Gianluca Fusai

An exact analytical solution for discrete barrier options

Abstract.In the present paper we provide an analytical solution for pricing discrete barrier options in the Black-Scholes framework. We reduce the valuation problem to a Wiener-Hopf equation that can be solved analytically. We are able to give explicit expressions for the Greeks of the contract. The results from our formulae are compared with those from other numerical methods available in the literature. Very good agreement is obtained, although evaluation using the present method is substantially quicker than the alternative methods presented.

Pricing Asian options via Fourier and Laplace transforms

By means of Fourier and Laplace transforms, we obtain a simple expression for the double transform (with respect to the logarithm of the strike and timeto-maturity) of the price of continuously monitored Asian options. The double transform is expressed in terms of gamma functions only. The computation of the price requires a multivariate numerical inversion. We show that the numerical inversion can be performed with great accuracy and low computational cost.

Pricing Discretely Monitored Asian Options by Maturity Randomization

We present a new methodology based on maturity randomization to price discretely monitored arithmetic Asian options when the underlying asset evolves according to a generic Levy process. Our randomization technique considers the option expiry to be a random variable distributed according to a geometric distribution of a parameter independent of the underlying process. This allows one to transform the pricing backward procedure into a set of independent integral equations. Numerical procedures for a fast and accurate solution of the pricing problem are provided.

Pricing of Occupation Time Derivatives: Continuous and Discrete Monitoring

In the present work we use different numerical methods (multidimensional inverse Laplace transform, numerical solution of a PDE by finite difference scheme, Monte Carlo simulation) for pricing occupation time derivatives in order to examine the effect of continuous and discrete time monitoring of the underlying asset. In particular we treat the problem of the numerical inversion of a multidimensional Laplace transform and we show that it can be performed very fast and with great accuracy. We conduct also an analysis of the numerical method for the solution of the PDE with discrete monitoring and we show that the proposed method avoids unwanted oscillations in the solution arising near the monitoring dates due to the updating of the occupation time.

General Closed-Form Basket Option Pricing Bounds

This article presents lower and upper bounds on the prices of basket options for a general class of continuous-time financial models. The techniques we propose are applicable whenever the joint characteristic function of the vector of log-returns is known. Moreover, the basket value is not required to be positive. We test our new price approximations on different multivariate models, allowing for jumps and stochastic volatility. Numerical examples are discussed and benchmarked against Monte Carlo simulations. All bounds are general and do not require any additional assumption on the characteristic function, so our methods may be employed also to non-affine models. All bounds involve the computation of one-dimensional Fourier transforms; hence, they do not suffer from the curse of dimensionality and can be applied also to high-dimensional problems where most existing methods fail. In particular, we study two kinds of price approximations: an accurate lower bound based on an approximating set and a fast bounded approximation based on the arithmetic-geometric mean inequality. We also show how to improve Monte Carlo accuracy by using one of our bounds as a control variate.

Dynamic value at risk under optimal and suboptimal portfolio policies

Abstract At present, all value at risk (VaR) implementations – i.e., all risk measures of the “maximum loss at a given level of confidence” type – are based on the assumption that the portfolio mix will not change before the VaR horizon. This hypothesis may be unrealistic, especially when the VaR horizon is established by the regulators (BIS). At the opposite, we measure VaR dynamically, i.e., taking into consideration portfolio mix adjustments over time: adjustments do not occur continuously, since they are costly. We allow both optimal rebalancing policies, which entail changing the portfolio mix whenever it is too far from the optimal one, and suboptimal policies, which mean adjusting at pre-fixed dates. We show that in both cases usual VaR measures underestimate portfolio losses, even if the underlying returns are normal. We study the dependence of the misestimate on the VaR horizon, the initial portfolio mix and the risk aversion of the portfolio manager, which in turn determines the frequency of interventions. The bias can be more relevant over one day than over longer horizons and even if the initial portfolio is nearly optimal. We also perform backtesting and estimate a “coherent” risk measure, namely conditional VaR, which confirms the inappropriateness of the usual, static VaR.

Spitzer Identity, Wiener-Hopf Factorization and Pricing of Discretely Monitored Exotic Options

The Wiener-Hopf factorization of a complex function arises in a variety of fields in applied mathematics such as probability, finance, insurance, queuing theory, radio engineering and fluid mechanics. The factorization fully characterizes the distribution of functionals of a random walk or a Levy process, such as the maximum, the minimum and hitting times. Here we propose a constructive procedure for the computation of the Wiener-Hopf factors, valid for both single and double barriers, based on the combined use of the Hilbert and the z-transform. The numerical implementation can be simply performed via the fast Fourier transform and the Euler summation. Given that the information in the Wiener-Hopf factors is strictly related to the distributions of the first passage times, as a concrete application in mathematical finance we consider the pricing of discretely monitored exotic options, such as lookback and barrier options, when the underlying price evolves according to an exponential Levy process. We show that the computational cost of our procedure is independent of the number of monitoring dates and the error decays exponentially with the number of grid points.

The Wiener–Hopf Technique and Discretely Monitored Path-Dependent Option Pricing

Fusai, Abrahams, and Sgarra (2006) employed the Wiener–Hopf technique to obtain an exact analytic expression for discretely monitored barrier option prices as the solution to the Black–Scholes partial differential equation. The present work reformulates this in the language of random walks and extends it to price a variety of other discretely monitored path-dependent options. Analytic arguments familiar in the applied mathematics literature are used to obtain fluctuation identities. This includes casting the famous identities of Baxter and Spitzer in a form convenient to price barrier, first-touch, and hindsight options. Analyzing random walks killed by two absorbing barriers with a modified Wiener–Hopf technique yields a novel formula for double-barrier option prices. Continuum limits and continuity correction approximations are considered. Numerically, efficient results are obtained by implementing Pade approximation. A Gaussian Black–Scholes framework is used as a simple model to exemplify the techniques, but the analysis applies to Levy processes generally.

Z-Transform and preconditioning techniques for option pricing

In the present paper, we convert the usual n-step backward recursion that arises in option pricing into a set of independent integral equations by using a z-transform approach. In order to solve these equations, we consider different quadrature procedures that transform the integral equation into a linear system that we solve by iterative algorithms and we study the benefits of suitable preconditioning techniques. We show the relevance of our procedure in pricing options (such as plain vanilla, lookback, single and double barrier options) when the underlying evolves according to an exponential Lévy process.

General Optimized Lower and Upper Bounds for Discrete and Continuous Arithmetic Asian Options

We propose an accurate method for pricing arithmetic Asian options on the discrete or continuous average in a general model setting by means of a lower bound approximation. In particular, we derive analytical expressions for the lower bound in the Fourier domain. This is then recovered by a single univariate inversion and sharpened using an optimization technique. In addition, we derive an upper bound to the error from the lower bound price approximation. Our proposed method can be applied to computing the prices and price sensitivities of Asian options with fixed or floating strike price, discrete or continuous averaging, under a wide range of stochastic dynamic models, including exponential Levy models, stochastic volatility models, and the constant elasticity of variance diffusion. Our extensive numerical experiments highlight the notable performance and robustness of our optimized lower bound for different test cases.