Asian Basket Options Research Articles

Density estimates and short-time asymptotics for a hypoelliptic diffusion process

We study a system of n differential equations, each in dimension d. Only the first equation is forced by a Brownian motion and the dependence structure is such that, under a local weak Hörmander condition, the noise propagates to the whole system. We prove upper bounds for the transition density (heat kernel) and its derivatives of any order. Then we give precise short-time asymptotics of the density at a suitable central limit time scale. Both these results account for the different non-diffusive scales of propagation in the various components. Finally, we provide a valuation formula for short-maturity at-the-money Asian basket options under correlated local volatility dynamics.

An efficient acceleration Monte Carlo simulation for pricing Asian option under variance gamma process by splitting

ABSTRACTThis paper proposes a hybrid acceleration method for pricing Asian options with arithmetic average under variance gamma process. We split the payoff of Asian option into two parts, the first part is a small probability event, so the importance sampling method is naturally used to reduce variance of simulation efficiently. The second part can be simplified as a semi-closed form, which also be considered as a conditional expectation formula, so it may reduce variance of simulations. Then the importance sampling method is also adapted to reduce variance of simulations further. To save computation cost for determining the optimal parameters in the importance sampling, a moment matching-based technique is proposed. Numerical results are given to show the high efficiency of our method. The idea used in this paper may also be applicable to price Asian option and Basket option under other time-changed Brownian motion and Heston models.

An Extension of the Chaos Expansion Approximation for the Pricing of Exotic Basket Options

Funahashi and Kijima (in press, A chaos expansion approach for the pricing of contingent claims, Journal of Computational Finance) have proposed an approximation method based on the Wiener–Ito chaos expansion for the pricing of European-style contingent claims. In this paper, we extend the method to the multi-asset case with general local volatility structure for the pricing of exotic basket options such as Asian basket options. Through ample numerical experiments, we show that the accuracy of our approximation remains quite high even for a complex basket option with long maturity and high volatility.

Parallel Randomized Quasi-Monte Carlo Simulation for Asian Basket Option Pricing

High-dimensional derivatives pricing, such as Asian basket options, poses great computational challenges in practice. In this paper, parallel Randomized Quasi-Monte Carlo (RQMC) simulation method is investigated to tackle this kind of intractable problems. By using the intrinsic nature of “embarrassingly parallel”, parallel RQMC algorithm for Asian basket option pricing is effectively implemented using MPI. Numerical experiments are performed on supercomputer DeepComp6800 and the parallel performance is then analyzed. Our parallel algorithm has exerted perfect performance and good scalability. Parallel computing has greatly improved the computational efficiency of derivatives pricing.

Pricing and Hedging Asian Basket Options with Quasi-Monte Carlo Simulations

In this article we consider the problem of pricing and hedging high-dimensional Asian basket options by Quasi-Monte Carlo simulations. We assume a Black–Scholes market with time-dependent volatilities, and we compute the deltas by means of the Malliavin Calculus as an extension of the procedures employed by Kohatsu-Higa and Montero (Physica A 320:548–570, 2003). Efficient path-generation algorithms, such as Linear Transformation and Principal Component Analysis, exhibit a high computational cost in a market with time-dependent volatilities. To face this challenge we then introduce a new and faster Cholesky algorithm for block matrices that makes the Linear Transformation more convenient. We also propose a new-path generation technique based on a Kronecker Product Approximation. Our procedure shows the same accuracy as the Linear Transformation used for the computation of deltas and prices in the case of correlated asset returns, while requiring a shorter computational time. All these techniques can be easily employed for stochastic volatility models based on the mixture of multi-dimensional dynamics introduced by Brigo et al. (2004a, Risk 17(5):97–101, b).

Implementing quasi-Monte Carlo simulations with linear transformations

The curse of dimensionality limits the accuracy of the quasi-Monte Carlo (QMC) method in high-dimensional problems. Imai and Tan (Proceedings of the 2002 winter simulation conference, 2002; Monte Carlo and Quasi-Monte Carlo Methods 2002, pp 275–292, Springer, Berlin, 2004; J Comput Finance 10(2):129–155, 2007) have proposed a dimension reduction technique, named linear transformation (LT), aiming to improve the efficiency of the QMC method. We investigate this approach in detail and make it more convenient. We implement a faster QR decomposition that considerably reduces the computational burden. The efficacy of our algorithm is illustrated by considering two high-dimensional option pricing problems: Asian basket options in the Black–Scholes (BS) model and Asian options in the Cox–Ingersoll–Ross (CIR) model. We employ a QMC generator only for the components selected by the LT construction and use Latin hypercube sampling (LHS) for all the others. Finally, we compare our results to those obtained by different random number generators and standard algorithms; subsequently, we benchmark our computational times against those presented in Imai and Tan (J Comput Finance 10(2):129–155, 2007).

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.

Efficient quasi-Monte simulations for pricing high-dimensional path-dependent options

Quasi-Monte Carlo method (QMC) is an efficient technique for numerical integration. QMC provides a lower convergence rate, O(lndn/n), than the standard Monte Carlo (MC), \({O( 1/\sqrt{n})}\) , where n is the number of simulations and d the nominal problem dimension. However, some studies in the literature have claimed that the QMC performs better than the MC method for d < 20/30 because of its dependence on d. Caflisch et al. (J Comput Finance 1(1):27–46, 1997) have proposed to extend the QMC superiority by ANOVA considerations. To this aim, we consider the Asian basket option pricing problem, where d is much higher than 30, by QMC simulation. We investigate the applicability of several path-generation constructions that have been proposed to overtake the dimensional drawback. We employ the principal component analysis, the linear transformation, the Kronecker product approximation and test their performance both in terms of computational cost and accuracy. Finally, we compare the results with those obtained by the standard MC.

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.

A general dimension reduction technique for derivative pricing

For a trajectory simulated from s standardized independent normal variates ε = (ε1, . . . , εs )1, the payoff of a European option can be represented as max[g(ε), 0], where the function g(ε) is assumed to be differentiable and it relates to the nature of the derivative securities. In this paper, we develop a new simulation technique by introducing an orthogonal class of transformation to ε so that the function g is instead generated from g(Aε), where A is an s-dimensional orthogonal matrix. The matrix A is optimally determined so that the effective dimension of the underlying function is minimized, thereby enhancing the quasi-Monte Carlo (QMC) method. The proposed simulation approach has the advantage of greater generality and has a wide range of applications as long as the problem of interest can be represented by g(ε). The flexibility of our proposed technique is illustrated by applying it to two high-dimensional applications: Asian basket options and European call options with a stochastic volatility model. We benchmark our proposed method against well-known efficient simulation algorithms that have been advocated in these applications. The numerical results demonstrate that our proposed technique can be an extremely powerful simulation method when combined with QMC.

Partially exact and bounded approximations for arithmetic Asian options

This paper considers the pricing of European Asian options in the Black-Scholes framework. All approaches we consider are readily extendable to the case of an Asian basket option. We consider three methods for evaluating the price of an Asian option, and contribute to all three. Firstly, we show the link between the approaches of Rogers and Shi [1995], Andreasen [1999], Hoogland and Neumann [2000] and Veceř [2001]. For the latter formulation we propose two reductions, which increase the numerical stability and reduce the calculation time. Secondly, we show how a closed-form expression can be derived for Curran’s and Rogers and Shi’s lower bound for the general case of multiple underlyings. Thirdly, we considerably sharpen Thompson’s [1999a,b] upper bound such that it is tighter than all known upper bounds. Finally, we consider analytical approximations and combine the traditional moment matching approximations with Curran’s conditioning approach. The resulting class of partially exact and bounded approximations can be proven to lie between a sharp lower and upper bound. In numerical examples we demonstrate that they outperform all current state-of-the-art bounds and approximations.

Partially Exact and Bounded Approximations for Arithmetic Asian Options

This paper considers the pricing of European Asian options in the Black-Scholes framework. All approaches we consider are readily extendable to the case of an Asian basket option. Firstly we consider the partial differential equation approach to the pricing of Asian options. We show the link between the approaches of Rogers and Shi [1995], Andreasen [1999], Hoogland and Neumann [2000] and Vecer [2001]. For the latter two formulations we propose two reductions, which increase the numerical stability and reduce the calculation time. Secondly, we show how a closed-form expression can be derived for Rogers and Shi's lower bound for the general case of multiple underlyings. Thirdly, we sharpen Thompson's [1999a,b] upper bound for the value of an Asian option. This is important for the practically relevant case of options with long maturities. Numerical results show that when the strike price is not extremely high, the resulting upper bound is tighter than recently introduced upper bounds in studies by Nielsen and Sandmann [2003] and Vanmaele et al. [2005]. Finally, we consider analytical approximations for the value of an Asian option. A much heard criticism on moment-matching approaches is that the error in the approximation is not known beforehand. We combine the traditional moment-matching approaches (e.g. Levy [1992]) with the conditioning approaches (e.g. Curran [1994]) and introduce a class of analytical approximations, which can be proven to lie between a sharp lower and upper bound. In numerical examples the accuracy of these new approximations is demonstrated. The approximations are found to outperform all of the current state-of-the-art upper bounds and approximations.

The design and pricing of fixed‐ and moving‐window contracts: An application of Asian‐Basket option pricing methods to the hog‐finishing sector

AbstractAsian‐Basket‐type moving‐window contracts are an increasingly used risk‐management tool in the North American hog sector. The moving‐window contract is decomposed into a portfolio of a long Asian‐Basket put and a short Asian‐Basket call option. A projected break‐even price is used to determine the floor price, and then Monte Carlo simulation methods are used to price both a moving‐ and a fixed‐window contract. These methods provide unbiased pricing of fixed‐ and moving‐window hog‐finishing contracts of 1‐year duration. © 2003 Wiley Periodicals, Inc. Jrl Fut Mark 23:1047–1073, 2003

Livestock Revenue Insurance

AbstractThis study outlines several possible structures for livestock revenue insurance. The policies take the form of an exotic option, an Asian basket option. The actuarially fair premiums for these policies are equal to the prices of the options they represent. Because of the complexity of pricing Asian basket options, we combined two techniques for pricing options to reach the actuarially fair premiums. Projected premiums, producer welfare, and program efficiency are evaluated for the insurance products and existing market tools. Using efficiency ratios and certainty equivalent returns, we compare the insurance policies to strategies involving existing futures and options. © 2001 John Wiley &amp; Sons, Inc. Jrl Fut Mark 21:553–580, 2001