39 research outputs found
The characteristic function of Gaussian stochastic volatility models: an analytic expression
Stochastic volatility models based on Gaussian processes, like fractional
Brownian motion, are able to reproduce important stylized facts of financial
markets such as rich autocorrelation structures, persistence and roughness of
sample paths. This is made possible by virtue of the flexibility introduced in
the choice of the covariance function of the Gaussian process. The price to pay
is that, in general, such models are no longer Markovian nor semimartingales,
which limits their practical use. We derive, in two different ways, an explicit
analytic expression for the joint characteristic function of the log-price and
its integrated variance in general Gaussian stochastic volatility models. Such
analytic expression can be approximated by closed form matrix expressions
stemming from Wishart distributions. This opens the door to fast approximation
of the joint density and pricing of derivatives on both the stock and its
realized variance using Fourier inversion techniques. In the context of rough
volatility modeling, our results apply to the (rough) fractional Stein-Stein
model and provide the first analytic formulae for option pricing known to date,
generalizing that of Stein-Stein, Sch{\"o}bel-Zhu and a special case of Heston
Lifting the Heston model
International audienceHow to reconcile the classical Heston model with its rough counterpart? We introduce a lifted version of the Heston model with n multi-factors, sharing the same Brownian motion but mean reverting at different speeds. Our model nests as extreme cases the classical Heston model (when n = 1), and the rough Heston model (when n goes to infinity). We show that the lifted model enjoys the best of both worlds: Markovianity and satisfactory fits of implied volatility smiles for short maturities with very few parameters. Further, our approach speeds up the calibration time and opens the door to time-efficient simulation schemes
Affine Volterra processes
We introduce affine Volterra processes, defined as solutions of certain
stochastic convolution equations with affine coefficients. Classical affine
diffusions constitute a special case, but affine Volterra processes are neither
semimartingales, nor Markov processes in general. We provide explicit
exponential-affine representations of the Fourier-Laplace functional in terms
of the solution of an associated system of deterministic integral equations of
convolution type, extending well-known formulas for classical affine
diffusions. For specific state spaces, we prove existence, uniqueness, and
invariance properties of solutions of the corresponding stochastic convolution
equations. Our arguments avoid infinite-dimensional stochastic analysis as well
as stochastic integration with respect to non-semimartingales, relying instead
on tools from the theory of finite-dimensional deterministic convolution
equations. Our findings generalize and clarify recent results in the literature
on rough volatility models in finance
Gaussian Agency problems with memory and Linear Contracts
Can a principal still offer optimal dynamic contracts that are linear in
end-of-period outcomes when the agent controls a process that exhibits memory?
We provide a positive answer by considering a general Gaussian setting where
the output dynamics are not necessarily semi-martingales or Markov processes.
We introduce a rich class of principal-agent models that encompasses dynamic
agency models with memory. From the mathematical point of view, we develop a
methodology to deal with the possible non-Markovianity and non-semimartingality
of the control problem, which can no longer be directly solved by means of the
usual Hamilton-Jacobi-Bellman equation. Our main contribution is to show that,
for one-dimensional models, this setting always allows for optimal linear
contracts in end-of-period observable outcomes with a deterministic optimal
level of effort. In higher dimension, we show that linear contracts are still
optimal when the effort cost function is radial and we quantify the gap between
linear contracts and optimal contracts for more general quadratic costs of
efforts
Equilibrium in Functional Stochastic Games with Mean-Field Interaction
We consider a general class of finite-player stochastic games with mean-field
interaction, in which the linear-quadratic cost functional includes linear
operators acting on controls in . We propose a novel approach for deriving
the Nash equilibrium of the game explicitly in terms of operator resolvents, by
reducing the associated first order conditions to a system of stochastic
Fredholm equations of the second kind and deriving their closed form solution.
Furthermore, by proving stability results for the system of stochastic Fredholm
equations we derive the convergence of the equilibrium of the -player game
to the corresponding mean-field equilibrium. As a by-product we also derive an
-Nash equilibrium for the mean-field game, which is valuable in
this setting as we show that the conditions for existence of an equilibrium in
the mean-field limit are less restrictive than in the finite-player game.
Finally we apply our general framework to solve various examples, such as
stochastic Volterra linear-quadratic games, models of systemic risk and
advertising with delay, and optimal liquidation games with transient price
impact.Comment: 48 page
Gaussian Agency problems with memory and Linear Contracts
Can a principal still offer optimal dynamic contracts that are linear in end-of-period outcomes when the agent controls a process that exhibits memory? We provide a positive answer by considering a general Gaussian setting where the output dynamics are not necessarily semi-martingales or Markov processes. We introduce a rich class of principal-agent models that encompasses dynamic agency models with memory. From the mathematical point of view, we develop a methodology to deal with the possible non-Markovianity and non-semimartingality of the control problem, which can no longer be directly solved by means of the usual Hamilton-Jacobi-Bellman equation. Our main contribution is to show that, for one-dimensional models, this setting always allows for optimal linear contracts in end-of-period observable outcomes with a deterministic optimal level of effort. In higher dimension, we show that linear contracts are still optimal when the effort cost function is radial and we quantify the gap betweenlinear contracts and optimal contracts for more general quadratic costs of efforts
The Laplace transform of the integrated Volterra Wishart process
International audienceWe establish an explicit expression for the conditional Laplace transform of the integrated Volterra Wishart process in terms of a certain resolvent of the covariance function. The core ingredient is the derivation of the conditional Laplace transform of general Gaussian processes in terms of Fredholm's determinant and resolvent. Furthermore , we link the characteristic exponents to a system of non-standard infinite dimensional matrix Riccati equations. This leads to a second representation of the Laplace transform for a special case of convolution kernel. In practice, we show that both representations can be approximated by either closed form solutions of conventional Wishart distributions or finite dimensional matrix Riccati equations stemming from conventional linear-quadratic models. This allows fast pricing in a variety of highly flexible models, ranging from bond pricing in quadratic short rate models with rich autocorrelation structures, long range dependence and possible default risk, to pricing basket options with covariance risk in multivariate rough volatility models
Weak existence and uniqueness for affine stochastic Volterra equations with L1-kernels
International audienceWe provide existence, uniqueness and stability results for affine stochastic Volterra equations with -kernels and jumps. Such equations arise as scaling limits of branching processes in population genetics and self-exciting Hawkes processes in mathematical finance. The strategy we adopt for the existence part is based on approximations using stochastic Volterra equations with -kernels combined with a general stability result. Most importantly, we establish weak uniqueness using a duality argument on the Fourier--Laplace transform via a deterministic Riccati--Volterra integral equation. We illustrate the applicability of our results on Hawkes processes and a class of hyper-rough Volterra Heston models with a Hurst index