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At least 19 recordsLinked to original sources

arXiv2015

Pricing and Risk Management with High-Dimensional Quasi Monte Carlo and Global Sensitivity Analysis

We review and apply Quasi Monte Carlo (QMC) and Global Sensitivity Analysis (GSA) techniques to pricing and risk management (greeks) of representative financial instruments of increasing complexity. We compare QMC vs standard Monte Carlo (MC) results in great detail, using high-dimensional Sobol' low discrepancy sequences, different discretization methods, and specific analyses of convergence, performance, speed up, stability, and error optimization for finite differences greeks. We find that our QMC outperforms MC in most cases, including the highest-dimensional simulations and greeks calculations, showing faster and more stable convergence to exact or almost exact results. Using GSA, we are able to fully explain our findings in terms of reduced effective dimension of our QMC simulation, allowed in most cases, but not always, by Brownian bridge discretization. We conclude that, beyond pricing, QMC is a very promising technique also for computing risk figures, greeks in particular, as it allows to reduce the computational effort of high-dimensional Monte Carlo simulations typical of modern risk management.

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arXiv2010

Proving Regularity of the Minimal Probability of Ruin via a Game of Stopping and Control

We reveal an interesting convex duality relationship between two problems: (a) minimizing the probability of lifetime ruin when the rate of consumption is stochastic and when the individual can invest in a Black-Scholes financial market; (b) a controller-and-stopper problem, in which the controller controls the drift and volatility of a process in order to maximize a running reward based on that process, and the stopper chooses the time to stop the running reward and rewards the controller a final amount at that time. Our primary goal is to show that the minimal probability of ruin, whose stochastic representation does not have a classical form as does the utility maximization problem (i.e., the objective's dependence on the initial values of the state variables is implicit), is the unique classical solution of its Hamilton-Jacobi-Bellman (HJB) equation, which is a non-linear boundary-value problem. We establish our goal by exploiting the convex duality relationship between (a) and (b).

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arXiv2008

Mutual Fund Theorems when Minimizing the Probability of Lifetime Ruin

We show that the mutual fund theorems of Merton (1971) extend to the problem of optimal investment to minimize the probability of lifetime ruin. We obtain two such theorems by considering a financial market both with and without a riskless asset for random consumption. The striking result is that we obtain two-fund theorems despite the additional source of randomness from consumption.

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arXiv2007

Optimal cross hedging for insurance derivatives

We consider insurance derivatives depending on an external physical risk process, for example a temperature in a low dimensional climate model. We assume that this process is correlated with a tradable financial asset. We derive optimal strategies for exponential utility from terminal wealth, determine the indifference prices of the derivatives, and interpret them in terms of diversification pressure. Moreover we check the optimal investment strategies for standard admissibility criteria. Finally we compare the static risk connected with an insurance derivative to the reduced risk due to a dynamic investment into the correlated asset. We show that dynamic hedging reduces the risk aversion in terms of entropic risk measures by a factor related to the correlation.

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arXiv2007

A Model for Counterparty Risk with Geometric Attenuation Effect and the Valuation of CDS

In this paper, a geometric function is introduced to reflect the attenuation speed of impact of one firm's default to its partner. If two firms are competitions (copartners), the default intensity of one firm will decrease (increase) abruptly when the other firm defaults. As time goes on, the impact will decrease gradually until extinct. In this model, the joint distribution and marginal distributions of default times are derived by employing the change of measure, so can we value the fair swap premium of a CDS.

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arXiv2007

Valuations and dynamic convex risk measures

This paper approaches the definition and properties of dynamic convex risk measures through the notion of a family of concave valuation operators satisfying certain simple and credible axioms. Exploring these in the simplest context of a finite time set and finite sample space, we find natural risk-transfer and time-consistency properties for a firm seeking to spread its risk across a group of subsidiaries.

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arXiv2008

Maturity-independent risk measures

The new notion of maturity-independent risk measures is introduced and contrasted with the existing risk measurement concepts. It is shown, by means of two examples, one set on a finite probability space and the other in a diffusion framework, that, surprisingly, some of the widely utilized risk measures cannot be used to build maturity-independent counterparts. We construct a large class of maturity-independent risk measures and give representative examples in both continuous- and discrete-time financial models.

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arXiv2007

Cash Sub-additive Risk Measures and Interest Rate Ambiguity

A new class of risk measures called cash sub-additive risk measures is introduced to assess the risk of future financial, nonfinancial and insurance positions. The debated cash additive axiom is relaxed into the cash sub additive axiom to preserve the original difference between the numeraire of the current reserve amounts and future positions. Consequently, cash sub-additive risk measures can model stochastic and/or ambiguous interest rates or defaultable contingent claims. Practical examples are presented and in such contexts cash additive risk measures cannot be used. Several representations of the cash sub-additive risk measures are provided. The new risk measures are characterized by penalty functions defined on a set of sub-linear probability measures and can be represented using penalty functions associated with cash additive risk measures defined on some extended spaces. The issue of the optimal risk transfer is studied in the new framework using inf-convolution techniques. Examples of dynamic cash sub-additive risk measures are provided via BSDEs where the generator can locally depend on the level of the cash sub-additive risk measure.

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arXiv2007

Optimal intertemporal risk allocation applied to insurance pricing

We present a general approach to the pricing of products in finance and insurance in the multi-period setting. It is a combination of the utility indifference pricing and optimal intertemporal risk allocation. We give a characterization of the optimal intertemporal risk allocation by a first order condition. Applying this result to the exponential utility function, we obtain an essentially new type of premium calculation method for a popular type of multi-period insurance contract. This method is simple and can be easily implemented numerically. We see that the results of numerical calculations are well coincident with the risk loading level determined by traditional practices. The results also suggest a possible implied utility approach to insurance pricing.

q-fin.PR↗

arXiv2007

Incorporating exchange rate risk into PDs and asset correlations

Intuitively, the default risk of a single borrower is higher when her or his assets and debt are denominated in different currencies. Additionally, the default dependence of borrowers with assets and debt in different currencies should be stronger than in the one-currency case. By combining well-known models by Merton (1974), Garman and Kohlhagen (1983), and Vasicek (2002) we develop simple representations of PDs and asset correlations that take into account exchange rate risk. From these results, consistency conditions can be derived that link the changes in PD and asset correlation and do not require knowledge of hard-to-estimate parameters like asset value volatility.

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arXiv2007

Arbitrage free cointegrated models in gas and oil future markets

In this article we present a continuous time model for natural gas and crude oil future prices. Its main feature is the possibility to link both energies in the long term and in the short term. For each energy, the future returns are represented as the sum of volatility functions driven by motions. Under the risk neutral probability, the motions of both energies are correlated Brownian motions while under the historical probability, they are cointegrated by a Vectorial Error Correction Model. Our approach is equivalent to defining the market price of risk. This model is free of arbitrage: thus, it can be used for risk management as well for option pricing issues. Calibration on European market data and numerical simulations illustrate well its behavior.

q-fin.ST↗

arXiv2010

Good deal bounds induced by shortfall risk

We shall provide in this paper good deal pricing bounds for contingent claims induced by the shortfall risk with some loss function. Assumptions we impose on loss functions and contingent claims are very mild. We prove that the upper and lower bounds of good deal pricing bounds are expressed by convex risk measures on Orlicz hearts. In addition, we obtain its representation with the minimal penalty function. Moreover, we give a representation, for two simple cases, of good deal bounds and calculate the optimal strategies when a claim is traded at the upper or lower bounds of its good deal pricing bound.

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arXiv2008

Feasibility of Portfolio Optimization under Coherent Risk Measures

It is shown that the axioms for coherent risk measures imply that whenever there is an asset in a portfolio that dominates the others in a given sample (which happens with finite probability even for large samples), then this portfolio cannot be optimized under any coherent measure on that sample, and the risk measure diverges to minus infinity. This instability was first discovered on the special example of Expected Shortfall which is used here both as an illustration and as a prompt for generalization.

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arXiv2009

Calibration of transparency risks: a note

The aim of this research is to give a simple framework to evaluate/quantize the "transparency" of a firm. We assume that the process of the firm value is only observable once in a while but is strongly correlated with the stock price which is observable and tradable. This hybrid type structure make the transparency "observable". The implication of the present study is that the depth of the shock to the market caused by the precise accounting information does reflect the degree of transparency. Furthermore, it can be quantized resorting to the calibration method.

q-fin.RM↗

arXiv2008

Convex Risk Measures: Lebesgue Property on one Period and Multi Period Risk Measures and Application in Capital Allocation Problem

In this work we study the Lebesgue property for convex risk measures on the space of bounded càdlàg random processes ($\mathcal{R}^\infty$). Lebesgue property has been defined for one period convex risk measures in \cite{Jo} and earlier had been studied in \cite{De} for coherent risk measures. We introduce and study the Lebesgue property for convex risk measures in the multi period framework. We give presentation of all convex risk measures with Lebesgue property on bounded càdlàg processes. To do that we need to have a complete description of compact sets of $\mathcal{A}^1$. The main mathematical contribution of this paper is the characterization of the compact sets of $\mathcal{A}^p$ (including $\mathcal{A}^1$). At the final part of this paper, we will solve the Capital Allocation Problem when we work with coherent risk measures.

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arXiv2008

Deterministic definition of the capital risk

In this paper we propose a look at the capital risk problem inspired by deterministic, known from classical mechanics, problem of juggling. We propose capital equivalents to the Newton's laws of motion and on this basis we determine the most secure form of credit repayment with regard to maximisation of profit. Then we extend the Newton's laws to models in linear spaces of arbitrary dimension with the help of matrix rates of return. The matrix rates describe the evolution of multidimensional capital and they are sensitive to both quantitative changes of individual elements and flows between them. This allows us for simultaneous analysis of evolution of complex capital in both continuous and discrete time models.

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arXiv2008

Minimizing the Probability of Ruin when Consumption is Ratcheted

We assume that an agent's rate of consumption is {\it ratcheted}; that is, it forms a non-decreasing process. Given the rate of consumption, we act as financial advisers and find the optimal investment strategy for the agent who wishes to minimize his probability of ruin.

q-fin.RM↗

arXiv2008

Heterogeneous credit portfolios and the dynamics of the aggregate losses

We study the impact of contagion in a network of firms facing credit risk. We describe an intensity based model where the homogeneity assumption is broken by introducing a random environment that makes it possible to take into account the idiosyncratic characteristics of the firms. We shall see that our model goes behind the identification of groups of firms that can be considered basically exchangeable. Despite this heterogeneity assumption our model has the advantage of being totally tractable. The aim is to quantify the losses that a bank may suffer in a large credit portfolio. Relying on a large deviation principle on the trajectory space of the process, we state a suitable law of large number and a central limit theorem useful to study large portfolio losses. Simulation results are provided as well as applications to portfolio loss distribution analysis.

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