Beatrice Acciaio

• Professor (Full) at ETH Zurich

In this paper, we provide a quantitative analysis of the concept of arbitrage, that allows us to deal with model uncertainty without imposing the no‐arbitrage condition. In markets that admit “small arbitrage,” we can still make sense of the problems of pricing and hedging. The pricing measures here will be such that asset price processes are close...

The adapted Wasserstein distance is a metric for quantifying distributional uncertainty and assessing the sensitivity of stochastic optimization problems on time series data. A computationally efficient alternative to it, is provided by the entropically regularized adapted Wasserstein distance. Suffering from similar shortcomings as classical optim...

We build a time-causal variational autoencoder (TC-VAE) for robust generation of financial time series data. Our approach imposes a causality constraint on the encoder and decoder networks, ensuring a causal transport from the real market time series to the fake generated time series. Specifically, we prove that the TC-VAE loss provides an upper bo...

We study the contracting problem in the persistent private information model of Williams [17], in which an agent provides a report of a privately observed path to the principal, who in turn pays the agent in the least expensive way that induces truthful reporting. We first argue that, in the case of persistent information, the contract in [17] does...

We introduce graphs associated to transport problems between discrete marginals, that allow to characterize the set of all optimizers given one primal optimizer. In particular, we establish that connectivity of those graphs is a necessary and sufficient condition for uniqueness of the dual optimizers. Moreover, we provide an algorithm that can effi...

The aim of this workshop was to convene experts for fostering the discussion and the development of innovative approaches in insurance and financial mathematics. New challenges like price instability, huge insurance claims and climate change are affecting the markets, while at the same time the possibility of using large volumes of data and continu...

We regard the optimal reinsurance problem as an iterated optimal transport problem between a (known) initial and an (unknown) resulting risk exposure of the insurer. We also provide conditions that allow to characterize the support of optimal treaties, and show how this can be used to deduce the shape of the optimal contract, reducing the task to a...

We introduce graphs associated to transport problems between discrete marginals, that allow to characterize the set of all optimizers given one primal optimizer. In particular, we establish that connectivity of those graphs is a necessary and sufficient condition for uniqueness of the dual optimizers. Moreover, we provide an algorithm that can effi...

Several problems in stochastic analysis are defined through their geometry, and preserving that geometric structure is essential to generating meaningful predictions. Nevertheless, how to design principled deep learning (DL) models capable of encoding these geometric structures remains largely unknown. We address this open problem by introducing a...

We consider empirical measures of $\mathbb{R}^{d}$-valued stochastic process in finite discrete-time. We show that the adapted empirical measure introduced in the recent work \cite{backhoff2022estimating} by Backhoff et al. in compact spaces can be defined analogously on $\mathbb{R}^{d}$, and that it converges almost surely to the underlying measur...

In this paper we provide a quantitative analysis to the concept of arbitrage, that allows to deal with model uncertainty without imposing the no-arbitrage condition. In markets that admit ``small arbitrage", we can still make sense of the problems of pricing and hedging. The pricing measures here will be such that asset price processes are close to...

We provide a short proof of the intriguing characterisation of the convex order given by Wiesel and Zhang.

From ecology to atmospheric sciences, many academic disciplines deal with data characterized by intricate spatio-temporal complexities, the modeling of which often requires specialized approaches. Generative models of these data are of particular interest, as they enable a range of impactful downstream applications like simulation or creating synth...

Several problems in stochastic analysis are defined through their geometry, and preserving that geometric structure is essential to generating meaningful predictions. Nevertheless, how to design principled deep learning (DL) models capable of encoding these geometric structures remains largely unknown. We address this open problem by introducing a...

From ecology to atmospheric sciences, many academic disciplines deal with data characterized by intricate spatio-temporal complexities, the modeling of which often requires specialized approaches. Generative models of these data are of particular interest, as they enable a range of impactful downstream applications like simulation or creating synth...

We consider a model‐independent pricing problem in a fixed‐income market and show that it leads to a weak optimal transport problem as introduced by Gozlan et al. We use this to characterize the extremal models for the pricing of caplets on the spot rate and to establish a first robust super‐replication result that is applicable to fixed‐income mar...

Causal Optimal Transport (COT) results from imposing a temporal causality constraint on classic optimal transport problems, which naturally generates a new concept of distances between distributions on path spaces. The first application of the COT theory for sequential learning was given in Xu et al. (2020), where COT-GAN was introduced as an adver...

We consider a model-independent pricing problem in a fixed-income market and show that it leads to a weak optimal transport problem as introduced by Gozlan et al. We use this to characterize the extremal models for the pricing of caplets on the spot rate and to establish a first robust super-replication result that is applicable to fixed-income mar...

We introduce COT-GAN, an adversarial algorithm to train implicit generative models optimized for producing sequential data. The loss function of this algorithm is formulated using ideas from Causal Optimal Transport (COT), which combines classic optimal transport methods with an additional temporal causality constraint. Remarkably, we find that thi...

We consider a large population dynamic game in discrete time. The peculiarity of the game is that players are characterized by time-evolving types, and so reasonably their actions should not anticipate the future values of their types. When interactions between players are of mean-field kind, we relate Nash equilibria for such games to an asymptoti...

It has often been stated that, within the class of continuous stochastic volatility models calibrated to vanillas, the price of a VIX future is maximized by the Dupire local volatility model. In this article we prove that this statement is incorrect: we build a continuous stochastic volatility model in which a VIX future is strictly more expensive...

It has often been stated that, within the class of continuous stochastic volatility models calibrated to vanillas, the price of a VIX future is maximized by the Dupire local volatility model. In this article we prove that this statement is incorrect: we build a continuous stochastic volatility model in which a VIX future is strictly more expensive...

We study Mean Field stochastic control problems where the cost function and the state dynamics depend upon the joint distribution of the controlled state and the control process. We prove suitable versions of the Pontryagin stochastic maximum principle, both in necessary and in sufficient form, which extend the known conditions to this general fram...

Semi-static trading strategies make frequent appearances in mathematical finance, where dynamic trading in a liquid asset is combined with static buy-and-hold positions in options on that asset. We show that the space of outcomes of such strategies can have very poor closure properties when all European options for a fixed date $T$ are available fo...

The martingale part in the semimartingale decomposition of a Brownian motion with respect to an enlargement of its filtration, is an anticipative mapping of the given Brownian motion. In analogy to optimal transport theory, we define causal transport plans in the context of enlargement of filtrations, as the Kantorovich counterparts of the aforemen...

The martingale part in the semimartingale decomposition of a Brownian motion with respect to an enlargement of its filtration, is an anticipative mapping of the given Brownian motion. In analogy to optimal transport theory, we define causal transport plans in the context of enlargement of filtrations, as the Kantorovich counterparts of the aforemen...

In this paper, we consider the pricing and hedging of a financial derivative for an insider trader, in a model-independent setting. In particular, we suppose that the insider wants to act in a way which is independent of any modelling assumptions, but that she observes market information in the form of the prices of vanilla call options on the asse...

In this paper, we consider the pricing and hedging of a financial derivative for an insider trader, in a model-independent setting. In particular, we suppose that the insider wants to act in a way which is independent of any modelling assumptions, but that she observes market information in the form of the prices of vanilla call options on the asse...

Semi-static trading strategies make frequent appearances in mathematical finance, where dynamic trading in a liquid asset is combined with static buy-and-hold positions in options on that asset. We show that the space of outcomes of such strategies can have very poor closure properties when all European options for a fixed date $T$ are available fo...

In this paper we study the problem of minimizing the area for the chord-convex sets of given size, that is, the sets for which each bisecting chord is a segment of length at least 2. This problem has been already studied and solved in the framework of convex sets, though nothing has been said in the non-convex case. We introduce here the relevant c...

We consider a continuous-time financial market that consists of securities available for dynamic trading, and securities only available for static trading. We work in a robust framework where a set of non-dominated models is given. The concept of semi-static completeness is introduced: it corresponds to having exact replication by means of semi-sta...

We consider a continuous-time financial market that consists of securities available for dynamic trading, and securities only available for static trading. We work in a robust framework where a set of non-dominated models is given. The concept of semi-static completeness is introduced: it corresponds to having exact replication by means of semi-sta...

We characterize the family of non-negative max-continuous local martingales vanishing at infinity via their times of maximum. This is an extension to the case of general filtrations of a result proved by Nikeghbali and Yor [NY06] for continuous filtrations.

In a general semimartingale financial model, we study the stability of the No Arbitrage of the First Kind (View the MathML sourceNA1) (or, equivalently, No Unbounded Profit with Bounded Risk) condition under initial and under progressive filtration enlargements. In both cases, we provide a simple and general condition which is sufficient to ensure...

While it is reasonable to assume that convex combinations on the level of random variables lead to a reduction of risk (diversification effect), this is no more true on the level of distributions. In the latter case, taking convex combinations corresponds to adding a risk factor. Hence, whereas asking for convexity of risk functions defined on rand...

We propose a Fundamental Theorem of Asset Pricing and a Super-Replication Theorem in a model-independent framework. We prove these theorems in the setting of finite, discrete time and a market consisting of a risky asset S as well as options written on this risky asset. As a technical condition, we assume the existence of a traded option with a sup...

We present a unified approach to Doob's $L^p$ maximal inequalities for $1\leq p<\infty$. The novelty of our method is that these martingale inequalities are obtained as consequences of elementary deterministic counterparts. The latter have a natural interpretation in terms of robust hedging. Moreover, our deterministic inequalities lead to new vers...

We consider conditional convex risk measures on L p and show their robust representation in a standard way. Such measures are used as evaluation functionals for optimal portfolio selection in a Black&Scholes setting. We study this problem focusing on the conditional Average Value at Risk and the conditional entropic risk measure and compare the r...

We prove that in a discrete-time market model the lower arbitrage bound of an American contingent claim is itself an arbitrage-free price if and only if it corresponds to the price of the claim optimally exercised under some equivalent martingale measure.

We propose a reduced-form model for credit risk in a multivariate setting. The default intensities are linear combinations of three independent affine jump-diffusion processes representing the intensities of general, sectoral and idiosyncratic credit events. The model can be efficiently calibrated to term structures of default probabilities and con...

We study the risk assessment of uncertain cash flows in terms of dynamic convex risk measures for processes as introduced in Cheridito, Delbaen, and Kupper (2006). These risk measures take into account not only the amounts but also the timing of a cash flow. We discuss their robust representation in terms of suitably penalized probability measures...

This paper gives an overview of the theory of dynamic convex risk measures for random variables in discrete time setting. We summarize robust representation results of conditional convex risk measures, and we characterize various time consistency properties of dynamic risk measures in terms of acceptance sets, penalty functions, and by supermarting...

We investigate the problem of optimal risk sharing between agents endowed with cash-invariant choice functions which are law-invariant with respect to different reference probability measures. We motivate a discrete setting both from an operational and a theoretical point of view, and give sufficient conditions for the existence of Pareto optimal a...

We model agents’ preferences by cash-invariant concave functionals defined on L ∞, and formulate the optimal risk allocation problem as their infimal-convolution. We study the case of agents whose choice functionals are law-invariant with respect to different probability measures and show how, in this case, the value function preserves a desirable...

The paper proposes a reduced-form model for credit risk in a multivariate setting. The default intensities are linear combinations of three independent affine jump-diffusion pro-cesses that can be interpreted as the intensities of general, sectoral and idiosyncratic credit events. The model is rather flexible and can be efficiently calibrated to te...

We consider the problem of sharing pooled risks among n economic agents endowed with non-necessarily monotone monetary functionals. In this framework, results of characterization and existence of optimal solutions are easily obtained as extensions from the convex risk measures setting. Moreover, the introduction of the best monotone approximation o...

In this paper, we consider the problem of maximizing the expected utility of terminal wealth in the framework of incomplete financial markets. In particular, we analyze the case where an economic agent, who aims at such an optimization, achieves infinite wealth with strictly positive probability. By convex duality theory, this is shown to be equiva...

We prove the existence of radial solutions of the quasilinear elliptic equation div(A(|Du|)Du) + f (u) = 0 in R n , n > 1, where f is either negative or positive for small u > 0, possibly singular at u = 0, and growths subcritically for large u. Our proofs use only elementary arguments based on a variational identity. No differentiability assumptio...

Recently, Jouini et al. (2005) studied the problem of optimal sharing of aggregate risks between two economic agents endowed with monetary utility func- tions. Strongly inspired by this paper, we consider the analogous problem when ad- mitting any number of agents characterized by non-necessarily monotone choice cri- terions, and generalize some of...

We study the survival probability of an homogeneous group of economic agents by adopting the reduced-form approach and assuming an ane evolution of the default intensities. We consider both the cases of continuous and discrete-times observations, and propose a modified likelihood function. We discuss the estimation of the parameters of interest and...