General lattice methods for arithmetic Asian options

Complete Analytical Solution of the American Style Option Pricing with Constant and Stochastic Volatilities: A Probability Density Function Approach

We consider a seasonal mean‐reverting model for energy commodity prices with jumps and Heston‐type stochastic volatility, and three nested models for comparison. By exploiting the affine form of the log‐spot models, we develop a general valuation framework for futures and discrete arithmetic Asian options. We investigate five major petroleum commodities from Europe (Brent crude oil, gasoil) and US (light sweet crude oil, gasoline, heating oil) and analyse the effects of the competing fitted spot models in futures pricing, Asian options pricing and hedging. We find evidence that price jumps and stochastic volatility are important features of the petroleum price dynamics.

Static super-replicating strategies for a class of exotic options

The goal of this chapter is to give a concise account of the connection between Bessel processes and the integral of geometric Brownian motion. The latter appears in the pricing of Asian options. Bessel processes are defined and some of their properties are given. The known expressions for the probability density function of the integral of geometric Brownian motion are stated, and other related results are given, in particular the Geman and Yor (1993) Laplace transform for Asian option prices.

A general framework is proposed for pricing both continuously and discretely monitored Asian options under one-dimensional Markov processes. For each type (continuously monitored or discretely monitored), we derive the double transform of the Asian option price in terms of the unique bounded solution to a related functional equation. In the special case of continuous-time Markov chain (CTMC), the functional equation reduces to a linear system that can be solved analytically via matrix inversion. Thus the Asian option prices under a one-dimensional Markov process can be obtained by first constructing a CTMC to approximate the targeted Markov process model, and then computing the Asian option prices under the approximate CTMC by numerically inverting the double transforms. Numerical experiments indicate that our pricing method is accurate and fast under popular Markov process models, including the CIR model, the CEV model, Merton’s jump diffusion model, the double-exponential jump diffusion model, the variance gamma model, and the CGMY model.

In this paper, we study the impact of introducing uncertainty volatility into Asian options pricing, with emphasis on the use of Hamilton-Jacobi-Bellman (HJB) equation. The traditional Asian option pricing model usually assumes that volatility is known and constant, but in the actual market, volatility is often uncertain and volatile. This paper first reviews the pricing theory of Asian options, and then introduces the hypothesis of uncertain volatility. By constructing the HJB equation based on uncertainty volatility, a new pricing method is proposed and verified by numerical simulation. The results show that after the introduction of uncertain volatility, the price range of Asian options expands significantly, reflecting higher market uncertainty and risk.

In This Issue

This paper presents a comprehensive comparative analysis of Monte Carlo simulation and binomial tree models for pricing geometric Asian options. In the process of evaluations, the Tesla (TSLA) stock is the underlying asset for the options, and the modified Black-Scholes model serves as the benchmark for comparisons of pricing accuracy, Greeks calculation. Besides, this research also analyzes the difference between the two methods in terms of computational efficiency. According to the simulations, the binomial tree model shows a higher accuracy compared to the Monte Carlo simulation. For Greeks calculations, both methods illustrate consistency in Delta and Gamma values, and the binomial tree model shows almost perfect Vega precision. At the same time, the Monte Carlo simulation offers computational advantages with near-linear complexity, making it ideal for real-time applications. The binomial tree model, despite exponential complexity growth, remains ideal for high-precision requirements. Therefore, these results offer a guideline for the application of these two methods, in which one can determine the usage based on the tradeoff between accuracy and computational speed.

In this paper, we consider Asian options with counterparty risk under stochastic volatility models. We propose a simple way to construct stochastic volatility models through the market factor channel. In the proposed framework, we obtain an explicit pricing formula of Asian options with counterparty risk and illustrate the effects of systematic risk on Asian option prices. Specially, the U‐shaped and inverted U‐shaped curves appear when we keep the total risk of the underlying asset and the issuer's assets unchanged, respectively.

Complete Analytical Solution of the Asian Option Pricing and Asian Option Value-at-Risk Problems: A Probability Density Function Approach

The valuation of arithmetic Asian options with mean reversion and jump clustering

Are There Critical Levels of Stochastic Volatility for Early Option Exercise?

We use Malliavin calculus and the Clark–Ocone formula to derive the hedging strategy of an arithmetic Asian Call option in general terms. Furthermore we derive an expression for the density of the integral over time of a geometric Brownian motion, which allows us to express hedging strategy and price of the Asian option as an analytic expression. Numerical computations which are based on this expression are provided.

In this article we propose a method to compute the density of the arithmetic average of a Markov process. This approach is then applied to the pricing of average rate options (Asian options). It is demonstrated that as long as a closed form formula is available for the discount bond price when the underlying process is treated as the riskless interest rate, analytical formulas for the density function of the arithmetic average and the Asian option price can be derived. This includes the affine class of term structure models. The Cox et al. (1985) square root interest rate process is used as an example. When the underlying process follows a geometric Brownian motion, a very efficient numerical method is proposed for computing the density function of the average. Extensions of the techniques to the cases of multiple state variables are also discussed.

Background/Objectives: The main objective of this paper is to present an n-step binomial model which can be used to price an option under any exotic conditions. Methods/Statistical Analysis: Mathematical models have been presented using which an n-step binomial model can be developed. The model can be used for estimating price of options under n of number exotic conditions that influence the option price. Findings: Pricing of exotic options like Asian, American etc., undertaken through Binomial Trees using only one-step considering maximum and minimum values that can be taken by the underling at the maturity leads to a rough approximation of the option price. The approximation is possible by assuming stock price movements to be in one or two binomial steps during the life of the option. A binomial tree extended to an N-step Model can be used to price various exotic options. A study of the convergence in European option price with respect to Number of steps (N) and variation in price of Asian and American options with respect to confidence factor (k) (proxy for implied volatility) using the maximum and minimum boundaries on the value of k gives the investors the ability to change the value of k so that they can have their own opinions concerning the risk-neutral probability distribution.