Hedging Parameters Research Articles

A simple efficient approximation to price basket stock options with volatility smile

This paper develops a new approach to obtain the price and risk sensitivities of basket options which have a volatility smile. Using this approach, the Black–Scholes model and the Stochastic Volatility Inspired model have been used to obtain an approximate analytical pricing formula for basket options with a volatility smile. It is found that our approximate formula is quite accurate by comparing it with Monte Carlo simulations. It is also proved the option value of our approach is consistent with the option value generated by Levy’s and Gentle’s approaches for typical ranges of volatility. Further, we give a theoretical proof that the option values from Levy’s and Gentle’s works are the upper bound and the lower bound, respectively, for our option value. The calibration procedure and a practical example are provided. The main advantage of our approach is that it provides accurate and easily implemented basket option prices with volatility smile and hedge parameters and avoids the need to use time-consuming numerical procedures such as Monte Carlo simulation.

A decomposition approach via Fourier sine transform for valuing American knock-out options with rebates

We present an innovative decomposition approach for computing the price and the hedging parameters of American knock-out options with a time-dependent rebate. Our approach is built upon: (i) the Fourier sine transform applied to the partial differential equation with a finite time-dependent spatial domain that governs the option price, and (ii) the decomposition technique that partitions the price of the option into that of the European counterpart and an early exercise premium. Our analytic representations can generalize a number of existing decomposition formulas for some European-style and American-style options. A complexity analysis of the method, together with numerical results, show that the proposed approach is significantly more efficient than the state-of-the-art adaptive finite difference methods, especially in dealing with spot prices near the barrier. Numerical results are also examined in order to provide new insight in the significant effects of the rebate on the option price, the hedging parameters, and the optimal exercise boundary.

A simulation-optimisation approach for supply chain network design under supply and demand uncertainties

We investigate a three-echelon stochastic supply chain network design problem. The problem requires selecting suppliers, determining warehouses locations and sizing, as well as the material flows. The objective is to minimise the total expected cost. An important feature of the investigated problem is that both the supply and the demand are uncertain. We solve this problem using a simulation-optimisation approach that is based on a novel hedging strategy that aims at capturing the randomness of the uncertain parameters. To determine the optimal hedging parameters, the search process is guided by particle swarm optimisation procedure. We present the results of extensive computational experiments that were conducted on a large set of instances and that provide evidence that the proposed hedging strategy constitutes an effective viable solution approach.

A Two-Step Approach for Analytical Optimal Hedging with Two Triggers

Hedging is widely used to mitigate severe water shortages in the operation of reservoirs during droughts. Rationing is usually instituted with one hedging policy, which is based only on one trigger, i.e., initial storage level or current water availability. It may perform poorly in balancing the benefits of a release during the current period versus those of carryover storage during future droughts. This study proposes a novel hedging rule to improve the efficiency of a reservoir operated to supply water, in which, based on two triggers, hedging is initiated with three different hedging sub-rules through a two-step approach. In the first step, the sub-rule is triggered based on the relationship between the initial reservoir storage level and the level of the target rule curve or the firm rule curve at the end of the current period. This step is mainly concerned with increasing the water level or not in the current period. Hedging is then triggered under the sub-rule based on current water availability in the second step, in which the trigger implicitly considers both initial and ending reservoir storage levels in the current period. Moreover, the amount of hedging is analytically derived based on the Karush–Kuhn–Tucker (KKT) conditions. In addition, the hedging parameters are optimized using the improved particle swarm optimization (IPSO) algorithm coupled with a rule-based simulation. A single water-supply reservoir located in Hubei Province in central China is selected as a case study. The operation results show that the proposed rule is reasonable and significantly improves the reservoir operation performance for both long-term and critical periods relative to other operation policies, such as the standard operating policy (SOP) and the most commonly used hedging rules.

An integral equation approach for the valuation of American-style down-and-out calls with rebates

In this paper, an integral equation approach is adopted to price American-style down-and-out calls. Instead of using the probability theory as used in the literature, we use the continuous Fourier sine transform to solve the partial differential equation system governing the option prices. As a way of validating our approach, we show that the “early exercise premium representation” for American-style down-and-out calls without rebate can be re-derived by using our approach. We then examine the case that time-dependent rebates are included in the contract of American-style down-and-out calls. As a result, a more general integral representation for the price of an American-style down-and-out call, with the presence of an extra term associated with the rebate, is obtained. Our numerical method based on the newly-derived integral representation appears to be efficient in computing the price and the hedging parameters for American-style down-and-out calls with rebates. In addition, significant effects of rebates on the option prices and the optimal exercise boundaries are illustrated through selected numerical results.

A practical finite difference method for the three-dimensional Black–Scholes equation

In this paper, we develop a fast and accurate numerical method for pricing of the three-asset equity-linked securities options. The option pricing model is based on the Black–Scholes partial differential equation. The model is discretized by using a non-uniform finite difference method and the resulting discrete equations are solved by using an operator splitting method. For fast and accurate calculation, we put more grid points near the singularity of the nonsmooth payoff function. To demonstrate the accuracy and efficiency of the proposed numerical method, we compare the results of the method with those from Monte Carlo simulation in terms of computational cost and accuracy. The numerical results show that the cost of the proposed method is comparable to that of the Monte Carlo simulation and it provides more stable hedging parameters such as the Greeks.

MODIFICATION OF DELTA FOR CHOOSER OPTIONS

Correctly used financial derivatives can help investors increase their expected returns and minimize their exposure to risk. To ensure the specific needs of investors, a large number of different types of non-standard exotic options is used. Chooser option is one of them. It is an option that gives its holder the right to choose at some predetermined future time whether the option will be a standard call or put with predetermined strike price and maturity time. Although the chooser options are more expensive than standard European-style options, in many cases they are a more suitable instrument for investors in hedging their portfolio value. For an effective use of the chooser option as a hedging instrument, it is necessary to check the values of the Greek parameters delta and gamma for the options. Especially, if the value of the parameter gamma is too large, hedging of the portfolio value using only parameter delta is insufficient and brings high transaction costs because the portfolio has to be reviewed relatively often. Therefore, in this article, a modification of delta-hedging as well as using the value of parameter gamma is suggested. Error of the delta modification is analyzed and compared with the error of widely used parameter delta. Typical patterns for the modified hedging parameter variation with various time to choose time for chooser options are also presented in this article.

Malay

The principle of two unilateral wa‘d or wa‘dan is derived directly from the word unilateral wa‘d which refers to two separate wa‘d made by two parties. This principle is seen to has a room for exploration as there is no injunction of its usage in Islamic financial transaction, as is the case with hedging. The question is, what is a recommended exploration of two unilateral wa‘d in hedging? This research aims to explore the usage of wa‘dan in hedging and then analyzes it based on hedging parameters. This study found that the princple of two unilateral wa‘d has room to be explored further as it is not similar to muwa‘adah principle which is forbidden by most of the Islamic jurists due to its similarity to contract. This study also comes out with a suggestion for the usage of two unilateral wa‘d in hedging, that is one party gives wa‘d to do sarf contract and then other party will gives a wa‘d to do commodity murabahah contract. The suggested structure of two unilateral wa‘d is not similar to muwa‘adah in three main aspects, namely separation of promise, different trigger event and the different outcome. Therefore two unilateral wa‘d is viable to be adapted in Islamic hedging contract as long as it is consistent with the parameter hedging in terms of structuring, procedure and usage. Keywords: Two unilateral wa‘d; application; hedging DOI: http://dx.doi.org/10.17576/islamiyyat-2015-3701-01

Fourier transform algorithms for pricing and hedging discretely sampled exotic variance products and volatility derivatives under additive processes

We develop efficient fast Fourier transform algorithms for pricing and hedging discretely sampled variance products and volatility derivatives under additive processes (time-inhomogeneous Levy processes). Our numerical algorithms are non-trivial versions of the Fourier space time stepping method to nonlinear path dependent payoff structures, like those in variance products and volatility derivatives. The exotic path dependency associated with the discretely sampled realized variance is captured in the numerical procedure by updating two path dependent state variables across monitoring dates. The time stepping procedure between successive monitoring dates can be performed using fast Fourier transform calculations without the usual tedious time stepping calculations in typical finite difference algorithms. We also derive effective numerical procedures that compute the hedge parameters of variance products and volatility derivatives. Numerical tests on pricing various variance products and volatility derivatives were performed that illustrate efficiency, accuracy, reliability and robustness of the proposed Fourier transform algorithms.

A Volatility-of-Volatility Expansion of the Option Prices in the SABR Stochastic Volatility Model

We study the SABR stochastic volatility model with the volatility-of-volatility parameter ν. We provide a method to expand the price CSABR(S, K, ν, σ, τ ) of a European call in this model as a Taylor series in ν, CSABR(S, K, ν, σ, τ ) = CBS(S,K, σ, τ ) νC1 ν2C2 . . . νkCk O(νk 1), where CBS is the usual Black-Scholes pricing formula. As an illustration of the method, we compute the terms C1 and C2. The formulas for the correction terms C1 and C2 are fully explicit and do not require any integration or any special functions (unlike CBS). We also provide approximate formulas for the implied volatility and for the hedging parameter
Δ. Formulas for other greeks can be obtained similarly. We provide numerical tests to show the accuracy of our formula for small ν and τ being not too large.

Fourier Transform Algorithms for Pricing and Hedging Discretely Sampled Exotic Variance Products and Volatility Derivatives under Additive Processes

We develop efficient fast Fourier transform algorithms for pricing and hedging discretely sampled variance products and volatility derivatives under additive processes (time-inhomogeneous Levy processes). Our numerical algorithms are non-trivial versions of the Fourier space time stepping method to nonlinear path dependent payoff structures, like those in variance products and volatility derivatives. The exotic path dependency associated with the discretely sampled realized variance is captured in the numerical procedure by updating two path dependent state variables across monitoring dates. The time stepping procedure between successive monitoring dates can be performed using fast Fourier transform calculations without the usual tedious time stepping calculations in typical nite di erence algorithms. We also derive effective numerical procedures that compute the hedge parameters of variance products and volatility derivatives. Numerical tests on pricing various variance products and volatility derivatives were performed that illustrate eciency, accuracy, reliability and robustness of the proposed Fourier transform algorithms.

Pricing and Hedging of Cliquet Options and Locally Capped Contracts

This paper provides a new approach for pricing and hedging popular highly path-dependent equity-linked contracts. We illustrate our technique with two examples: the locally capped contracts (a popular design on the exchange-listed retail investment contracts on the American Stock Exchange) and the cliquet option (extensively sold by insurance companies). Wilmott [Wilmott Magazine, 2002, pp. 78--83] describes these types of contracts as the “height of fashion in the world of equity derivatives.” Existing literature proposes methods based on partial differential equations, Monte Carlo techniques, or Fourier analysis. We show that there exist semi-closed-form expressions of their prices as well as of the hedging parameters.

A new fourth-order numerical scheme for option pricing under the CEV model

The empirically observed negative relationship between a stock price and its return volatility can be captured by the constant elasticity of variance option pricing model. For European options, closed form expressions involve the non-central chi-square distribution whose computation can be slow when the elasticity factor is close to one, volatility is low or time to maturity is small. We present a fast numerical scheme based on a high-order compact discretisation which accurately computes the option price. Various numerical examples indicate that for comparable computational times, the option price computed with the scheme has higher accuracy than the Crank–Nicolson numerical solution. The scheme accurately computes the hedging parameters and is stable for strongly negative values of the elasticity factor.

The Evaluation of Financial Assets with Autocorrelations in Returns

In this paper, we will describe an analytical solution to a problem of pricing financial assets with autocorrelations in returns. We will develop a continuous diffusion model for the case of autocorrelation in stock returns, obtain the European call option pricing formula written on a stock with autocorrelation in returns and show that even small levels of predictability due to autocorrelation can give a substantial deviation from the results obtained by Black-Sholes formula. Also, we will calculate the modified sensitivities of the value of European call option and show how in risk management widely used option hedging parameters depend on assumptions made about correlation in underlying asset returns. Finally, we will show convergence time for the stationary solution of the derived continuous diffusion model and test its distribution.

Pricing and Hedging of Quantile Options in a Flexible Jump Diffusion Model

This paper proposes a Laplace-transform-based approach to price the fixed-strike quantile options as well as to calculate the associated hedging parameters (delta and gamma) under a hyperexponential jump diffusion model, which can be viewed as a generalization of the well-known Black-Scholes model and Kou's double exponential jump diffusion model. By establishing a relationship between floating- and fixed-strike quantile option prices, we can also apply this pricing and hedging method to floating-strike quantile options. Numerical experiments demonstrate that our pricing and hedging method is fast, stable, and accurate.

Pricing and Hedging of Quantile Options in a Flexible Jump Diffusion Model

This paper proposes a Laplace-transform-based approach to price the fixed-strike quantile options as well as to calculate the associated hedging parameters (delta and gamma) under a hyperexponential jump diffusion model, which can be viewed as a generalization of the well-known Black-Scholes model and Kou's double exponential jump diffusion model. By establishing a relationship between floating- and fixed-strike quantile option prices, we can also apply this pricing and hedging method to floating-strike quantile options. Numerical experiments demonstrate that our pricing and hedging method is fast, stable, and accurate.

An Improved Binomial Method using Cell Averages for Option Pricing

We present an improved binomial method for pricing financial deriva-tives by using cell averages. After non-overlapping cells are introduced around each node in the binomial tree, the proposed method calculates cell averages of payoffs at expiry and then performs the backward valuation process. The price of the derivative and its hedging parameters such as Greeks on the valuation date are then computed using the compact scheme and Richardson extrapolation. The simulation results for European and American barrier options show that the pro-posed method gives much more accurate price and Greeks than other recent lattice methods with less computational effort.

A COST-EFFECTIVE MODIFICATION OF THE TRINOMIAL METHOD FOR OPTION PRICING

A new method for option pricing based on the trinomial tree method is introduced. The new method calculates the local average of option prices around a node at each time, instead of computing prices at each node of the trinomial tree. Local averaging has a smoothing effect to reduce oscillations of the tree method and to speed up the convergence. The option price and the hedging parameters are then obtained by the compact scheme and the Richardson extrapolation. Computational results for the valuation of European and American vanilla and barrier options show superiority of the proposed scheme to several existing tree methods.

Gestión ambiental de los servicios públicos domiciliarios: seguimiento para el departamento de Cundinamarca

Este artículo presenta el estado actual del sector de servicios públicos domiciliarios del departamento de Cundinamarca, teniendo en cuenta únicamente acueducto, alcantarillado y manejo de residuos sólidos, considerados determinantes de la calidad de vida de la población, vistos principalmente a partir de parámetros como cobertura e inversión en la prestación de estos servicios. Se toma como referente teórico la Nueva Economía Institucional, considerando que son las instituciones las que pueden aportar al cambio en beneficio del sector; el marco regulatorio lo establece la Constitución Política de 1991, la Ley 142 de 1994 para el control social y administrativo de los servicios públicos domiciliarios (SSPPDD), la Ley 143 de 1994 de procedimientos básicos desde la recolección hasta la disposición final de los desechos, entre otros. Se concluye que es determinante involucrar a todos los estamentos de la sociedad en la solución del problema de la prestación de los SSPPDD referidos, por ejemplo, mediante la conformación de los Comités de Desarrollo y Control Social, que permitan organizar la participación comunitaria para la vigilancia y la fiscalización de la gestión de las empresas que presten el servicio, sean de carácter privado, oficial o mixto, así como los Fondos de Solidaridad y Redistribución de ingresos en el caso de los Acueductos Municipales.

On coordinate transformation and grid stretching for sparse grid pricing of basket options

We evaluate two coordinate transformation techniques in combination with grid stretching for pricing basket options in a sparse grid setting. The sparse grid technique is a basic technique for solving a high-dimensional partial differential equation. By creating a small hypercube sub-grid in the ‘composite’ sparse grid we can also determine hedge parameters accurately. We evaluate these techniques for multi-asset examples with up to five underlying assets in the basket.