Showing papers on "Binomial options pricing model published in 2007"
Book•
30 Nov 2007
TL;DR: The Black-Scholes model as mentioned in this paper is a discrete-time formalism for estimating martingales and arbitrage opportunities in the stock market with continuous-time processes, and it has been applied to American options.
Abstract: INTRODUCTION DISCRETE-TIME MODELS Discrete-time formalism Martingales and arbitrage opportunities Complete markets and option pricing Problem: Cox, Ross and Rubinstein model OPTIMAL STOPPING PROBLEM AND AMERICAN OPTIONS Stopping time The Snell envelope Decomposition of supermartingales Snell envelope and Markov chains Application to American options BROWNIAN MOTION AND STOCHASTIC DIFFERENTIAL EQUATIONS General comments on continuous-time processes Brownian motion Continuous-time martingales Stochastic integral and Ito calculus Stochastic differential equations THE BLACK-SCHOLES MODEL Description of the model Change of probability: Representation of martingales Pricing and hedging options in the Black-Scholes model American options Implied volatility and local volatility models The Black-Scholes model with dividends and call/put symmetry Problems OPTION PRICING AND PARTIAL DIFFERENTIAL EQUATIONS European option pricing and diffusions Solving parabolic equations numerically American options INTEREST RATE MODELS Modeling principles Some classical models ASSET MODELS WITH JUMPS Poisson process Dynamics of the risky asset Martingales in a jump-diffusion model Pricing options in a jump-diffusion model CREDIT RISK MODELS Structural models Intensity-based models Copulas SIMULATION AND ALGORITHMS FOR FINANCIAL MODELS Simulation and financial models Introduction to variance reduction methods Computer experiments APPENDIX Normal random variables Conditional expectation Separation of convex sets BIBLIOGRAPHY INDEX Exercises appear at the end of each chapter.
658 citations
Posted Content•
TL;DR: In this article, a fast and accurate method for pricing early exercise and certain exotic options in computational finance is presented, which is based on a quadrature technique and relies heavily on Fourier transformations.
Abstract: A fast and accurate method for pricing early exercise and certain exotic options in computational finance is presented. The method is based on a quadrature technique and relies heavily on Fourier transformations. The main idea is to reformulate the well-known risk-neutral valuation formula by recognising that it is a convolution. The resulting convolution is dealt with numerically by using the Fast Fourier Transform (FFT). This novel pricing method, which we dub the Convolution method, CONV for short, is applicable to a wide variety of payoffs and only requires the knowledge of the characteristic function of the model. As such the method is applicable within exponentially Levy models, including the exponentially affine jump-diffusion models. For an M-times exercisable Bermudan option, the overall complexity is O(MN log(N)) with N grid points used to discretise the price of the underlying asset. It is shown how to price American options efficiently by applying Richardson extrapolation to the prices of Bermudan options.
145 citations
TL;DR: In this paper, the pricing of options when the volatility of the underlying asset depends upon a hidden Markov process which takes discrete values is studied, and the difference in option prices with and without regime switching is substantial for lookback options and more moderate for European and Asian options.
Abstract: This paper studies the pricing of options when the volatility of the underlying asset depends upon a hidden Markov process which takes discrete values. It is assumed that the regime switching process is generated by time-independent rate parameters and is independent of the Brownian motion. We derive the coupled Black–Scholes-type partial differential equations that govern the dynamics of several exotic options. These include European, Asian and lookback options. The difference in option prices with and without regime switching is substantial for lookback options and more moderate for European and Asian options.
139 citations
TL;DR: The main idea is to reformulate the well-known risk-neutral valuation formula by recognizing that it is a convolution, and the resulting convolution is dealt with numerically by using the fast Fourier transform.
Abstract: A fast and accurate method for pricing early exercise and certain exotic options in computational finance is presented. The method is based on a quadrature technique and relies heavily on Fourier transformations. The main idea is to reformulate the well-known risk-neutral valuation formula by recognising that it is a convolution. The resulting convolution is dealt with numerically by using the Fast Fourier Transform (FFT). This novel pricing method, which we dub the Convolution method, CONV for short, is applicable to a wide variety of payoffs and only requires the knowledge of the characteristic function of the model. As such the method is applicable within exponentially Levy models, including the exponentially affine jump-diffusion models. For an M -times exercisable Bermudan option, the overall complexity is O(MN log(N)) with N grid points used to discretise the price of the underlying asset. It is shown how to price American options efficiently by applying Richardson extrapolation to the prices of Bermudan options.
114 citations
01 Jan 2007
TL;DR: In this article, the authors focus on the following issues related to jump-diffusion models for asset pricing in financial engineering: the controversy over tailweight of distributions, identifying a risk-neutral pricing measure by using the rational expectations equilibrium, using Laplace transforms to pricing options, including European call/put options, pathdependent options, such as barrier and lookback options, Difficulties associated with the partial integro-differential equations related to barrier-crossing problems, and analytical approximations for finite-horizon American options with jump risk.
Abstract: In this survey we shall focus on the following issues related to jump-diffusion models for asset pricing in financial engineering (1) The controversy over tailweight of distributions (2) Identifying a risk-neutral pricing measure by using the rational expectations equilibrium (3) Using Laplace transforms to pricing options, including European call/put options, path-dependent options, such as barrier and lookback options (4) Difficulties associated with the partial integro-differential equations related to barrier-crossing problems (5) Analytical approximations for finite-horizon American options with jump risk (6) Multivariate jump-diffusion models
106 citations
TL;DR: In this article, a new adaptive simulation based algorithm for the numerical solution of optimal stopping problems in discrete time is proposed, which recursively computes the so-called continuation values.
Abstract: Under the assumption of no-arbitrage, the pricing of American and Bermudan options can be casted into optimal stopping problems. We propose a new adaptive simulation based algorithm for the numerical solution of optimal stopping problems in discrete time. Our approach is to recursively compute the so-called continuation values. They are defined as regression functions of the cash flow, which would occur over a series of subsequent time periods, if the approximated optimal exercise strategy is applied. We use nonparametric least squares regression estimates to approximate the continuation values from a set of sample paths which we simulate from the underlying stochastic process. The parameters of the regression estimates and the regression problems are chosen in a data-dependent manner. We present results concerning the consistency and rate of convergence of the new algorithm. Finally, we illustrate its performance by pricing high-dimensional Bermudan basket options with strangle-spread payoff based on the average of the underlying assets.
61 citations
TL;DR: In this article, the utility of the real options approach to irrigation dam investment analysis is demonstrated, where the authors calculate the option values of selected options that may be available to managers of irrigation dam investments.
Abstract: This paper demonstrates the utility of the real options approach to irrigation dam investment analysis. The main objective is to show how to calculate the option values of selected options that may be available to managers of irrigation dam investments. The paper provides an empirical application, which compares an irrigation dam investment using the static Net Present Value (NPV) model and the real options approach and shows how it can be adopted to model uncertainty and managerial flexibility in dam management. Four management options are used for the real options approach: an option to delay the investment, an option to enlarge the dam, an option to abandon the dam, and multiple options that evaluated all three options together. All options were evaluated using the binomial option pricing model, where water values are assumed to follow a multiplicative binomial process. The analysis show that although the traditional NPV approach accepted the investment as profitable the option approach provided better results showing that all three options were highly valuable if exercised. When real options are considered, the traditional NPV model for assessing the profitability of a dam investment may fail to provide an adequate decision-making framework because it does not properly value management’s ability to adjust to shocks in the economy, as well as risks and uncertainty.
48 citations
TL;DR: In this paper, the authors described the valuation scheme of European, barrier, and Asian options of single asset by using radial basis function approximation, and the option prices are governed with Black-Scholes equation.
Abstract: This paper describes the valuation scheme of European, barrier, and Asian options of single asset by using radial basis function approximation. The option prices are governed with Black–Scholes equation. The equation is discretized with Crank–Nicolson scheme and then, the option price is approximated with the radial basis functions with unknown parameters. In the European and the barrier options, the prices are governed with Black–Scholes equation. The governing option of the Asian option, however, is different from them of the others. In that case, one has to adopt the other radial basis functions than that for the original Black–Scholes equation. Finally, numerical results are compared with theoretical and finite difference solutions in order to confirm the validity of the present formulation.
36 citations
TL;DR: In this paper, a binomial lattice approach is proposed for valuing options whose payoff depends on multiple state variables following correlated geometric Brownian processes, which is step by step consistent with the continuous time diffusions, and a change of basis of the asset span, to transform asset prices into uncorrelated processes.
Abstract: A binomial lattice approach is proposed for valuing options whose payoff depends on multiple state variables following correlated geometric Brownian processes. The proposed approach relies on two simple ideas: a log‐transformation of the underlying processes, which is step by step consistent with the continuous‐time diffusions, and a change of basis of the asset span, to transform asset prices into uncorrelated processes. An additional transformation is applied to approximate driftless dynamics. Even if these features are simple and straightforward to implement, it is shown that they significantly improve the efficiency of the multi‐dimensional binomial algorithm. A thorough test of efficiency is provided compared with most popular binomial and trinomial lattice approaches for multi‐dimensional diffusions. Although the order of convergence is the same for all lattice approaches, the proposed method shows improved efficiency.
35 citations
TL;DR: The analysis provides insights into the fine structure of convergence of the general binomial model to the Black-Scholes formula and the numerical results suggest that the GCRR models with various modifications are efficient for pricing a range of options.
Abstract: This paper generalizes the seminal Cox-Ross-Rubinstein (CRR) binomial model by adding a stretch parameter. The generalized CRR (GCRR) model allows us to fine-tune (via the stretch parameter) the lattice structure so as to efficiently price a range of options, such as barrier options. Our analysis provides insights into the fine structure of convergence of the general binomial model to the Black-Scholes formula. We also discuss how to improve the rate of convergence or the oscillatory behavior of the GCRR model. The numerical results suggest that the GCRR models with various modifications are efficient for pricing a range of options.
31 citations
17 Sep 2007
TL;DR: In this paper, a transform approach for regime-switching Levy models is presented, which is applicable to a wide class of path-dependent options (such as Bermudan, barrier and shout options) and options on multiple assets.
Abstract: Although jump-diffusion and Levy models have been widely used in industry, the resulting pricing partial-integro differential equations poses various difficulties for valuation. Diverse finite-difference schemes for solving the problem have been introduced in the literature. Invariably, the integral and diffusive terms are treated asymmetrically, large jumps are truncated and the methods are difficult to extend to higher dimensions. We present a new efficient transform approach for regime-switching Levy models which is applicable to a wide class of path-dependent options (such as Bermudan, barrier, and shout options) and options on multiple assets.
TL;DR: A strongly stable and highly accurate method for pricing exotic options based on Pade schemes and also utilizes partial fraction decomposition to address issues regarding accuracy and computational efficiency is developed.
Abstract: In this paper we develop a strongly stable ( L -stable) and highly accurate method for pricing exotic options. The method is based on Pade schemes and also utilizes partial fraction decomposition to address issues regarding accuracy and computational efficiency. Due to non-smooth payoffs, which cause discontinuities in the solution (or its derivatives), standard A -stable methods are prone to produce large and spurious oscillations in the numerical solutions which would mislead to estimating options accurately. The proposed method does not suffer these drawbacks while being easy to implement on concurrent processors. Numerical results are presented for digital options, butterfly spread and barrier options in one and two assets. In addition, the methods are tested on the Heston stochastic volatility model.
TL;DR: In this article, the authors provide a review and synthesis of 11 alternative versions of the binomial option pricing model for pricing options on lognormally distributed assets, showing their commonalities and differences and demonstrating how 11 diverse models all produce the same result in the limit.
Abstract: The finance literature has revealed no fewer than 11 alternative versions of the binomial option pricing model for pricing options on lognormally distributed assets. These models are derived under a variety of assumptions and in some cases require unnecessary information. This paper provides a review and synthesis of these models, showing their commonalities and differences and demonstrating how 11 diverse models all produce the same result in the limit. Some of the models admit arbitrage with a finite number of time steps and some fail to capture the correct volatility. This paper also examines the convergence properties of each model and finds that none exhibit consistently superior performance over the others. Finally, it demonstrates how a general model that accepts any arbitrage-free risk neutral probability will reproduce the Black-Scholes-Merton model in the limit.
TL;DR: In this paper, 20 different implementation methodologies for each of 11 different choices of parameters of binomial trees and investigate the speed of convergence for pricing American put options numerically were investigated.
Abstract: We study 20 different implementation methodologies for each of 11 different choices of parameters of binomial trees and investigate the speed of convergence for pricing American put options numerically. We conclude that the most effective methods involve using truncation, Richardson extrapolation and sometimes smoothing. We do not recommend use of a European option as a control. The most effective trees are the Tian third order moment matching tree and a new tree designed to minimize oscillations.
TL;DR: In this article, the authors evaluate vulnerable American options based on the two-point Geske and Johnson method and provide analytical pricing formulas for European and multi-exercisable options under risk-neutral measures.
Abstract: This article evaluates vulnerable American options based on the two-point Geske and Johnson method. In accordance with the Martingale approach, we provide analytical pricing formulas for European and multi-exercisable options under risk-neutral measures. Employing Richardson’s extrapolation gets the values of vulnerable American options. To demonstrate the accuracy of our proposed method, we use numerical examples to compare the values of vulnerable American options from our proposed method with the benchmark values from the least-square Monte Carlo simulation method. We also perform sensitivity analyses for vulnerable American options and show how the prices of vulnerable American options vary with the correlation between the underlying assets and the option writer’s assets.
TL;DR: In this article, a nonparametric method to accurately price American option contingent claims is proposed, which uses only historical stock price data, not option price data to generate the American option price.
Abstract: We introduce a nonparametric method to accurately price American style contingent claims. This method uses only historical stock price data, not option price data, to generate the American option price. We test the accuracy of this method in a controlled experimental environment under both Black & Scholes (1973) and Heston (1993) assumptions and perform an error-metric analysis. These numerical experiments demonstrate that this method is an accurate and precise method of pricing American options under a variety of market conditions.
TL;DR: Pironneau and Pironneau as mentioned in this paper used the Dupire-like identity to compute all financial options based on a single asset at once and also for the calibration of models.
TL;DR: In this paper, the authors consider how an insurer should invest in order to hedge the maturity guarantees inherent in participating policies and use a binomial tree model to show how this risk can be hedged, and hence calculate the fair value of the contract at the outset.
Abstract: In this paper we consider how an insurer should invest in order to hedge the maturity guarantees inherent in participating policies. Many papers have considered the case where the guarantee is increased each year according to the performance of an exogenously given reference portfolio subject to some guaranteed rate. However, in this paper we will consider the more realistic case whereby the reference portfolio is replaced by the insurer’s own investments which are controlled completely at the discretion of the insurer’s management. Hence in our case any change in the insurer’s investment strategy leads to a change in the underlying value process of the participating contract. We use a binomial tree model to show how this risk can be hedged, and hence calculate the fair value of the contract at the outset.
TL;DR: The Esscher transform is employed to determine an equivalent martingale measure for option valuation and the impact of the high-order effect of the states of the economy on the prices of some path-dependent exotic options, such as Asian options, lookback options, and barrier options are investigated.
Abstract: We consider the pricing of exotic options when the price dynamics of the underlying
risky asset are governed
by a discrete-time Markovian regime-switching process driven by an observable, high-order
Markov model (HOMM). We assume that the market interest rate, the drift, and the volatility of the underlying risky asset's return switch over time according to the states of the HOMM, which are interpreted as the states of an economy. We will then employ the well-known
tool in actuarial science, namely, the Esscher transform to determine an equivalent martingale measure for option valuation. Moreover, we will also investigate the impact of the high-order effect of the states of the economy on the prices of some path-dependent exotic options, such as Asian options, lookback options, and barrier options.
TL;DR: In this paper, a new family of binomial trees as approximations to the Black-Scholes model is introduced and the existence of complete asymptotic expansions for the prices of vanilla European options is demonstrated and the first three terms are explicitly computed.
Abstract: A new family of binomial trees as approximations to the Black-Scholes model is introduced. For this class of trees, the existence of complete asymptotic expansions for the prices of vanilla European options is demonstrated and the first three terms are explicitly computed. As special cases, a tree with third order convergence is constructed and the conjecture of Leisen and Reimer that their tree has second order convergence is proven.
Posted Content•
TL;DR: In this article, the authors apply the Lie-algebraic technique for the valuation of moving barrier options with time-dependent parameters, where the value of the underlying asset is assumed to follow the constant elasticity of variance (CEV) process.
Abstract: In this paper we apply the Lie-algebraic technique for the valuation of moving barrier options with time-dependent parameters. The value of the underlying asset is assumed to follow the constant elasticity of variance (CEV) process. By exploiting the dynamical symmetry of the pricing partial differential equations, the new approach enables us to derive the analytical kernels of the pricing formulae straightforwardly, and thus provides an efficient way for computing the prices of the moving barrier options. The method is also able to provide tight upper and lower bounds for the exact prices of CEV barrier options with fixed barriers. In view of the CEV model being empirically considered to be a better candidate in equity option pricing than the traditional Black-Scholes model, our new approach could facilitate more efficient comparative pricing and precise risk management in equity derivatives with barriers by incorporating term-structures of interest rates, volatility and dividend into the CEV option valuation model.
TL;DR: In this article, the authors proposed a new approach to this problem using spectral methods, which enables them to evaluate numerically the joint distribution between the underlying stock price and the realized variance, which in turn gives a way of pricing consistently the European options, general accrued variance payoffs as well as forward-starts and VIX options.
Abstract: In the first quarter of 2006 Chicago Board Options Exchange (CBOE) introduced, as one of the listed products, options on its implied volatility index (VIX). This opened the challenge of developing a pricing framework that can simultaneously handle European options, forward-starts, options on the realized variance and options on the VIX. In this paper we propose a new approach to this problem using spectral methods. We define a stochastic volatility model with jumps and local volatility, which is almost stationary, and calibrate it to the European options on the S&P 500 for a broad range of strikes and maturities. We then extend the model, by lifting the corresponding Markov generator, to keep track of relevant path information, namely the realized variance. The lifted generator is too large a matrix to be diagonalized numerically. We overcome this difficulty by developing a new semi-analytic algorithm for block-diagonalisation. This method enables us to evaluate numerically the joint distribution between the underlying stock price and the realized variance which in turn gives us a way of pricing consistently the European options, general accrued variance payoffs as well as forward-starts and VIX options.
TL;DR: Using the numerical analysis and the notion of viscosity solution, it is proved the uniform convergence of the trinomial tree method for European/American options and the accuracy and efficiency are shown.
TL;DR: The main result consists in the explicit calculation of the hedge price for a variety of the rainbow options including option delivering the best of J risky assets and cash, calls on the maximum of J perilous assets and the multiple-strike options.
Abstract: The general approach for the pricing of rainbow (or colored) options with fixed transaction costs is developed from the game theoretic point of view. The evolution of the underlying common stocks is considered in discrete time. The main result consists in the explicit calculation of the hedge price for a variety of the rainbow options including option delivering the best of J risky assets and cash, calls on the maximum of J risky assets and the multiple-strike options. The results obtained can be also used in the framework of real options.
TL;DR: In this article, the value of an American exchange option when the underlying assets follow jump-diffusion processes is estimated based on combining a European exchange option and a Bermudan exchange option with two exercise dates by using Richardson extrapolation.
Abstract: A way to estimate the value of an American exchange option when the underlying assets follow jump-diffusion processes is presented. The estimate is based on combining a European exchange option and a Bermudan exchange option with two exercise dates by using Richardson extrapolation as proposed by R. Geske and H. Johnson (1984). Closed-form solutions for the values of European and Bermudan exchange options are derived. Several numerical examples are presented, illustrating that the early exercise feature may have a significant economic value. The results presented should have potential for pricing over-the-counter options and in particular for pricing real options. © 2007 Wiley Periodicals, Inc. Jrl Fut Mark 27:257–273, 2007
TL;DR: In this article, a general numerical method for pricing American options in regime-switching jump-diffusion models of stock dynamics with stochastic interest rates and/or volatility is developed.
Abstract: A general numerical method for pricing American options in regime-switching jump-diffusion models of stock dynamics with stochastic interest rates and/or volatility is developed. Time derivative and infinitesimal generator of the process for factors that determine the dynamics of the interest rate and/or volatility are discretized. The result is a sequence of embedded perpetual options in a Markov-modulated Levy model. Options in this sequence are solved using an iteration method based on the Wiener-Hopf factorization. An explicit algorithm for the case of positive stochastic interest rates driven by a process of the Ornstein-Uhlenbeck type is derived. Efficiency of the method is illustrated with numerical examples.
01 Jan 2007
TL;DR: A sample shows an implementation of the binomial model in CUDA, a very important problem encountered in financial engineering since the creation of organized option trading in 1973.
Abstract: vpodlozhnyuk Initial release 1.0 2007/04/05 Mharris Grammar and clarity fixes. Abstract The pricing of options is a very important problem encountered in financial engineering since the creation of organized option trading in 1973. As more computation has been applied to finance-related problems, finding efficient ways to implement option pricing models on modern architectures has become more important. This sample shows an implementation of the binomial model in CUDA.
TL;DR: In this article, the credit risk and valuation model for the European options in the OTC markets, incorporating a practical default trigger mechanism, were developed based on the first passage time models, and the default probability and the pricing formulae of the options were obtained by using partial differential equation (PDE) techniques, especially Green's function.
Abstract: In the over-the-counter (OTC) markets, the options traded are always subject to credit risk. Therefore the counterparty’s credit risk is a striking factor when pricing options, whereas it is not considered in the classic Black-Scholes models. Based on the first passage time models, this paper develops the credit risk and valuation model for the European options in the OTC markets, incorporating a practical default trigger mechanism. The default probability and the pricing formulae of the OTC options are obtained by using partial differential equation (PDE) techniques, especially Green’s function.
TL;DR: This work presents a new approach to pricing discretely monitored barrier options based on numerical approximation of the transition probability density associated with the stochastic differential equation describing the stock price dynamics, and provides accurate results in less than one second whenever a contract expires in a year or less.
Abstract: Barrier options are financial derivative contracts that are activated or deactivated according to the crossing of specified barriers by an underlying asset price. Exact models for pricing barrier options assume continuous monitoring of the underlying dynamics, usually a stock price. Barrier options in traded markets, however, nearly always assume less frequent observation, e.g. daily or weekly. These situations require approximate solutions to the pricing problem. We present a new approach to pricing such discretely monitored barrier options that may be applied in many realistic situations. In particular, we study daily monitored up-and-out call options of the European type with a single underlying stock. The approach is based on numerical approximation of the transition probability density associated with the stochastic differential equation describing the stock price dynamics, and provides accurate results in less than one second whenever a contract expires in a year or less. The flexibility of the method permits more complex underlying dynamics than the Black and Scholes paradigm, and its relative simplicity renders it quite easy to implement.
TL;DR: The convergence rate of the Binomial Tree Scheme is obtained with a penalization approximation to an American put option modelled as a variational inequality.
Abstract: An American put option can be modelled as a variational inequality. With a penalization approximation to this variational inequality, the convergence rate $$O\big((\Delta x)^{2/3}\big)$$ of the Binomial Tree Scheme is obtained in this paper.