Geometric Asian Options Research Articles

Pricing Asian options under the mixed fractional Brownian motion with jumps

The mixed fractional Brownian motion (mfBm) has gained popularity in finance because it can effectively model long-range dependence, self-similarity, and is arbitrage-free. This paper focuses on mfBm with jumps modeled by the Poisson process and derives an analytical formula for valuing geometric Asian options. Additionally, approximate closed-form solutions for pricing arithmetic Asian options and arithmetic Asian power options are obtained. Numerical examples are provided to demonstrate the accuracy of these formulas, which rely on a convenient approximation of the option strike price. The proposed approximation demonstrates significantly higher computational efficiency compared to Monte Carlo simulation.

Commodity Asian option pricing and simulation in a 4-factor model with jump clusters

Mean reversion, stochastic volatility, convenience yield and presence of jump clustering are well documented salient features of commodity markets, where Asian options are very popular. We propose a model which takes into account all these stylized features. We first state our model under the historical measure, then, after introducing a structure preserving change of measure, we provide a risk-neutral version of the same model and we show how to price geometric and arithmetic Asian options. To this end, we derive semi-closed formulas for the geometric Asian options price and develop a computationally efficient simulation scheme for the price process, allowing to price the arithmetic counterparts using control variate technique. Finally, we propose a simple econometric experiment to document presence of jump clusters in commodity prices and evaluate the performances of the proposed simulation scheme on some parameter sets calibrated on real data.

Pricing of the geometric Asian options under a multifactor stochastic volatility model

This paper focuses on the pricing of continuous geometric Asian options (GAOs) under a new multifactor stochastic volatility model. The model considers fast and slow mean reverting factors of volatility, where slow volatility factor is approximated by a quadratic arc. The asymptotic expansion of the price function is assumed, and the first order price approximation is derived using the perturbation techniques for both floating and fixed strike GAOs. Much simplified pricing formulae for the GAOs are obtained in this multifactor stochastic volatility framework. The zeroth order term in the price approximation is the modified Black–Scholes price for the GAOs. This modified price is expressed in terms of the Black–Scholes price for the GAOs. The accuracy of the approximate option pricing formulae is established, and also verified numerically by comparing the model prices with the Monte Carlo simulation prices and the Black–Scholes prices for the GAOs. The model parameter is estimated by capturing the volatility smiles. The sensitivity analysis is also performed to investigate the effect of underlying parameters on the approximated prices.

On the Valuation of Discrete Asian Options in High Volatility Environments

ABSTRACT In this paper, we are concerned with the Monte Carlo valuation of discretely sampled arithmetic and geometric average options in the Black-Scholes model and the stochastic volatility model of Heston in high volatility environments. To this end, we examine the limits and convergence rates of asset prices in these models when volatility parameters tend to infinity. We observe, on the one hand, that asset prices, as well as their arithmetic means converge to zero almost surely, while the respective expectations are constantly equal to the initial asset price. On the other hand, the expectation of geometric means of asset prices converges to zero. Moreover, we elaborate on the direct consequences for option prices based on such means and illustrate the implications of these findings for the design of efficient Monte-Carlo valuation algorithms. As a suitable control variate, we need among others the price of such discretely sampled geometric Asian options in the Heston model, for which we derive a closed-form solution.

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 geometric asian power options in the sub-fractional brownian motion environment

This paper aims of obtaining the closed form expressions for the prices of the geometric Asian options and power options when the payoff function is a power function. After discussing the option pricing in the sub-fractional Brownian motion environment, by the fractional Ito^ formula which is based on the theory of stochastic differential equation, the sub-fractional Ito^ formula is derived. Furthermore, the solution of the stochastic differential equation satisfied by stock prices is obtained. The stock price process is modeled well with the driving force as the sub-fractional Brownian motion. The empirical results show that the fitting effect is better than the Brownian motion.

High Volatility Limits of Asset Prices with Applications to Option Pricing

In this paper, we are concerned with the Monte Carlo valuation of discretely sampled arithmetic and geometric average options in the Black-Scholes model and the stochastic volatility model of Heston in high volatility environments. To this end, we examine the limits and convergence rates of asset prices in these models when volatility parameters tend to infinity. We observe on the one hand that asset prices as well as their arithmetic means converge to zero almost surely, while the respective expectations are constantly equal to the initial asset price. On the other hand, the expectation of geometric means of asset prices converges to zero. Moreover, we elaborate on the direct consequences for option prices based on such means and illustrate the implications of these findings for the design of efficient Monte-Carlo valuation algorithms. As a suitable control variate, we need among others the price of such discretely sampled geometric Asian options in the Heston model, for which we derive a closed-form solution.

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.

Option pricing of geometric Asian options in a subdiffusive Brownian motion regime

In this paper, pricing problem of the geometric Asian option in a subdiffusive Brownian motion regime is discussed. The subdiffusive property is manifested by the random periods of time, during which the asset price does not change. Subdiffusive partial differential equations for geometric Asian option are derived by using delta-hedging strategy. Explicit formula for geometric Asian option is obtained by using partial differential equation method. Furthermore, numerical studies are performed to illustrate the performance of our proposed pricing model.

AN EFFICIENT VARIANCE REDUCTION-BASED SIMULATION ALGORITHM FOR PRICING ARITHMETIC ASIAN OPTIONS

This paper proposes a new hybrid algorithm to price the arithmetic Asian options under the geometric Brownian motion (GBM). The proposed algorithm is based on the control variate technique, such that the control variable is a combination of the barrier arithmetic Asian option and the geometric Asian option, which each one will be estimated by the importance sampling and the control variate techniques, respectively. Besides, we drive a conditional expectation for the estimator that it can reduce variance of simulations. The merits of the proposed algorithm for pricing arithmetic Asian options are illustrated by several examples.

Asian options pricing in Hawkes-type jump-diffusion models

In this paper we propose a method for pricing Asian options in market models with the risky asset dynamics driven by a Hawkes process with exponential kernel. For these processes the couple $$ (\lambda (t), X(t) ) $$ is affine, this property allows to extend the general methodology introduced by Hubalek et al. (Quant Finance 17:873–888, 2017) for Geometric Asian option pricing to jump-diffusion models with stochastic jump intensity. Although the system of ordinary differential equations providing the characteristic function of the related affine process cannot be solved in closed form, a COS-type algorithm allows to obtain the relevant quantities needed for options valuation. We describe, by means of graphical illustrations, the dependence of Asian options prices by the main parameters of the driving Hawkes process. Finally, by using Geometric Asian options values as control variates, we show that Arithmetic Asian options prices can be computed in a fast and efficient way by a standard Monte Carlo method.

An asymptotic expansion method for geometric Asian options pricing under the double Heston model

Abstract The purpose of the paper is to provide an efficient method for the continuously monitored geometric Asian options under the double Heston model. By introducing two small parameters, we slightly modify the double Heston model. With singular and regular perturbation techniques, we derive the first-order asymptotic expansions for pricing geometric Asian options with fixed and floating strikes and provide the convergence of the asymptotic formulae. We also provide the Greeks of geometric Asian options. Numerical results verify the efficiency of the pricing method. We calibrate the modified model to real markets and examine the impacts of two-factor volatilities on geometric Asian option prices.

Estimation of Ask and Bid Prices for Geometric Asian Options

Traditional derivative pricing theories usually focus on the risk-neutral price or the equilibrium price. However, in highly competitive financial markets, we observed two prices which are called bid and ask prices; then the unique risk-neutral price fails to hold. In this paper, within the framework of conic finance, we provide a useful approach to evaluate the ask and bid prices of geometric Asian options and obtain the explicit formulas for the ask and bid prices. Finally, numerical examples show that the higher the market liquidity parameter γ, the wider the spread and hence the less the liquidity.

SMALL-TIME ASYMPTOTICS IN GEOMETRIC ASIAN OPTIONS FOR A STOCHASTIC VOLATILITY JUMP-DIFFUSION MODEL

The aim of this paper is to study the small time to maturity of the behavior of the geometric Asian option price and implied volatility under a general stochastic volatility model with Lévy process. The volatility process does not need to be a diffusion or a Markov process, but the future average volatility in the model is a nonadapted process. An anticipating Itô formula for Lévy process and the decomposition of the price (Hull–White formula) are obtained using the Malliavin calculus techniques. The decomposition formula is applied to find the small-time limit of the geometric Asian option price and the implied volatility for the model in at-the-money and out-of-the-money cases.

Geometric Asian Options Pricing under the Double Heston Stochastic Volatility Model with Stochastic Interest Rate

This paper presents an extension of double Heston stochastic volatility model by incorporating stochastic interest rates and derives explicit solutions for the prices of the continuously monitored fixed and floating strike geometric Asian options. The discounted joint characteristic function of the log‐asset price and its log‐geometric mean value is computed by using the change of numeraire and the Fourier inversion transform technique. We also provide efficient approximated approach and analyze several effects on option prices under the proposed model. Numerical examples show that both stochastic volatility and stochastic interest rate have a significant impact on option values, particularly on the values of longer term options. The proposed model is suitable for modeling the longer time real‐market changes and managing the credit risks.

Asian Options Pricing in Hawkes-Type Jump-Diffusion Models

In this paper we propose a method for pricing Asian options in market models with the risky asset dynamics driven by a Hawkes process with exponential kernel. For these processes the couple (λ(t), X(t)) is affine, this property allows to extend the general methodology introduced by Hubalek, Keller-Ressel and Sgarra for Geometric Asian option pricing to jump-diffusion models with stochastic jump intensity. Although the system of ordinary differential equations providing the characteristic function of the related affine process cannot be solved in closed form, a COS-type algorithm allows to obtain the relevant quantities needed for options valuation. We describe, by means of graphical illustrations, the dependence of Asian options prices by the main parameters of the driving Hawkes process. Finally, by using Geometric Asian options values as control variates, we show that Arithmetic Asian options prices can be computed in a fast and efficient way by a standard Monte Carlo method.

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.

Numerical pricing of geometric asian options with barriers

In this article, a semianalytical method for pricing of barrier options is described and applied in the context of Asian options with geometric mean. The efficiency of the method is tested and compared with two finite difference methods.

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.