This paper describes a practical simulation-based algorithm, which we call the Stochastic Grid Bundling Method (SGBM) for pricing multi-dimensional Bermudan (i.e. discretely exercisable) options. The ...method generates a direct estimator of the option price, an optimal early-exercise policy as well as a lower bound value for the option price. An advantage of SGBM is that the method can be used for fast approximation of the Greeks (i.e., derivatives with respect to the underlying spot prices, such as delta, gamma, etc.) for Bermudan-style options. Computational results for various multi-dimensional Bermudan options demonstrate the simplicity and efficiency of the algorithm proposed.
Least-squares Monte Carlo generates regression-based continuation value estimators that are heteroscedastic. Fabozzi et al. (2017) propose weighted least-squares regression to correct for this. We ...show that heteroscedastic-corrected estimators are more accurate than uncorrected estimators far from the exercise boundary and where the exercise decision is obvious. However, the corrected estimators do not translate into improved exercise decisions and hence correcting has little effect on option price estimates. This holds when using alternative specifications for the correction and when implementing an iterative method. We conclude that correcting for heteroscedasticity does not result in more efficient prices and generally should be avoided.
•Regression-based continuation value estimators are heteroscedastic.•Heteroscedasticity is most prevalent far from the exercise boundary.•Corrected estimators do not lead to improved exercise decisions.•Correcting for heteroscedasticity has little effect on option price estimators.
This paper presents a high-order deferred correction algorithm combined with penalty iteration for solving free and moving boundary problems, using a fourth-order finite difference method. Typically, ...when free boundary problems are solved on a fixed computational grid, the order of the solution is low due to the discontinuity in the solution at the free boundary, even if a high-order method is used. Using a detailed error analysis, we observe that the order of convergence of the solution can be increased to fourth-order by solving successively corrected finite difference systems, where the corrections are derived from the previously computed lower order solutions. The corrections are applied solely to the right-hand side, and leave the finite difference matrix unchanged. The penalty iterations converge quickly given a good initial guess. We demonstrate the accuracy and efficiency of our algorithm using several examples. Numerical results show that our algorithm gives fourth-order convergence for both the solution and the free boundary location. We also test our algorithm on the challenging American put option pricing problem. Our algorithm gives the expected high-order convergence.
In this study, we develop a semi-analytic method to evaluate American options under a two-state regime-switching economy. The two free boundaries corresponding to the states divide the pricing domain ...into two regions: a common continuation region and a transition region. Non-linear partial differential equation (PDE) systems are derived under the Black–Scholes framework for each region. The Laplace transform method is used to solve the PDE systems. Equations for determining the optimal exercise prices are obtained analytically and solved numerically in the Laplace space. A numerical inversion technique is then used to obtain the free boundaries and the option prices in the original time space. The results of various examples show that our technique is efficient and accurate.
•Semi-analytic pricing of American options under a regime-switching economy.•Volatility regimes governed by two-state Markov chains.•Derivation of coupled nonlinear equations for the moving (optimal exercise) boundaries.•Easy to implement procedure for the computation of optimal prices and option values.•Efficient and accurate method extendable to pricing of other financial derivatives.
Many financial assets, such as currencies, commodities, and equity stocks, exhibit both jumps and stochastic volatility, which are especially prominent in the market after the financial crisis. Some ...strategic decision making problems also involve American-style options. In this paper, we develop a novel, fast and accurate method for pricing American and barrier options in regime switching jump diffusion models. By blending regime switching models and Markov chain approximation techniques in the Fourier domain, we provide a unified approach to price Bermudan, American options and barrier options under general stochastic volatility models with jumps. The models considered include Heston, Hull–White, Stein–Stein, Scott, the 3/2 model, and the recently proposed 4/2 model and the α-Hypergeometric model with general jump amplitude distributions in the return process. Applications include the valuation of discretely monitored contracts as well as continuously monitored contracts common in the foreign exchange markets. Numerical results are provided to demonstrate the accuracy and efficiency of the proposed method.
The uncertainty associated with option price predictions has largely been overlooked in the literature. This paper aims to fill this gap by quantifying such uncertainty using conformal prediction. ...Conformal prediction is a model-agnostic procedure that constructs prediction intervals, ensuring valid coverage in finite samples without relying on distributional assumptions. Through the simulation of synthetic option prices, we find that conformal prediction generates prediction intervals for gradient boosting machines with an empirical coverage close to the nominal level. Conversely, non-conformal prediction intervals exhibit empirical coverage levels that fall short of the nominal target. In other words, they fail to contain the actual option price more frequently than expected for a given coverage level. As anticipated, we also observe a decrease in the width of prediction intervals as the size of the training data increases. However, we uncover significant variations in the width of these intervals across different options. Specifically, out-of-the-money options and those with a short time-to-maturity exhibit relatively wider prediction intervals. Then, we perform an empirical study using American call and put options on individual stocks. We find that the empirical results replicate those obtained in the simulation experiment.
•Conformal prediction quantifies well the uncertainty of option price predictions.•Conformal prediction intervals have empirical coverage near the nominal level.•Non-conformal prediction intervals have empirical coverage below the nominal level.•Large variations in prediction intervals are found for American call and put options.•Larger intervals are found for out-of-the-money and short time-to-maturity options.
In an incomplete market construction and by no-arbitrage assumption, the American options pricing problem under the jump-diffusion regime-switching process is formulated by a variational form of ...coupled partial integro-differential equations. In this paper, a valuation algorithm is developed for American options when the dynamics of underlying assets follow the regime-switching jump-diffusion processes. Using the fact that the price of an American option under jump-diffusion regime-switching processes is formulated by a collection of coupled variational partial integro-differential equations with the free boundary characteristic, we combine the moving least-squares approximation with an operator splitting method to treat American constraints. Numerical experiments with American options under three, five, and seven regimes demonstrate the efficiency and effectiveness of our computational scheme for pricing American options under the regime-switching models.
•Applying the meshfree moving least-squares collocation for options pricing problem under the jump-diffusion regime-switching processes.•Applying the proposed method for pricing American options with the free boundary feature.•Computing hedge parameters of American options and early exercise boundary of American options with little extra cost via the proposed method.•Proposing numerical test problems for American options pricing with high regimes-switching (till seven regimes) assumption.
We discuss applications of the Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) to numerically solve partial differential equations arising from stochastic models in ...quantitative finance. In particular, we develop a framework for solving Black–Scholes-type equations by transforming them into advection–diffusion problems. The equations are then numerically integrated backward in time using an iterative explicit finite-difference approach, in which the Fickian term is represented as an additional advective term. We discuss the correspondence between transport phenomena and financial models, uncovering the possibility of expressing the no-arbitrage principle as a conservation law. We show second-order accuracy in time and space of the embraced numerical scheme. This is done via a convergence analysis comparing MPDATA numerical solutions with classic Black–Scholes analytical formulæ for the valuation of European options. We demonstrate in addition a way of applying MPDATA to solve the free boundary problem (leading to a linear complementarity problem) for the valuation of American options. We finally comment on the potential of MPDATA methods with respect to more complex models typically used in quantitative finance.
•Deriving the buyer’s and seller’s exercise regions.•Using American style derivatives with a stochastic maturity.•Deriving the equations for the exercise boundaries.•Finding the fair game call option ...price.•Giving some numerical examples.
The purpose of this paper is to examine the problem of pricing discounted perpetual game call options. In addition to the properties of the American options, the game options give the seller the right to cancel the contract at some chosen from him moment. As a compensation for this, he has to pay some amount above the usual payment. We assume that this penalty payment is a constant. We examine the case without maturity – the exercise can be made in every future moment. We first derive the optimal exercise regions for the buyer and the seller and then calculate the fair option price. Our approach is based on some American style derivatives with a stochastic maturity date.
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