95 research outputs found
Measure-valued processes for energy markets
No full textWe introduce a framework that allows to employ (non-negative) measure-valued processes for energy market modeling, in particular for electricity and gas futures. Interpreting the process' spatial structure as time to maturity, we show how the Heath-Jarrow-Morton approach (see Heath et al. (1992)) can be translated to this framework, thus guaranteeing arbitrage free modeling in infinite dimensions,
while allowing for the incorporation of important stylized facts, in particular stochastic discontinuities, i.e. jumps or spikes at pre-specified (deterministic) dates. We derive an analog to the HJM-drift condition and then treat in a Markovian setting existence of non-negative measure-valued diffusions that satisfy this condition. To analyze mathematically convenient classes we build on Cuchiero et al. (2024) and consider measure-valued polynomial and affine diffusions,
where we can precisely specify the diffusion part in terms of continuous
functions satisfying certain admissibility conditions. For calibration purposes these functions
can then be parameterized by neural networks yielding measure-valued analogs of neural SPDEs.
By combining Fourier
approaches or the moment formula with stochastic gradient descent methods, this then allows for tractable calibration procedures which we also test by way of example on market data
Infinite dimensional polynomial processes
No full textWe introduce polynomial processes taking values in an arbitrary Banach space B via their infinitesimal generator L and the associated martingale problem. We obtain two representations of the (conditional) moments in terms of solutions of a system of ODEs on the truncated tensor algebra of dual respectively bidual spaces. We illustrate how the well-known moment formulas for finite-dimensional or probability-measure-valued polynomial processes can be deduced in this general framework. As an application, we consider polynomial forward variance curve models which appear in particular as Markovian lifts of (rough) Bergomi-type volatility models. Moreover, we show that the signature process of a d -dimensional Brownian motion is polynomial and derive its expected value via the polynomial approach
Risk measures under model uncertainty: A Bayesian viewpoint
No full textRisk measures, model risk, robust finance, mixture probability measure,
Bayesian methods in financ
Holomorphic jump-diffusions
No full textWe introduce a class of jump-diffusions, called holomorphic, of which the well-known classes of affine and polynomial processes are particular instances. The defining property concerns the extended generator, which is required to map a (subset of) holomorphic functions to themselves. This leads to a representation of the expectation of power series of the process’ marginals via a potentially infinite dimensional linear ODE. We apply the same procedure by considering exponentials of holomorphic functions, leading to a class of processes named affine-holomorphic for which a representation for quantities as the characteristic function of power series is provided. Relying on powerful results from complex analysis, we obtain sufficient conditions on the process’ characteristics which guarantee the holomorphic and affine-holomorphic properties and provide applications to several classes of jump-diffusions
Propagation of minimality in the supercooled Stefan problem
No full textSupercooled Stefan problems describe the evolution of the boundary between
the solid and liquid phases of a substance, where the liquid is assumed to be
cooled below its freezing point. Following the methodology of Delarue,
Nadtochiy and Shkolnikov, we construct solutions to the one-phase
one-dimensional supercooled Stefan problem through a certain McKean-Vlasov
equation, which allows to define global solutions even in the presence of
blow-ups. Solutions to the McKean-Vlasov equation arise as mean-field limits of
particle systems interacting through hitting times, which is important for
systemic risk modeling. Our main contributions are: (i) we prove a general
tightness theorem for the Skorokhod M1-topology which applies to processes that
can be decomposed into a continuous and a monotone part. (ii) We prove
propagation of chaos for a perturbed version of the particle system for general
initial conditions. (iii) We prove a conjecture of Delarue, Nadtochiy and
Shkolnikov, relating the solution concepts of so-called minimal and physical
solutions, showing that minimal solutions of the McKean-Vlasov equation are
physical whenever the initial condition is integrable.Comment: To appear in Annals of Applied Probabilit
Signature-based models: theory and calibration
No full textWe consider asset price models whose dynamics are described by linear functions of the (time extended) signature of a primary underlying process, which can range from a (market-inferred) Brownian motion to a general multidimensional continuous semimartingale. The framework is universal in the sense that classical models can be approximated arbitrarily well and that the model’s parameters can be learned from all sources of available data by simple methods. We provide conditions guaranteeing absence of arbitrage as well as tractable option pricing formulas for so-called sig-payoffs, exploiting the polynomial nature of generic primary processes. One of our main focus lies on calibration, where we consider both time-series and implied volatility surface data, generated from classical stochastic volatility models and also from S&P 500 index market data. For both tasks the linearity of the model turns out to be the crucial tractability feature which allows to get fast and accurate calibrations results
Universal approximation theorems for continuous functions of càdlàg paths and Lévy-type signature models
No full textWe prove two versions of a universal approximation theorem that allow to approximate continuous functions of ca`dla`g (rough) paths via linear functionals of their time-extended signature, one with respect to the Skorokhod J1-topology and the other one with respect to (a rough path version of) the Skorokhod M1-topology. Our main motivation to treat this question comes from signature-based models for finance that allow for the inclusion of jumps. Indeed, as an important application, we define a new class of universal signature models based on an augmented L ́evy process, which we call L ́evy-type signature models. They extend continuous signature models for asset prices as proposed e.g. by Arribas et al. (2020) in several directions, while still preserving universality and tractability properties. To analyze this, we first show that the signature process of a generic multivariate L ́evy process is a polynomial process on the extended tensor algebra and then use this for pricing and hedging approaches within L ́evy-type signature models
Signature SDEs from an affine and polynomial perspective
No full textSignature stochastic differential equations (SDEs) constitute a large class of stochas- tic processes, here driven by Brownian motions, whose characteristics are entire or real- analytic functions of their own signature, i.e. of iterated integrals of the process with itself, and allow therefore for a generic path dependence. We show that their pro- longation with the corresponding signature is an affine and polynomial process taking values in subsets of group-like elements of the extended tensor algebra. By relying on the duality theory for affine and polynomial processes we obtain explicit formulas in terms of novel and proper notions of converging power series for the Fourier-Laplace transform and the expected value of entire functions of the signature process. The coefficients of these power series are solutions of extended tensor algebra valued Ric- cati and linear ordinary differential equations (ODEs), respectively, whose vector fields can be expressed in terms of the entire characteristics of the corresponding SDEs. In other words, we construct a class of stochastic processes, which is universal within Itˆo processes with path-dependent characteristics and which allows for a relatively explicit characterization of the Fourier-Laplace transform and hence the full law on path space. We also analyze the special case of one-dimensional signature SDEs, which correspond to classical SDEs with real-analytic characteristics. Finally, the practical feasibility of this affine and polynomial approach is illustrated by several numerical examples
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