arXiv subjects

Damiano Brigo

Publications and source records attributed to Damiano Brigo.

At least 19 recordsLinked to original sources

arXiv2026

Exact calibration of structural models via time-change

In this note, we propose a general structural approach to model a default time $τ$ as the first-passage time (FPT) of a (``firm-value'') process $S$ below a (``debt'') barrier $K$ that comply with a pre-specified survival probability curve $G(t)=\Pr(τ>t)$. Following an idea of Mbaye and Vrins (Mathematical Finance, 2022) applied to reduced-form models, our approach consists in two steps: choose a latent FPT model driven by a barrier $\tilde{K}$ and process $\tilde{S}$, and time-change those using a deterministic clock $Θ$ to get $K_t=\tilde{K}_{Θ(t)}$ and $S_t:=\tilde{S}_{Θ(t)}$, leading to the final FTP model $(K,S,Θ)$. As the market curve $G$ and the latent model $(\tilde{K},\tilde{S})$ are assumed to be given, the calibration step simply consists in finding the clock $Θ$ such that the distribution of the FPT of $S$ below $K$ coincides with the survival curve $G$. We show that this is achievable for a broad class of specified curves $G$ and latent FTP models. The calibration amounts to a simple inversion of a function, which is almost immediate provided that the latent model is tractable enough. In particular, we show that the AT1P model of Brigo, Morini and Tarenghi - which is able to reproduce a broad range of CDS term-structures - can be regarded as the FPT of a time-changed drifted Brownian motion to a constant barrier: $\tilde{V}_t=μt+W_t$ and $\tilde{K}_t=k<0$. This connection offers an elegant interpretation for the instantaneous volatility function featured in AT1P and yields an immediate calibration of the latter to perfectly match a target survival curve.

q-fin.MF↗

arXiv2026

Local Stochastic Rough Volatility: Pathwise Filtering and the Conditional Density Equation

This article studies the conditional-density equation and its pathwise transformation in local stochastic rough volatility models, with rough Heston (rHeston) as the main explicit example. Under the stated common-filtration, measurability, predictability and spatial-regularity assumptions, we show that the Ito-Wentzell random-PDE reduction of the conditional density SPDE remains valid under local stochastic rough volatility. After fixing a common-environment realization and the associated stochastic flow, the transformed equation becomes a deterministic PDE with path-dependent coefficients. This yields a pathwise Fokker--Planck formulation that connects naturally with Rao--Blackwellized calibration. In the pure rough Heston case, the transformed coefficients simplify and the conditional density admits an explicit lognormal form.

q-fin.MF↗

arXiv2026

On the boundaries, asymptotic law and Bernoulli-Doob representation of homogeneous bounded martingales

We study homogeneous diffusion martingales evolving in a bounded state space $D=[a,b]$, where $a$ and $b$ are zeros of the diffusion coefficient. We call a process of the form $Z_t=\mathbb{E}[B\mid\mathcal{F}_t]$, with $B$ a Bernoulli random variable, a Bernoulli-Doob martingale. Our main results establish a complete equivalence: every such diffusion martingale is a Bernoulli-Doob martingale (Theorem 2) and, conversely, every continuous time-homogeneous Markov Bernoulli-Doob martingale on a Brownian filtration arises from such a diffusion (Theorem 3). The intuitive reason is that a bounded martingale has constant expectation while accumulating variance, so it converges to the maximum-variance distribution with given mean and range, namely the Bernoulli. We further show that this Bernoulli limit is truly asymptotic: for any fixed finite horizon $T$, the probability of not yet having reached the boundary is strictly positive (Theorem 4), even when the individual boundaries are accessible. We clarify the relationship between Feller's boundary classification, the pathwise SDE framework, and the martingale constraint, showing that the martingale property forces absorption at any attainable boundary. The theory is illustrated with the $Φ$-martingale, the Jacobi martingale, and applications to credit-risk modelling.

math.PR↗

arXiv2024

Deep learning interpretability for rough volatility

Deep learning methods have become a widespread toolbox for pricing and calibration of financial models. While they often provide new directions and research results, their `black box' nature also results in a lack of interpretability. We provide a detailed interpretability analysis of these methods in the context of rough volatility - a new class of volatility models for Equity and FX markets. Our work sheds light on the neural network learned inverse map between the rough volatility model parameters, seen as mathematical model inputs and network outputs, and the resulting implied volatility across strikes and maturities, seen as mathematical model outputs and network inputs. This contributes to building a solid framework for a safer use of neural networks in this context and in quantitative finance more generally.

q-fin.CP↗

arXiv2022

Projections of SDEs onto Submanifolds

In [ABF19] the authors define three projections of Rd-valued stochastic differential equations (SDEs) onto submanifolds: the Stratonovich, Ito-vector and Ito-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 $\mathbb R^d$-coordinates. We use these to show that the Ito-vector and Ito-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 [ABF19], which is formulated in terms of weak and strong Ito-Taylor expansions. In the final section we exhibit examples showing how the three projections can differ, and explore alternative notions of optimality.

math.PR↗

arXiv2022

Optimal Projection Filters

We present the two new notions of projection of a stochastic differential equation (SDE) onto a submanifold, as developed in Armstrong, Brigo e Rossi Ferrucci (2019, 2018): the Ito-vector and Ito-jet projections. This allows one to systematically and optimally develop low dimensional approximations to high dimensional SDEs using differential geometric techniques. Our new projections are based on optimality arguments and yield a well-defined ``optimal'' approximation to the original SDE in the mean-square sense. We also show that the earlier Stratonovich projection satisfies an optimality criterion that is more ad hoc and less natural than the criteria satisfied by the new projections. As an application, we consider approximating the solution of the non-linear filtering problem within a given manifold of densities, using either the Hellinger or $L^2$ direct metrics and related Information Geometry structures on the space of densities. The Stratonovich projection had yielded the projection filters studied in Brigo, Hanzon and Le Gland (1998, 1999), while the new projections lead to the optimal projection filters. The optimal projection filters have been introduced in Armstrong, Brigo e Rossi Ferrucci (2019), where numerical examples for the Gaussian case are given and where they are compared to more traditional nonlinear filters.

math.PR↗

arXiv2022

Non-average price impact in order-driven markets

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

q-fin.TR↗

arXiv2021

Mild to classical solutions for XVA equations under stochastic volatility

We extend the valuation of contingent claims in presence of default, collateral and funding to a random functional setting and characterise pre-default value processes by martingales. Pre-default value semimartingales can also be described by BSDEs with random path-dependent coefficients and martingales as drivers. En route, we generalise previous settings by relaxing conditions on the available market information, allowing for an arbitrary default-free filtration and constructing a broad class of default times. Moreover, under stochastic volatility, we characterise pre-default value processes via mild solutions to parabolic semilinear PDEs and give sufficient conditions for mild solutions to exist uniquely and to be classical.

math.PR↗

arXiv2021

Price Impact on Term Structure

We introduce a first theory of price impact in presence of an interest-rates term structure. We explain how one can formulate instantaneous and transient price impact on bonds with different maturities, including a cross price impact that is endogenous to the term structure. We connect the introduced impact to classic no-arbitrage theory for interest rate markets, showing that impact can be embedded in the pricing measure and that no-arbitrage can be preserved. We present pricing examples in presence of price impact and numerical examples of how impact changes the shape of the term structure. Finally, to show that our approach is applicable we solve an optimal execution problem in interest rate markets with the type of price impact we developed in the paper.

q-fin.TR↗

arXiv2021

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 hedging models without using probability. We present a model that is consistent with two notions of volatility: a historical volatility consistent with statistical analysis, and an implied volatility consistent with options priced with the model. The latter will be also the quadratic variation of the model, a pathwise property. 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 is associated with a purely pathwise lift of the stock dynamics involving no probability and no semimartingale theory in particular, leading to option models without probability. Finally, an intermediate result by Bender et al. (2008) is recalled. Using semimartingale theory, Bender et al. showed that one could obtain option prices based only on the semimartingale quadratic variation of the model, a pathwise property, and highlighted the difference between historical and implied volatility. All three works confirm the idea that while historical volatility is a statistical quantity, implied volatility is a pathwise one. This leads to a 20 years mini-anniversary of pathwise pricing through 1998, 2008 and 2018, which is rather fitting for a talk presented at the conference for the 45 years of the Black, Scholes and Merton option pricing paradigm.

q-fin.PR↗

arXiv2021

Interpretability in deep learning for finance: a case study for the Heston model

Deep learning is a powerful tool whose applications in quantitative finance are growing every day. Yet, artificial neural networks behave as black boxes and this hinders validation and accountability processes. Being able to interpret the inner functioning and the input-output relationship of these networks has become key for the acceptance of such tools. In this paper we focus on the calibration process of a stochastic volatility model, a subject recently tackled by deep learning algorithms. We analyze the Heston model in particular, as this model's properties are well known, resulting in an ideal benchmark case. We investigate the capability of local strategies and global strategies coming from cooperative game theory to explain the trained neural networks, and we find that global strategies such as Shapley values can be effectively used in practice. Our analysis also highlights that Shapley values may help choose the network architecture, as we find that fully-connected neural networks perform better than convolutional neural networks in predicting and interpreting the Heston model prices to parameters relationship.

q-fin.PR↗

arXiv2020

The importance of dynamic risk constraints for limited liability operators

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 asset performance, 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.

q-fin.PM↗

arXiv2020

The ineffectiveness of coherent risk measures

We show that coherent risk measures are ineffective in curbing the behaviour of investors with limited 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 whether such $ρ$-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 constraints are ineffective in these markets. We find that the answer depends heavily upon the probability model selected by the 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 results can be applied to value at risk. By contrast, we show that reasonable expected utility constraints are effective in any arbitrage-free market.

q-fin.RM↗

arXiv2020

Non-Geometric Rough Paths on Manifolds

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, and coordinate-free (but connection-dependent) definitions of the rough integral of cotangent bundle-valued controlled paths, and of RDEs 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 SDEs on manifolds [É89]. We proceed to present the extrinsic counterparts to our local formulae, and show how these extend the work in [CDL15] to the setting of non-geometric rough paths and controlled integrands more general than 1-forms. In the last 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 TM, which figures in an Itô correction term in the parallelism RDE; such connection, which is not needed in the geometric/Stratonovich setting, is required to satisfy properties which guarantee well-definedness, linearity, and optionally isometricity of parallel transport. We conclude by providing numerous examples, some accompanied by numerical simulations, which explore the additional subtleties introduced by our change in perspective.

math.CA↗

arXiv2020

Option pricing models without probability: a rough paths approach

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 generalise 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.

q-fin.MF↗

arXiv2020

Mechanics of good trade execution in the framework of linear temporary market impact

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.

q-fin.TR↗

arXiv2019

Static vs Adaptive Strategies for Optimal Execution with Signals

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.

q-fin.TR↗

arXiv2019

On the consistency of jump-diffusion dynamics for FX rates under inversion

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

q-fin.MF↗