Geometric Average Asian Option Research Articles

Pricing of geometric average Asian option under the sub-diffusion Merton interest rate model

. Asian option is an essentially new type of option. In existing option pricing models, the Brownian motion is generally the stochastic driving source of changes in the underlying asset price. This article explores the pricing problem of geometric average Asian option under the sub-diffusive Merton interest rate model. The results obtained the partial differential equations satisfied by the geometric average Asian option via constructing portfolios and utilizing the delta hedging technique. Finally, the findings provide display expressions for the geometric average Asian option under the sub-diffusive Merton interest rate model, combined with boundary conditions and variable substitutions.

Analytic approximations for European-style Asian spread options

<abstract><p>Spread option is a exotic option, which allows investors to simultaneously take positions in two correlated underlying assets and profit from their price difference over some spread. This option provides stable investment opportunities for practitioners in unpredictable and complex financial markets. However, investors of the spread option may face problems caused by manipulating the two underlying assets' prices near the expiry, compared to plain vanilla options. To overcome such disadvantages, we propose Asian-spread options, which are linked to the price difference between two average prices of two underlying assets over the life of the option, and exhibit the original properties of standard spread options. In this paper, using distribution-approximating and moment-matching approaches, lower bounds of prices for the European spread option on the geometric average Asian option and arithmetic average Asian option are obtained in the classical Black-Scholes model. We verified the pricing accuracy of the proposed Asian-spread options by comparing our solutions with those obtained by Monte Carlo simulations. Finally, we analyzed the influence of stock price, maturity date, and some model parameters on option price and delta value through numerical examples. Numerical results showed that the lower bounds had a very high precision.</p></abstract>

Pricing Asian options with stochastic convenience yield and jumps

We price Asian options on commodity futures contracts in the presence of stochastic convenience yield, stochastic interest rates and jumps in the commodity spot price. In the case of no jumps, we obtain a closed-form solution for a geometric average Asian option. This analytic result enables us to employ this option as a suitable control variate when pricing the corresponding arithmetic average Asian option. Discussion of further applications and comparative statics are presented. To cover the case with jumps, we condition on the jump times first and then average over the sequences of jump times.

Estimation of the Bid-Ask Prices for the European Discrete Geometric Average and Arithmetic Average Asian Options

Conic finance is a new and exciting development in quantitative finance, which is widely applied to several topics in finance. The theory of conic finance extends the law of one price to the law of two prices, which yields closed forms for bid-ask prices of European options. In this paper, within the framework of conic finance, we derive effective, explicit, approximate formulas to estimate the bid-ask prices for the European discrete geometric average and arithmetic average Asian options. Finally, we give two examples to demonstrate and validate that the approximate closed-form solutions are efficient and accurate.

Pricing Asian Options with Stochastic Convenience Yield and Jumps

We price Asian options on commodity futures contracts in the presence of stochastic convenience yield, stochastic interest rates and jumps in the commodity spot price. We obtain a closed-form solution for the case of a geometric average option without the presence of jumps, both for continuous and discrete averaging. This analytic result enables us to employ the geometric average option as suitable control variate for pricing the corresponding arithmetic average Asian option via Monte Carlo simulation. We observe a significant reduction in the size of confidence intervals in relation to plain Monte Carlo. Varying the parameters of the model, we obtain a good understanding as to how Asian option prices change accordingly in relation to a stochastic convenience yield. We then consider a variation of the model by adding stochastic jumps to the commodity spot price dynamics. By conditioning on the jump times first and then averaging over the sequences of jump times, we show that our control variate still provides a significant variance reduction, even though a closed-form solution for a geometric average Asian option in the presence of jumps is unavailable.

Stochastic Mesh Method for Optimal Stopping Problems

This paper considers the application of the stochastic mesh method in solving the multidimensional optimal stopping problem for a diffusion process with nonlinear payoff functions. A special discretization scheme of the diffusion process is presented to solve the problem in the case of geometric average Asian option payoff functions. This discretization scheme makes it possible to eliminate singularities in transition probabilities. Next, two estimates are given of the solution of the problem by the stochastic mesh method for the case of the stochastic mesh transition probabilities defined as averaged densities. The consistency of the estimates is proven. It is shown that the variance of the estimates is inversely proportional to the number of points in each mesh layer. The result extends the application area of the stochastic mesh method and methods for treating Asian options. A numerical example of the result of applying the obtained estimates to the call and put options compared to the obtained option prices using a regular mesh is presented.

Pricing Asian Options with Stochastic Convenience Yield and Jumps

We price Asian options on commodity futures contracts in the presence of stochastic convenience yield, stochastic interest rates and jumps in the commodity spot price. We obtain a closed-form solution for the case of a geometric average option without the presence of jumps, both for continuous and discrete averaging. This analytic result enables us to employ the geometric average option as suitable control variate for pricing the corresponding arithmetic average Asian option via Monte Carlo simulation. We observe a significant reduction in the size of confidence intervals in relation to plain Monte Carlo. Varying the parameters of the model, we obtain a good understanding as to how Asian option prices change accordingly in relation to a stochastic convenience yield. We then consider a variation of the model by adding stochastic jumps to the commodity spot price dynamics. By conditioning on the jump times first and then averaging over the sequences of jump times, we show that our control variate still provides a significant variance reduction, even though a closed-form solution for a geometric average Asian option in the presence of jumps is unavailable.

随机波动率模型下离散几何平均亚式障碍期权定价

随机波动率模型下离散几何平均亚式障碍期权定价

Geometric Average Asian Option Pricing with Paying Dividend Yield under Non-Extensive Statistical Mechanics for Time-Varying Model.

This paper is dedicated to the study of the geometric average Asian call option pricing under non-extensive statistical mechanics for a time-varying coefficient diffusion model. We employed the non-extensive Tsallis entropy distribution, which can describe the leptokurtosis and fat-tail characteristics of returns, to model the motion of the underlying asset price. Considering that economic variables change over time, we allowed the drift and diffusion terms in our model to be time-varying functions. We used the formula, Feynman–Kac formula, and ansatz to obtain a closed-form solution of geometric average Asian option pricing with a paying dividend yield for a time-varying model. Moreover, the simulation study shows that the results obtained by our method fit the simulation data better than that of Zhao et al. From the analysis of real data, we identify the best value for q which can fit the real stock data, and the result shows that investors underestimate the risk using the Black–Scholes model compared to our model.

The evaluation of geometric Asian power options under time changed mixed fractional Brownian motion

In this paper, the geometric Asian option pricing problem is investigated under the assumption that the underlying stock price is assumed following a mixed fractional subdiffusive Black–Scholes model, and the geometric average Asian option pricing formula is derived under this assumption. We then apply the results to value Asian power options on the stocks that pay constant dividends when the payoff is a power function. Finally, lower bound of Asian options and some special cases are provided.

Non-Gaussian Closed Form Solutions for Geometric Average Asian Options in the Framework of Non-Extensive Statistical Mechanics.

In this paper we consider pricing problems of the geometric average Asian options under a non-Gaussian model, in which the underlying stock price is driven by a process based on non-extensive statistical mechanics. The model can describe the peak and fat tail characteristics of returns. Thus, the description of underlying asset price and the pricing of options are more accurate. Moreover, using the martingale method, we obtain closed form solutions for geometric average Asian options. Furthermore, the numerical analysis shows that the model can avoid underestimating risks relative to the Black-Scholes model.

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.

Asian Option Pricing with Transaction Costs and Dividends under the Fractional Brownian Motion Model

The pricing problem of geometric average Asian option under fractional Brownian motion is studied in this paper. The partial differential equation satisfied by the option’s value is presented on the basis of no-arbitrage principle and fractional formula. Then by solving the partial differential equation, the pricing formula and call-put parity of the geometric average Asian option with dividend payment and transaction costs are obtained. At last, the influences of Hurst index and maturity on option value are discussed by numerical examples.

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.

Asian Option Pricing with Monotonous Transaction Costs under Fractional Brownian Motion

Geometric-average Asian option pricing model with monotonous transaction cost rate under fractional Brownian motion was established. The method of partial differential equations was used to solve this model and the analytical expressions of the Asian option value were obtained. The numerical experiments show that Hurst exponent of the fractional Brownian motion and transaction cost rate have a significant impact on the option value.

Sensitivity analysis for averaged asset price dynamics with gamma processes

The main purpose of this paper is to derive unbiased Monte Carlo estimators of various sensitivity indices for an averaged asset price dynamics governed by the gamma Lévy process. The key idea is to apply a scaling property of the gamma process with respect to the Esscher density transform parameter. Our framework covers not only the continuous Asian option, but also European, discrete Asian, average strike Asian, weighted average, spread options, and geometric average Asian options. Numerical results are provided to illustrate the effectiveness of our formulas in Monte Carlo simulations, relative to finite difference approximation.

Variance reduction for Monte Carlo methods to evaluate option prices under multi-factor stochastic volatility models

We present variance reduction methods for Monte Carlo simulations to evaluate European and Asian options in the context of multiscale stochastic volatility models. European option price approximations, obtained from singular and regular perturbation analysis [Fouque J P, Papanicolaou G, Sircar R and Solna K 2003 Multiscale stochastic volatility asymptotics, SIAM J. Multiscale Modeling and Simulation 2], are used in importance sampling techniques, and their efficiencies are compared. Then we investigate the problem of pricing arithmetic average Asian options (AAOs) by Monte Carlo simulations. A two-step strategy is proposed to reduce the variance where geometric average Asian options (GAOs) are used as control variates. Due to the lack of analytical formulas for GAOs under stochastic volatility models, it is then necessary to consider efficient Monte Carlo methods to estimate the unbiased means of GAOs. The second step consists in deriving formulas for approximate prices based on perturbation techniques, and in computing GAOs by using importance sampling. Numerical results illustrate the efficiency of our method.