Basket options valuation for a local volatility jump–diffusion model with the asymptotic expansion method

In this paper we discuss the basket options valuation for a jump–diffusion model. The underlying asset prices follow some correlated local volatility diffusion processes with systematic jumps. We derive a forward partial integral differential equation (PIDE) for general stochastic processes and use the asymptotic expansion method to approximate the conditional expectation of the stochastic variance associated with the basket value process. The numerical tests show that the suggested method is fast and accurate in comparison with the Monte Carlo and other methods in most cases.     

Approximate basket options valuation for a jump-diffusion model

In this paper we discuss the approximate basket options valuation for a jump-diffusion model. The underlying asset prices follow some correlated diffusion processes with idiosyncratic and systematic jumps. We suggest a new approximate pricing formula which is the weighted sum of Roger and Shi’s lower bound and the conditional second moment adjustments. We show that the approximate value is always within the lower and upper bounds of the option and is very sharp in our numerical tests.     

Optimal premium policy of an insurance firm: Full and partial information

Herein, we study the optimization problem faced by an insurance firm who can control its cash-balance dynamics by adjusting the underlying premium rate. The firm’s objective is to minimize the total deviation of its cash-balance process to some pre-set target levels by selecting an appropriate premium policy. Our problem is totally new and has three distinguishable features: (1) both full and partial information cases are investigated here; (2) the state is subject to terminal constraint; (3) a forward-backward stochastic differential equation formulation is given which is more systematic and mathematically advanced. This formulation also enables us to continue further research in a generalized stochastic recursive control framework (see Duffie and Epstein (1992), El Karoui et al. (2001), etc.). The optimal premium policy with the associated optimal objective functional are completely and explicitly derived. In addition, a backward separation technique adaptive to forward-backward stochastic systems with the state constraint is presented as an efficient and convenient alternative to the traditional Wonham’s (1968) separation principle in our partial information setup. Some concluding remarks are also given here.     

Perpetual Options on Multiple Underlyings

Abstract

We study three classes of perpetual option with multiple uncertainties and American-style exercise boundaries, using a partial differential equation-based approach. A combination of accurate numerical techniques and asymptotic analyses is implemented, with each approach informing and confirming the other. The first two examples we study are a put basket option and a call basket option, both involving two stochastic underlying assets, whilst the third is a (novel) class of real option linked to stochastic demand and costs (the details of the modelling for this are described in the paper). The Appendix addresses the issue of pricing American-style perpetual options involving (just) one stochastic underlying, but in which the volatility is also modelled stochastically, using the Heston (1993) framework.     

Modelling Credit Risk in the Jump Threshold Framework

ABSTRACT

The jump threshold framework for credit risk modelling developed by Garreau and Kercheval enjoys the advantages of both structural- and reduced-form models. In their article, the focus is on multidimensional default dependence, under the assumptions that stock prices follow an exponential Lévy process (i.i.d. log returns) and that interest rates and stock volatility are constant. Explicit formulas for default time distributions and basket credit default swap (CDS) prices are obtained when the default threshold is deterministic, but only in terms of expectations when the default threshold is stochastic. In this article, we restrict attention to the one-dimensional, single-name case in order to obtain explicit closed-form solutions for the default time distribution when the default threshold, interest rate and volatility are all stochastic. When the interest rate and volatility processes are affine diffusions and the stochastic default threshold is properly chosen, we provide explicit formulas for the default time distribution, prices of defaultable bonds and CDS premia. The main idea is to make use of the Duffie–Pan–Singleton method of evaluating expectations of exponential integrals of affine diffusions.     

基于t-Copula的一篮子信用违约互换定价模型

为了刻画分布函数的厚尾特征和违约的传染性,构建了单因子t-Copula模型,以此研究一篮子信用违约互换(BDS)的定价问题。依据风险中性定价原理和顺序统计量方法,分别得到了第k次违约和n个参照实体中m个受保护的BDS价格的解析式.为了说明定价模型的有效性,用随机模拟方法分析了相应的数值算例.     

On the construction of non-affine jump-diffusion models

We describe a method for construction of jump analogues of certain one-dimensional diffusion processes satisfying solvable stochastic differential equations. The method is based on the reduction of the original stochastic differential equations to the ones with linear diffusion coefficients, which are reducible to the associated ordinary differential equations, by using the appropriate integrating factor processes. The analogues are constructed by means of adding the jump components linearly into the reduced stochastic differential equations. We illustrate the method by constructing jump analogues of several diffusion processes and expand the notion of market price of risk to the resulting non-affine jump-diffusion models.     

A stochastic approach for simulating spatially inhomogeneous coagulation dynamics in the gelation regime

We present a stochastic approach for the simulation of coagulation–diffusion dynamics in the gelation regime. The method couples the mass flow algorithm for coagulation processes with a stochastic variant of the diffusion-velocity method in a discretized framework. The simulation of the stochastic processes occurs according to an optimized implementation of the principle of grouping the possible events. A full simulation of a particle system driven by coagulation–diffusion dynamics is performed with a high degree of accuracy. This allows a qualitative and quantitative analysis of the behaviour of the system. The performance of the method becomes more evident especially in the gelation regime, where the computations become usually very time consuming.     

Sampling and reconstruction for shift-invariant stochastic processes

In this paper, combining stochastic processes with shift-invariant spaces, we introduce shift-invariant stochastic processes. It is a general case of the classical band-limited stochastic processes and a kind of non-band-limited stochastic processes. Two sampling theorems are obtained for the shift-invariant stochastic processes. The results for band-limited stochastic processes and shift-invariant spaces are generalized by our new results.     

用变分方法研究篮球投篮最佳出手角

运用变分方法对投篮最佳出手角的确定进行了研究,从抗角度偏差与抗速度误差及出手最省力三个方面,对出手角的优化进行了讨论,得到了较好的,并且符合实际的精确定量结果.     

非时齐马氏过程的随机单调性

本文研究了非时齐马氏过程的随机单调性问题.利用时齐的马氏过程随机单调性的相关证明方法,加以改进,获得了非时齐马氏过程随机单调性的显式判定方法,并进一步将这一充分性条件推广为等价条件.     

On the robustness of longevity risk pricing

For longevity bond pricing, the most popular methods contain the risk-neutral method, the Wang transform and the Sharpe ratio rule. This paper studies robustness of these three methods and investigates connections and differences among them through theoretic analysis and numerical illustrations. We adopt the dynamic mortality models with jumps to capture the permanent effects caused by unexpected factors and allow the correlation between mortality and interest rate be nonzero. The analysis is based on four typical mortality models, including the mean-reverting models and the non mean-reverting ones. Our work may provide a guidance for participants on choice of pricing methods.     

Error bounds in approximations of random sums using gamma-type operators

In this work we deal with approximations of compound distributions, that is, distribution functions of random sums. More specifically, we obtain a discrete compound distribution by replacing each summand in the initial random sum by a discrete random variable whose probability mass function is related to a well-known inversion formula for Laplace transforms [cf. Feller, W., 1971. An Introduction to Probability Theory and its Applications, vol. II, second edn. Wiley, New York]. Our aim is to show the advantages that this method has in the context of compound distributions. In particular we give accurate error bounds for the distance between the initial random sum and its approximation when the individual summands are mixtures of gamma distributions.     

Difference Approximation of Stochastic Elastic Equation Driven by Infinite Dimensional Noise

An explicit difference scheme is described, analyzed and tested for numerically approximating stochastic elastic equation driven by infinite dimensional noise. The noise processes are approximated by piecewise constant random processes and the integral formula of the stochastic elastic equation is approximated by a truncated series. Error analysis of the numerical method yields estimate of convergence rate. The rate of convergence is demonstrated with numerical experiments.     

Multivariate option pricing using copulae

The complexity of financial products significantly increased in the past 10 years. In this paper, we investigate the pricing of basket options and more generally of complex exotic contracts depending on multiple indices. Our approach assumes that the underlying assets evolve as dependent GARCH(1, 1) processes. The dependence among the assets is modeled using a copula based on pair‐copula constructions. Unlike most previous studies on this topic, we do not assume that the dependence observed between historical asset prices is similar to the dependence under the risk‐neutral probability. The method is illustrated with US market data on basket options written on two or three international indices. Copyright © 2012 John Wiley & Sons, Ltd.     

Generation of strongly non-Gaussian stochastic processes by iterative scheme upgrading phase and amplitude contents

Random excitations, such as wind velocity, always exhibit non-Gaussian features. Sample realisations of stochastic processes satisfying given features should be generated, in order to perform the dynamical analysis of structures under stochastic loads based on the Monte Carlo simulation. In this paper, an efficient method is proposed to generate stationary non-Gaussian stochastic processes. It involves an iterative scheme that produces a class of sample processes satisfying the following conditions. (1) The marginal cumulative distribution function of each sample process is perfectly identical to the prescribed one. (2) The ensemble-averaged power spectral density function of these non-Gaussian sample processes is as close to the prescribed target as possible. In this iterative scheme, the underlying processes are generated by means of the spectral representation method that recombines the upgraded power spectral density function with the phase contents of the new non-Gaussian processes in the latest iteration. Numerical examples are provided to demonstrate the capabilities of the proposed approach for four typical non-Gaussian distributions, some of which deviate significantly from the Gaussian distribution. It is found that the estimated power spectral density functions of non-Gaussian processes are close to the target ones, even for the extremely non-Gaussian case. Furthermore, the capability of the proposed method is compared to two other methods. The results show that the proposed method performs well with convergence speed, accuracy, and random errors of power spectral density functions.     

A Dirichlet space associated with consistent networks on the ring of <Emphasis Type="Italic">p</Emphasis>-adic Integers

In this article, we will give a method of reconstruction of a random walk in the ring of p-adic integers. Paying attention to a structural importance in the self similarity, we will perform the construction by means of modified method for constructing canonical stochastic processes on fractals in the Euclidean space. As a result, we will obtain an important subfamily of Albeverio and Karwowski’s stochastic processes with a self-similar randomness.     

Application of stochastic point processes in mechanics

Stochastic point processes are the mathematical tools relevant to all problems where the phenomena have the nature of a random train of events. Applications may be found in structural dynamics where some stochastic excitations may be adequately idealized as random trains of impulses or general pulses. An example of application in mechanics of materials is the stochastic model of the grain growth processes in polycrystalline nanomaterials. Based on the stochastic differential equations formulation, analysis methods such as the moment equations method or the method of equation for the response probability density are dealt with. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal decision policy for real options under general Markovian dynamics

The Least-Squares Monte Carlo Method (LSM) has become the standard tool to solve real options modeled as an optimal switching problem. The method has been shown to deliver accurate valuation results under complex and high dimensional stochastic processes; however, the accuracy of the underlying decision policy is not guaranteed. For instance, an inappropriate choice of regression functions can lead to noisy estimates of the optimal switching boundaries or even continuation/switching regions that are not clearly separated. As an alternative to estimate these boundaries, we formulate a simulation-based method that starts from an initial guess of them and then iterates until reaching optimality. The algorithm is applied to a classical mine under a wide variety of underlying dynamics for the commodity price process. The method is first validated under a one-dimensional geometric Brownian motion and then extended to general Markovian processes. We consider two general specifications: a two-factor model with stochastic variance and a rich jump structure, and a four-factor model with stochastic cost-of-carry and stochastic volatility. The method is shown to be robust, stable, and easy-to-implement, converging to a more profitable strategy than the one obtained with LSM.     

ESTIMATING A STOCHASTIC MODEL OF POPULATION DYNAMICS WITH AN APPLICATION TO KANGAROOS

In this study, we derive stochastic models of population dynamics and devise a new method of estimating the models. The models allow growth and harvest to be nonlinear functions of stochastic processes and the error terms to be nonlinear and heteroskedastic. Ordinary least-squares estimates would be biased and inefficient and generalized least-squares estimates cannot be calculated. Therefore, we implement nonlinear maximum likelihood methods to find unbiased and efficient estimates of parameters. The method is applied to the population dynamics of kangaroos in South Australia. Aerial survey data of kangaroo numbers are combined with harvest, effort and rainfall data to estimate the growth and harvest functions and the variances of the stochastic processes which drive the system. Results suggest that growth and harvest should be modeled as functions of stochastic processes and that observations on kangaroo numbers are critical for estimating population dynamics. The results also indicate that the estimation method works well and is a viable alternative to ARIMA and GARCH models, particularly for small data sets.