Geometric Asian Research Articles

Generalized Barndorff-Nielsen and Shephard Model and Discretely Monitored Option Pricing

This paper proposes a generalization of the Generalized Barndorff-Nielsen and Shephard model, in which the log return on an asset price is governed by a Levy process with stochastic volatility modeled by a non-Gaussian Ornstein-Uhlenbeck process. Under the generalized model, we derive a closed-form expression for the multivariate characteristic function of the intertemporal joint distribution of the underlying log return. Then, we also investigate asymptotic behavior of the log return and its variance. Moreover, we evaluate discretely monitored path-dependent derivatives such as geometric Asian, forward start, barrier, fade-in, and lookback options as well as European options.

Static Hedging of Geometric Average Asian Options with Standard Options

In this article, we first establish a theorem that represents the price of an Asian option in terms of standard European options with a shorter term and different strikes. Then using Gauss–Hermite numerical integration, we discretize our theorem so as to use Monte Carlo simulation to examine the error of the static hedging under the Black–Scholes model and the Merton jump-diffusion model. For ease of comparison, we also provide the error of the dynamic hedging. The numerical results show that the static hedging strategy performs better than the dynamic one under both models.

The Mellin Transform Method as an Alternative Analytic Solution for the Valuation of Geometric Asian Option

This paper presents the Mellin transform method as an alternative analytic solution for the valuation of geometric Asian option. Asian options are options in which the variable is the average price over a period of time. The analytical solution of the Black-Scholes partial differential equation for Asian option is known as an explicit formula, this is due to the fact that the geometric average of a set of lognormal random variables is lognormally distributed. We derive a closed form solution for a continuous geometric Asian option by means of the partial differential equations. We also provide an alternative method for solving geometric Asian options partial differential equations using the Mellin transform method.

Hedging strategies for discretely monitored Asian options under Lévy processes

In this work, we consider a variance-optimal hedging strategy for discretely sampled geometric Asian options, under exponential Levy dynamics. Since it is difficult to hedge these instruments perfectly, here we choose to maximize a quadratic utility function and give the expressions of hedging strategies explicitly, based on the derived Follmer-Schweizer decomposition of the contingent claim of geometric Asian options monitored at discrete times. The expression of its corresponding error is also given.

Evaluation of Geometric Asian Power Options under Fractional Brownian Motion

Modern option pricing techniques are often considered among the most mathematical complex of all applied areas of financial mathematics. In particular, the fractional Brownian motion is proper to model the stock dynamics for its long-range dependence. In this paper, we evaluate the price of geometric Asian options under fractional Brownian motion framework. Furthermore, the options are generalized to those with the added feature whose payoff is a power function. Based on the equivalent martingale theory, a closed form solution has been derived under the risk neutral probability.

Optimal system, symmetry reductions and new closed form solutions for the geometric average Asian options

In this paper, the Lie group analysis method is applied to the geometric average Asian option pricing equation in financial problems. Firstly, the complete Lie symmetry group and infinitesimal generators of this equation are derived. Then the optimal system with one parameter for the Lie symmetry algebra are obtained, which gives the possibility to describe a complete set of invariant solutions to the pricing equation. Finally, based on the optimal system the symmetry reductions and corresponding closed form solutions for the pricing equation are proposed.

Efficient Pricing of European-Style Asian Options under Exponential Lévy Processes Based on Fourier Cosine Expansions

We propose an efficient pricing method for arithmetic and geometric Asian options under exponential Levy processes based on Fourier cosine expansions and Clenshaw–Curtis quadrature. The pricing method is developed for both European style and American-style Asian options and for discretely and continuously monitored versions. In the present paper we focus on the European-style Asian options. The exponential convergence rates of Fourier cosine expansions and Clenshaw–Curtis quadrature reduces the CPU time of the method to milliseconds for geometric Asian options and a few seconds for arithmetic Asian options. The method’s accuracy is illustrated by a detailed error analysis and by various numerical examples.

Pricing Path-Dependent Options with Discrete Monitoring Under Time-Changed Levy Processes

This paper proposes a pricing method for path-dependent derivatives with discrete monitoring when an underlying asset price is driven by a time-changed Levy process. The key to our method is to derive a backward recurrence relation for computing the multivariate characteristic functions of the intertemporal joint distribution of time-changed Levy processes. Using the derived representation of the characteristic function we obtain semi-analytical pricing formulas for geometric Asian, forward start, barrier, fader, and lookback options, all of which are discretely monitored.

Pricing of geometric Asian options under Heston's stochastic volatility model

In this work, it is assumed that the underlying asset price follows Heston's stochastic volatility model and explicit solutions for the prices of geometric Asian options with fixed and floating strikes are derived. This approach has to deal with the derivation of the generalized joint Fourier transform of a square-root process and of three different weighted integrals of the square-root process with constant, linear and quadratic weights. Numerical implementation results for the complicated expressions are presented, together with the computational stability and efficiency of the method.

Option Pricing Under Compound Bernoulli and Poisson Perturbations of AR(1) Log-Price Process

Compound Poisson and compound Bernoulli perturbations are introduced in addition to normal errors in the AR(1) model for discrete time version of log-price process. The formula for expected present value of European Call Option is extended to these cases. Other than European option, the valuation of Asian option is also considered and formulas for Geometric Asian option for AR(1) log-price process incorporating compound Poisson and compound Bernoulli jumps also obtained.

PRICING ASIAN OPTIONS IN AFFINE GARCH MODELS

We derive recursive relationships for the m.g.f. of the geometric average of the underlying within some affine Garch models [Heston and Nandi (2000), Christoffersen et al. (2006), Bellini and Mercuri (2007), Mercuri (2008)] and use them for the semi-analytical valuation of geometric Asian options. Similar relationships are obtained for low order moments of the arithmetic average, that are used for an approximated valuation of arithmetic Asian options based on truncated Edgeworth expansions, following the approach of Turnbull and Wakeman (1991). In both cases the accuracy of the semi-analytical procedure is assessed by means of a comparison with Monte Carlo prices. The results are quite good in the geometric case, while in the arithmetic case the proposed methodology seems to work well only in the Heston and Nandi case.

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.

Quasi-Monte Carlo Methods in Financial Engineering: An Equivalence Principle and Dimension Reduction

Quasi-Monte Carlo (QMC) methods are playing an increasingly important role in the pricing of complex financial derivatives. For models in which the prices of the underlying assets are driven by Brownian motions, the performance of QMC methods is known to depend crucially on the construction of Brownian motions. This paper focuses on the impact of various constructions. Although the Brownian bridge (BB) construction often yields very good results, as Papageorgiou pointed out, there are financial derivatives for which the BB construction performs badly [Papageorgiou, A. 2002. The Brownian bridge does not offer a consistent advantage in quasi-Monte Carlo integration. J. Complexity 18(1) 171–186]. In this paper we first extend Papageorgiou's analysis to establish an equivalence principle: if the BB construction (or any other construction) is the preferred construction for a particular financial derivative, then for any other construction, there is another financial derivative for which the latter construction is the preferred one. In this sense, all methods of construction are equivalent and no method is consistently superior to others; it all depends on the particular financial derivative. We then show how to find a good construction for a particular class of financial derivatives. In practice, our strategy is to find a good construction for an “easy” problem and then apply it to more complicated problems related to the easy one. This strategy is applied to the arithmetic Asian options (including Bermudan Asian options) based on the weighted average of the stock prices. We do this by studying a simpler problem, namely, the geometric Asian option, for which the best construction is easily available, and applying it to the arithmetic Asian option. Numerical experiments confirm the success of this strategy: whereas in QMC all the common methods (the standard method, BB, and principal component analysis) may lose their power in some situations, the new method behaves very well in all cases. Further large variance reduction can be achieved in combination with a control variate. The new method can be interpreted as a practical way of reducing the effective dimension for some class of functions.

Graphical Asian options

AbstractWe discuss the problem of pricing Asian options in the Black–Scholes model using CUDA on a graphics processing unit. We survey some of the issues with GPU programming and discuss code design and memory usage. We show that by using a Quasi Monte Carlo simulation with a geometric Asian option as a control variate, it is possible to get prices that are accurate to 2E‐4 within a fiftieth of a second. Copyright © 2010 Wilmott Magazine Ltd.

Path integral approach to Asian options in the Black–Scholes model

We derive a closed-form solution for the price of an average strike as well as an average price geometric Asian option, by making use of the path integral formulation. Our results are compared to a numerical Monte Carlo simulation. We also develop a pricing formula for an Asian option with a barrier on a control process, combining the method of images with a partitioning of the set of paths according to the average along the path. This formula is exact when the correlation is zero, and is approximate when the correlation increases.

Pricing discretely monitored Asian options under Lévy processes

We present methodologies to price discretely monitored Asian options when the underlying evolves according to a generic Lévy process. For geometric Asian options we provide closed-form solutions in terms of the Fourier transform and we study in particular these formulas in the Lévy-stable case. For arithmetic Asian options we solve the valuation problem by recursive integration and derive a recursive theoretical formula for the moments to check the accuracy of the results. We compare the implementation of our method to Monte Carlo simulation implemented with control variates and using different parametric Lévy processes. We also discuss model risk issues.

Free boundary and optimal stopping problems for American Asian options

We give a complete and self-contained proof of the existence of a strong solution to the free boundary and optimal stopping problems for pricing American path-dependent options. The framework is sufficiently general to include geometric Asian options with nonconstant volatility and recent path-dependent volatility models.

Pricing Geometric Asian Options under the CEV Process

This paper discusses the pricing of geometric Asian options when the underlying stock follows the constant elasticity of variance (CEV) process. We build a binomial tree method to estimate the CEV process and use it to price geometric Asian options. We find that the binomial tree method for the lognormal case can effectively solve the computational problems arising from the inherent complexities of geometric Asian options when the stock price follows the CEV process. We present numerical results to demonstrate the validity and the convergence of the approach for the different parameter values set in the CEV process.

Geometric Asian options: valuation and calibration with stochastic volatility

This paper studies continuously sampled geometricAsian options (GAO) in a stochastic volatility economy. Theunderlying asset price is assumed to follow a geometric Brownianmotion with stochastic volatility driven by a mean-revertingprocess. Semi-analytical pricing formulae for GAO are derived ina fast mean-reverting stochastic volatility economy by the meansof a perturbation method. The effects of stochastic volatility onaveraging type options are examined. A unified regressionapproach is proposed to capture smiles of some geometric Asianoptions and European options in one shot.

One-state variable binomial models for European-/ American-style geometric Asian options

Geometric Asian options are path-dependent options whose payoffs depend on the geometric average of the underlying asset prices. Following Cox et al (1979) arbitrage arguments, We develop one-state variable binomial models for the options on the basis of the idea of Cheuk and Vorst (1997). The models are more efficient and faster than those lattice methods proposed by Hull and White (1993), Ritchken et al (1993), Barraquand and Pudet (1996) and Cho and Lee (1997). We also establish the equivalence of the models and difference schemes.