Hedging Parameters Research Articles

Valuing foreign currency options with a mean‐reverting process: a study of Hong Kong dollar

AbstractThe theoretical prediction on targeted exchange rates expects mean reversion of the exchange rates. There is some empirical evidence to support this prediction. This paper presents a model for valuing European foreign exchange options in which the forward foreign exchange rate follows a mean‐reverting lognormal process. The corresponding closed‐form solutions for the option valuation are derived. The mean‐reverting process has material impact on the foreign exchange rate option values and their hedge parameters. This tends to decrease the value of a simple put or call. On the other hand, the process also keeps the exchange rate in a small range around the mean level. As this is the region in which an option's intrinsic value is high because of the level of its strike price, there is also a tendency for option values to be enhanced compared with the values of the Black–Scholes model. The numerical results using the forward exchange rates of the Hong Kong dollar and market data of their options show that both of these effects are important for the realistic choices of parameter values. As the dynamics of targeted exchange rates may not follow the standard lognormal process as described by the Black–Scholes model, the mean‐reverting option‐pricing model may be considered for the valuation of options and estimation of associated hedge parameters on targeted exchange rates. Copyright © 2007 John Wiley & Sons, Ltd.

Pricing Exotic Barrier Options with Finite Differences

A barrier option is a derivative instrument whose payoff is dependent on the path of the underlying security up to maturity. We design a pricing system using Finite Differences to investigate the properties of and price options with exotic barrier features. The system determines the payoff, delta and gamma functions of these options with accuracy and speed, and can therefore provide quality pricing and hedging parameters to the professional trading community.

On smoothing of the Crank–Nicolson scheme and higher order schemes for pricing barrier options

Most option pricing problems have nonsmooth payoffs or discontinuous derivatives at the exercise price. Discrete barrier options have not only nonsmooth payoffs but also time dependent discontinuities. In pricing barrier options, certain aspects are triggered if the asset price becomes too high or too low. Standard smoothing schemes used to solve problems with nonsmooth payoff do not work well for discrete barrier options because of discontinuities introduced in the time domain when each barrier is applied. Moreover, these unwanted oscillations become worse when estimating the hedging parameters, e.g., Delta and Gamma. We have an improved smoothing strategy for the Crank–Nicolson method which is unique in achieving optimal order convergence for barrier option problems. Numerical experiments are discussed for one asset and two asset problems. Time evolution graphs are obtained for one asset problems to show how option prices change with respect to time. This smoothing strategy is then extended to higher order methods using diagonal ( m , m ) —Padé main schemes under a smoothing strategy of using as damping schemes the ( 0 , 2 m - 1 ) subdiagonal Padé schemes.

A NEW ANALYTICAL APPROXIMATION FORMULA FOR THE OPTIMAL EXERCISE BOUNDARY OF AMERICAN PUT OPTIONS

In this paper, a new analytical formula as an approximation to the value of American put options and their optimal exercise boundary is presented. A transform is first introduced to better deal with the terminal condition and, most importantly, the optimal exercise price which is an unknown moving boundary and the key reason that valuing American options is much harder than valuing its European counterparts. The pseudo-steady-state approximation is then used in the performance of the Laplace transform, to convert the systems of partial differential equations to systems of ordinary differential equations in the Laplace space. A simple and elegant formula is found for the optimal exercise boundary as well as the option price of the American put with constant interest rate and volatility. Other hedge parameters as the derivatives of this solution are also presented.

Currency barrier option pricing with mean reversion

AbstractThis article develops a barrier option pricing model in which the exchange rate follows a mean‐reverting lognormal process. The corresponding closed‐form solutions for the barrier options with time‐dependent barriers are derived. The numerical results show that barrier option values and the corresponding hedge parameters under the proposed model are different from those based on the Black‐Scholes model. For an up‐and‐out call, the mean‐reverting process keeps the exchange rate in a small range around the mean level. When the mean level is below the barrier but above the strike price, the risk of the call to be knocked out is reduced and its option value is enhanced compared with the value under the Black‐Scholes model. The parameters of the mean‐reverting lognormal process therefore have a material impact on the valuation of currency barrier options and their hedge parameters. © 2006 Wiley Periodicals, Inc. Jrl Fut Mark 26:939–958, 2006

Efficient Computation of Hedging Parameters for Discretely Exercisable Options

We propose an algorithm to calculate confidence intervals for the values of hedging parameters of discretely exercisable options using Monte-Carlo simulation. The algorithm is based on a combination of the duality formulation of the optimal stopping problem for pricing discretely exercisable options and Monte-Carlo estimation of hedging parameters for European options. We show that the width of the confidence interval for a hedging parameter decreases, with an increase in the computer budget, asymptotically at the same rate as the width of the confidence interval for the price of the option. The method can handle arbitrary payoff functions, general diffusion processes, and a large number of random factors. We also present a fast, heuristic, alternative method and use our method to evaluate its accuracy.

Catastrophe options with stochastic interest rates and compound Poisson losses

We analyze the pricing and hedging of catastrophe put options under stochastic interest rates with losses generated by a compound Poisson process. Asset prices are modeled through a jump-diffusion process which is correlated to the loss process. We obtain explicit closed form formulae for the price of the option, and the hedging parameters Delta, Gamma and Rho. The effects of stochastic interest rates and variance of the loss process on the option’s price are illustrated through numerical experiments. Furthermore, we carry out a simulation analysis to hedge a short position in the catastrophe put option by using a Delta–Gamma–Rho neutral self-financing portfolio. We find that accounting for stochastic interest rates, through Rho hedging, can significantly reduce the expected conditional loss of the hedged portfolio.

Efficient Computation of Hedging Parameters for Discretely Exercisable Options

We propose an algorithm to calculate confidence intervals for the values of hedging parameters of discretely exercisable options using Monte Carlo simulation. The algorithm is based on a combination of the duality formulation of the optimal stopping problem for pricing discretely exercisable options and Monte Carlo estimation of hedging parameters for European options. We show that the width of the confidence interval for a hedging parameter decreases, with an increase in the computer budget, asymptotically at the same rate as the width of the confidence interval for the price of the option. The method can handle arbitrary payoff functions, general diffusion processes, and a large number of random factors. We also present a fast, heuristic, alternative method and use our method to evaluate its accuracy.

Hedging Using Different Models

Although the reconstructed local volatility function can give an accurate approximation to the market volatility which leads to more efficient hedging, using the implied volatility calculated from a carefully chosen model is more ready to be accepted for the sake of easy implementation. We compare the dynamic hedging performance of the constant elasticity of variance model family with constant volatility Black-Scholes model. By assuming that the underlying stock follows a specific diffusion process, we use hypothetic examples to illustrate that Black-Scholes model yields the largest hedging parameter and the greater standard error of relative hedging error. Our observation shows that the constant volatility model can result in significant hedging error, and by allowing for varying volatility, the constant elasticity of variance model is more adaptive.

Selective hedging with market views and risk limits: the case of Hydro-Quebec

This study presents how Hydro-Quebec manages its short-term financial risks. The quantitative hedging model is articulated over forward and volatility premia and constrained by a yearly risk limit provided by the firm's Finance Committee. The hedging solutions provide the optimal linear hedging parameters and option strike levels. The second part of the paper is devoted to a theoretical analysis of the sensitivity of the optimal solutions to changes in the premia, by means of the general implicit function theorem.

Finite hedging in field theory models of interest rates.

We use path integrals to calculate hedge parameters and efficacy of hedging in a quantum field theory generalization of the Heath, Jarrow, and Morton [Robert Jarrow, David Heath, and Andrew Morton, Econometrica 60, 77 (1992)] term structure model, which parsimoniously describes the evolution of imperfectly correlated forward rates. We calculate, within the model specification, the effectiveness of hedging over finite periods of time, and obtain the limiting case of instantaneous hedging. We use empirical estimates for the parameters of the model to show that a low-dimensional hedge portfolio is quite effective.

Numerical pricing of discrete barrier and lookback options via Laplace transforms

URL: www.thejournalofcomputationalfinance.com Most contracts of barrier and lookback options specify discrete monitoring policies. However, unlike their continuous counterparts, discrete barrier and lookback options essentially have no analytical solution. For a broad class of models, including the classical Brownian model and jump-diffusion models, we show that the Laplace transforms of discrete barrier and lookback options can be obtained via a recursion involving only analytical formulae of standard European call and put options, thanks to Spitzer’s formula. The Laplace transforms can be numerically inverted to get option prices fast and accurately. Furthermore, the same method can be used to compute the hedging parameters (the greeks) of these products.

Alternative Neural Network Approach for Option Pricing and Hedging

Since its introduction in 1973, the Black-Scholes model has found increasingly more resistance in application. In order to use Black-Scholes to price any option, one needs to know the implied volatility surface. The existence of such surface is an evidence of misspecification of the model. In this case, the problem is with the assumption of a geometric Brownian motion for the stock price process. There is strong empirical evidence that stocks do not follow such process. However, no agreement has been reached on what is the best distribution to use. Neural Network approaches the problem very differently. Essentially, a Neural Network is a non-parametric estimation technique. It does not make any distributional assumption regarding the underlying variable. Instead, it puts up a formula with a set of unknown parameters and let the optimization routine search for the parameters best fitted to the desired results. Hutchinson-Lo-Poggio (1994) showed that it is indeed possible to use a Neural Network to price S&P futures options. In this paper, we will continue with this line of research. Specifically, we will examine the best way to set up and train a Neural Network for option pricing and hedging. We will also investigate if a Neural Network could produce better hedging parameters than the standard option pricing model. We use S&P futures options data covering the period 1990–2000.

A Quantum Field Theory Term Structure Model Applied to Hedging

A quantum field theory generalization, Baaquie [1], of the Heath, Jarrow and Morton (HJM) [10] term structure model parsimoniously describes the evolution of imperfectly correlated forward rates. Field theory also offers powerful computational tools to compute path integrals which naturally arise from all forward rate models. Specifically, incorporating field theory into the term structure facilitates hedge parameters that reduce to their finite factor HJM counterparts under special correlation structures. Although investors are unable to perfectly hedge against an infinite number of term structure perturbations in a field theory model, empirical evidence using market data reveals the effectiveness of a low dimensional hedge portfolio.

Convergence remedies for non-smooth payoffs in option pricing

Discontinuities in the payoff function (or its derivatives) can cause inaccuracies for numerical schemes when pricing financial contracts. In particular, large errors may occur in the estimation of the hedging parameters. Three methods of dealing with discontinuities are discussed in this paper: averaging the initial data, shifting the grid, and a projection method. By themselves, these techniques are not sufficient to restore expected behaviour. However, when combined with a special timestepping method, high accuracy is achieved. Examples are provided for one and two factor option pricing problems.

Hedging with Energy: Simple Error Bounds for Mis-Specified Diffusions

In the setting of diffusion models for price evolution, we suggest an easily implementable approximate evaluation formula for measuring the errors in option pricing and hedging implied in volatility mis-specification. The main tool we use in this paper is a fundamental inequality for the L-2 norm of the solution, and the derivatives of the solution, of a partial differential equation (the so called inequality). This result allows us to give bounds to the errors implied in the use of approximate models for option valuation and hedging. When statistical or a-priori information is available on the volatility, the error measure can be minimized w.r.t. the parameters of the approximating model. In this case the usual approximate hedging procedure, where hedging parameters are computed using an approximate model calibrated to observed prices (e.g. implied volatility), is sub optimal. The method suggested in this paper can help in conjugating statistical estimation of the volatility function derived from flexible but computationally cumbersome statistical models, with the use of analytically tractable approximate models calibrated using error estimates derived from energy inequalities. The performance of the new method is compared with the standard implied volatility hedging method in an example where the true model is a lognormal mixture while the approximating model is the standard lognormal model.

단일 저수지의 위험도 평가기준을 고려한 가뭄대비 Hedging Rule 개발

가뭄상황 또는 가뭄이 임박한 상황에서의 저수지 운영은 수요관리개념(단계별 급수)에서의 운영률을 필요로 한다. 본 연구는 저수지 갈수대응 차원에서 수문상황에 따른 단계별 방류량 감소를 고려할 수 있는 heding 효과를 고려한 단일 저수지 운영률 개발을 목표로 하였다. hedging 효과를 고려한 최적운영률 결정에는 혼합정수계획기법이 적용되었으며, 정식화단계에는 Shih 등(1994)의 hedging효과를 고려한 운영률을 개선하여 정식화 요소에 Hashimoto 등(1982)의 위험도 평가기준을 포함시켰다. 또한 hedging항의 비선형 해석을 수행하기 위하여 축차 선형계획기법을 도입ㆍ정식화에 적용하였다. 본 hedging운영률의 적용결과 대청다목적댐에 대하여 hedging 매개변수론 산정하였으며, 이를 통하여 각 월별 갈수대응 제한공급 시점 저수량(trigger volume)을 산정할 수 있었다. During drought or impending drought period, the reservoir operation method is required to incorporate demand-management policy rule. The objective of this study is focused to the development of demand reduction rule by incorporating hedging-effect for a single reservoir system. To improve the performance measure of the objective function and constraints, we could incorporate three risk performance criteria proposed by Hashimoto et al. (1982) by mixed-integer programming and also incorporate successive linear programming to overcome nonlinear hedging term from the previous study(Shih et al., 1994). To verify this model, this hedging rule was applied to the Daechung multi-purpose dam. As a result, we could evaluate optimal hedging parameters and monthly trigger volumes.

Multi-moment Approximate Option Pricing Models: A General Comparison (Part 1)

After the seminal paper of Jarrow and Rudd (1982), several authors have proposed to use different statistical series expansion to price options when the risk-neutral density is asymmetric and leptokurtic. Amongst them, one can distinguish the Gram-Charlier Type A series expansion (Corrado and Su, 1996-b and 1997-b), log-normal Gram-Charlier series expansion (Jarrow and Rudd, 1982) and Edgeworth series expansion (Rubinstein, 1998). The purpose of this paper is to compare these different multi-moment approximate option pricing models. We first recall the link between risk-neutral densities and moments in a general statistical series expansion framework. We then derive analytical formulae for these different four-moment approximate option pricing models, namely, the Jarrow and Rudd (1982), Corrado and Su (1996-b and 1997-b) and Rubinstein (1998) models. We investigate in particular the conditions that ensure the respect of the martingale restriction (see Longstaff, 1995) and compare with option pricing models such as Black and Scholes (1973) and Hermite polynomial models (see Madan and Milne, 1994, Abken et al.,1996). We also get for these approximate option pricing models analytical expressions of implied probability distribution, implied volatility smile functions and several hedging parameters of interest, such as the Psi and the Chi that measure respectively the changes in the option price with respect to the changes in kurtosis and skewness.

Credit risk: The case of First Interstate Bankcorp

Structural models for pricing credit risk can be used to forecast the spread on risky bonds and for hedging credit risk. This article examines the forecasting accuracy of the Black–Scholes–Merton (BSM) model of risky debt using a data set consisting of weekly bond data for First Interstate Bancorp over the period January 1986–August 1993. In addition, structural model hedge parameters and credit spread options are tested for their effectiveness in hedging the increasing credit risk premium on First Interstate Bancorp debt. Credit spread options in combination with a duration hedge offer the best hedging strategy, reducing the standard deviation of the hedging error by a minimum of 84%.

The GARCH-stable option pricing model

An option pricing model is developed based on a generalized autoregressive conditional heteroskedastic (GARCH) asset return process with stable Paretian innovations. Our approach is based on the locally risk-neutral valuation relationship. Methods for maximum likelihood estimation of GARCH-stable processes are presented as well as empirical results for the DAX index. Finally, the results of Monte Carlo simulations evaluating prices of European call options, implied volatility, delta hedging parameters, and value at risk are presented.