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arXiv2016

Boundary-degenerate elliptic operators and Holder continuity for solutions to variational equations and inequalities

The Heston stochastic volatility process, which is widely used as an asset price model in mathematical finance, is a paradigm for a degenerate diffusion process where the degeneracy in the diffusion coefficient is proportional to the square root of the distance to the boundary of the half-plane. The generator of this process with killing, called the elliptic Heston operator, is a second-order, degenerate-elliptic partial differential operator whose coefficients have linear growth in the spatial variables and where the degeneracy in the operator symbol is proportional to the distance to the boundary of the half-plane. With the aid of weighted Sobolev spaces, we prove supremum bounds, a Harnack inequality, and Hölder continuity near the boundary for solutions to variational equations defined by the elliptic Heston operator, as well as Hölder continuity up to the boundary for solutions to variational inequalities defined by the elliptic Heston operator. In mathematical finance, solutions to obstacle problems for the elliptic Heston operator correspond to value functions for perpetual American-style options on the underlying asset.

math.AP↗

arXiv2011

Symmetries of the Black-Scholes equation

We determine the algebra of isovectors for the Black--Scholes equation. As a consequence, we obtain some previously unknown families of transformations on the solutions.

q-fin.CP↗

arXiv2011

Fat Tails Quantified and Resolved: A New Distribution to Reveal and Characterize the Risk and Opportunity Inherent in Leptokurtic Data

I report a new statistical distribution formulated to confront the infamous, long-standing, computational/modeling challenge presented by highly skewed and/or leptokurtic ("fat- or heavy-tailed") data. The distribution is straightforward, flexible and effective. Even when working with far fewer data points than are routinely required, it models non-Gaussian data samples, from peak center through far tails, within the context of a single probability density function (PDF) that is valid over an extremely broad range of dispersions and probability densities. The distribution is a precision tool to characterize the great risk and the great opportunity inherent in fat-tailed data.

q-fin.ST↗

arXiv2011

Copula-based Hierarchical Aggregation of Correlated Risks. The behaviour of the diversification benefit in Gaussian and Lognormal Trees

The benefits of diversifying risks are difficult to estimate quantitatively because of the uncertainties in the dependence structure between the risks. Also, the modelling of multidimensional dependencies is a non-trivial task. This paper focuses on one such technique for portfolio aggregation, namely the aggregation of risks within trees, where dependencies are set at each step of the aggregation with the help of some copulas. We define rigorously this procedure and then study extensively the Gaussian Tree of quite arbitrary size and shape, where individual risks are normal, and where the Gaussian copula is used. We derive exact analytical results for the diversification benefit of the Gaussian tree as a function of its shape and of the dependency parameters. Such a "toy-model" of an aggregation tree enables one to understand the basic phenomena's at play while aggregating risks in this way. In particular, it is shown that, for a fixed number of individual risks, "thin" trees diversify better than "fat" trees. Related to this, it is shown that hierarchical trees have the natural tendency to lower the overall dependency with respect to the dependency parameter chosen at each step of the aggregation. We also show that these results hold in more general cases outside the gaussian world, and apply notably to more realistic portfolios (LogNormal trees). We believe that any insurer or reinsurer using such a tool should be aware of these systematic effects, and that this awareness should strongly call for designing trees that adequately fit the business. We finally address the issue of specifying the full joint distribution between the risks. We show that the hierarchical mechanism does not require nor specify the joint distribution, but that the latter can be determined exactly (in the Gaussian case) by adding conditional independence hypotheses between the risks and their sums.

q-fin.RM↗

arXiv2011

Numerical Solutions of Optimal Risk Control and Dividend Optimization Policies under A Generalized Singular Control Formulation

This paper develops numerical methods for finding optimal dividend pay-out and reinsurance policies. A generalized singular control formulation of surplus and discounted payoff function are introduced, where the surplus is modeled by a regime-switching process subject to both regular and singular controls. To approximate the value function and optimal controls, Markov chain approximation techniques are used to construct a discrete-time controlled Markov chain with two components. The proofs of the convergence of the approximation sequence to the surplus process and the value function are given. Examples of proportional and excess-of-loss reinsurance are presented to illustrate the applicability of the numerical methods.

q-fin.CP↗

arXiv2011

ADI finite difference schemes for the Heston-Hull-White PDE

In this paper we investigate the effectiveness of Alternating Direction Implicit (ADI) time discretization schemes in the numerical solution of the three-dimensional Heston-Hull-White partial differential equation, which is semidiscretized by applying finite difference schemes on nonuniform spatial grids. We consider the Heston-Hull-White model with arbitrary correlation factors, with time-dependent mean-reversion levels, with short and long maturities, for cases where the Feller condition is satisfied and for cases where it is not. In addition, both European-style call options and up-and-out call options are considered. It is shown through extensive tests that ADI schemes, with a proper choice of their parameters, perform very well in all situations - in terms of stability, accuracy and efficiency.

q-fin.CP↗

arXiv2011

Privacy-Preserving Methods for Sharing Financial Risk Exposures

Unlike other industries in which intellectual property is patentable, the financial industry relies on trade secrecy to protect its business processes and methods, which can obscure critical financial risk exposures from regulators and the public. We develop methods for sharing and aggregating such risk exposures that protect the privacy of all parties involved and without the need for a trusted third party. Our approach employs secure multi-party computation techniques from cryptography in which multiple parties are able to compute joint functions without revealing their individual inputs. In our framework, individual financial institutions evaluate a protocol on their proprietary data which cannot be inverted, leading to secure computations of real-valued statistics such a concentration indexes, pairwise correlations, and other single- and multi-point statistics. The proposed protocols are computationally tractable on realistic sample sizes. Potential financial applications include: the construction of privacy-preserving real-time indexes of bank capital and leverage ratios; the monitoring of delegated portfolio investments; financial audits; and the publication of new indexes of proprietary trading strategies.

q-fin.RM↗

arXiv2011

On Markovian solutions to Markov Chain BSDEs

We study (backward) stochastic differential equations with noise coming from a finite state Markov chain. We show that, for the solutions of these equations to be `Markovian', in the sense that they are deterministic functions of the state of the underlying chain, the integrand must be of a specific form. This allows us to connect these equations to coupled systems of ODEs, and hence to give fast numerical methods for the evaluation of Markov-Chain BSDEs.

math.PR↗

arXiv2012

Optimal dual martingales, their analysis and application to new algorithms for Bermudan products

In this paper we introduce and study the concept of optimal and surely optimal dual martingales in the context of dual valuation of Bermudan options, and outline the development of new algorithms in this context. We provide a characterization theorem, a theorem which gives conditions for a martingale to be surely optimal, and a stability theorem concerning martingales which are near to be surely optimal in a sense. Guided by these results we develop a framework of backward algorithms for constructing such a martingale. In turn this martingale may then be utilized for computing an upper bound of the Bermudan product. The methodology is pure dual in the sense that it doesn't require certain input approximations to the Snell envelope. In an Itô-Lévy environment we outline a particular regression based backward algorithm which allows for computing dual upper bounds without nested Monte Carlo simulation. Moreover, as a by-product this algorithm also provides approximations to the continuation values of the product, which in turn determine a stopping policy. Hence, we may obtain lower bounds at the same time. In a first numerical study we demonstrate the backward dual regression algorithm in a Wiener environment at well known benchmark examples. It turns out that the method is at least comparable to the one in Belomestny et. al. (2009) regarding accuracy, but regarding computational robustness there are even several advantages.

q-fin.CP↗

arXiv2011

Adaptive Simulation of the Heston Model

Recent years have seen an increased level of interest in pricing equity options under a stochastic volatility model such as the Heston model. Often, simulating a Heston model is difficult, as a standard finite difference scheme may lead to significant bias in the simulation result. Reducing the bias to an acceptable level is not only challenging but computationally demanding. In this paper we address this issue by providing an alternative simulation strategy -- one that systematically decreases the bias in the simulation. Additionally, our methodology is adaptive and achieves the reduction in bias with "near" minimum computational effort. We illustrate this feature with a numerical example.

q-fin.CP↗

arXiv2012

Bivariate Semi-Markov Process for Counterparty Credit Risk

We consider the problem of constructing an appropriate multivariate model for the study of the counterparty credit risk in credit rating migration problem. For this financial problem different multivariate Markov chain models were proposed. However the markovian assumption may be inappropriate for the study of the dynamic of credit ratings which typically show non markovian like behaviour. In this paper we develop a semi-Markov approach to the study of the counterparty credit risk by defining a new multivariate semi-Markov chain model. Methods are given for computing the transition probabilities, reliability functions and the price of a risky Credit Default Swap.

math.PR↗

arXiv2011

Looking for grass-root sources of systemic risk: the case of "cheques-as-collateral" network

The global financial system has become highly connected and complex. Has been proven in practice that existing models, measures and reports of financial risk fail to capture some important systemic dimensions. Only lately, advisory boards have been established in high level and regulations are directly targeted to systemic risk. In the same direction, a growing number of researchers employ network analysis to model systemic risk in financial networks. Current approaches are concentrated on interbank payment network flows in national and international level. This work builds on existing approaches to account for systemic risk assessment in micro level. Particularly, we introduce the analysis of intra-bank financial risk interconnections, by examining the real case of "cheques-as-collateral" network for a major Greek bank. Our model offers useful information about the negative spillovers of disruption to a financial entity in a bank's lending network and could complement existing credit scoring models that account only for idiosyncratic customer's financial profile. Most importantly, the proposed methodology can be employed in many segments of the entire financial system, providing a useful tool in the hands of regulatory authorities in assessing more accurate estimates of systemic risk.

q-fin.RM↗

arXiv2011

Arbitrage-free Self-organizing Markets with GARCH Properties: Generating them in the Lab with a Lattice Model

We extend our studies of a quantum field model defined on a lattice having the dilation group as a local gauge symmetry. The model is relevant in the cross-disciplinary area of econophysics. A corresponding proposal by Ilinski aimed at gauge modeling in non-equilibrium pricing is realized as a numerical simulation of the one-asset version. The gauge field background enforces minimal arbitrage, yet allows for statistical fluctuations. The new feature added to the model is an updating prescription for the simulation that drives the model market into a self-organized critical state. Taking advantage of some flexibility of the updating prescription, stylized features and dynamical behaviors of real-world markets are reproduced in some detail.

q-fin.CP↗

arXiv2011

Dual representations for general multiple stopping problems

In this paper, we study the dual representation for generalized multiple stopping problems, hence the pricing problem of general multiple exercise options. We derive a dual representation which allows for cashflows which are subject to volume constraints modeled by integer valued adapted processes and refraction periods modeled by stopping times. As such, this extends the works by Schoenmakers (2010), Bender (2011a), Bender (2011b), Aleksandrov and Hambly (2010), and Meinshausen and Hambly (2004) on multiple exercise options, which either take into consideration a refraction period or volume constraints, but not both simultaneously. We also allow more flexible cashflow structures than the additive structure in the above references. For example some exponential utility problems are covered by our setting. We supplement the theoretical results with an explicit Monte Carlo algorithm for constructing confidence intervals for the price of multiple exercise options and exemplify it by a numerical study on the pricing of a swing option in an electricity market.

q-fin.CP↗

arXiv2012

High-order short-time expansions for ATM option prices under the CGMY model

The short-time asymptotic behavior of option prices for a variety of models with jumps has received much attention in recent years. In the present work, a novel second-order approximation for ATM option prices under the CGMY Lévy model is derived, and then extended to a model with an additional independent Brownian component. Our results shed light on the connection between both the volatility of the continuous component and the jump parameters and the behavior of ATM option prices near expiration. In case of an additional Brownian component, the second-order term, in time-t, is of the form $ d_{2} t^{(3-Y)/2}$, with the coefficient $d_{2}$ depending only on the overall jump intensity parameter C and the tail-heaviness parameter Y. This extends the known result that the leading term is $(σ/\sqrt{2π})t^{1/2}$, where $σ$ is the volatility of the continuous component. In contrast, under a pure-jump CGMY model, the dependence on the two parameters C and Y is already reflected in the leading term, which is of the form $d_{1} t^{1/Y}$. Information on the relative frequency of negative and positive jumps appears only in the second-order term, which is shown to be of the form $d_{2} t$ and whose order of decay turns out to be independent of Y. The third-order asymptotic behavior of the option prices as well as the asymptotic behavior of the corresponding Black-Scholes implied volatilities are also addressed. Our numerical results show that in most cases the second-order term significantly outperform the first-order approximation.

q-fin.CP↗

arXiv2011

Monte Carlo methods via a dual approach for some discrete time stochastic control problems

We consider a class of discrete time stochastic control problems motivated by some financial applications. We use a pathwise stochastic control approach to provide a dual formulation of the problem. This enables us to develop a numerical technique for obtaining an estimate of the value function which improves on purely regression based methods. We demonstrate the competitiveness of the method on the example of a gas storage valuation problem.

math.PR↗

arXiv2011

Efficient simulation and calibration of general HJM models by splitting schemes

We introduce efficient numerical methods for generic HJM equations of interest rate theory by means of high-order weak approximation schemes. These schemes allow for QMC implementations due to the relatively low dimensional integration space. The complexity of the resulting algorithm is considerably lower than the complexity of multi-level MC algorithms as long as the optimal order of QMC-convergence is guaranteed. In order to make the methods applicable to real world problems, we introduce and use the setting of weighted function spaces, such that unbounded payoffs and unbounded characteristics of the equations in question are still allowed. We also provide an implementation, where we efficiently calibrate an HJM equation to caplet data.

math.PR↗

arXiv2011

Dependent default and recovery: MCMC study of downturn LGD credit risk model

There is empirical evidence that recovery rates tend to go down just when the number of defaults goes up in economic downturns. This has to be taken into account in estimation of the capital against credit risk required by Basel II to cover losses during the adverse economic downturns; the so-called "downturn LGD" requirement. This paper presents estimation of the LGD credit risk model with default and recovery dependent via the latent systematic risk factor using Bayesian inference approach and Markov chain Monte Carlo method. This approach allows joint estimation of all model parameters and latent systematic factor, and all relevant uncertainties. Results using Moody's annual default and recovery rates for corporate bonds for the period 1982-2010 show that the impact of parameter uncertainty on economic capital can be very significant and should be assessed by practitioners.

q-fin.RM↗