Imperial College London


Faculty of Natural SciencesDepartment of Mathematics

Chair in Mathematical Finance







805Weeks BuildingSouth Kensington Campus





Publication Type

141 results found

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

Brigo D, Morini M, Dangers of Bilateral Counterparty Risk: the fundamental impact of closeout conventions

We analyze the practical consequences of the bilateral counterparty riskadjustment. We point out that past literature assumes that, at the moment ofthe first default, a risk-free closeout amount will be used. We argue that thelegal (ISDA) documentation suggests in many points that a substitution closeoutshould be used. This would take into account the risk of default of thesurvived party. We show how the bilateral counterparty risk adjustment changesstrongly when a substitution closeout amount is considered. We model the twoextreme cases of default independence and co-monotonicity, which highlight prosand cons of both risk free and substitution closeout formulations, and allow usto interpret the outcomes as dramatic consequences on default contagion.Finally, we analyze the situation when collateral is present.

Scholarly edition

Brigo D, Hanzon B, On three filtering problems arising in mathematical finance

Three situations in which filtering theory is used in mathematical financeare illustrated at different levels of detail. The three problems originatefrom the following different works: 1) On estimating the stochastic volatilitymodel from observed bilateral exchange rate news, by R. Mahieu, and P.Schotman; 2) A state space approach to estimate multi-factors CIR models of theterm structure of interest rates, by A.L.J. Geyer, and S. Pichler; 3)Risk-minimizing hedging strategies under partial observation in pricingfinancial derivatives, by P. Fischer, E. Platen, and W. J. Runggaldier; In thefirst problem we propose to use a recent nonlinear filtering technique based ongeometry to estimate the volatility time series from observed bilateralexchange rates. The model used here is the stochastic volatility model. Thefilters that we propose are known as projection filters, and a brief derivationof such filters is given. The second problem is introduced in detail, and apossible use of different filtering techniques is hinted at. In fact thefilters used for this problem in 2) and part of the literature can beinterpreted as projection filters and we will make some remarks on how moregeneral and possibly more suitable projection filters can be constructed. Thethird problem is only presented shortly.

Scholarly edition

Brigo D, Dalessandro A, Neugebauer M, Triki Fet al., A Stochastic Processes Toolkit for Risk Management

In risk management it is desirable to grasp the essential statisticalfeatures of a time series representing a risk factor. This tutorial aims tointroduce a number of different stochastic processes that can help in graspingthe essential features of risk factors describing different asset classes orbehaviors. This paper does not aim at being exhaustive, but gives examples anda feeling for practically implementable models allowing for stylised featuresin the data. The reader may also use these models as building blocks to buildmore complex models, although for a number of risk management applications themodels developed here suffice for the first step in the quantitative analysis.The broad qualitative features addressed here are {fat tails} and {meanreversion}. We give some orientation on the initial choice of a suitablestochastic process and then explain how the process parameters can be estimatedbased on historical data. Once the process has been calibrated, typicallythrough maximum likelihood estimation, one may simulate the risk factor andbuild future scenarios for the risky portfolio. On the terminal simulateddistribution of the portfolio one may then single out several risk measures,although here we focus on the stochastic processes estimation preceding thesimulation of the risk factors Finally, this first survey report focuses onsingle time series. Correlation or more generally dependence across riskfactors, leading to multivariate processes modeling, will be addressed infuture work.

Scholarly edition

Brigo D, The general mixture-diffusion SDE and its relationship with an uncertain-volatility option model with volatility-asset decorrelation

In the present paper, given an evolving mixture of probability densities, wedefine a candidate diffusion process whose marginal law follows the sameevolution. We derive as a particular case a stochastic differential equation(SDE) admitting a unique strong solution and whose density evolves as a mixtureof Gaussian densities. We present an interesting result on the comparisonbetween the instantaneous and the terminal correlation between the obtainedprocess and its squared diffusion coefficient. As an application tomathematical finance, we construct diffusion processes whose marginal densitiesare mixtures of lognormal densities. We explain how such processes can be usedto model the market smile phenomenon. We show that the lognormal mixturedynamics is the one-dimensional diffusion version of a suitable uncertainvolatility model, and suitably reinterpret the earlier correlation result. Weexplore numerically the relationship between the future smile structures ofboth the diffusion and the uncertain volatility versions.

Scholarly edition

Brigo D, Constant Maturity Credit Default Swap Pricing with Market Models

In this work we derive an approximated no-arbitrage market valuation formulafor Constant Maturity Credit Default Swaps (CMCDS). We move from the CDSoptions market model in Brigo (2004), and derive a formula for CMCDS that isthe analogous of the formula for constant maturity swaps in the default freeswap market under the LIBOR market model. A "convexity adjustment"-likecorrection is present in the related formula. Without such correction, or withzero correlations, the formula returns an obvious deterministic-credit-spreadexpression for the CMCDS price. To obtain the result we derive a joint dynamicsof forward CDS rates under a single pricing measure, as in Brigo (2004).Numerical examples of the "convexity adjustment" impact complete the paper.

Scholarly edition

Brigo D, Pallavicini A, Torresetti R, Default correlation, cluster dynamics and single names: The GPCL dynamical loss model

We extend the common Poisson shock framework reviewed for example in Lindskogand McNeil (2003) to a formulation avoiding repeated defaults, thus obtaining amodel that can account consistently for single name default dynamics, clusterdefault dynamics and default counting process. This approach allows one tointroduce significant dynamics, improving on the standard "bottom-up"approaches, and to achieve true consistency with single names, improving onmost "top-down" loss models. Furthermore, the resulting GPCL model hasimportant links with the previous GPL dynamical loss model in Brigo,Pallavicini and Torresetti (2006a,b), which we point out. Model extensionsallowing for more articulated spread and recovery dynamics are hinted at.Calibration to both DJi-TRAXX and CDX index and tranche data across attachmentsand maturities shows that the GPCL model has the same calibration power as theGPL model while allowing for consistency with single names

Working paper

Brigo D, Counterparty Risk FAQ: Credit VaR, PFE, CVA, DVA, Closeout, Netting, Collateral, Re-hypothecation, WWR, Basel, Funding, CCDS and Margin Lending

We present a dialogue on Counterparty Credit Risk touching on Credit Value atRisk (Credit VaR), Potential Future Exposure (PFE), Expected Exposure (EE),Expected Positive Exposure (EPE), Credit Valuation Adjustment (CVA), DebitValuation Adjustment (DVA), DVA Hedging, Closeout conventions, Netting clauses,Collateral modeling, Gap Risk, Re-hypothecation, Wrong Way Risk, Basel III,inclusion of Funding costs, First to Default risk, Contingent Credit DefaultSwaps (CCDS) and CVA restructuring possibilities through margin lending. Thedialogue is in the form of a Q&A between a CVA expert and a newly hiredcolleague.

Scholarly edition

Brigo D, Capponi A, Bilateral counterparty risk valuation with stochastic dynamical models and application to Credit Default Swaps

We introduce the general arbitrage-free valuation framework for counterparty risk adjustments in presence of bilateral default risk, including default of the investor. We illustrate the symmetry in the valuation and show that the adjustment involves a long position in a put option plus a short position in a call option, both with zero strike and written on the residual net value of the contract at the relevant default times. We allow for correlation between the default times of the investor, counterparty and underlying portfolio risk factors. We use arbitrage-free stochastic dynamical models. We then specialize our analysis to Credit Default Swaps (CDS) as underlying portfolio, generalizing the work of Brigo and Chourdakis (2008) [5] who deal with unilateral and asymmetric counterparty risk. We introduce stochastic intensity models and a trivariate copula function on the default times exponential variables to model default dependence. Similarly to [5], we find that both default correlation and credit spread volatilities have a relevant and structured impact on the adjustment. Differently from [5], the two parties will now agree on the credit valuation adjustment. We study a case involving British Airways, Lehman Brothers and Royal Dutch Shell, illustrating the bilateral adjustments in concrete crisis situations.

Scholarly edition

Brigo D, Mercurio F, Discrete Time vs Continuous Time Stock-price Dynamics and implications for Option Pricing

In the present paper we construct stock price processes with the samemarginal log-normal law as that of a geometric Brownian motion and also withthe same transition density (and returns' distributions) between any twoinstants in a given discrete-time grid. We then illustrate how option pricesbased on such processes differ from Black and Scholes', in that option pricescan be either arbitrarily close to the option intrinsic value or arbitrarilyclose to the underlying stock price. We also explain that this is due to theparticular way one models the stock-price process in between the grid timeinstants which are relevant for trading. The theoretical result concerningscalar stochastic differential equations with prescribed diffusion coefficientwhose densities evolve in a prescribed exponential family, on which part of thepaper is based, is presented in detail.

Scholarly edition

Brigo D, El-Bachir N, Credit Derivatives Pricing with a Smile-Extended Jump Stochastic Intensity Model

We present a two-factor stochastic default intensity and interest rate model for pricing single-name default swaptions. The specific positive square root processes considered fall in the relatively tractable class of affine jump diffusions while allowing for inclusion of stochastic volatility and jumps in default swap spreads. The parameters of the short rate dynamics are first calibrated to the interest rates markets, before calibrating separately the default intensity model to credit derivatives market data. A few variants of the model are calibrated in turn to market data, and different calibration procedures are compared. Numerical experiments show that the calibrated model can generate plausible volatility smiles. Hence, the model can be calibrated to a default swap term structure and few default swaptions, and the calibrated parameters can be used to value consistently other default swaptions (different strikes and maturities, or more complex structures) on the same credit reference name.

Scholarly edition

Morini M, Brigo D, Arbitrage-free Pricing of Credit Index Options: The no-armageddon pricing measure and the role of correlation after the subprime crisis

In this work we consider three problems of the standard market approach topricing of credit index options: the definition of the index spread is notvalid in general, the usually considered payoff leads to a pricing which is notalways defined, and the candidate numeraire one would use to define a pricingmeasure is not strictly positive, which would lead to a non-equivalent pricingmeasure. We give a general mathematical solution to the three problems, based on anovel way of modeling the flow of information through the definition of a newsubfiltration. Using this subfiltration, we take into account consistently thepossibility of default of all names in the portfolio, that is neglected in thestandard market approach. We show that, while the related mispricing can benegligible for standard options in normal market conditions, it can becomehighly relevant for different options or in stressed market conditions. In particular, we show on 2007 market data that after the subprime creditcrisis the mispricing of the market formula compared to the no arbitrageformula we propose has become financially relevant even for the liquidCrossover Index Options.

Scholarly edition

Brigo D, El-Bachir N, An exact formula for default swaptions' pricing in the SSRJD stochastic intensity model

We develop and test a fast and accurate semi-analytical formula for single-name default swaptions in the context of a shifted square root jump diffusion (SSRJD) default intensity model. The model can be calibrated to the CDS term structure and a few default swaptions, to price and hedge other credit derivatives consistently. We show with numerical experiments that the model implies plausible volatility smiles.

Scholarly edition

Brigo D, Tarenghi M, Credit Default Swap Calibration and Counterparty Risk Valuation with a Scenario based First Passage Model

In this work we develop a tractable structural model with analytical defaultprobabilities depending on a random default barrier and possibly randomvolatility ideally associated with a scenario based underlying firm debt. Weshow how to calibrate this model using a chosen number of reference CreditDefault Swap (CDS) market quotes. In general this model can be seen as apossible extension of the time-varying AT1P model in Brigo and Tarenghi (2004).The calibration capability of the Scenario Volatility/Barrier model (SVBAT1P),when keeping time-constant volatility, appears inferior to the one of AT1P withtime-varying deterministic volatility. The SVBAT1P model, however, maintainsthe benefits of time-homogeneity and can lead to satisfactory calibrationresults, as we show in a case study where we compare different choices onscenarios and parameters. Similarly to AT1P, SVBAT1P is suited to pricinghybrid equity/credit derivatives and to evaluate counterparty risk in equitypayoffs, and more generally to evaluate hybrid credit/equity payoffs. Weconsider the equity return swap in Brigo and Tarenghi (2004) and show itsvaluation under SVBAT1P with the same CDS and equity calibration input usedearlier for AT1P, and further we hint at equity default swap valuation in theconclusions.

Scholarly edition

Brigo D, Tarenghi M, Credit Default Swap Calibration and Equity Swap Valuation under Counterparty Risk with a Tractable Structural Model

In this paper we develop a tractable structural model with analytical defaultprobabilities depending on some dynamics parameters, and we show how tocalibrate the model using a chosen number of Credit Default Swap (CDS) marketquotes. We essentially show how to use structural models with a calibrationcapability that is typical of the much more tractable credit-spread basedintensity models. We apply the structural model to a concrete calibration caseand observe what happens to the calibrated dynamics when the CDS-implied creditquality deteriorates as the firm approaches default. Finally we provide atypical example of a case where the calibrated structural model can be used forcredit pricing in a much more convenient way than a calibrated reduced formmodel: The pricing of counterparty risk in an equity swap.

Scholarly edition

Brigo D, Nordio C, Liquidity-adjusted Market Risk Measures with Stochastic Holding Period

Within the context of risk integration, we introduce in risk measurementstochastic holding period (SHP) models. This is done in order to obtain a`liquidity-adjusted risk measure' characterized by the absence of a fixed timehorizon. The underlying assumption is that - due to changes on market liquidityconditions - one operates along an `operational time' to which the P&L processof liquidating a market portfolio is referred. This framework leads to amixture of distributions for the portfolio returns, potentially allowing forskewness, heavy tails and extreme scenarios. We analyze the impact of possibledistributional choices for the SHP. In a multivariate setting, we hint at thepossible introduction of dependent SHP processes, which potentially lead to nonlinear dependence among the P&L processes and therefore to tail dependenceacross assets in the portfolio, although this may require drastic choices onthe SHP distributions. We also find that increasing dependence as measured byKendall's tau through common SHP's appears to be unfeasible. We finally discusspotential developments following future availability of market data.

Scholarly edition

El-Bachir N, Brigo D, An analytically tractable time-changed jump-diffusion default intensity model

We present a stochastic default intensity model where the intensity follows a tractable jump-diffusion process obtained by applying a deterministic change of time to a non mean-reverting square root jump-diffusion process. The model generates higher implied volatilities for default swaptions than mean-reverting versions, consistent with volatility levels observed on the market.

Scholarly edition

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