A functional approach to pricing complex barrier options

In this article, a new method for pricing contingent claims, which is particularly well suited for options with complex barrier and volatility structures, is introduced. The approach is based on a high-precision approximation of the Feynman–Kac equation with distributed approximating functionals. The method is particularly well suited for long maturity valuation problems, and it is shown to be faster and more accurate than conventional solution schemes.     

A perturbative approach to Bermudan options pricing with applications

In this paper we address the problem of the valuation of Bermudan option derivatives in the framework of multi-factor interest rate models. We propose a solution in which the exercise decision entails a properly defined series expansion. The method allows for the fast computation of both a lower and an upper bound for the option price, and a tight control of its accuracy, for a generic Markovian interest rate model. In particular, we show detailed computations in the case of the Bond Market Model. As examples we consider the case of a zero coupon Bermudan option and a coupon bearing Bermudan option; in order to demonstrate the wide applicability of the proposed methodology we also consider the case of a last generation payoff, a Bermudan option on a CMS spread bond.     

A simple approach for pricing barrier options with time-dependent parameters

Abstract

In this paper we present a simple and easy-to-use method for computing accurate estimates (in closed form) of Black-Scholes barrier option prices with time-dependent parameters. This new approach is also able to provide tight upper and lower bounds (in closed form) for the exact barrier option prices.     

A dynamic programming approach for pricing CDS and CDS options

We propose a flexible framework for pricing single-name knock-out credit derivatives. Examples include Credit Default Swaps (CDSs) and European, American and Bermudan CDS options. The default of the underlying reference entity is modelled within a doubly stochastic framework where the default intensity follows a CIR++ process. We estimate the model parameters through a combination of a cross sectional calibration-based method and a historical estimation approach. We propose a numerical procedure based on dynamic programming and a piecewise linear approximation to price American-style knock-out credit options. Our numerical investigation shows consistency, convergence and efficiency. We find that American-style CDS options can complete the credit derivatives market by allowing the investor to focus on spread movements rather than on the default event.     

An equilibrium approach to pricing foreign currency options

The paper presents a modified version of the Garman-Kohlhagen formula for pricing European currency options. The equilibrium approach deviates from the no-arbitrage approach by allowing domestic and foreign interest rates and their dynamics to be determined endogenously in the model. By using the relations between exchange rate dynamics and the dynamics of interest rates, I provide a new characterisation of the relevant volatilities for European currency option pricing, which only depends on parameters describing the variability of the log-exchange rate. The implications of the model for the valuation of American currency options and optimal exercise strategies are examined by applying numerical methods.     

A risk-based approach for pricing American options under a generalized Markov regime-switching model

This paper considers a risk-based approach for pricing an American contingent claim in an incomplete market described by a continuous-time, Markov, regime-switching jump-diffusion model. We formulate the valuation problem as a stochastic differential game and use dynamic programming. Verification theorems for the Hamilton–Jacobi–Bellman–Issacs (HJBI) variational inequalities of the games are used to determine the seller's and buyer's prices and optimal exercise strategies.     

Which approach to accounting for employee stock options best reflects market pricing?

We use a residual income valuation framework to compare equity valuation implications of four approaches to employee stock options (ESOs) accounting: APB 25 “recognize nothing”, SFAS 123 (revised) “recognize ESO expense”, FASB Exposure Draft “recognize and expense ESO asset” and “recognize ESO asset and liability”. Theoretical analysis shows only grant date recognition of an asset and liability, and subsequent marking-to-market of the liability, results in accounting numbers that capture the dilution effects of ESOs on current shareholder value. Out-of-sample equity market value prediction tests and in-sample comparisons of model explanatory power also support the “recognize ESO asset and liability” method.     

Pseudospectral methods for pricing options

Models with two or more risk sources have been widely applied in option pricing in order to capture volatility smiles and skews. However, the computational cost of implementing these models can be large—especially for American-style options. This paper illustrates how numerical techniques called ‘pseudospectral’ methods can be used to solve the partial differential and partial integro-differential equations that apply to these multifactor models. The method offers significant advantages over finite-difference and Monte Carlo simulation schemes in terms of accuracy and computational cost.     

An endogenous volatility approach to pricing and hedging call options with transaction costs

Standard delta hedging fails to exactly replicate a European call option in the presence of transaction costs. We study a pricing and hedging model similar to the delta hedging strategy with an endogenous volatility parameter for the calculation of delta over time. The endogenous volatility depends on both the transaction costs and the option strike prices. The optimal hedging volatility is calculated using the criterion of minimizing the weighted upside and downside replication errors. The endogenous volatility model with equal weights on the up and down replication errors yields an option premium close to the Leland [J. Finance, 1985 Leland, HE. 1985. Option pricing and replication with transaction costs. J. Finance, 40: 1283–1301. [Crossref], [Web of Science ®] , [Google Scholar], 40, 1283–1301] heuristic approach. The model with weights being the probabilities of the option's moneyness provides option prices closest to the actual prices. Option prices from the model are identical to the Black–Scholes option prices when transaction costs are zero. Data on S&P 500 index cash options from January to June 2008 illustrate the model.     

A linearly implicit predictor–corrector scheme for pricing American options using a penalty method approach

Pricing of an American option is complicated since at each time we have to determine not only the option value but also whether or not it should be exercised (early exercise constraint). This makes the valuation of an American option a free boundary problem. Typically at each time there is a particular value of the asset, which marks the boundary between two regions: to one side one should hold the option and to other side one should exercise it. Assuming that investors act optimally, the value of an American option cannot fall below the value that would be obtained if it were exercised early. Effectively, this means that the American option early exercise feature transforms the original linear pricing partial differential equation into a nonlinear one. We consider a penalty method approach in which the free and moving boundary is removed by adding a small and continuous penalty term to the Black–Scholes equation; consequently,the problem can be solved on a fixed domain. Analytical solutions of the Black–Scholes model of American option problems are seldom available and hence such derivatives must be priced by stable and efficient numerical techniques. Standard numerical methods involve the need to solve a system of nonlinear equations, evolving from the finite difference discretization of the nonlinear Black–Scholes model, at each time step by a Newton-type iterative procedure. We implement a novel linearly implicit scheme by treating the nonlinear penalty term explicitly, while maintaining superior accuracy and stability properties compared to the well-known θ-methods.     

A direct solution method for pricing options involving the maximum process

One often encounters options involving not only the stock price, but also its running maximum. We provide, in a fairly general setting, explicit solutions for optimal stopping problems concerned with a diffusion process and its running maximum. Our approach is to use excursion theory for Markov processes and rewrite the original two-dimensional problem as an infinite number of one-dimensional ones. Our method is rather direct without presupposing the existence of an optimal threshold or imposing a smooth-fit condition. We present a systematic solution method by illustrating it through classical and new examples.     

Discussion of “Which approach to accounting for employee stock options best reflects market pricing?”

The objective of the Landsman, Peasnel, Pope and Yeah paper (in this issue) is to compare, for current shareholders, the value relevance of four methods of accounting for employee stock options (ESOs). My discussion provides a unifying framework for the theoretical analyses and the link between the theoretical analyses and the empirical investigation.     

A continuous time approach to the pricing of bonds

This paper develops an arbitrage model of the term structure of interest rates based on the assumptions that the whole term structure at any point in time may be expressed as a function of the yields on the longest and shortest maturity default free instruments and that these two yields follow a Gauss-Wiener process. Arbitrage arguments are used to derive a partial differential equation which must be satisfied by the values of all default free bonds. The joint stochastic process for the two yields is estimated using Canadian data and the model is used to price a sample of Government of Canada bonds.     

A model-free approach to multivariate option pricing

Review of Derivatives Research - We propose a novel model-free approach to extract a joint multivariate distribution, which is consistent with options written on individual stocks as well as on...     

A simple approach to interest-rate option pricing

A simple introduction to contingent claim valuation of riskyassets in a discrete time, stochastic interest-rate economyis provided. Taking the term structure of interest rates asexogenous, closed-form solutions are derived for European optionswritten on (i) Treasury bills, (ii) interest-rate forward contracts,(iii) interest-rate futures contracts, (iv) Treasury bonds,(v) interest-rate caps, (vi) stock options, (vii) equity forwardcontracts, (viii) equity futures contracts, (ix) Eurodollarliabilities, and (x) foreign exchange contracts.     

A moment expansion approach to option pricing

In this paper we present a new methodology for option pricing. The main idea consists of representing a generic probability distribution function (PDF) by an expansion around a given, simpler, PDF (typically a Gaussian function) by matching moments of increasing order. Because, as shown in the literature, the pricing of path-dependent European options can often be reduced to recursive (or nested) one-dimensional integral calculations, the moment expansion (ME) approach leads very quickly to excellent numerical solutions. In this paper, we present the basic ideas of the method and the relative applications to a variety of contracts, mainly: Asian, reverse cliquet and barrier options. A comparison with other numerical techniques is also presented.     

Two-dimensional risk-neutral valuation relationships for the pricing of options

The Black–Scholes model is based on a one-parameter pricing kernel with constant elasticity. Theoretical and empirical results suggest declining elasticity and, hence, a pricing kernel with at least two parameters. We price European-style options on assets whose probability distributions have two unknown parameters. We assume a pricing kernel which also has two unknown parameters. When certain conditions are met, a two-dimensional risk-neutral valuation relationship exists for the pricing of these options: i.e. the relationship between the price of the option and the prices of the underlying asset and one other option on the asset is the same as it would be under risk neutrality. In this class of models, the price of the underlying asset and that of one other option take the place of the unknown parameters.      

A novel approach to the valuation of American options

We introduce a practical numerical method to the valuation of American options. The new feature is the exact reformulation of the problems over very small regions. Numerical examples and analyses show that our algorithm leads to very fast and highly accurate results.     

An analytical approximation for pricing VWAP options

This paper proposes a unified approximation method for various options whose pay-offs depend on the volume weighted average price (VWAP). Despite their popularity in practice, very few pricing models have been developed in the literature. Also, in previous works, the underlying asset process has been restricted to a geometric Brownian motion. In contrast, our method is applicable to the general class of continuous Markov processes such as local volatility models. Moreover, our method can be used for any type of VWAP options with fixed-strike, floating-strike, continuously sampled, discretely sampled, forward-start and in-progress transactions.