Valuing American Put Options Using Chebyshev Polynomial Approximation

This paper suggests a simple valuation method based on Chebyshev approximation at Chebyshev nodes to value American put options. It is similar to the approach taken in Sullivan (2000), where the option`s continuation region function is estimated by using a Chebyshev polynomial. However, in contrast to Sullivan (2000), the functional is fitted by using Chebyshev nodes. The suggested method is flexible, easy to program and efficient, and can be extended to price other types of derivative instruments. It is also applicable in other fields, providing efficient solutions to complex systems of partial differential equations. The paper also describes an alternative method based on dynamic programming and backward induction to approximate the option value in each time period

Application of Operator Splitting Methods in Finance

Financial derivatives pricing aims to find the fair value of a financial contract on an underlying asset. Here we consider option pricing in the partial differential equations framework. The contemporary models lead to one-dimensional or multidimensional parabolic problems of the convection-diffusion type and generalizations thereof. An overview of various operator splitting methods is presented for the efficient numerical solution of these problems. Splitting schemes of the Alternating Direction Implicit (ADI) type are discussed for multidimensional problems, e.g. given by stochastic volatility (SV) models. For jump models Implicit-Explicit (IMEX) methods are considered which efficiently treat the nonlocal jump operator. For American options an easy-to-implement operator splitting method is described for the resulting linear complementarity problems. Numerical experiments are presented to illustrate the actual stability and convergence of the splitting schemes. Here European and American put options are considered under four asset price models: the classical Black-Scholes model, the Merton jump-diffusion model, the Heston SV model, and the Bates SV model with jumps

Anterior Prefrontal Cortex Contributes to Action Selection through Tracking of Recent Reward Trends

The functions of prefrontal cortex remain enigmatic, especially for its anterior sectors, putatively ranging from planning to self-initiated behavior, social cognition, task switching, and memory. A predominant current theory regarding the most anterior sector, the frontopolar cortex (FPC), is that it is involved in exploring alternative courses of action, but the detailed causal mechanisms remain unknown. Here we investigated this issue using the lesion method, together with a novel model-based analysis. Eight patients with anterior prefrontal brain lesions including the FPC performed a four-armed bandit task known from neuroimaging studies to activate the FPC. Model-based analyses of learning demonstrated a selective deficit in the ability to extrapolate the most recent trend, despite an intact general ability to learn from past rewards. Whereas both brain-damaged and healthy controls used comparisons between the two most recent choice outcomes to infer trends that influenced their decision about the next choice, the group with anterior prefrontal lesions showed a complete absence of this component and instead based their choice entirely on the cumulative reward history. Given that the FPC is thought to be the most evolutionarily recent expansion of primate prefrontal cortex, we suggest that its function may reflect uniquely human adaptations to select and update models of reward contingency in dynamic environments

The Effect of Non-Smooth Payoffs on the Penalty Approximation of American Options

This article combines various methods of analysis to draw a comprehensive picture of penalty approximations to the value, hedge ratio, and optimal exercise strategy of American options. While convergence of the penalised solution for sufficiently smooth obstacles is well established in the literature, sharp rates of convergence and particularly the effect of gradient discontinuities (i.e., the omni-present `kinks' in option payoffs) on this rate have not been fully analysed so far. This effect becomes important not least when using penalisation as a numerical technique. We use matched asymptotic expansions to characterise the boundary layers between exercise and hold regions, and to compute first order corrections for representative payoffs on a single asset following a diffusion or jump-diffusion model. Furthermore, we demonstrate how the viscosity theory framework in [Jakobsen, 2006] can be applied to this setting to derive upper and lower bounds on the value. In a small extension to [Bensoussan & Lions, 1982], we derive weak convergence rates also for option sensitivities for convex payoffs under jump-diffusion models. Finally, we outline applications of the results, including accuracy improvements by extrapolation.Comment: 34 Pages, 10 Figure

Assessing national values to protect the health of the Great Barrier Reef

The Great Barrier Reef (GBR) is a vast iconic environmental asset covering an area of approximately 35 million hectares. It is valued by people all over Australia, as well as overseas. Non-market values for the GBR will comprise both use and non-use values. The values of people who live closer to the GBR and who can visit it more frequently are likely to be higher than those who live further away. The aim of this study was to estimate the values to protect the health of the GBR at the national level and to examine the effects of distance decay on valuation estimates. A split-sample choice-modelling experiment was conducted in six locations: a regional town within the GBR catchment area (Townsville); Brisbane, the state capital approximately 450 km from the southern limit of the GBR; and four other capital cities (Sydney, Melbourne, Adelaide and Perth) ranging from 730 km to over 3600 km from Brisbane. The results indicate that the total national value for a 1 per cent improvement in the health of the GBR ranges from between a low of approximately $433.6 million to a high of$811.3 million, depending on the underlying assumptions made. There was some evidence of distance decay in values. Most decline occurred once outside the home state, and little further decline once away from the east coast. There was no evidence to suggest any difference in patterns of use and non-use values. The values of the potential future users were most influential in determining WTP estimates.Environmental Economics and Policy,