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preprint2016arXiv

Numerical analysis of an extended structural default model with mutual liabilities and jump risk

We consider a structural default model in an interconnected banking network as in Lipton [International Journal of Theoretical and Applied Finance, 19(6), 2016], with mutual obligations between each pair of banks. We analyse the model numerically for two banks with jumps in their asset value processes. Specifically, we develop a finite difference method for the resulting two-dimensional partial integro-differential equation, and study its stability and consistency. We then compute joint and marginal survival probabilities, as well as prices of credit default swaps (CDS), first-to-default swaps (FTD), credit and debt value adjustments (CVA and DVA). Finally, we calibrate the model to market data and assess the impact of jump risk.

preprint2016arXiv

Reduced Order Models for Pricing European and American Options under Stochastic Volatility and Jump-Diffusion Models

European options can be priced by solving parabolic partial(-integro) differential equations under stochastic volatility and jump-diffusion models like Heston, Merton, and Bates models. American option prices can be obtained by solving linear complementary problems (LCPs) with the same operators. A finite difference discretization leads to a so-called full order model (FOM). Reduced order models (ROMs) are derived employing proper orthogonal decomposition (POD). The early exercise constraint of American options is enforced by a penalty on subset of grid points. The presented numerical experiments demonstrate that pricing with ROMs can be orders of magnitude faster within a given model parameter variation range.

preprint2016arXiv

Mathematical Foundations of Realtime Equity Trading. Liquidity Deficit and Market Dynamics. Automated Trading Machines

We postulates, and then show experimentally, that liquidity deficit is the driving force of the markets. In the first part of the paper a kinematic of liquidity deficit is developed. The calculus-like approach, which is based on Radon--Nikodym derivatives and their generalization, allows us to calculate important characteristics of observable market dynamics. In the second part of the paper this calculus is used in an attempt to build a dynamic equation in the form: future price tend to the value maximizing the number of shares traded per unit time. To build a practical automated trading machine P&L dynamics instead of price dynamics is considered. This allows a trading automate resilient to catastrophic P&L drains to be built. The results are very promising, yet when all the fees and trading commissions are taken into account, are close to breakeven. In the end of the paper important criteria for automated trading systems are presented. We list the system types that can and cannot make money on the market. These criteria can be successfully applied not only by automated trading machines, but also by a human trader.

preprint2015arXiv

Realized Volatility Analysis in A Spin Model of Financial Markets

We calculate the realized volatility in the spin model of financial markets and examine the returns standardized by the realized volatility. We find that moments of the standardized returns agree with the theoretical values of standard normal variables. This is the first evidence that the return dynamics of the spin financial market is consistent with the view of the mixture-of-distribution hypothesis that also holds in the real financial markets.

preprint2014arXiv

Analysis of Spin Financial Market by GARCH Model

A spin model is used for simulations of financial markets. To determine return volatility in the spin financial market we use the GARCH model often used for volatility estimation in empirical finance. We apply the Bayesian inference performed by the Markov Chain Monte Carlo method to the parameter estimation of the GARCH model. It is found that volatility determined by the GARCH model exhibits "volatility clustering" also observed in the real financial markets. Using volatility determined by the GARCH model we examine the mixture-of-distribution hypothesis (MDH) suggested for the asset return dynamics. We find that the returns standardized by volatility are approximately standard normal random variables. Moreover we find that the absolute standardized returns show no significant autocorrelation. These findings are consistent with the view of the MDH for the return dynamics.

preprint2016arXiv

Mean-Reverting Portfolio Design via Majorization-Minimization Method

This paper considers the mean-reverting portfolio design problem arising from statistical arbitrage in the financial markets. The problem is formulated by optimizing a criterion characterizing the mean-reversion strength of the portfolio and taking into consideration the variance of the portfolio and an investment budget constraint at the same time. An efficient algorithm based on the majorization-minimization (MM) method is proposed to solve the problem. Numerical results show that our proposed mean-reverting portfolio design method can significantly outperform every underlying single spread and the benchmark method in the literature.

preprint2016arXiv

GPU Computing in Bayesian Inference of Realized Stochastic Volatility Model

The realized stochastic volatility (RSV) model that utilizes the realized volatility as additional information has been proposed to infer volatility of financial time series. We consider the Bayesian inference of the RSV model by the Hybrid Monte Carlo (HMC) algorithm. The HMC algorithm can be parallelized and thus performed on the GPU for speedup. The GPU code is developed with CUDA Fortran. We compare the computational time in performing the HMC algorithm on GPU (GTX 760) and CPU (Intel i7-4770 3.4GHz) and find that the GPU can be up to 17 times faster than the CPU. We also code the program with OpenACC and find that appropriate coding can achieve the similar speedup with CUDA Fortran.

preprint2015arXiv

Box-Cox transformation of firm size data in statistical analysis

Firm size data usually do not show the normality that is often assumed in statistical analysis such as regression analysis. In this study we focus on two firm size data: the number of employees and sale. Those data deviate considerably from a normal distribution. To improve the normality of those data we transform them by the Box-Cox transformation with appropriate parameters. The Box-Cox transformation parameters are determined so that the transformed data best show the kurtosis of a normal distribution. It is found that the two firm size data transformed by the Box-Cox transformation show strong linearity. This indicates that the number of employees and sale have the similar property as a firm size indicator. The Box-Cox parameters obtained for the firm size data are found to be very close to zero. In this case the Box-Cox transformations are approximately a log-transformation. This suggests that the firm size data we used are approximately log-normal distributions.

preprint2016arXiv

LSV models with stochastic interest rates and correlated jumps

Pricing and hedging exotic options using local stochastic volatility models drew a serious attention within the last decade, and nowadays became almost a standard approach to this problem. In this paper we show how this framework could be extended by adding to the model stochastic interest rates and correlated jumps in all three components. We also propose a new fully implicit modification of the popular Hundsdorfer and Verwer and Modified Craig-Sneyd finite-difference schemes which provides second order approximation in space and time, is unconditionally stable and preserves positivity of the solution, while still has a linear complexity in the number of grid nodes.

preprint2016arXiv

Model reduction for calibration of American options

American put options are among the most frequently traded single stock options, and their calibration is computationally challenging since no closed-form expression is available. Due to the higher flexibility in comparison to European options, the mathematical model involves additional constraints, and a variational inequality is obtained. We use the Heston stochastic volatility model to describe the price of a single stock option. In order to speed up the calibration process, we apply two model reduction strategies. Firstly, a reduced basis method (RBM) is used to define a suitable low-dimensional basis for the numerical approximation of the parameter-dependent partial differential equation ($μ$PDE) model. By doing so the computational complexity for solving the $μ$PDE is drastically reduced, and applications of standard minimization algorithms for the calibration are significantly faster than working with a high-dimensional finite element basis. Secondly, so-called de-Americanization strategies are applied. Here, the main idea is to reformulate the calibration problem for American options as a problem for European options and to exploit closed-form solutions. Both reduction techn

preprint2011arXiv

Computing Economic Equilibria by a Homotopy Method

In this paper the possibility of computing equilibrium in pure exchange and production economies by a homotopy method is investigated. The performance of the algorithm is tested on examples with known equilibria taken from the literature on general equilibrium models and numerical results are presented. In computing equilibria, economy will be specified by excess demand function.

preprint2015arXiv

A Robust Version of Convex Integral Functionals

We study the pointwise supremum of convex integral functionals $\mathcal{I}_{f,γ}(ξ)= \sup_{Q} \left( \int_Ωf(ω,ξ(ω))Q(dω)-γ(Q)\right)$ on $L^\infty(Ω,\mathcal{F},\mathbb{P})$ where $f:Ω\times\mathbb{R}\rightarrow\overline{\mathbb{R}}$ is a proper normal convex integrand, $γ$ is a proper convex function on the set of probability measures absolutely continuous w.r.t. $\mathbb{P}$, and the supremum is taken over all such measures. We give a pair of upper and lower bounds for the conjugate of $\mathcal{I}_{f,γ}$ as direct sums of a common regular part and respective singular parts; they coincide when $\mathrm{dom}(γ)=\{\mathbb{P}\}$ as Rockafellar's result, while both inequalities can generally be strict. We then investigate when the conjugate eliminates the singular measures, which a fortiori yields the equality in bounds, and its relation to other finer regularity properties of the original functional and of the conjugate.

preprint2016arXiv

Calibration to American Options: Numerical Investigation of the de-Americanization

American options are the reference instruments for the model calibration of a large and important class of single stocks. For this task, a fast and accurate pricing algorithm is indispensable. The literature mainly discusses pricing methods for American options that are based on Monte Carlo, tree and partial differential equation methods. We present an alternative approach that has become popular under the name de-Americanization in the financial industry. The method is easy to implement and enjoys fast run-times. Since it is based on ad hoc simplifications, however, theoretical results guaranteeing reliability are not available. To quantify the resulting methodological risk, we empirically test the performance of the de-Americanization method for calibration. We classify the scenarios in which de-Americanization performs very well. However, we also identify the cases where de-Americanization oversimplifies and can result in large errors.

preprint2016arXiv

Interplay between endogenous and exogenous fluctuations in financial markets

We address microscopic, agent based, and macroscopic, stochastic, modeling of the financial markets combining it with the exogenous noise. The interplay between the endogenous dynamics of agents and the exogenous noise is the primary mechanism responsible for the observed long-range dependence and statistical properties of high volatility return intervals. By exogenous noise we mean information flow or/and order flow fluctuations. Numerical results based on the proposed model reveal that the exogenous fluctuations have to be considered as indispensable part of comprehensive modeling of the financial markets.

preprint2016arXiv

A Finite Volume - Alternating Direction Implicit Approach for the Calibration of Stochastic Local Volatility Models

Calibration of stochastic local volatility (SLV) models to their underlying local volatility model is often performed by numerically solving a two-dimensional non-linear forward Kolmogorov equation. We propose a novel finite volume (FV) discretization in the numerical solution of general 1D and 2D forward Kolmogorov equations. The FV method does not require a transformation of the PDE. This constitutes a main advantage in the calibration of SLV models as the pertinent PDE coefficients are often nonsmooth. Moreover, the FV discretization has the crucial property that the total numerical mass is conserved. Applying the FV discretization in the calibration of SLV models yields a non-linear system of ODEs. Numerical time stepping is performed by the Hundsdorfer-Verwer ADI scheme to increase the computational efficiency. The non-linearity in the system of ODEs is handled by introducing an inner iteration. Ample numerical experiments are presented that illustrate the effectiveness of the calibration procedure.

preprint2016arXiv

Sparse grid high-order ADI scheme for option pricing in stochastic volatility models

We present a sparse grid high-order alternating direction implicit (ADI) scheme for option pricing in stochastic volatility models. The scheme is second-order in time and fourth-order in space. Numerical experiments confirm the computational efficiency gains achieved by the sparse grid combination technique.

preprint2016arXiv

Magic points in finance: Empirical integration for parametric option pricing

We propose an offline-online procedure for Fourier transform based option pricing. The method supports the acceleration of such essential tasks of mathematical finance as model calibration, real-time pricing, and, more generally, risk assessment and parameter risk estimation. We adapt the empirical magic point interpolation method of Barrault, Nguyen, Maday and Patera (2004) to parametric Fourier pricing. In the offline phase, a quadrature rule is tailored to the family of integrands of the parametric pricing problem. In the online phase, the quadrature rule then yields fast and accurate approximations of the option prices. Under analyticity assumptions the pricing error decays exponentially. Numerical experiments in one dimension confirm our theoretical findings and show a significant gain in efficiency, even for examples beyond the scope of the theoretical results.

preprint2016arXiv

Essentially high-order compact schemes with application to stochastic volatility models on non-uniform grids

We present high-order compact schemes for a linear second-order parabolic partial differential equation (PDE) with mixed second-order derivative terms in two spatial dimensions. The schemes are applied to option pricing PDE for a family of stochastic volatility models. We use a non-uniform grid with more grid-points around the strike price. The schemes are fourth-order accurate in space and second-order accurate in time for vanishing correlation. In our numerical convergence study we achieve fourth-order accuracy also for non-zero correlation. A combination of Crank-Nicolson and BDF-4 discretisation is applied in time. Numerical examples confirm that a standard, second-order finite difference scheme is significantly outperformed.

preprint2016arXiv

Pricing Bounds for VIX Derivatives via Least Squares Monte Carlo

Derivatives on the Chicago Board Options Exchange volatility index (VIX) have gained significant popularity over the last decade. The pricing of VIX derivatives involves evaluating the square root of the expected realised variance which cannot be computed by direct Monte Carlo methods. Least squares Monte Carlo methods can be used but the sign of the error is difficult to determine. In this paper, we propose new model independent upper and lower pricing bounds for VIX derivatives. In particular, we first present a general stochastic duality result on payoffs involving concave functions. This is then applied to VIX derivatives along with minor adjustments to handle issues caused by the square root function. The upper bound involves the evaluation of a variance swap, while the lower bound involves estimating a martingale increment corresponding to its hedging portfolio. Both can be achieved simultaneously using a single linear least square regression. Numerical results show that the method works very well for VIX futures, calls and puts under a wide range of parameter choices.

preprint2016arXiv

Numerical study of splitting methods for American option valuation

This paper deals with the numerical approximation of American-style option values governed by partial differential complementarity problems. For a variety of one- and two-asset American options we investigate by ample numerical experiments the temporal convergence behaviour of three modern splitting methods: the explicit payoff approach, the Ikonen-Toivanen approach and the Peaceman-Rachford method. In addition, the temporal accuracy of these splitting methods is compared to that of the penalty approach.

preprint2015arXiv

Double-jump stochastic volatility model for VIX: evidence from VVIX

The paper studies the continuous-time dynamics of VIX with stochastic volatility and jumps in VIX and volatility. Built on the general parametric affine model with stochastic volatility and jump in logarithm of VIX, we derive a linear relation between the stochastic volatility factor and VVIX index. We detect the existence of co-jump of VIX and VVIX and put forward a double-jump stochastic volatility model for VIX through its joint property with VVIX. With VVIX index as a proxy for the stochastic volatility, we use MCMC method to estimate the dynamics of VIX. Comparing nested models on VIX, we show the jump in VIX and the volatility factor is statistically significant. The jump intensity is also statedependent. We analyze the impact of jump factor on the VIX dynamics.

preprint2015arXiv

A computational spectral approach to interest rate models

The Polynomial Chaos Expansion (PCE) technique recovers a finite second order random variable exploiting suitable linear combinations of orthogonal polynomials which are functions of a given stochas- tic quantity ξ, hence acting as a kind of random basis. The PCE methodology has been developed as a mathematically rigorous Uncertainty Quantification (UQ) method which aims at providing reliable numerical estimates for some uncertain physical quantities defining the dynamic of certain engineering models and their related simulations. In the present paper we exploit the PCE approach to analyze some equity and interest rate models considering, without loss of generality, the one dimensional case. In particular we will take into account those models which are based on the Geometric Brownian Motion (gBm), e.g. the Vasicek model, the CIR model, etc. We also provide several numerical applications and results which are discussed for a set of volatility values. The latter allows us to test the PCE technique on a quite large set of different scenarios, hence providing a rather complete and detailed investigation on PCE-approximation's features and properties, such as the convergence of sta

preprint2016arXiv

On the difference between locally risk-minimizing and delta hedging strategies for exponential Lévy models

We discuss the difference between locally risk-minimizing and delta hedging strategies for exponential Lévy models, where delta hedging strategies in this paper are defined under the minimal martingale measure. We give firstly model-independent upper estimations for the difference. In addition we show numerical examples for two typical exponential Lévy models: Merton models and variance gamma models.

preprint2016arXiv

Sensitivity and Computational Complexity in Financial Networks

Modern financial networks exhibit a high degree of interconnectedness and determining the causes of instability and contagion in financial networks is necessary to inform policy and avoid future financial collapse. In the American Economic Review, Elliott, Golub and Jackson proposed a simple model for capturing the dynamics of complex financial networks. In Elliott, Golub and Jackson's model, each institution in the network can buy underlying assets or percentage shares in other institutions (cross-holdings) and if any institution's value drops below a critical threshold value, its value suffers an additional failure cost. This work shows that even in simple model put forward by Elliott, Golub and Jackson there are fundamental barriers to understanding the risks that are inherent in a network. First, if institutions are not required to maintain a minimum amount of self-holdings, an $ε$ change in investments by a single institution can have an arbitrarily magnified influence on the net worth of the institutions in the system. This sensitivity result shows that if institutions have small self-holdings, then estimating the market value of an institution requires almost perfect

255 works