PIDE and Solution Related to Pricing of Lévy Driven Arithmetic Type Floating Asian Options

We propose a stochastic model to develop a pricing partial integro-differential equation (PIDE) and its Fourier transform expression for floating Asian options based on the Itô-Lévy calculus. The stock price is driven by a class of infinite activity Lévy processes leading to the market inherently incomplete, and dynamic hedging is no longer risk free. We first develop a PIDE for floating Asian options, and apply the Fourier transform to derive a pricing expression. Our main contribution is to develop a PIDE with its closed form pricing expression for the contract. The procedure is easy to implement for all class of Lévy processes. Finally, the model is calibrated with the market data and its accuracy is presented.     

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.     

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.     

Option Pricing and Hedging under a Markov Switching Lévy Process Model

In this paper,we consider a Markov switching Lévy process model in which the underlying risky assets are driven by the stochastic exponential of Markov switching Lévy process and then apply the model to option pricing and hedging.In this model,the market interest rate,the volatility of the underlying risky assets and the N-state compensator,depend on unobservable states of the economy which are modeled by a continuous-time Hidden Markov process.We use the MEMM(minimal entropy martingale measure) as the equivalent martingale measure.The option price using this model is obtained by the Fourier transform method.We obtain a closed-form solution for the hedge ratio by applying the local risk minimizing hedging.     

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 stable local radial basis function method for option pricing problem under the Bates model

We propose a local mesh‐free method for the Bates–Scott option pricing model, a 2D partial integro‐differential equation (PIDE) arising in computational finance. A Wendland radial basis function (RBF) approach is used for the discretization of the spatial variables along with a linear interpolation technique for the integral operator. The resulting set of ordinary differential equations (ODEs) is tackled via a time integration method. A potential advantage of using RBFs is the small number of discrete equations that need to be solved. Computational experiments are presented to illustrate the performance of the contributed approach.     

Lie-algebraic approach for pricing moving barrier options with time-dependent parameters

In this paper we apply the Lie-algebraic technique for the valuation of moving barrier options with time-dependent parameters. The value of the underlying asset is assumed to follow the constant elasticity of variance (CEV) process. By exploiting the dynamical symmetry of the pricing partial differential equations, the new approach enables us to derive the analytical kernels of the pricing formulae straightforwardly, and thus provides an efficient way for computing the prices of the moving barrier options. The method is also able to provide tight upper and lower bounds for the exact prices of CEV barrier options with fixed barriers. In view of the CEV model being empirically considered to be a better candidate in equity option pricing than the traditional Black-Scholes model, our new approach could facilitate more efficient comparative pricing and precise risk management in equity derivatives with barriers by incorporating term-structures of interest rates, volatility and dividend into the CEV option valuation model.     

Continuous time mean‐variance optimal portfolio allocation under jump diffusion: An numerical impulse control approach

We present efficient partial differential equation methods for continuous time mean‐variance portfolio allocation problems when the underlying risky asset follows a jump‐diffusion. The standard formulation of mean‐variance optimal portfolio allocation problems, where the total wealth is the underlying stochastic process, gives rise to a one‐dimensional (1D) nonlinear Hamilton–Jacobi–Bellman (HJB) partial integrodifferential equation (PIDE) with the control present in the integrand of the jump term, and thus is difficult to solve efficiently. To preserve the efficient handling of the jump term, we formulate the asset allocation problem as a 2D impulse control problem, 1D for each asset in the portfolio, namely the bond and the stock. We then develop a numerical scheme based on a semi‐Lagrangian timestepping method, which we show to be monotone, consistent, and stable. Hence, assuming a strong comparison property holds, the numerical solution is guaranteed to converge to the unique viscosity solution of the corresponding HJB PIDE. The correctness of the proposed numerical framework is verified by numerical examples. We also discuss the effects on the efficient frontier of realistic financial modeling, such as different borrowing and lending interest rates, transaction costs, and constraints on the portfolio, such as maximum limits on borrowing and solvency. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 664–698, 2014     

Mean‐variance hedging with basis risk

Basis risk arises in a number of financial and insurance risk management problems when the hedging assets do not perfectly match the underlying asset in a hedging program. Notable examples in insurance include the hedging for longevity risks, weather index–based insurance products, variable annuities, etc. In the presence of basis risk, a perfect hedging is impossible, and in this paper, we adopt a mean‐variance criterion to strike a balance between the expected hedging error and its variability. Under a time‐dependent diffusion model setup, explicit optimal solutions are derived for the hedging target being either a European option or a forward contract. The solutions are obtained by a delicate application of the linear quadratic control theory, the method of backward stochastic differential equation, and Malliavin calculus. A numerical example is presented to illustrate our theoretical results and their interesting implications.     

Adaptive Optimal Allocation in Stratified Sampling Methods

In this paper, we propose a stratified sampling algorithm in which the random drawings made in the strata to compute the expectation of interest are also used to adaptively modify the proportion of further drawings in each stratum. These proportions converge to the optimal allocation in terms of variance reduction and our stratified estimator is asymptotically normal with asymptotic variance equal to the minimal one. Numerical experiments confirm the efficiency of our algorithm. For the pricing of arithmetic average Asian options in the Black and Scholes model, the variance is divided by a factor going from 1.1 to 50.4 (depending on the option type and the moneyness) in comparison with the standard allocation procedure, while the increase in computation time does not overcome 1%.     

Variational Solutions of the Pricing PIDEs for European Options in Lévy Models

Abstract

One of the fundamental problems in financial mathematics is to develop efficient algorithms for pricing options in advanced models such as those driven by Lévy processes. Essentially there are three approaches in use. These are Monte Carlo, Fourier transform and partial integro-differential equation (PIDE)-based methods. We focus our attention here on the latter. There is a large arsenal of numerical methods for efficiently solving parabolic equations that arise in this context. Especially Galerkin and Galerkin-inspired methods have an impressive potential. In order to apply these methods, what is required is a formulation of the equation in the weak sense.

The contribution of this paper is therefore to analyse weak solutions of the Kolmogorov backward equations which are related to prices of European options in (time-inhomogeneous) Lévy models and to establish a precise link between the prices and the weak solutions of these equations. The resulting relation is a Feynman–Kac representation of the solution as a conditional expectation. Our special concern is to provide a framework that is able to cover both, the common types of European options and a wide range of advanced models in which these derivatives are priced.

An application to financial models requires in particular to admit pure jump processes such as generalized hyperbolic processes as well as unbounded domains of the equation. In order to deal at the same time with the typical pay-offs that can arise, the weak formulation of the equation is based on exponentially weighted Sobolev–Slobodeckii spaces. We provide a number of examples of models that are covered by this general framework. Examples of options for which such an analysis is required are calls, puts, digital and power options as well as basket options.     

Dynamic conic hedging for competitiveness

The paper provides a new hedging methodology permitting systematic hedging choices with wide applications. Dynamic concave bid price, and convex ask price functionals from the recent literature are employed to construct new hedging strategies termed dynamic conic hedging. The primary focus of these strategies is to adopt positions maximizing a nonlinear conditional expectation expressed recursively as a concave current bid price for the one step ahead risk held or minimizing the convex current ask price for the risk promised. Risk management and hedging then have a new market value enhancing perspective different from the classical forms of risk mitigation, local variance minimization, or even expected utility maximization.     

Asset pricing for an affine jump‐diffusion model using an FD method of lines on nonuniform meshes

We present a novel numerical scheme for the valuation of options under a well‐known jump‐diffusion model. European option pricing for such a case satisfies a 1 + 2 partial integro‐differential equation (PIDE) including a double integral term, which is nonlocal. The proposed approach relies on nonuniform meshes with a focus on the discontinuous and degenerate areas of the model and applying quadratically convergent finite difference (FD) discretizations via the method of lines (MOL). A condition for observing the time stability of the fully discretized problem is given. Also, we report results of numerical experiments.     

MC方差减小技术在算术平均亚式外汇期权定价中的应用

建立了利率和汇率波动率均为随机情形下算术平均亚式外汇期权的定价模型.由于其定价问题求解十分困难,运用蒙特卡罗(Monte Carlo)方法并结合控制变量方差减小技术进行模拟,有效地减小了模拟方差,得到了期权定价问题的数值结果.     

A Mellin transform approach to pricing barrier options under stochastic elasticity of variance

This article considers a problem of evaluating barrier option prices when the underlying dynamics are driven by stochastic elasticity of variance (SEV). We employ asymptotic expansions and Mellin transform to evaluate the option prices. The approach is able to efficiently handle barrier options in a SEV framework and produce explicitly a semi-closed form formula for the approximate barrier option prices. The formula is an expansion of the option price in powers of the characteristic amplitude scale and variation time of the elasticity and it can be calculated easily by taking the derivatives of the Black–Scholes price for a barrier option with respect to the underlying price and computing the one-dimensional integrals of some linear combinations of the Greeks with respect to time. We confirm the accuracy of our formula via Monte-Carlo simulation and find the SEV effect on the Black–Scholes barrier option prices.     

A Stochastic Programming Model for Currency Option Hedging

In this paper we use a stochastic programming approach to develop currency option hedging models which can address problems with multiple random factors in an imperfect market. The portfolios considered in our model are rebalanced at the end of each time period, and reinvestments are allowed during the hedging process. These sequential decisions (reinvestments) are based on the evolution of random parameters such as exchange rates, interest rates, etc. We also allow the inclusion of a variety of instruments in the hedging portfolio, including short term derivative securities, short term options, and futures. These instruments help generate strategies that provide good liquidity and low trade intensity. One of the important features of the model is that it incorporates constraints on sensitivity measures such as Delta and Gamma. By ensuring that these hedge parameters track a desired trajectory (e.g., the parameters of a target option), the new model provides investment strategies that are robust with respect to the perturbations measured by Delta and Gamma. In order to manage the explosion of scenarios due to multiple random factors, we incorporate sampling within a scenario aggregation algorithm. We illustrate that when compared with other myopic hedging methods in imperfect markets, the new stochastic programming model can provide better performance. Our examples also illustrate stochastic programming as a practical computational tool for realistic hedging problems.     

关于CIR模型中即期利率的条件密度及贴现债券定价(英文)

本文对CIR模型运用Kolgomorov向后方程给出即期利率的条件密度所满足的偏微分方程 ,并用谱方法将该密度表达成一个Laguerre型的正交级数之和 ,借助一个经验级数和公式 ,我们得出了该和式的第一类修正贝塞尔函数表达式 .同时 ,我们猜测贴现债券价格关于即期利率的特殊表达式 ,并通过求解两个一阶常微分方程验证了我们的猜测 .在文章的结尾 ,我们给出了贴现债券价格的条件期望和方差以反映必要的矩信息 .     

The Efficient Computation of Fourier Transforms on Semisimple Algebras

We present a general diagrammatic approach to the construction of efficient algorithms for computing a Fourier transform on a semisimple algebra. This extends previous work wherein we derive best estimates for the computation of a Fourier transform for a large class of finite groups. We continue to find efficiencies by exploiting a connection between Bratteli diagrams and the derived path algebra and construction of Gel’fand–Tsetlin bases. Particular results include highly efficient algorithms for the Brauer, Temperley–Lieb, and Birman–Murakami–Wenzl algebras.     

A fast numerical approach to option pricing with stochastic interest rate,stochastic volatility and double jumps

This study proposes a pricing model through allowing for stochastic interest rate and stochastic volatility in the double exponential jump-diffusion setting. The characteristic function of the proposed model is then derived. Fast numerical solutions for European call and put options pricing based on characteristic function and fast Fourier transform (FFT) technique are developed. Simulations show that our numerical technique is accurate, fast and easy to implement, the proposed model is suitable for modeling long-time real-market changes. The model and the proposed option pricing method are useful for empirical analysis of asset returns and risk management in firms.