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.     

Bounds for in-progress floating-strike Asian options using symmetry

This paper studies symmetries between fixed and floating-strike Asian options and exploits this symmetry to derive an upper bound for the price of a floating-strike Asian. This bound only involves fixed-strike Asians and vanillas, and can be computed simply given one of the many efficient methods for pricing fixed-strike Asian options. The bound coincides with the true price until after the averaging has begun and again at maturity. The bound is compared to benchmark prices obtained via Monte Carlo simulation in numerical examples. D. Hobson is supported by an Advanced Fellowship from the EPSRC. V. Henderson is partially supported by the NSF under grant DMI 0447990.     

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.     

双随机跳扩散模型下亚式期权的定价

研究了双随机跳扩散模型下的亚式期权的定价问题.首先引入一个双随机跳扩散过程.然后通过测度变换消除了亚式期权定价中的路经依赖性问题.最后利用鞅定价方法和Ito引理得到了跳扩散模型下的亚式期权价格必须满足的一个积微分方程.通过数值求解该积微分方程就可以得到了亚式期权的价格,供投资者参考.     

Bounds for Asian basket options

In this paper we propose pricing bounds for European-style discrete arithmetic Asian basket options in a Black and Scholes framework. We start from methods used for basket options and Asian options. First, we use the general approach for deriving upper and lower bounds for stop-loss premia of sums of non-independent random variables as in Kaas et al. [Upper and lower bounds for sums of random variables, Insurance Math. Econom. 27 (2000) 151–168] or Dhaene et al. [The concept of comonotonicity in actuarial science and finance: theory, Insurance Math. Econom. 31(1) (2002) 3–33]. We generalize the methods in Deelstra et al. [Pricing of arithmetic basket options by conditioning, Insurance Math. Econom. 34 (2004) 55–57] and Vanmaele et al. [Bounds for the price of discrete sampled arithmetic Asian options, J. Comput. Appl. Math. 185(1) (2006) 51–90]. Afterwards we show how to derive an analytical closed-form expression for a lower bound in the non-comonotonic case. Finally, we derive upper bounds for Asian basket options by applying techniques as in Thompson [Fast narrow bounds on the value of Asian options, Working Paper, University of Cambridge, 1999] and Lord [Partially exact and bounded approximations for arithmetic Asian options, J. Comput. Finance 10 (2) (2006) 1–52]. Numerical results are included and on the basis of our numerical tests, we explain which method we recommend depending on moneyness and time-to-maturity.     

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 .     

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.     

NUMERICAL ANALYSIS ON BINOMIAL TREE METHODS FOR AMERICAN LOOKBACK OPTIONS

1　IntroductionLookback options are path-dependent options whose payoffs depend on the maximumor the minimum of the underlying asset price during the life of the options( see[6] [1 0 ][1 4] ) .Here the maximum or minimum realized asset price may be monitored either con-tinuously or discretely.An American lookback call( put) option allows to be exercised atany time prior to expiry and gives the holder the rightto buy( sell) atthe historical mini-mum( maximum) of the underlying asset price on ex…     

Regret aversion and annuity risk in defined contribution pension plans

The high value of the implicit option to choose a retirement date at which interest rates are particularly high and life annuities relatively cheap, leads to the possibility to introduce regret aversion in the retirement investment decision of defined contribution plan participants. As a remedy for regret aversion in retirement investment decisions, this paper develops and prices a lookback option on a life annuity contract. We determine a closed-form option value under the restriction that the option holder invests risklessly during the time to maturity of the option and without the guarantee that the exact amount of retirement wealth is converted into a life annuity at retirement. Thereafter the investment restriction is relaxed and the guarantee of exact conversion is imposed and the option is priced via Monte Carlo simulations in an economic environment with a stochastic discount factor. Option price sensitivities are determined via the pricing of alternative options. We find that the price of a lookback option, with a maturity of three years, amounts to 8%–9% of the wealth at the option issuance date. The option price is highly sensitive to the exercise price of the option, i.e. pricing alternative options (e.g. Asian) substantially lowers the price. Time to maturity and interest rate volatility are other important option price drivers. Asset allocation decisions and initial interest rates hardly affect the option price.     

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.     

Two Efficient Parameterized Boundaries for Veneers Asian Option Pricing PDE

In this paper, we derive two general parameterized boundaries of finite difference scheme for Ve?e???s PDE which is used to price both fixed and floating strike Asian options. Using these two boundaries, we can deal with all kinds of situations, especially, some extreme cases, like overhigh volatility, very small volatility, etc, under which the Asian option is usually mispriced in many existing numerical methods. Numerical results show that our boundaries are pretty efficient.     

DG method for the numerical pricing of two-asset European-style Asian options with fixed strike

The evaluation of option premium is a very delicate issue arising from the assumptions made under a financial market model, and pricing of a wide range of options is generally feasible only when numerical methods are involved. This paper is based on our recent research on numerical pricing of path-dependent multi-asset options and extends these results also to the case of Asian options with fixed strike. First, we recall the three-dimensional backward parabolic PDE describing the evolution of European-style Asian option contracts on two assets, whose payoff depends on the difference of the strike price and the average value of the basket of two underlying assets during the life of the option. Further, a suitable transformation of variables respecting this complex form of a payoff function reduces the problem to a two-dimensional equation belonging to the class of convection-diffusion problems and the discontinuous Galerkin (DG) method is applied to it in order to utilize its solving potentials. The whole procedure is accompanied with theoretical results and differences to the floating strike case are discussed. Finally, reference numerical experiments on real market data illustrate comprehensive empirical findings on Asian options.     

Convergence of the binomial tree method for Asian options in jump-diffusion models

The binomial tree methods (BTM), first proposed by Cox, Ross and Rubinstein [J. Cox, S. Ross, M. Rubinstein, Option pricing: A simplified approach, J. Finan. Econ. 7 (1979) 229-264] in diffusion models and extended by Amin [K.I. Amin, Jump diffusion option valuation in discrete time, J. Finance 48 (1993) 1833-1863] to jump-diffusion models, is one of the most popular approaches to pricing options. In this paper, we present a binomial tree method for Asian options in jump-diffusion models and show its equivalence to certain explicit difference scheme. Employing numerical analysis and the notion of viscosity solution, we prove the uniform convergence of the binomial tree method for European-style and American-style Asian options.     

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.     

Bounds for the price of a European-style Asian option in a binary tree model

Inspired by the ideas of Rogers and Shi [J. Appl. Prob. 32 (1995) 1077], Chalasani et al. [J. Comput. Finance 1(4) (1998) 11] derived accurate lower and upper bounds for the price of a European-style Asian option with continuous averaging over the full lifetime of the option, using a discrete-time binary tree model. In this paper, we consider arithmetic Asian options with discrete sampling and we generalize their method to the case of forward starting Asian options. In this case with daily time steps, the method of Chalasani et al. is still very accurate but the computation can take a very long time on a PC when the number of steps in the binomial tree is high. We derive analytical lower and upper bounds based on the approach of Kaas et al. [Insurance: Math. Econ. 27 (2000) 151] for bounds for stop-loss premiums of sums of dependent random variables, and by conditioning on the value of underlying asset at the exercise date. The comonotonic upper bound corresponds to an optimal superhedging strategy. By putting in less information than Chalasani et al. the bounds lose some accuracy but are still very good and they are easily computable and moreover the computation on a PC is fast. We illustrate our results by different numerical experiments and compare with bounds for the Black and Scholes model [J. Pol. Econ. 7 (1973) 637] found in another paper [Bounds for the price of discretely sampled arithmetic Asian options, Working paper, Ghent University, 2002]. We notice that the intervals of Chalasani et al. do not always lie within the Black and Scholes intervals. We have proved that our bounds converge to the corresponding bounds in the Black and Scholes model. Our numerical illustrations also show that the hedging error is small if the Asian option is in the money. If the option is out of the money, the price of the superhedging strategy is not as adequate, but still lower than the straightforward hedge of buying one European option with the same exercise price.     

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.     

Pricing and hedging Asian basket spread options

Asian options, basket options and spread options have been extensively studied in the literature. However, few papers deal with the problem of pricing general Asian basket spread options. This paper aims to fill this gap. In order to obtain prices and Greeks in a short computation time, we develop approximation formulae based on comonotonicity theory and moment matching methods. We compare their relative performances and explain how to choose the best approximation technique as a function of the Asian basket spread characteristics. We also give explicitly the Greeks for our proposed methods. In the last section we extend our results to options denominated in foreign currency.     

跳跃扩散型离散算术平均亚式期权的近似价格公式

在标的资产价格遵循跳跃扩散过程条件下 ,研究没有封闭形式解的离散算术平均亚式期权 ,运用二阶 Edgeworth逼近得到离散算术平均亚式期权的近似价格公式 .     

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.     

Moment matching approximation of Asian basket option prices

In this paper we propose some moment matching pricing methods for European-style discrete arithmetic Asian basket options in a Black & Scholes framework. We generalize the approach of [M. Curran, Valuing Asian and portfolio by conditioning on the geometric mean price, Management Science 40 (1994) 1705-1711] and of [G. Deelstra, J. Liinev, M. Vanmaele, Pricing of arithmetic basket options by conditioning, Insurance: Mathematics & Economics 34 (2004) 55-57] in several ways. We create a framework that allows for a whole class of conditioning random variables which are normally distributed. We moment match not only with a lognormal random variable but also with a log-extended-skew-normal random variable. We also improve the bounds of [G. Deelstra, I. Diallo, M. Vanmaele, Bounds for Asian basket options, Journal of Computational and Applied Mathematics 218 (2008) 215-228]. Numerical results are included and on the basis of our numerical tests, we explain which method we recommend depending on moneyness and time-to-maturity.