Finance and Stochastics最新文献

Guyon and Lekeufack (Quant. Finance 23:1221–1258, 2023) recently proposed a path-dependent volatility model and documented its excellent performance in fitting market data and capturing stylised facts. The instantaneous volatility is modelled as a linear combination of two processes; one is an integral of weighted past price returns and the other is the square root of an integral of weighted past squared volatility. Each weighting is built using two exponential kernels reflecting long and short memory. Mathematically, the model is a coupled system of four stochastic differential equations. Our main result is the wellposedness of this system: the model has a unique strong (non-explosive) solution for all parameter values. We also study the positivity of the resulting volatility process and the martingale property of the associated exponential price process.

In this paper, we show that Hilbert space-valued stochastic models are robust with respect to perturbations, due to measurement or approximation errors, in the underlying volatility process. Within the class of stochastic-volatility-modulated Ornstein–Uhlenbeck processes, we quantify the error induced by the volatility in terms of perturbations in the parameters of the volatility process. We moreover study the robustness of the volatility process itself with respect to finite-dimensional approximations of the driving compound Poisson process and semigroup generator, respectively, when considering operator-valued Barndorff-Nielsen and Shephard stochastic volatility models. We also give results on square root approximations. In all cases, we provide explicit bounds for the induced error in terms of the approximation of the underlying parameter. We discuss some applications to robustness of prices of options on forwards and volatility.

This paper introduces stochastic covariance models in Hilbert spaces with stationary affine instantaneous covariance processes. We explore the applications of these models in the context of forward curve dynamics within fixed-income and commodity markets. The affine instantaneous covariance process is defined on positive self-adjoint Hilbert–Schmidt operators, and we prove the existence of a unique limit distribution for subcritical affine processes, provide convergence rates of the transition kernels in the Wasserstein distance of order (p in [1,2]), and give explicit formulas for the first two moments of the limit distribution. Our results allow us to introduce affine stochastic covariance models in the stationary covariance regime and to investigate the behaviour of the implied forward volatility for large forward dates in commodity forward markets.

We propose an extension of the model introduced by Barndorff-Nielsen and Shephard, based on stochastic processes of Ornstein–Uhlenbeck type taking values in Hilbert spaces and including the leverage effect. We compute explicitly the characteristic function of the log-return and the volatility processes. By introducing a measure change of Esscher type, we provide a relation between the dynamics described with respect to the historical and the risk-neutral measures. We discuss in detail the application of the proposed model to describe the commodity forward curve dynamics in a Heath–Jarrow–Morton framework, including the modelling of forwards with delivery period occurring in energy markets and the pricing of options. For the latter, we show that a Fourier approach can be applied in this infinite-dimensional setting, relying on the attractive property of conditional Gaussianity of our stochastic volatility model. In our analysis, we study both arithmetic and geometric models of forward prices and provide appropriate martingale conditions in order to ensure arbitrage-free dynamics.

Dybvig (1988a, 1988b) solves in a complete market setting the problem of finding a payoff that is cheapest possible in reaching a given target distribution (“cost-efficient payoff”). In the presence of ambiguity, the distribution of a payoff is, however, no longer known with certainty. We study the problem of finding the cheapest possible payoff whose worst-case distribution stochastically dominates a given target distribution (“robust cost-efficient payoff”) and determine solutions under certain conditions. We study the link between “robust cost-efficiency” and the maxmin expected utility setting of Gilboa and Schmeidler (1989), as well as more generally in a possibly nonexpected robust utility setting. Specifically, we show that solutions to maxmin robust expected utility are necessarily robust cost-efficient. We illustrate our study with examples involving uncertainty both on the drift and on the volatility of the risky asset.

This paper concerns a local volatility model in which the volatility takes two possible values, and the specific value depends on whether the underlying price is above or below a given threshold. The model is known, and a number of results have been obtained for it. In particular, option pricing formulas and a power-law behaviour of the implied volatility skew have been established in the case when the threshold is taken at the money. In this paper, we derive an alternative representation of option pricing formulas. In addition, we obtain an approximation of option prices by the corresponding Black–Scholes prices. Using this approximation streamlines obtaining the aforementioned behaviour of the skew. Our approach is based on the natural relationship of the model with skew Brownian motion and consists of the systematic use of the joint distribution of this stochastic process and some of its functionals.

We consider the problem of finding (Pareto-)optimal allocations of risk among finitely many agents. The associated individual risk measures are law-invariant, but with respect to agent-dependent and potentially heterogeneous reference probability measures. Moreover, we assume that the individual risk assessments are consistent with the respective second-order stochastic dominance relations, but remain agnostic about their convexity. A simple sufficient condition for the existence of Pareto optima is provided. The proof combines local comonotonic improvement with a Dieudonné-type argument, which also establishes a link of the optimal allocation problem to the realm of “collapse to the mean” results.

Motivated by the challenges related to the calibration of financial models, we consider the problem of numerically solving a singular McKean–Vlasov equation

where (W) is a Brownian motion and (v) is an adapted diffusion process. This equation can be considered as a singular local stochastic volatility model. While such models are quite popular among practitioners, its well-posedness has unfortunately not yet been fully understood and in general is possibly not guaranteed at all. We develop a novel regularisation approach based on the reproducing kernel Hilbert space (RKHS) technique and show that the regularised model is well posed. Furthermore, we prove propagation of chaos. We demonstrate numerically that a thus regularised model is able to perfectly replicate option prices coming from typical local volatility models. Our results are also applicable to more general McKean–Vlasov equations.

We show how inter-asset dependence information derived from market prices of options can lead to improved model-free price bounds for multi-asset derivatives. Depending on the type of the traded option, we either extract correlation information or derive restrictions on the set of admissible copulas that capture the inter-asset dependences. To compute the resulting price bounds for some multi-asset options of interest, we apply a modified martingale optimal transport approach. Several examples based on simulated and real market data illustrate the improvement of the obtained price bounds and thus provide evidence for the relevance and tractability of our approach.

We start with a stochastic control problem where the control process is of finite variation (possibly with jumps) and acts as integrator both in the state dynamics and in the target functional. Problems of such type arise in the stream of literature on optimal trade execution pioneered by Obizhaeva and Wang (J. Financ. Mark. 16:1–32, 2013) (models with finite resilience). We consider a general framework where the price impact and the resilience are stochastic processes. Both are allowed to have diffusive components. First we continuously extend the problem from processes of finite variation to progressively measurable processes. Then we reduce the extended problem to a linear–quadratic (LQ) stochastic control problem. Using the well-developed theory on LQ problems, we describe the solution to the obtained LQ one and translate it back to the solution for the (extended) initial trade execution problem. Finally, we illustrate our results by several examples. Among other things, the examples discuss the Obizhaeva–Wang model with random (terminal and moving) targets, the necessity to extend the initial trade execution problem to a reasonably large class of progressively measurable processes (even going beyond semimartingales), and the effects of diffusive components in the price impact process and/or the resilience process.