American-style Options Research Articles

Perpetual Options on Multiple Underlyings

We study three classes of perpetual option with multiple uncertainties and American-style exercise boundaries, using a partial differential equation-based approach. A combination of accurate numerical techniques and asymptotic analyses is implemented, with each approach informing and confirming the other. The first two examples we study are a put basket option and a call basket option, both involving two stochastic underlying assets, whilst the third is a (novel) class of real option linked to stochastic demand and costs (the details of the modelling for this are described in the paper). The Appendix addresses the issue of pricing American-style perpetual options involving (just) one stochastic underlying, but in which the volatility is also modelled stochastically, using the Heston (1993) framework.

Option pricing under regime-switching jump–diffusion models

We present an explicit formula and a multinomial approach for pricing contingent claims under a regime-switching jump–diffusion model. The explicit formula, obtained as an expectation of Merton-type formulae for jump–diffusion processes, allows to compute the price of European options in the case of a two-regime economy with lognormal jumps, while the multinomial approach allows to accommodate an arbitrary number of regimes and a generic jump size distribution, and is suitable for pricing American-style options. The latter algorithm discretizes log-returns in each regime independently, starting from the highest volatility regime where a recombining multinomial lattice is established. In the remaining regimes, lattice nodes are the same but branching probabilities are adjusted. Derivative prices are computed by a backward induction scheme.

Bayesian analysis of equity-linked savings contracts with American-style options

In this paper, a full Bayesian procedure is developed and implemented for the market consistent valuation of a fairly general class of equity-linked savings contracts. The developed procedure allows several combinations of contract properties. For example, the contract returns may include a guaranteed interest rate and a bonus depending on the yield of a total return equity index. Especially, the contract may include an American-style path-dependent surrender option. The underlying asset and interest rate processes are estimated using the Markov chain Monte Carlo method, and their simulation is based on their posterior predictive distribution, which is, however, adjusted to give risk-neutral dynamics. Financial guarantees and equity-linked components are common in many life insurance products. From the insurance company’s viewpoint, this paper provides a realistic and flexible modelling tool for product design and risk analysis. The focus is on a novel application of advanced theoretical and computational methods, which enable us to deal with a fairly realistic valuation framework and to address model and parameter error issues. Our empirical results support the use of elaborated instead of stylized models for asset dynamics in practical applications.

Pricing and static hedging of American-style options under the jump to default extended CEV model

This paper prices (and hedges) American-style options through the static hedge approach (SHP) proposed by Chung and Shih (2009) and extends the literature in two directions. First, the SHP approach is generalized to the jump to default extended CEV (JDCEV) model of Carr and Linetsky (2006), and plain-vanilla American-style options on defaultable equity are priced. The robustness and efficiency of the proposed pricing solutions are compared with the optimal stopping approach offered by Nunes (2009), under both the JDCEV framework and the nested constant elasticity of variance (CEV) model of Cox (1975), using different elasticity parameter values. Second, the early exercise boundary near expiration is derived under the JDCEV model.

Complete Analytical Solution of the American Style Option Pricing with Constant and Stochastic Volatilities: A Probability Density Function Approach

The first ever explicit formulation of the concept of an option’s probability density functions has been introduced in our publications “Breakthrough in Understanding Derivatives and Option Based Hedging - Marginal and Joint Probability Density Functions of Vanilla Options - True Value-at-Risk and Option Based Hedging Strategies”, “Complete Analytical Solution of the Asian Option Pricing and Asian Option Value-at-Risk Problems. A Probability Density Function Approach” and “Complete Analytical Solution of the Heston Model for Option Pricing and Value-At-Risk Problems. A Probability Density Function Approach.” Please see links http://ssrn.com/abstract= 2489601, http://ssrn.com/abstract= 2546430, http://ssrn.com/abstract= 2549033). In this paper we report unique analytical results for pricing American Style Options in the presence of both constant and stochastic volatility (Heston model), enabling complete analytical resolution of all problems associated with American Style Options considered within the Heston Model. Our discovery of the probability density function for American and European Style Options with constant and stochastic volatilities enables exact closed-form analytical results for their expected values (prices) for the first time without depending on approximate numerical methods. Option prices, i.e. their expected values, are just the first moments. All higher moments are as easily represented in closed form based on our probability density function, but are not calculable by extensions of other numerical methods now used to represent the first moment. Our formulation of the density functions for options with American and European Style execution rights with constant and stochastic volatility (Heston model) is expressive enough to enable derivation for the first time ever of corollary closed-form analytical results for such Value-At-Risk characteristics as the probabilities that options with different execution rights, with constant or stochastic volatility, will be below or above any set of thresholds at termination. Such assessments are absolutely out of reach of current published methods for treating options.All numerical evaluations based on our analytical results are practically instantaneous and absolutely accurate.

Some control variates for exotic options

There are no known exact formulas for the valuation of a number of exotic options, and this is particularly true for options under discrete monitoring and for American style options. Therefore, one usually recourses to a Monte Carlo Simulation approach, amongst other numerical methods, to estimate the value of these options. The problem which then arises with this method is one of variance reduction. Control variates are often used, and we present some results for the optimization of these control variables, for the valuation of Asian and lookback options. An inequality on functions of correlations useful for comparing estimators in variance reduction procedures is also provided.

Option valuation under a regime-switching constant elasticity of variance process

We investigate the pricing of both European and American-style options when the price dynamics of the underlying risky assets are governed by a Markov-modulated constant elasticity of variance process. Both probabilistic and partial differential equation approaches are considered in deriving the value of a European-style option. For the case of an American-style option, we consider a probabilistic approach and derive an integral representation for the early exercise premium.

Adjusting Volatility to Discrete Cash Dividends

This article studies the modelling of discrete cash dividends. We distinguish the escrowed dividend model from the stock price process jumping downwards at ex-dividend dates. We show how dividend jumps impact the second moment of the stock terminal distribution. This distinctive feature precludes usual implied volatility data from being directly input in such a type of model, since they are extracted from another pricing framework. A preliminary volatility adjustment to discrete cash dividends is then described. This global adjustment leads to a refinement of the local volatility surface. This refinement appears to be quite natural theoretically speaking. It proves also to be very efficient numerically, at least for the simplest products such as American-style options.

A Forward Monte Carlo Method for American Options Pricing

This study proposes a forward Monte Carlo method for the pricing of American options. The main advantage of this method is that it does not use backward induction as required by other methods. Instead, the proposed approach relies on a wise determination about whether a simulated stock price has entered the exercise region. The validity of the proposed method is supported by the mathematical proofs for the vanilla cases. With some adaption, it is shown that this forward method can be extended to price other American style options such as chooser and exchange options. This study demonstrates the effectiveness of the proposed approach using a series of numerical examples, revealing significant improvements in numerical efficiency and accuracy in contrast with the standard regression‐based method of Longstaff and Schwartz (2001). © 2012 Wiley Periodicals, Inc. Jrl Fut Mark 33:369‐395, 2013

Joint Characteristic Function of Stock Log-Price and Squared Volatility in the Bates Model and Its Asset Pricing Applications

The model of Bates specifies a rich, flexible structure of stock dynamics suitable for applications in finance and economics, including valuation of derivative securities. This paper analytically derives a closed-form expression for the joint conditional characteristic function of a stock’s log-price and squared volatility under the model dynamics. The use of the function, based on inverting it, is illustrated on examples of pricing European-, Bermudan-, and American-style options. The discussed approach for European-style derivatives improves on the option formula of Bates. The suggested approach for American-style derivatives, based on a compound-option technique, offers an alternative solution to existing finite-difference methods.

Early Exercise Boundary for American Type of Floating Strike Asian Option and Its Numerical Approximation

In this article, we generalize and analyse the model for pricing American-style Asian options proposed by Hansen and Jørgensen (2000) by including a continuous dividend rate q and a general method of averaging the floating strike. We focus on the qualitative and quantitative analysis of the early exercise boundary. The first-order expansion in terms of of the early exercise boundary close to expiry is constructed. We furthermore propose an efficient numerical algorithm for determining the early exercise boundary position based on the front-fixing method. Construction of the algorithm is based on a solution to a non-local parabolic partial differential equation for the transformed variable representing the synthesized portfolio. Various numerical results and comparisons of our numerical method and the method developed by Dai and Kwok (2006) are presented.

GPGPUs in computational finance: massive parallel computing for American style options

SUMMARYThe pricing of American style and multiple exercise options is a very challenging problem in mathematical finance. One usually employs a least squares Monte Carlo approach (Longstaff–Schwartz method) for the evaluation of conditional expectations, which arise in the backward dynamic programming principle for such optimal stopping or stochastic control problems in a Markovian framework. Unfortunately, these least squares MC approaches are rather slow and allow, because of the dependency structure in the backward dynamic programming principle, no parallel implementation neither on the MC level nor on the time layer level of this problem. We therefore present in this paper a quantization method for the computation of the conditional expectations that allows a straightforward parallelization on the MC level. Moreover, we are able to develop for first‐order autoregressive processes a further parallelization in the time domain, which makes use of faster memory structures and therefore maximizes parallel execution. Furthermore, we discuss the generation of random numbers in parallel on a GPGPU architecture, which is the crucial tool for the application of massive parallel computing architectures in mathematical finance. Finally, we present numerical results for a CUDA implementation of these methods. It will turn out that such an implementation leads to an impressive speed‐up compared with a serial CPU implementation. Copyright © 2011 John Wiley & Sons, Ltd.

PERPETUAL CANCELLABLE AMERICAN CALL OPTION

This paper examines the valuation of a generalized American‐style option known as a game‐style call option in an infinite time horizon setting. The specifications of this contract allow the writer to terminate the call option at any point in time for a fixed penalty amount paid directly to the holder. Valuation of a perpetual game‐style put option was addressed by Kyprianou (2004) in a Black‐Scholes setting on a nondividend paying asset. Here, we undertake a similar analysis for the perpetual call option in the presence of dividends and find qualitatively different explicit representations for the value function depending on the relationship between the interest rate and dividend yield. Specifically, we find that the value function is not convex when r > d. Numerical results show the impact this phenomenon has upon the vega of the option.

OPTIMAL LIQUIDATION OF DERIVATIVE PORTFOLIOS

We consider the problem facing a risk averse agent who seeks to liquidate or exercise a portfolio of (infinitely divisible) perpetual American style options on a single underlying asset. The optimal liquidation strategy is of threshold form and can be characterized explicitly as the solution of a calculus of variations problem. Apart from a possible initial exercise of a tranche of options, the optimal behavior involves liquidating the portfolio in infinitesimal amounts, but at times which are singular with respect to calendar time. We consider a number of illustrative examples involving CRRA and CARA utility, stocks, and portfolios of options with different strikes, and a model where the act of exercising has an impact on the underlying asset price.

HEDGING SWING OPTIONS

We study models for electricity pricing and derivatives in the context of a deregulated market setting. In particular we value swing options, since these are the electricity derivatives that attract the most attention from market participants. These are American style options in that they allow for multiple exercises subject to a set of constraints on the consumption process. Through the use of a penalty function, we generalize the problem by allowing for the consumption restrictions to be broken. We characterize the price function as a stochastic optimal control problem, and show that the option is exercised in a bang-bang fashion. The value of the swing option is the solution to a backward stochastic differential equation, and we show how European calls, along with forward contracts, can be used to hedge them.

Evaluating natural resource projects with embedded options and limited reserves

This study proposes a Dynamic Option Simulation (DOS) approach to evaluate natural resource investment projects that contain several embedded options and limited reserves. To construct a practical pricing model, DOS combines simulation and dynamic programming techniques that can value natural resource investments with multi-variable, early exercise, various embedded options and finite reserve properties. A copper mine project offers a demonstration; the mine holder may temporarily close, reopen and abandon the mine at specific times before the expiration date. The mine holder may also accelerate mining speed to be optimal, referred to as the acceleration option. Numerical analyses of the copper price and interest rate effects on the copper mine value suggest that DOS can efficiently assess complex, multiple-variable, American-style real options problems.

Correction: Sequential Monte Carlo pricing of American-style options under stochastic volatility models

Correction: Sequential Monte Carlo pricing of American-style options under stochastic volatility models

Memory-Reduction Method for Pricing American-Style Options under Exponential Lévy Processes

AbstractThis paper concerns the Monte Carlo method in pricing American-style options under the general class of exponential Lévy models. Traditionally, one must store all the intermediate asset prices so that they can be used for the backward pricing in the least squares algorithm. Therefore the storage requirement grows like , where m is the number of time steps and n is the number of simulated paths. In this paper, we propose a simulation method where the storage requirement is only . The total computational cost is less than twice that of the traditional method. For machines with limited memory, one can now enlarge m and n to improve the accuracy in pricing the options. In numerical experiments, we illustrate the efficiency and accuracy of our method by pricing American options where the log-prices of the underlying assets follow typical Lévy processes such as Brownian motion, lognormal jump-diffusion process, and variance gamma process.

American options and callable bonds under stochastic interest rates and endogenous bankruptcy

A new characterization of the American-style option is proposed under a very general multifactor Markovian and diffusion framework. The efficiency of the proposed pricing solutions is shown to depend only on the use of a viable valuation method for the corresponding European-style option and for the transition density of the model’s state variables. Under a Gauss-Markov stochastic interest rates setup, these new American option pricing solutions are shown to offer a much better accuracy-efficiency trade-off than the approximations already available in the literature. This result is also used to price callable corporate bonds under an endogenous bankruptcy structural approach, by decomposing the option to call or default into a European put on the firm value plus two early exercise premium components.

Implementation of the Longstaff and Schwartz American Option Pricing Model on FPGA

American style options are widely used financial products, whose pricing is a challenging problem due to their path dependency characteristic. Finite difference methods and tree-based methods can be used for American option pricing. However, the major drawback of these methods is that they can often only handle one or two sources of uncertainty; for more state variables they become computationally prohibitive, with computation times typically increasing exponentially with the number of state variables. Alternative solutions are the extended Monte Carlo methods, such as the Least-Squares Monte Carlo (LSMC) method suggested by Longstaff and Schwartz, which uses of regression to estimate continuation values from simulated paths. In this paper, we present an FPGA hardware architecture for the acceleration of the LSMC method, with Quasi-Monte Carlo path generation. Our FPGA hardware implementation on a Xilinx Virtex-4 XC4VFX100 chip achieves 25× and 18× speed-ups in the path generation and regression steps, respectively, compared to an equivalent pure software implementation captured in C++ and run on an Intel Xeon 2.8 GHz CPU. This provides overall speed-up of 20× compared to a CPU-based implementation. Power measurements also show that our FPGA implementation is 54× more energy efficient than the pure software implementation.