Early-exercise Options Research Articles

An efficient algorithm for Bermudan barrier option pricing

An efficient option pricing method based on Fourier-cosine expansions was presented by Fang and Oosterlee for European options in 2008, and later, this method was also used by them to price early-exercise options and barrier options respectively, in 2009. In this paper, this method is applied to price discretely American barrier options in which the monitored dates are many times more than the exercise dates. The corresponding algorithm is presented to practical option pricing. Numerical experiments show that this algorithm works very well and efficiently for different exponential Levy asset models.

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 Schwartzʼ 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.

Enhanced policy iteration for American options via scenario selection

Kolodko and Schoenmakers (2006) and Bender and Schoenmakers (2006) introduced a policy iteration that allows the achievement of a tight lower approximations of the price for early exercise options via a nested Monte Carlo simulation in a Markovian setting. In this paper we enhance the algorithm by a scenario selection method. It is demonstrated by numerical examples that the scenario selection can significantly reduce the number of inner simulations actually performed, and thus can greatly speed up the method (by up to a factor of 15 in some examples). Moreover, it is shown that the modified algorithm retains the desirable properties of the original, such as the monotone improvement property, termination after a finite number of iteration steps, and numerical stability.

A Fast and Accurate FFT-Based Method for Pricing Early-Exercise Options under Lévy Processes

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 recognizing that it is a convolution. The resulting convolution is dealt with numerically by using the fast Fourier transform. This novel pricing method, which we dub the convolution method, is applicable to a wide variety of payoffs and requires only the knowledge of the characteristic function of the model. As such, the method is applicable within many regular affine models, among which is the class of exponential Levy models. For an $M$-times exercisable Bermudan option, the overall complexity is $O(MN\log_2(N))$, with $N$ grid points used to discretize the price of the underlying asset. American options are priced efficiently by applying Richardson extrapolation to the prices of Bermudan options.

Enhanced Finite-Difference Techniques for Early-Exercise Options on Single and Multiple Underlyings

This paper develops improvements to existing finite-difference methods for pricing derivatives (on single and also multiple underlying factors) which contain early-exercise features. Both fixed-grid methods (based on Projected Successive Over Relaxation) and moving-grid methods (based on boundary-fitted coordinates), are considered exploiting some prior (but limited) knowledge of the topology of the pricing in time and underlying price(s). These techniques can be quite readily applied to ensure pricing is both highly accurate and very efficient.

Stochastic models of election timing

Under the democratic systems of government instilled in many sovereign states, the party in government maintains a constitutional right to call an early election. While the constitution states that there is a maximum period between elections, early elections are frequently called. This right to call an early election gives the government a control to maximize its remaining life in power. The optimal control for the government is found by locating an exercise boundary that indicates whether or not a premature election should be called. This problem draws upon the body of literature on optimal stopping problems and stochastic control. Morgan Poll’s two-party-preferred data are used to model the behaviour of the poll process and a mean reverting Stochastic Differential Equation (SDE) is fitted to these data. Parameters of this SDE are estimated using the Maximum Likelihood Estimation (MLE) Method. Analytic analysis of the SDE for the poll process is given and it will be proven that there is a unique solution to the SDE subject to some conditions. In the first layer, a discrete time model is developed by considering a binary control for the government, viz. calling an early election or not. A comparison between a three-year and a four-year maximum term is also given. A condition when the early exercise option is removed, which leads to a fixed term government such as in the USA is also considered. In the next layer, the possibility for the government to use some control tools that are termed as ‘boosts’ to induce shocks to the opinion polls by making timely policy announcements or economic actions is also considered. These actions will improve the government’s popularity and will have some impacts upon the early-election exercise boundary. An extension is also given by allowing the government to choose the size of its ‘boosts’ to maximize its expected remaining life in power. In the next layer, a continuous time model for this election timing is developed by using a martingale approach and Ito’s Lemma which leads to a problem of solving a partial differential equation (PDE) along with some boundary conditions. Another condition considered is when the government can only call an election and the opposition can apply ‘boosts’ to raise its popularity or just to pull government’s popularity down. The ultimate case analysed is when both the government and the opposition can use ‘boosts’ and the government still has option to call an early election. In these two cases a game theory approach is employed and results are given in terms of the expected remaining life in power and the probability of calling and using ‘boosts’ at every time step and at certain level of popularity.

A Fast and Accurate FFT-Based Method for Pricing Early-Exercise Options Under Levy Processes

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.

Upper Bounds for American Options

This paper provides a fuller characterization of the analytical upper bounds for American options than has been available to date. We establish properties required of analytical upper bounds without any direct reliance on the exercise boundary. A class of generalized European claims on the same underlying asset is then proposed as upper bounds. This set contains the existing closed form bounds of Margrabe, (1978) and Chen and Yeh (2002) as special cases and allows randomization of the maturity payoff. Owing to the European nature of the bounds, across-strike arbitrage conditions on option prices seem to carry over to the bounds. Among other things, European option spreads may be viewed as ratio positions on the early exercise option. To tighten the upper bound, we propose a quasi-bound that holds as an upper bound for most situations of interest and seems to offer considerable improvement over the currently available closed form bounds. As an approximation, the discounted value of Chen and Yeh's (2002) bound holds some promise. We also discuss implications for parametric and nonparametric empirical option pricing. Sample option quotes for the European (XEO) and the American (OEX) options on the S&P 100 Index appear well behaved with respect to the upper bound properties but the bid–ask spreads are too wide to permit a synthetic short position in the early exercise option.