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The invariant distribution of wealth and employment status in a small open economy with precautionary savings
We study optimal savings in continuous time with exogenous transitions between employment and unemployment as the only source of uncertainty in a small open economy. We prove the existence of an optimal consumption path. We exploit that the dynamics of consumption and wealth between jumps can be expressed as a Fuchsian system. We derive conditions under which an invariant joint distribution for the state variables, i.e., wealth and labour market status, exists and is unique. We also provide conditions under which the distribution of these variables converges to the invariant distribution. Our analysis relies on the notion of T-processes and applies results on the stability of Markovian processes from Meyn and Tweedie (1993a, b,c). © 2019 The Author(s)
A functional limit theorem for limit order books
We consider a stochastic model for the dynamics of the two-sided limit order book (LOB). For the joint dynamics of best bid and ask prices and the standing buy and sell volume densities, we derive a functional limit theorem, which states that our LOB model converges to a continuous-time limit when the order arrival rates tend to infinity, the impact of an individual order arrival on the book as well as the tick size tend to zero. The limits of the standing buy and sell volume densities are described by two linear stochastic partial differential equations, which are coupled with a two-dimensional reflected Brownian motion that is the limit of the best bid and ask price processes.
Smoothing the payoff for efficient computation of basket option prices
We consider the problem of pricing basket options in a multivariate Black Scholes or Variance Gamma model. From a numerical point of view, pricing such options corresponds to moderate and high dimensional numerical integration problems with nonsmooth integrands. Due to this lack of regularity, higher order numerical integration techniques may not be directly available, requiring the use of methods like Monte Carlo specifically designed to work for non-regular problems. We propose to use the inherent smoothing property of the density of the underlying in the above models to mollify the payoff function by means of an exact conditional expectation. The resulting conditional expectation is unbiased and yields a smooth integrand, which is amenable to the efficient use of adaptive sparse grid cubature. Numerical examples indicate that the high-order method may perform orders of magnitude faster compared to Monte Carlo or Quasi Monte Carlo in dimensions up to 25.
Asymptotics for at the money local vol basket options
We consider a basket or spread option on based on a multi-dimensional local volatility model. Bayer and Laurence [Comm. Pure. Appl. Math., to appear] derived highly accurate analytic formulas for prices and implied volatilities of such options when the options are not at the money. We now extend these results to the ATM case. Moreover, we also derive similar formulas for the local volatility of the basket.
A regularity structure for rough volatility
A new paradigm has emerged recently in financial modeling: rough (stochastic) volatility. First observed by Gatheral et al. in high-frequency data, subsequently derived within market microstructure models, rough volatility captures parsimoniously key-stylized facts of the entire implied volatility surface, including extreme skews (as observed earlier by Alòs et al.) that were thought to be outside the scope of stochastic volatility models. On the mathematical side, Markovianity and, partially, semimartingality are lost. In this paper, we show that Hairer's regularity structures, a major extension of rough path theory, which caused a revolution in the field of stochastic partial differential equations, also provide a new and powerful tool to analyze rough volatility models.
Pricing American options by exercise rate optimization
We present a novel method for the numerical pricing of American options based on Monte Carlo simulation and the optimization of exercise strategies. Previous solutions to this problem either explicitly or implicitly determine so-called optimal exercise regions, which consist of points in time and space at which a given option is exercised. In contrast, our method determines the exercise rates of randomized exercise strategies. We show that the supremum of the corresponding stochastic optimization problem provides the correct option price. By integrating analytically over the random exercise decision, we obtain an objective function that is differentiable with respect to perturbations of the exercise rate even for finitely many sample paths. The global optimum of this function can be approached gradually when starting from a constant exercise rate. Numerical experiments on vanilla put options in the multivariate Black-Scholes model and a preliminary theoretical analysis underline the efficiency of our method, both with respect to the number of time-discretization steps and the required number of degrees of freedom in the parametrization of the exercise rates. Finally, we demonstrate the flexibility of our method through numerical experiments on max call options in the classical Black-Scholes model, and vanilla put options in both the Heston model and the non-Markovian rough Bergomi model.
Numerical smoothing with hierarchical adaptive sparse grids and quasi-Monte Carlo methods for efficient option pricing
When approximating the expectation of a functional of a stochastic process, the efficiency and performance of deterministic quadrature methods, such as sparse grid quadrature and quasi-Monte Carlo (QMC) methods, may critically depend on the regularity of the integrand. To overcome this issue and reveal the available regularity, we consider cases in which analytic smoothing cannot be performed, and introduce a novel numerical smoothing approach by combining a root finding algorithm with one-dimensional integration with respect to a single well-selected variable. We prove that under appropriate conditions, the resulting function of the remaining variables is a highly smooth function, potentially affording the improved efficiency of adaptive sparse grid quadrature (ASGQ) and QMC methods, particularly when combined with hierarchical transformations (i.e., Brownian bridge and Richardson extrapolation on the weak error). This approach facilitates the effective treatment of high dimensionality. Our study is motivated by option pricing problems, and our focus is on dynamics where the discretization of the asset price is necessary. Based on our analysis and numerical experiments, we show the advantages of combining numerical smoothing with the ASGQ and QMC methods over ASGQ and QMC methods without smoothing and the Monte Carlo approach.
Low-dimensional approximations of high-dimensional asset price models
We consider high-dimensional asset price models that are reduced in their dimension in order to reduce the complexity of the problem or the effect of the curse of dimensionality in the context of option pricing. We apply model order reduction (MOR) to obtain a reduced system. MOR has been previously studied for asymptotically stable controlled stochastic systems with zero initial conditions. However, stochastic differential equations modeling price processes are uncontrolled, have non-zero initial states and are often unstable. Therefore, we extend MOR schemes and combine ideas of techniques known for deterministic systems. This leads to a method providing a good pathwise approximation. After explaining the reduction procedure, the error of the approximation is analyzed and the performance of the algorithm is shown conducting several numerical experiments. Within the numerics section, the benefit of the algorithm in the context of option pricing is pointed out.
Forward-reverse EM algorithm for Markov chains
We develop an EM algorithm for estimating parameters that determine the dynamics of a discrete time Markov chain evolving through a certain measurable state space. As a key tool for the construction of the EM method we develop forward-reverse representations for Markov chains conditioned on a certain terminal state. These representations may be considered as an extension of the earlier work [1] on conditional diffusions. We present several experiments and consider the convergence of the new EM algorithm.
Pricing high-dimensional Bermudan options with hierarchical tensor formats
An efficient compression technique based on hierarchical tensors for popular option pricing methods is presented. It is shown that the ``curse of dimensionality" can be alleviated for the computation of Bermudan option prices with the Monte Carlo least-squares approach as well as the dual martingale method, both using high-dimensional tensorized polynomial expansions. This discretization allows for a simple and computationally cheap evaluation of conditional expectations. Complexity estimates are provided as well as a description of the optimization procedures in the tensor train format. Numerical experiments illustrate the favourable accuracy of the proposed methods. The dynamical programming method yields results comparable to recent Neural Network based methods.