Modelling the evolution of credit spreads using the Cox process within the HJM framework: A CDS option pricing model

In this paper a simulation approach for defaultable yield curves is developed within the Heath et al. (1992) framework. The default event is modelled using the Cox process where the stochastic intensity represents the credit spread. The forward credit spread volatility function is affected by the entire credit spread term structure. The paper provides the defaultable bond and credit default swap option price in a probability setting equipped with a subfiltration structure. The Euler–Maruyama stochastic integral approximation and the Monte Carlo method are applied to develop a numerical scheme for pricing. Finally, the antithetic variable technique is used to reduce the variance of credit default swap option prices.     

A finite volume approach for contingent claims valuation

This paper presents a finite volume approach for solving two-dimensionalcontingent claims valuation problems. The contingent claimsPDEs are in non-divergence form. The finite volume method ismore flexible than finite difference schemes which are oftendescribed in the finance literature and frequently used in practice.Moreover, the finite volume method naturally handles cases wherethe underlying partial differential equation becomes convectiondominated or degenerate. A compact method is developed whichuses a high-order flux limiter for the convection terms. Thispaper will demonstrate how a variety of two-dimensional valuationproblems can all be solved using the same approach. The generalityof the approach is in part due to the fact that changes causedby different model specifications are localized. Constraintson the solution are treated in a uniform manner using a penaltymethod. A variety of illustrative example computations are presented.     

Pricing catastrophe options with counterparty credit risk in a reduced form model

In this paper, we study the price of catastrophe options with counterparty credit risk in a reduced form model. We assume that the loss process is generated by a doubly stochastic Poisson process, the share price process is modeled through a jump-diffusion process which is correlated to the loss process, the interest rate process and the default intensity process are modeled through the Vasicek model. We derive the closed form formulae for pricing catastrophe options in a reduced form model. Furthermore, we make some numerical analysis on the explicit formulae.     

A latent process model for the pricing of corporate securities

We propose a structural model with a joint process of tangible assets (marker) and firm status for the pricing of corporate securities. The firm status is assumed to be latent or unobservable, and default occurs when the firm status process reaches a default threshold at the first time. The marker process is observable and assumed to be correlated with the latent firm status. The recovery upon default is a fraction of tangible assets at the time of default. Our model can evaluate both the corporate debt and equity to fit their market prices in a unified framework. When the two processes are perfectly correlated, our model is reduced to the seminal Black–Cox model. Numerical examples are given to support the usefulness of our model. A previous version of this paper was presented at the Tsukuba–Stanford workshop held at Stanford University on March 2006. The authors are grateful to participants of the workshop for helpful discussions.     

On the valuation of constant barrier options under spectrally one-sided exponential Lévy models and Carr''s approximation for American puts

This paper provides a general framework for pricing options with a constant barrier under spectrally one-sided exponential Lévy model, and uses it to implement of Carr's approximation for the value of the American put under this model. Simple analytic approximations for the exercise boundary and option value are obtained.     

A hidden Markov regime-switching model for option valuation

We investigate two approaches, namely, the Esscher transform and the extended Girsanov’s principle, for option valuation in a discrete-time hidden Markov regime-switching Gaussian model. The model’s parameters including the interest rate, the appreciation rate and the volatility of a risky asset are governed by a discrete-time, finite-state, hidden Markov chain whose states represent the hidden states of an economy. We give a recursive filter for the hidden Markov chain and estimates of model parameters using a filter-based EM algorithm. We also derive predictors for the hidden Markov chain and some related quantities. These quantities are used to estimate the price of a standard European call option. Numerical examples based on real financial data are provided to illustrate the implementation of the proposed method.     

A valuation model for firms with stochastic earnings

A model is proposed to value a firm with stochastic earnings. It is assumed that the earnings of the firm follow a time‐varying mean reverting stochastic process. It is shown that the value of the firm satisfies a boundary value problem of a second‐order partial differential equation, which can be solved numerically. Some special cases are discussed. An analytic solution is found for one special case. Moreover, it is shown that the analytic solution is consistent with a previous result obtained by other researchers. Numerical solutions are obtained for the other special cases. Finally, the model is also applied to value the debt issued by the firm.     

A micro-level claim count model with overdispersion and reporting delays

The accurate estimation of outstanding liabilities of an insurance company is an essential task. This is to meet regulatory requirements, but also to achieve efficient internal capital management. Over the recent years, there has been increasing interest in the utilisation of insurance data at a more granular level, and to model claims using stochastic processes. So far, this so-called ‘micro-level reserving’ approach has mainly focused on the Poisson process.In this paper, we propose and apply a Cox process approach to model the arrival process and reporting pattern of insurance claims. This allows for over-dispersion and serial dependency in claim counts, which are typical features in real data. We explicitly consider risk exposure and reporting delays, and show how to use our model to predict the numbers of Incurred-But-Not-Reported (IBNR) claims. The model is calibrated and illustrated using real data from the AUSI data set.     

A risk-based model for the valuation of pension insurance

In the US, defined benefit plans are insured by the Pension Benefit Guaranty Corporation (PBGC). Taking account of the fact that the PBGC covers only the residual deficits of the pension fund the sponsoring company is unable to cover and that the plans can be prematurely terminated, we consider a model that accounts for the joint dynamics of the pension fund’s and sponsoring firm’s assets in order to effectively determine the risk-based pension premium for the insurance provided by the PBGC. We obtain a closed-form pricing formula for this risk-based premium. Its magnitude depends highly on the investment portfolio of the pension fund and of the sponsoring company as well as the correlation between these two portfolios.     

A risk model with renewal shot-noise Cox process

In this paper we generalise the risk models beyond the ordinary framework of affine processes or Markov processes and study a risk process where the claim arrivals are driven by a Cox process with renewal shot-noise intensity. The upper bounds of the finite-horizon and infinite-horizon ruin probabilities are investigated and an efficient and exact Monte Carlo simulation algorithm for this new process is developed. A more efficient estimation method for the infinite-horizon ruin probability based on importance sampling via a suitable change of probability measure is also provided; illustrative numerical examples are also provided.     

A model form for exact -metrics

Any manifold with boundary can be equipped with a -metric which takes the form with respect to some product decomposition near the boundary, and positive definite on restriction to the tangent space of the boundary. Here we show the existence of a product decomposition such that is independent of modulo terms vanishing to infinite order at the boundary. The uniqueness of this decomposition is also examined.

     

Robust spectral method for numerical valuation of european options under Merton's jump‐diffusion model

We propose a novel numerical method based on rational spectral collocation and Clenshaw–Curtis quadrature methods together with the “” transformation for pricing European vanilla and butterfly spread options under Merton's jump‐diffusion model. Under certain assumptions, such model leads to a partial integro‐differential equation (PIDE). The differential and integral parts of the PIDE are approximated by the rational spectral collocation and the Clenshaw–Curtis quadrature methods, respectively. The application of spectral collocation method to the PIDE leads to a system of ordinary differential equations, which is solved using the implicit–explicit predictor–corrector (IMEX‐PC) schemes in which the diffusion term is integrated implicitly, whereas the convolution integral, reaction, advection terms are integrated explicitly. Numerical experiments illustrate that our approach is highly accurate and efficient for pricing financial options.Copyright © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1169–1188, 2014     

A semi-analytic pricing formula for lookback options under a general stochastic volatility model

In general, the pricing problems of exotic options in finance do not have analytic solutions under stochastic volatility and so it is hard to compute the option prices or at least it requires much of time to compute them. This paper investigates a semi-analytic pricing method for lookback options in a general stochastic volatility framework. The resultant formula is well connected to the Black–Scholes price that is the first term of a series expansion, which makes computing the option prices relatively efficient. Further, a convergence condition for the expansion is provided with an error bound.     

时间分数阶期权定价模型的一类有效差分方法

时间分数阶期权定价模型(时间分数阶Black-Scholes方程)数值解法的研究具有重要的理论意义和实际应用价值.对时间分数阶Black-Scholes方程构造了显-隐格式和隐-显差分格式,讨论了两类格式解的存在唯一性,稳定性和收敛性.理论分析证实,显-隐格式和隐-显格式均为无条件稳定和收敛的,两种格式具有相同的计算量.数值试验表明:显-隐和隐-显格式的计算精度与经典Crank-Nicolson(C-N)格式的计算精度相当,其计算效率(计算时间)比C-N格式提高30%.数值试验验证了理论分析,表明本文的显-隐和隐-显差分方法对求解时间分数阶期权定价模型是高效的,证实了时间分数阶Black-Scholes方程更符合实际金融市场.     

A novel repair model for imperfect maintenance

^** Email: shaomin.wu{at}reading.ac.uk Commonly used repair rate models for repairable systems in thereliability literature are renewal processes, generalised renewalprocesses or non-homogeneous Poisson processes. In additionto these models, geometric processes (GP) are studied occasionally.The GP, however, can only model systems with monotonously changing(increasing, decreasing or constant) failure intensities. Thispaper deals with the reliability modelling of failure processesfor repairable systems where the failure intensity shows a bathtub-typenon-monotonic behaviour. A new stochastic process, i.e. an extendedPoisson process, is introduced in this paper. Reliability indicesare presented, and the parameters of the new process are estimated.Experimental results on a data set demonstrate the validityof the new process.     

A marked Cox model for the number of IBNR claims: Theory

Incurred but not reported (IBNR) loss reserving is an important issue for Property & Casualty (P&C) insurers. To calculate IBNR reserve, one needs to model claim arrivals and then predict IBNR claims. However, factors such as temporal dependence among claim arrivals and environmental variation are often not incorporated in many of the current loss reserving models, which may greatly affect the accuracy of IBNR predictions.In this paper, we propose to model the claim arrival process together with its reporting delays as a marked Cox process. Our model is versatile in modeling temporal dependence, allowing also for natural interpretations. This paper focuses mainly on the theoretical aspects of the proposed model. We show that the associated reported claim process and IBNR claim process are both marked Cox processes with easily convertible intensity functions and marking distributions. The proposed model can also account for fluctuations in the exposure. By an order statistics property, we show that the corresponding discretely observed process preserves all the information about the claim arrivals. Finally, we derive closed-form expressions for both the autocorrelation function (ACF) and the distributions of the numbers of reported claims and IBNR claims. Model estimation and its applications are considered in a subsequent paper, Badescu et al. (2015b).     

A Generalized Renewal Equation for Perturbed Compound Poisson Processes with Two-Sided Jumps

Abstract

In this article, we study the discounted penalty at ruin in a perturbed compound Poisson model with two-sided jumps. We show that it satisfies a renewal equation under suitable conditions and consider an application of this renewal equation to study some perpetual American options. In particular, our renewal equation gives a generalization of the renewal equation in Gerber and Landry [2 Gerber , H.U. , and Landry , B. 1998 . On the discounted penalty at ruin in a jump-diffusion and the perpetual put option . Insurance: Mathematics and Economics 22 : 263 – 276 .[Crossref], [Web of Science ®] , [Google Scholar]] where only downward jumps are allowed.     

Analysis of the expected discounted penalty function for a general jump-diffusion risk model and applications in finance

In this paper, we extend the Cramér-Lundberg risk model perturbed by diffusion to incorporate the jumps of surplus investment return. Under the assumption that the jump of surplus investment return follows a compound Poisson process with Laplace distributed jump sizes, we obtain the explicit closed-form expression of the resulting Gerber-Shiu expected discounted penalty (EDP) function through the Wiener-Hopf factorization technique instead of the integro-differential equation approach. Especially, when the claim distribution is of Phase-type, the expression of the EDP function is simplified even further as a compact matrix-type form. Finally, the financial applications include pricing barrier option and perpetual American put option and determining the optimal capital structure of a firm with endogenous default.     

A uniform asymptotic expansion for stochastic volatility model in pricing multi‐asset European options

In this paper, we consider a stochastic volatility model for pricing multi‐asset European options that are widely used in the real world, under the assumption that the volatilities are driven by different OU processes. Using the singular perturbation method for multi‐parameter and the boundary layer theory, we derive a uniform asymptotic expansion for the option prices, as well as the uniform error estimates. Copyright © 2011 John Wiley & Sons, Ltd.     

A space-time clustering model for historical earthquakes

This paper describes a generalization of Hawkes' self-exciting process in which each event creates a process of offspring with conditional intensity governed by a diffusion kernel. The process may be described as a space-time branching process with immigration, the immigration representing a background series of independent events. The model can be fitted by likelihood methods. As an illustration it is fitted to the catalogue of historical Italian earthquakes.