Efficient pricing of discrete Asian options

Asian options are popular path-dependent financial derivatives. This paper uses lattices to price fixed-strike European-style Asian options that are discretely monitored. The algorithm proposed can also be applied to floating-strike Asian options as well because fixed-strike and floating-strike Asian options are related through an equation. The discretely monitored version is usually found in practice instead of the continuously monitored version usually encountered in the literature. This paper presents the first provably quadratic-time convergent lattice algorithm for pricing fixed-strike European-style discretely monitored Asian options. It is the most efficient lattice algorithm with convergence guarantees. The algorithm relies on the Lagrange multipliers to choose the number of states for each node of the lattice. Extensive numerical experiments and comparisons with many existing numerical methods confirm the performance claims and the competitiveness of our algorithm. This result places fixed-strike European-style discretely monitored Asian options in the same complexity class as vanilla options.     

GARCH模型中美式亚式期权价值的蒙特卡罗模拟算法

我们运用 Longstaff和 Schwartz最近提出的用蒙特卡罗模拟法计算美式期权的方法在 GARCH模型中求解美式亚式期权 ,我们的结果表明和其它数值方法相比 ,这个方法不仅有相当的精确度 ,而且使用简便并具有更广泛的适用性 ,对于 GARCH模型中运用格点法难以求解的浮动执行价格的美式亚式期权同样可以得到稳定解 .     

基于加权可能性均值的亚式期权模糊定价

主要探讨不确定环境下用模糊集理论处理亚式期权的定价问题.运用梯形模糊数来表示标的资产价格、无风险利率、红利率和波动率,建立了亚式期权的加权可能性均值模糊定价模型,得到连续几何和算术亚式期权的模糊价格公式.最后通过数值例子表明:亚式期权的加权可能性均值模糊定价模型具有很大的灵活性,更符合现实的不确定情况,具有较强的实用价值.     

Prices and sensitivities of Asian options: A survey

Asian options are hard to price both analytically and numerically. Even though they have been the focus of much attention in recent years, there is no single technique which is widely accepted to price Asian options for all choices of market parameters. For hedging purposes, the estimation of the price sensitivities is often as important as the evaluation of the prices themselves. This paper provides a survey of current methods for pricing Asian options and computing their sensitivities to the key input parameters. The methods discussed include: Monte Carlo simulation, the finite difference approach and various quasi analytical approaches and approximations. We discuss practical numerical issues that arise in implementing these methods. The paper compares the accuracy and efficiency of the different approaches and offers some general conclusions.     

拟蒙特卡罗法在亚洲期权定价中的应用

亚洲期权是场外交易中几种最受欢迎的新型期权之一,但它的价格却没有解析表达式,到目前为止,亚洲期权的定价仍是个公开问题.本文采用拟蒙特卡罗法中的Halton序列来估计它的价格,数值结果表明当观察点的个数N13时,它比蒙特卡罗法要好.本文还利用MATLAB程序生成了随机Halton序列,并将它与控制变量法结合起来估计亚洲期权的价格,估计值标准差的比较表明它在大多情况下比相应的蒙特卡罗法的估计效果要好.     

Asian option as a fixed-point

We characterize the price of an Asian option, a financial contract, as a fixed-point of a non-linear operator. In recent years, there has been interest in incorporating changes of regime into the parameters describing the evolution of the underlying asset price, namely the interest rate and the volatility, to model sudden exogenous events in the economy. Asian options are particularly interesting because the payoff depends on the integrated asset price. We study the case of both floating- and fixed-strike Asian call options with arithmetic averaging when the asset follows a regime-switching geometric Brownian motion with coefficients that depend on a Markov chain. The typical approach to finding the value of a financial option is to solve an associated system of coupled partial differential equations. Alternatively, we propose an iterative procedure that converges to the value of this contract with geometric rate using a classical fixed-point theorem.     

The pricing of Asian options under stochastic interest rates

The purpose of this paper is to analyse the effect of stochastic interest rates on the pricing of Asian options. It is shown that a stochastic, in contrast to a deterministic, development of the term structure of interest rates has a significant influence. The price of the underlying asset, e.g. a stock or oil, and the prices of bonds are assumed to follow correlated two-dimensional Itô processes. The averages considered in the Asian options are calculated on a discrete time grid, e.g. all closing prices on Wednesdays during the lifetime of the contract. The value of an Asian option will be obtained through the application of Monte Carlo simulation, and for this purpose the stochastic processes for the basic assets need not be severely restricted. However, to make comparison with published results originating from models with deterministic interest rates, we will stay within the setting of a Gaussian framework.     

Numerical solution of variational inequalities for pricing Asian options by higher order Lagrange–Galerkin methods

Asian options prices can be modelled in the Black–Scholes framework leading to two-factor models depending on the asset price, the average of the asset price and the time. They can also involve inequality constraints, as in the case of Amerasian options, leading to variational inequalities (VI). In the first section, we completely describe the pricing model for fixed-strike Eurasian and Amerasian options and list some properties satisfied by the option value function. Then, since no solutions in closed form are known, we deal with the numerical solution of the above problems proposing a general methodology: an iterative algorithm for the VI, combined with higher order Lagrange–Galerkin methods for partial differential equations. Finally, numerical results are shown.     

一种亚式风格可重置执行价格期权设计

本文设计了一种亚式风格的可重置执行价格期权;严格证明了可重置执行边界的存在性,以及连续区域与重置区域的单连通性;利用Hartman-Watson分布,写出了可重置期权的定价公式,并利用此公式给出了可重置执行边界的一种新的数值算法.     

具有固定敲定价格的算术平均亚式期权的计算

本文研究算术平均的欧式亚式期权.我们将充分利用偏微分方程的Fichera理论和边值问题的定解理论,求出了一个简单的近似解析表达式.经实际数据验算,有较满意的逼近结果,特别地,在部分区域内的计算效果好于文章[1].     

基于分数O-U随机过程的新型亚式期权的定价

本文运用衍生证券理论的最基本原理(△对冲和无套利原理),研究了一种新型亚式期权的定价问题,该类型期权因具有常数平均值久期而不同于标准化情形.假设标的资产(气温)由分数Ornstein-Uhlenbeck过程驱动,这样假设对天气衍生品来说是合理的.本文得到了这种新型亚式期权的动态定价方程.     

Robust Approximations for Pricing Asian Options and Volatility Swaps Under Stochastic Volatility

Abstract

We show that if the discounted Stock price process is a continuous martingale, then there is a simple relationship linking the variance of the terminal Stock price and the variance of its arithmetic average. We use this to establish a model-independent upper bound for the price of a continuously sampled fixed-strike arithmetic Asian call option, in the presence of non-zero time-dependent interest rates (Theorem 1.2). We also propose a model-independent lognormal moment-matching procedure for approximating the price of an Asian call, and we show how to apply these approximations under the Black–Scholes and Heston models (subsection 1.3). We then apply a similar analysis to a time-dependent Heston stochastic volatility model, and we show how to construct a time-dependent mean reversion and volatility-of-variance function, so as to be consistent with the observed variance swap curve and a pre-specified term structure for the variance of the integrated variance (Theorem 2.1). We characterize the small-time asymptotics of the first and second moments of the integrated variance (Proposition 2.2) and derive an approximation for the price of a volatility swap under the time-dependent Heston model ( Equation (52)), using the Brockhaus–Long approximation (Brockhaus, and Long, 2000 Brockhaus, O. and Long, D. 2000. Volatility Swaps made simple. Risk, 13(1) January: 92–96.  [Google Scholar]). We also outline a bootstrapping procedure for calibrating a piecewise-linear mean reversion level and volatility-of-volatility function (Subsection 2.3.2).     

Options on Realized Variance in Log-OU Models

Abstract

We study the pricing of options on realized variance in a general class of Log-OU (Ornstein–Ühlenbeck) stochastic volatility models. The class includes several important models proposed in the literature. Having as common feature the log-normal law of instantaneous variance, the application of standard Fourier–Laplace transform methods is not feasible. We derive extensions of Asian pricing methods, to obtain bounds, in particular, a very tight lower bound for options on realized variance.     

On the explicit evaluation of the Geometric Asian options in stochastic volatility models with jumps

In the present paper we provide a semiexplicit valuation formula for Geometric Asian options, with fixed and floating strike under continuous monitoring, when the underlying stock price process exhibits both stochastic volatility and jumps. More precisely, we shall work in the Barndorff-Nielsen and Shephard (BNS) model framework. We shall provide some numerical illustrations of the results obtained.     

Static super-replicating strategies for a class of exotic options

In this paper, we investigate static super-replicating strategies for European-type call options written on a weighted sum of asset prices. This class of exotic options includes Asian options and basket options among others. We assume that there exists a market where the plain vanilla options on the different assets are traded and hence their prices can be observed in the market. Both the infinite market case (where prices of the plain vanilla options are available for all strikes) and the finite market case (where only a finite number of plain vanilla option prices are observed) are considered. We prove that the finite market case converges to the infinite market case when the number of observed plain vanilla option prices tends to infinity.We show how to construct a portfolio consisting of the plain vanilla options on the different assets, whose pay-off super-replicates the pay-off of the exotic option. As a consequence, the price of the super-replicating portfolio is an upper bound for the price of the exotic option. The super-hedging strategy is model-free in the sense that it is expressed in terms of the observed option prices on the individual assets, which can be e.g. dividend paying stocks with no explicit dividend process known. This paper is a generalization of the work of Simon et al. [Simon, S., Goovaerts, M., Dhaene, J., 2000. An easy computable upper bound for the price of an arithmetic Asian option. Insurance Math. Econom. 26 (2–3), 175–184] who considered this problem for Asian options in the infinite market case. Laurence and Wang [Laurence, P., Wang, T.H., 2004. What’s a basket worth? Risk Mag. 17, 73–77] and Hobson et al. [Hobson, D., Laurence, P., Wang, T.H., 2005. Static-arbitrage upper bounds for the prices of basket options. Quant. Fin. 5 (4), 329–342] considered this problem for basket options, in the infinite as well as in the finite market case.As opposed to Hobson et al. [Hobson, D., Laurence, P., Wang, T.H., 2005. Static-arbitrage upper bounds for the prices of basket options. Quant. Fin. 5 (4), 329–342] who use Lagrange optimization techniques, the proofs in this paper are based on the theory of integral stochastic orders and on the theory of comonotonic risks.     

Simulation methods for valuing Asian option prices in a hyperbolic asset price model

In this paper, we present a Quasi-Monte Carlo approach for pricingEuropean-style Asian options, i.e. for options whose pay-offdepends on the average price of the underlying asset where theaverage is extended over a fixed period up to the maturity date.Following a recent development in mathematical finance, we assumethat the log returns of the asset are not normally but hyperbolicallydistributed. This hypothesis is approved by several authorswith different statistic tests on real financial data. The aimof this paper is to advance the hyperbolic model to the pricingof Asian options, since there only exist pricing formulae forplain vanilla options and some types of exotic options (e.g.power call options, barrier options) so far. We show how onecan obtain prices of general Asian options in such incompletemarkets in an efficient way.     

Semiparametric bounds of mean and variance for exotic options

Finding semiparametric bounds for option prices is a widely studied pricing technique. We obtain closed-form semiparametric bounds of the mean and variance for the pay-off of two exotic (Collar and Gap) call options given mean and variance information on the underlying asset price. Mathematically, we extended domination technique by quadratic functions to bound mean and variances. This work was supported by National Science Foundation of the United States (Grant Nos. DMS-0720977 and DMS-0805929)     

The discontinuous Galerkin method for discretely observed Asian options

Asian options represent an important subclass of the path-dependent contracts that are identified by payoff depending on the average of the underlying asset prices over the prespecified period of option lifetime. Commonly, this average is observed at discrete dates, and also, early exercise features can be admitted. As a result, analytical pricing formulae are not always available. Therefore, some form of a numerical approximation is essential for efficient option valuation. In this paper, we study a PDE model for pricing discretely observed arithmetic Asian options with fixed as well as floating strike for both European and American exercise features. The pricing equation for such options is similar to the Black-Scholes equation with 1 underlying asset, and the corresponding average appears only in the jump conditions across the sampling dates. The objective of the paper is to present the comprehensive methodological concept that forms and improves the valuation process. We employ a robust numerical procedure based on the discontinuous Galerkin approach arising from the piecewise polynomial generally discontinuous approximations. This technique enables a simple treatment of discrete sampling by incorporation of jump conditions at each monitoring date. Moreover, an American early exercise constraint is directly handled as an additional nonlinear source term in the pricing equation. The proposed solving procedure is accompanied by an empirical study with practical results compared to reference values.     

两个或多个几何平均价格的最小或最大值期权的定价

两个或多个几何平均价格的最小或最大值期权是金融领域极具应用前景的新型复合期权.提出了一种新方法,简单而巧妙地得到了两个几何平均价格的最小值期权价格的解析公式.将该法直接推广,首次得到多个几何平均价格的最小和最大值期权的解析公式.首次给出的数值算例表明两个几何平均价格的最小值期权要比相应的最大值期权便宜,而它们都要比两资产的最大值期权便宜.若考虑红利率,则它们两者的价格都会减少.     

Pricing American Asian options with higher moments in the underlying distribution

We develop a modified Edgeworth binomial model with higher moment consideration for pricing American Asian options. With lognormal underlying distribution for benchmark comparison, our algorithm is as precise as that of Chalasani et al. [P. Chalasani, S. Jha, F. Egriboyun, A. Varikooty, A refined binomial lattice for pricing American Asian options, Rev. Derivatives Res. 3 (1) (1999) 85–105] if the number of the time steps increases. If the underlying distribution displays negative skewness and leptokurtosis as often observed for stock index returns, our estimates can work better than those in Chalasani et al. [P. Chalasani, S. Jha, F. Egriboyun, A. Varikooty, A refined binomial lattice for pricing American Asian options, Rev. Derivatives Res. 3 (1) (1999) 85–105] and are very similar to the benchmarks in Hull and White [J. Hull, A. White, Efficient procedures for valuing European and American path-dependent options, J. Derivatives 1 (Fall) (1993) 21–31]. The numerical analysis shows that our modified Edgeworth binomial model can value American Asian options with greater accuracy and speed given higher moments in their underlying distribution.