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We propose a deep signature/log-signature FBSDE algorithm to solve forward-backward stochastic differential equations (FBSDEs) with state and path dependent features. By incorporating the deep signature/log-signature transformation into the recurrent neural network (RNN) model, our algorithm shortens the training time, improves the accuracy, and extends the time horizon comparing to methods in the existing literature. Moreover, our algorithms can be applied to a wide range of applications such as state and path dependent option pricing involving high-frequency data, model ambiguity, and stochastic games, which are linked to parabolic partial differential equations (PDEs), and path-dependent PDEs (PPDEs). Lastly, we also derive the convergence analysis of the deep signature/log-signature FBSDE algorithm.
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This paper offers a general and comprehensive definition of the day-of-the-week effect. Using symbolic dynamics, we develop a unique test based on ordinal patterns in order to detect it. This test uncovers the fact that the so-called day-of-the-week effect is partly an artifact of the hidden correlation structure of the data. We present simulations based on artificial time series as well. Whereas time series generated with long memory are prone to exhibit daily seasonality, pure white noise signals exhibit no pattern preference. Since ours is a non parametric test, it requires no assumptions about the distribution of returns so that it could be a practical alternative to conventional econometric tests. We made also an exhaustive application of the here proposed technique to 83 stock indices around the world. Finally, the paper highlights the relevance of symbolic analysis in economic time series studies.
This paper describes another extension of the Local Variance Gamma model originally proposed by P. Carr in 2008, and then further elaborated on by Carr and Nadtochiy, 2017 (CN2017), and Carr and Itkin, 2018 (CI2018). As compared with the latest version of the model developed in CI2018 and called the ELVG (the Expanded Local Variance Gamma model), here we provide two innovations. First, in all previous papers the model was constructed based on a Gamma time-changed {it arithmetic} Brownian motion: with no drift in CI2017, and with drift in CI2018, and the local variance to be a function of the spot level only. In contrast, here we develop a {it geometric} version of this model with drift. Second, in CN2017 the model was calibrated to option smiles assuming the local variance is a piecewise constant function of strike, while in CI2018 the local variance is a piecewise linear} function of strike. In this paper we consider 3 piecewise linear models: the local variance as a function of strike, the local variance as function of log-strike, and the local volatility as a function of strike (so, the local variance is a piecewise quadratic function of strike). We show that for all these new constructions it is still possible to derive an ordinary differential equation for the option price, which plays a role of Dupires equation for the standard local volatility model, and, moreover, it can be solved in closed form. Finally, similar to CI2018, we show that given multiple smiles the whole local variance/volatility surface can be recovered which does not require solving any optimization problem. Instead, it can be done term-by-term by solving a system of non-linear algebraic equations for each maturity which is fast.
This paper presents how to apply the stochastic collocation technique to assets that can not move below a boundary. It shows that the polynomial collocation towards a lognormal distribution does not work well. Then, the potentials issues of the related collocated local volatility model (CLV) are explored. Finally, a simple analytical expression for the Dupire local volatility derived from the option prices modelled by stochastic collocation is given.
We extend the approach of Carr, Itkin and Muravey, 2021 for getting semi-analytical prices of barrier options for the time-dependent Heston model with time-dependent barriers by applying it to the so-called $lambda$-SABR stochastic volatility model. In doing so we modify the general integral transform method (see Itkin, Lipton, Muravey, Generalized integral transforms in mathematical finance, World Scientific, 2021) and deliver solution of this problem in the form of Fourier-Bessel series. The weights of this series solve a linear mixed Volterra-Fredholm equation (LMVF) of the second kind also derived in the paper. Numerical examples illustrate speed and accuracy of our method which are comparable with those of the finite-difference approach at small maturities and outperform them at high maturities even by using a simplistic implementation of the RBF method for solving the LMVF.
Todays consumer goods markets are rapidly evolving with significant growth in the number of information media as well as the number of competitive products. In this environment, obtaining a quantitative grasp of heterogeneous interactions of firms and customers, which have attracted interest of management scientists and economists, requires the analysis of extremely high-dimensional data. Existing approaches in quantitative research could not handle such data without any reliable prior knowledge nor strong assumptions. Alternatively, we propose a novel method called complex Hilbert principal component analysis (CHPCA) and construct a synchronization network using Hodge decomposition. CHPCA enables us to extract significant comovements with a time lead/delay in the data, and Hodge decomposition is useful for identifying the time-structure of correlations. We apply this method to the Japanese beer market data and reveal comovement of variables related to the consumer choice process across multiple products. Furthermore, we find remarkable customer heterogeneity by calculating the coordinates of each customer in the space derived from the results of CHPCA. Lastly, we discuss the policy and managerial implications, limitations, and further development of the proposed method.
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