A PDE approach to risk measures of derivatives

This paper proposes a partial differential equation (PDE) approach to calculate coherent risk measures for portfolios of derivatives under the Black-Scholes economy. It enables us to define the risk measures in a dynamic way and to deal with American options in a relatively effective way. Our risk measure is based on the representation form of coherent risk measures. Through the use of some earlier results the PDE satisfied by the risk measures are derived. The PDE resembles the standard Black-Scholes type PDE which can be solved using standard techniques from the mathematical finance literature. Indeed, these results reveal that the PDE approach can provide practitioners with a more applicable and flexible way to implement coherent risk measures for derivatives in the context of the Black-Scholes model.     

债券定价及利率风险的市场价格的重构方法

假设利率变化的模型是由随机微分方程给出,则可以用推导Black-Scholes方程的方法来推出债券价格满足的偏微分方程,得到一个抛物型的偏微分方程.但是,在债券定价的方程中隐含有一个参数λ称为利率风险的市场价格.所谓债券定价的反问题,就是由不同到期时间的债券的现在价格来得到利率风险的市场价格.对随机利率模型下债券定价的正问题先给予介绍和差分数值求解方法,并介绍了反问题,且对反问题给出了数值方法.     

GENERAL BLACK-SCHOLES MODEL OF SECURITY VALUATION

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Numerical solution of European call option with dividends and variable volatility

In this paper, the effect of strike price, interest rate, dividends and maturities on European call option with dividends is discussed. The volatility for the data of ONGC Ltd. listed in National Stock Exchange, India, during 03-01-2000 to 30-03-2009 is forecasted by GJR-GARCH method. The option price and Greeks are determined by solving modified Black-Scholes partial differential equation by adjusting forecasted volatility at each grid point of finite difference method. It is observed that call option premium decreases as strike price and dividend increases but it increases as rate of interest and time of maturities increases. Hence call option is more profitable for a long maturity, high interest rate and low dividend.     

Indifference pricing and hedging in a multiple-priors model with trading constraints

This paper considers utility indifference valuation of derivatives under model uncertainty and trading constraints, where the utility is formulated as an additive stochastic differential utility of both intertemporal consumption and terminal wealth, and the uncertain prospects are ranked according to a multiple-priors model of Chen and Epstein (2002). The price is determined by two optimal stochastic control problems (mixed with optimal stopping time in the case of American option) of forward-backward stochastic differential equations. By means of backward stochastic differential equation and partial differential equation methods, we show that both bid and ask prices are closely related to the Black-Scholes risk-neutral price with modified dividend rates. The two prices will actually coincide with each other if there is no trading constraint or the model uncertainty disappears. Finally, two applications to European option and American option are discussed.     

分形布朗运动下有交易成本的外汇期权定价

研究了有交易成本的分形Black-Scholes外汇期权定价问题.基于汇率的分形布朗运动分布假设,运用分形布朗运动的性质和随机微积分方法,得到了欧式外汇期权价格所满足的偏微分方程.最后,建立离散时间条件下的非线性期权定价模型,并且通过解期权价格的偏微分方程给出了有交易成本的欧式外汇期权定价公式.     

Analytic and Experimental Studies of the Errors in Numerical Methods for the Valuation of Options

The value of a European option satisfies the Black-Scholes equation with appropriately specified final and boundary conditions.We transform the problem to an initial boundary value problem in dimensionless form.There are two parameters in the coefficients of the resulting linear parabolic partial differential equation.For a range of values of these parameters,the solution of the problem has a boundary or an initial layer.The initial function has a discontinuity in the first-order derivative,which leads to the appearance of an interior layer.We construct analytically the asymptotic solution of the equation in a finite domain.Based on the asymptotic solution we can determine the size of the artificial boundary such that the required solution in a finite domain in x and at the final time is not affected by the boundary.Also,we study computationally the behaviour in the maximum norm of the errors in numerical solutions in cases such that one of the parameters varies from finite (or pretty large) to small values,while the other parameter is fixed and takes either finite (or pretty large) or small values. Crank-Nicolson explicit and implicit schemes using centered or upwind approximations to the derivative are studied.We present numerical computations,which determine experimentally the parameter-uniform rates of convergence.We note that this rate is rather weak,due probably to mixed sources of error such as initial and boundary layers and the discontinuity in the derivative of the solution.     

分数布朗运动的美式障碍期权定价

假定标的股票服从分数布朗运动,应用二次近似法和偏微分方程方法求出了美式下降敲出看涨、看跌障碍期权价格近似解以及最佳实施边界.最后,通过显式差分法比较近似解的准确性,并分析Hurst参数对期权价格和最佳实施边界S*的影响.     

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.     

特殊形式的倒向随机微分方程在金融市场中的应用

首先,针对一类线性倒向随机微分方程,给出了g-鞅同鞅之间相互联系所满足的充分条件.通过该条件得到了经典的Black-Scholes模型下未定权益的公平价格过程以及最优增长投资策略的价格过程.其次,引入了带惩罚的非线性倒向随机微分方程,并通过惩罚比率的不同取值来讨论相关的经济学意义.     

Analytic models for parameter dependency in option price modelling

Options are a type of financial instrument classed as derivatives, as they derive their value from an underlying asset. The equations used to model the option price are often expressed as partial differential equations (PDEs). Once expressed in this form, a discretization method on a finite grid can be applied and the numerical valuation obtained. Remains the problem of writing down an (approximate) closed-form analytic model for the option price in function of all the variables and parameters, which is the main objective of this paper. At the same time we also consider the Greeks, which are the quantities representing the sensitivities of the price to a change in the underlying variables or parameters. Discrete values for these Greeks can again be derived, either directly from the differentiation matrices occurring in the option price PDE or by solving new but similar PDEs. Next, analytic models for the Greeks are computed in the same way as for the option price. As a prototype case, The Black-Scholes PDE for European call options is considered.     

求解非线性Black-Scholes模型的自适应小波精细积分法

针对非线性Black-Scholes方程,基于quasi-Shannon小波函数给出了一种求解非线性偏微分方程的自适应多尺度小波精细积分法.该方法首先利用插值小波理论构造了用于逼近连续函数的多尺度小波插值算子,利用该算子可以将非线性Black-Scholes方程自适应离散为非线性常微分方程组;然后将用于求解常微分方程组的精细积分法和小波变换的动态过程相结合,并利用非线性处理技术(如同伦分析技术)可有效求解非线性Black-Scholes方程.数值结果表明了该方法在数值精度和计算效率方面的优越性.     

规定水平下重置期权的有限差分解

利用Black—Scholes偏微分方程,结合重置期权与关卡期权的关系,建立了规定水平下的重置期权定价模型,最后运用C—N格式和θ法构造该模型的有限差分格式．     

分数次布朗运动的欧式障碍期权定价

假定标的股票服从分数次布朗运动,应用偏微分方程的方法求出下降敲出欧式看涨障碍期权价格显示解,以及看涨-看跌的平价关系式.最后,通过有限差分法比较了显示解的准确性,分析了Hurst参数对期权价格和风险特征参数的影响.     

分数布朗运动下几何平均亚式权证的定价

假设股票价格变化过程服从几何分数布朗运动,建立了分数布朗运动下的亚式期权定价模型.利用分数-It-公式,推导出分数布朗运动下亚式期权的价值所满足的含有三个变量偏微分方程.然后,引进适当的组合变量,将其定解问题转化为一个与路径无关的一维微分方程问题.进一步通过随机偏微分方程方法求解出分数布朗运动下亚式期权的定价公式.最后利用权证定价原理对稀释效用做出调整后,得到分数布朗运动下亚式股本权证定价公式.<正>~~     

The pricing of derivatives on assets with quadratic volatility

The basic model of financial economics is the Samuelson model of geometric Brownian motion because of the celebrated Black-Scholes formula for pricing the call option. The asset's volatility is a linear function of the asset value and the model guarantees positive asset prices. In this paper, it is shown that the pricing partial differential equation can be solved for level-dependent volatility which is a quadratic polynomial. If zero is attainable, both absorption and negative asset values are possible. Explicit formulae are derived for the call option: a generalization of the Black-Scholes formula for an asset whose volatiliy is affine, the formula for the Bachelier model with constant volatility, and new formulae in the case of quadratic volatility. The implied Black-Scholes volatilities of the Bachelier and the affine model are frowns, the quadratic specifications imply smiles.     

非仿射随机波动率的欧式障碍期权定价

研究非仿射随机波动率模型的欧式障碍期权定价问题时,首先介绍了非仿射随机波动率模型,其次利用投资组合和It^o引理,得到了该模型下扩展的Black-Schole偏微分方程.由于这个方程没有显示解,因此采用对偶蒙特卡罗模拟法计算欧式障碍期权的价格.最后,通过数值实例验证了算法的可行性和准确性.     

Risk Minimizing Option Pricing for a Class of Exotic Options in a Markov-Modulated Market

We address risk minimizing option pricing in a regime switching market where the floating interest rate depends on a finite state Markov process. The growth rate and the volatility of the stock also depend on the Markov process. Using the minimal martingale measure, we show that the locally risk minimizing prices for certain exotic options satisfy a system of Black-Scholes partial differential equations with appropriate boundary conditions. We find the corresponding hedging strategies and the residual risk. We develop suitable numerical methods to compute option prices.     

可违约债券在随机波动率假定下近似定价公式的求解

在假设标的资产价格的波动率是一个快速均值回复OU过程的函数的条件下,导出相应的可违约债券价格公式所应满足的偏微分方程,并利用Taylor级数展开得到一组Poisson方程.求解这些方程,得到非完全市场下固定补偿率的债券价格的近似表达式,然后在不同的补偿率规定上作了一些修正和推广.     

Calibration of options on a reduced basis

Calibration of models is an important step in financial engineering. However it can be costly, especially in view of the increasing complexity of the models.In this paper we explore the use of reduced basis as is done in fluid mechanics for the Navier-Stokes equations or as proposed by Maday, Patera and Turinici [Y. Maday et al., A priori convergence theory for reduced-basis approximations of single-parameter elliptic partial differential equations, J. Sci. Comput. 17 (1-4) (2002) 437-446]. It is shown that the method works well if we use convex combination of the basis functions instead of the more general linear combination; however, while this idea makes sense in view of the properties of the Black-Scholes equation, we have no proof to general linear combination; however, while this idea makes sense in view of the properties of the Black-Scholes equation, we have no proof to justify it mathematically.The paper presents a numerical investigation of the problem posed.