Bessel Processes and Asian Options

In this paper we obtain a simple, explicit integral form for the moment generating function of the reciprocal of the random variable defined by A t ( ν ) := ∫ 0 t exp ⁡ ( 2 B s + 2 ν s ) d s A^{(\nu )}_t := \int ^t _0 \exp (2B_s + 2 \nu s) ds , where B s B_s , s > 0 s>0 , is a one-dimensional Brownian motion starting from 0. In case ν = 1 \nu = 1 , the moment generating function has a particularly simple form.

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.

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.

Static super-replicating strategies for a class of exotic options

Asymmetric skew Bessel processes and their applications to finance

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

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.

In This Issue

General lattice methods for arithmetic Asian options

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

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.

Pricing Asian Options: a Comparison of Numerical and Simulation Approaches, Twenty Years Later

SummaryAsian options are popular path-dependent derivatives in the financial market. However, how to price them efficiently and accurately has been a longstanding research and practical problem. No known exact pricing formulas are available to price the Asian option. Although approximate pricing formulas exist, they lack accuracy guarantees. Asian options can be priced on the lattice. A lattice divides a time interval into n equal-length time steps. It is known that the value computed by the lattice converges to the true option value as n → ∞. Unfortunately, only subexponential-time algorithms are available if Asian options are to be priced on the lattice without approximations. Efficient approximation algorithms are available for the lattice. The best known in the literature is an O(n 3.5)-time approximation lattice algorithm and an O(n 3)-time approximation PDE algorithm. Our paper suggests an O(n 2.5)-time lattice algorithm. Our algorithm uses a novel technique based on the method of Lagrange multipliers to minimize the approximation error. Numerical results verify the accuracy and the excellent performance of our algorithm.

Equity-indexed annuities (EIAs) provide investors with a minimum rate of return and at the same time the opportunity of gaining a profit that is linked to the performance of an equity index. These properties make EIAs a popular product in the market. For modeling the equity index process and calculating the price of EIAs, as the maturity of EIAs usually is long, it is more reasonable to assume that the interest rate and the volatility of the equity index are stochastic processes. One simple way is to apply the regime-switching model, which allows these parameters depending on the market situation. However, the valuation of derivatives in such models is challenging, especially for the strong path-dependent options such as Asian options. A trinomial tree model is introduced to provide an efficient way to solve this problem. The valuation of Asian options is studied and extended to Asian-option-related EIAs.