Efficient numerical methods for pricing American options under stochastic volatility

Five numerical methods for pricing American put options under Heston's stochastic volatility model are described and compared. The option prices are obtained as the solution of a two‐dimensional parabolic partial differential inequality. A finite difference discretization on nonuniform grids leading to linear complementarity problems with M‐matrices is proposed. The projected SOR, a projected multigrid method, an operator splitting method, a penalty method, and a componentwise splitting method are considered. The last one is a direct method while all other methods are iterative. The resulting systems of linear equations in the operator splitting method and in the penalty method are solved using a multigrid method. The projected multigrid method and the componentwise splitting method lead to a sequence of linear complementarity problems with one‐dimensional differential operators that are solved using the Brennan and Schwartz algorithm. The numerical experiments compare the accuracy and speed of the considered methods. The accuracies of all methods appear to be similar. Thus, the additional approximations made in the operator splitting method, in the penalty method, and in the componentwise splitting method do not increase the error essentially. The componentwise splitting method is the fastest one. All multigrid‐based methods have similar rapid grid independent convergence rates. They are about two or three times slower that the componentwise splitting method. On the coarsest grid the speed of the projected SOR is comparable with the multigrid methods while on finer grids it is several times slower. ©John Wiley & Sons, Inc. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Optimal dynamic price selection under attraction choice models

We present a single-resource finite-horizon Markov decision process approach for a firm that seeks to maximize expected revenues by dynamically adjusting the menu of offered products and their prices to be selected from a finite set of alternative values predetermined as a matter of policy. Consumers choose among available products according to an attraction choice model, a special but widely applied class of discrete choice models.     

An iterative method for pricing American options under jump-diffusion models

We propose an iterative method for pricing American options under jump-diffusion models. A finite difference discretization is performed on the partial integro-differential equation, and the American option pricing problem is formulated as a linear complementarity problem (LCP). Jump-diffusion models include an integral term, which causes the resulting system to be dense. We propose an iteration to solve the LCPs efficiently and prove its convergence. Numerical examples with Kou?s and Merton?s jump-diffusion models show that the resulting iteration converges rapidly.     

Combined pricing and supply chain operations under price-dependent stochastic demand

In this study, we determined product prices and designed an integrated supply chain operations plan that maximized a manufacturer’s expected profit. The computational results of this study revealed that as the variance of the demand distribution increases, a manufacturer will increase its inventory to levels that are greater than the anticipated demand to prevent the potential loss of sales and will simultaneously raise product prices to obtain a greater profit. In the cost minimization approach, the manufacturer may earn the highest possible profits, as determined by the profit optimization approach, only if this firm precisely forecasts the mean market demand for its products. Greater inaccuracies in this forecast will produce lower levels of expected profit.     

On pricing derivatives under nonlinear time series models

In this note, we summarize some of our recent works on pricing derivative securities under nonlinear time series models. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Efficient smoothing of Crank-Nicolson method for pricing barrier options under stochastic volatility

Most of the option pricing problems have nonsmooth payoff. In barrier options certain aspects of the option 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 the barrier option because a discontinuity is introduced in the time domain each time a barrier is applied. An improved smoothing strategy is introduced for smoothing the A -stable Cranck-Nicolson scheme at each time when a barrier is applied. A partial differential equation (PDE) approach is utilized for the evaluation of complex option pricing models under stochastic volatility which brings major mathematical and computational challenges for estimation and stability of the estimates. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bayesian solution to pricing and inventory control under unknown demand distribution

This paper addresses the simultaneous determination of pricing and inventory control with learning. The Bayesian formulation of this model results in a dynamic program with a multi-dimension state-space. We show that the state-space of the Bayesian model can be reduced under some conditions and characterize the structure of the optimal policy.     

Stocking and pricing decisions under endogenous demand and reference point effects

In this paper, we study the newsvendor’s pricing and stocking decisions under reference point effects. The demand faced by the newsvendor is endogenous and the customers may also decide to procure the product from an outside option. We characterize the firm’s optimal pricing and stocking decisions. Our analysis reveals a threshold policy on the firm’s ordering and pricing decisions while considering the impact of reference point effects. We also find that as the level of optimism increases, the firm’s optimal ordering level decreases and optimal price increases. We further study the impact of loss aversion on the firm’s ordering and pricing decisions.     

Inventory control and pricing for perishable products under age and price dependent stochastic demand

Multistage stochastic programming (SP) with both endogenous and exogenous uncertainties is a novel problem in which some uncertain parameters are decision-dependent and others are independent of decisions. The main difficulty of this problem is that nonanticipativity constraints (NACs) make up a significantly large constraint set, growing very fast with the number of scenarios and leading to an intractable model. Usually, a lot of these constraints are redundant and hence, identification and elimination of redundant NACs can cause a significant reduction in the problem size. Recently, a polynomial time algorithm has been proposed in the literature which is able to identify all redundant NACs in an SP problem with only endogenous uncertainty. In this paper, however, we extend the algorithm proposed in the literature and present a new method which is able to make the upper most possible reduction in the number of NACs in any SP with both exogenous and endogenous uncertain parameters. Proving the validity of this method is another innovation of this study. Computational results confirm that the proposed approach can significantly reduce the problem size within a reasonable computation time.     

Option pricing under jump-diffusion models with mean-reverting bivariate jumps

We propose a jump-diffusion model where the bivariate jumps are serially correlated with a mean-reverting structure. Mathematical analysis of the jump accumulation process is given, and the European call option price is derived in analytical form. The model and analysis are further extended to allow for more general jump sizes. Numerical examples are provided to investigate the effects of mean-reversion in jumps on the risk-neutral return distributions, option prices, hedging parameters, and implied volatility smiles.     

Convergence rates of trinomial tree methods for option pricing under regime-switching models

Recently trinomial tree methods have been developed to option pricing under regime-switching models. Although these novel trinomial tree methods are shown to be accurate via numerical examples, it needs to give a rigorous proof of the accuracy which can theoretically guarantee the reliability of the computations. The aim of this paper is to prove the convergence rates (measure of the accuracy) of the trinomial tree methods for the option pricing under regime-switching models.     

Efficient calibration for problems in option pricing

The pricing of derivatives has gained considerable importance in the finance industry and leads to challenging problems in numerical optimization. We focus on the numerical solution of a stochastic model for option prices. In particular, we are concerned with the calibration of these models to real data, which leads to large scale optimization problems. We consider the numerical solution of these optimization problems and give some indication how to reduce the complexity of these problems. Special emphasis is devoted to a multi-layer strategy which is embedded into the optimization iteration. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Cash flow models for pricing mortgages

A number of recent trends have led academics and practitionersto question the separation in theory, practice and regulationbetween the insurance and banking industries. These trends haveincluded corporate integration and the creation of financialconglomerates, the creation of products such as credit derivativesand credit insurance, which involve underwriting similar risksin both banks and non-banks, and the recent development of morestandardized accounting practices. Until recently, the methodologiesfor pricing insurance and banking products were quite separate.This paper presents an approach to pricing mortgages that iscommonly used for pricing insurance products. Pricing the twoshould not be radically different, since the fundamental financialcharacteristics of the products offered are similar. The modelexplicitly identifies different elements of the cash flows tothe providers of equity capital and prices the loan to achievean appropriate risk-adjusted rate of return on equity capital.Expenses, size of loan, term of loan and special ‘features’(such as cash backs) can be as, or more, important than defaultrisk when pricing mortgages. Methodologies similar to the oneproposed are now used in the banking sector for analysing theprofitability of a new product prior to launch. However, creditscoring techniques would still generally be used in taking lendingdecisions in individual cases.     

Efficient pricing of commodity options with early-exercise under the Ornstein-Uhlenbeck process

We analyze the efficiency properties of a numerical pricing method based on Fourier-cosine expansions for early-exercise options. We focus on variants of Schwartz? model based on a mean reverting Ornstein-Uhlenbeck process, which is commonly used for modeling commodity prices. This process however does not possess favorable properties for the option pricing method of interest. We therefore propose an approximation of its characteristic function, so that the Fast Fourier Transform can be applied for highest efficiency.     

Option pricing under regime-switching models: Novel approaches removing path-dependence

A well-known approach for the pricing of options under regime-switching models is to use the regime-switching Esscher transform (also called regime-switching mean-correcting martingale measure) to obtain risk-neutrality. One way to handle regime unobservability consists in using regime probabilities that are filtered under this risk-neutral measure to compute risk-neutral expected payoffs. The current paper shows that this natural approach creates path-dependence issues within option price dynamics. Indeed, since the underlying asset price can be embedded in a Markov process under the physical measure even when regimes are unobservable, such path-dependence behavior of vanilla option prices is puzzling and may entail non-trivial theoretical features (e.g., time non-separable preferences) in a way that is difficult to characterize. This work develops novel and intuitive risk-neutral measures that can incorporate regime risk-aversion in a simple fashion and which do not lead to such path-dependence side effects. Numerical schemes either based on dynamic programming or Monte-Carlo simulations to compute option prices under the novel risk-neutral dynamics are presented.     

New formulations and valid inequalities for a bilevel pricing problem

Consider the problem of maximizing the toll revenue collected on a multi-commodity transportation network. This fits a bilevel framework where a leader sets tolls, while users respond by selecting cheapest paths to their destination. We propose novel formulations of the problem, together with valid inequalities yielding improved algorithms.     

A dimension and variance reduction Monte-Carlo method for option pricing under jump-diffusion models

We develop a highly efficient MC method for computing plain vanilla European option prices and hedging parameters under a very general jump-diffusion option pricing model which includes stochastic variance and multi-factor Gaussian interest short rate(s). The focus of our MC approach is variance reduction via dimension reduction. More specifically, the option price is expressed as an expectation of a unique solution to a conditional Partial Integro-Differential Equation (PIDE), which is then solved using a Fourier transform technique. Important features of our approach are (1) the analytical tractability of the conditional PIDE is fully determined by that of the Black–Scholes–Merton model augmented with the same jump component as in our model, and (2) the variances associated with all the interest rate factors are completely removed when evaluating the expectation via iterated conditioning applied to only the Brownian motion associated with the variance factor. For certain cases when numerical methods are either needed or preferred, we propose a discrete fast Fourier transform method to numerically solve the conditional PIDE efficiently. Our method can also effectively compute hedging parameters. Numerical results show that the proposed method is highly efficient.     

Dynamic pricing with real-time demand learning

In many service industries, the firm adjusts the product price dynamically by taking into account the current product inventory and the future demand distribution. Because the firm can easily monitor the product inventory, the success of dynamic pricing relies on an accurate demand forecast. In this paper, we consider a situation where the firm does not have an accurate demand forecast, but can only roughly estimate the customer arrival rate before the sale begins. As the sale moves forward, the firm uses real-time sales data to fine-tune this arrival rate estimation. We show how the firm can first use this modified arrival rate estimation to forecast the future demand distribution with better precision, and then use the new information to dynamically adjust the product price in order to maximize the expected total revenue. Numerical study shows that this strategy not only is nearly optimal, but also is robust when the true customer arrival rate is much different from the original forecast. Finally, we extend the results to four situations commonly encountered in practice: unobservable lost customers, time dependent arrival rate, batch demand, and discrete set of allowable prices.