the underlying for defaultrisk. We apply this technique to European put and call options and derive theanalytic formulas for EPE under WWR obtained with various approximationsof the drift adjustment. We give the results of numerical experiments basedon 4 parameter sets, and supply figures of the CVA based on both of the sug-gested proxys, comparing with CVA based on a 2D-Monte Carlo scheme andGaussian Copula resampling. We also show the CVA obtained by the formulasfrom Basel III. We observe that the Basel III formula does not account forthe credit-market correlation, while the Gaussian Copula resampling methodestimates a too large impact of this correlation. The two proxies account forthe credit-market correlation, and give results that are mostly similar to the2D-Monte Carlo results.

Bormetti G, Brigo D, Francischello M, et al., 2018, Impact of multiple curve dynamics in credit valuation adjustments under collateralization, Publisher: ROUTLEDGE JOURNALS, TAYLOR & FRANCIS LTD

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Brigo D, Rapisarda F, Sridi A, 2017, The multivariate mixture dynamics: consistent no-arbitrage single-asset and index volatility smiles, *IISE Transactions*, Vol: 50, Pages: 27-44, ISSN: 0740-817X

We introduce a new arbitrage-free multivariate dynamic asset pricing model that allows us to reconcile single name and index/basket volatility smiles using a tractable and explicit dependence structure that goes beyond instantaneous correlation. Each asset volatility smile is modeled according to a density-mixture dynamical model while the same property holds for the multivariate process of all assets, whose density is a mixture of multivariate basic densities. After introducing the model, we derive tractable index option smile formulas resulting from the model and related closed form solutions for multivariate densities taking the form of multivariate mixtures. Using Markovian projection techniques, we relate our model to a multivariate uncertain volatility model and show a consistency result with geometric baskets with hints on possible uses in investigating triangular relationships between foreign exchange rates and the related smiles in practice. We also derive closed form solutions for a number of terminal statistics of dependence, and derive a precise relationship with a simpler but less tractable model based on a basic instantaneous correlation structure. Finally, closed form solutions for volatility/assets correlations illuminating the relationship with the uncertain volatility model are introduced. The model tractability makes it particularly suited for calibration and risk management applications, where speed of calculations and tractability are essential. A few numerical examples on basket and spread options pricing conclude the paper.

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