Óscar Gutiérrez

Option valuation, time-changed processes and the fast Fourier transform

The pricing of options, in many frameworks, can be performed by inverting the transformed distribution function to obtain the elements involved in option prices; Heston (1993), Bates (1996) and Bakshi et al. (1997), among others, use this procedure, and Bakshi and Madan (2000) show how characteristic functions span the payoff universe of most derivatives. This method has an inconvenience: the integrands involved have a singularity at the origin, the fast Fourier transform (FFT) cannot be invoked and the highly oscillatory integrands converge very slowly. Given the computational advantages that the FFT algorithm implies, it is desirable to find expressions computable by FFT. Carr and Madan (1999) circumvent the obstacle dampening the integrands by means of an external factor. Following such a direction, Lewis (2001) analyses the problem when Lévy processes drive the returns uncertainty, and Lee (2004) generalizes the method and sets bounds for the error committed in pricing computations. The introduction of dampening factors is widely used in practical implementations when option valuation is required; see Carr et al. (2002), Carr and Wu (2003a, b), Bakshi et al. (2005) or Wu (2006). The present approach analyses the problem when the returns process is modeled as a time-changed Brownian motion with drift. It is important to note that the timechanged processes include pure-jump models, with either finite, or infinite activity (see Geman et al. 2001 for examples), as well as continuous models (a square-root stochastic volatility model is provided below, the ‘time-changed’ version of the Heston model). The method is particularly useful given that most of the successful models in option valuation can be thought of as time-changed models (Carr and Wu 2004). We begin by expressing the returns density function in an adequate manner for our purposes, which differs from the approach in the papers of Carr and Madan (1999), Lewis (2001) and Lee (2004), who transform option prices directly. The expression obtained allows us to write European option prices by means of simplified integral forms computable by FFT. The use of external dampening factors with option pricing purposes can thus be circumvented.

American option valuation using first-passage densities

This paper explores the advantages of pricing American options using the first-passage density of a Brownian motion to a curved barrier. First, we demonstrate that, under this approach, the exact computation of the optimal boundary becomes secondary. Consequently, a simple approximation to the optimal boundary suffices to obtain accurate prices. Moreover, the first-passage approach tends to give more accurate prices than the early-exercise-premium integral representation. We present two ways of implementing the approach. The first is based on an exact representation of the first-passage density. The second exploits the method of images, which gives us a family of barriers with first-passage densities given in closed form. Both methods are very easy to implement and give accurate prices. In particular, the images-based method is extremely accurate.

The Effect of Ownership Structure on Financial Stability: An Empirical Investigation

This paper analyzes the stability of financial systems with different bank’s ownership structures. We distinguish between profit oriented banks (commercial banks) and non-profit maximizing bank such as savings banks and cooperative banks (stakeholder banks). Using a country-level panel dataset over the period 1993-2007, and two different measures of financial stability at the systemic level (the country portfolio Z-score and a crisis dummy variable), we find consistent evidence that the presence of stakeholder banks in the financial sector increases systemic financial stability.

American option pricing by a method of error correction

ABSTRACT The real options approach often assumes that investment projects last indefinitely, which is an unrealistic assumption. When projects live finitely, valuation techniques from American option pricing are required. This article presents a method for pricing American options based on the first-passage approach to the problem. The key is to correct the error associated with the price obtained from a rough first approximation. The procedure leads to a significant reduction in error corresponding to the initial approximation. As a particular case of the method proposed, we derive a closed-form approximation of the option price. The existence of a closed-form approximating formula (that does not involve iterative methods) keeps the computational cost low. In terms of accuracy, the method can be compared to much more sophisticated methods. A tight lower bound (given in closed form) is also provided. The method is fast, accurate, flexible, and easy to implement. A spreadsheet suffices for practical implementation.

Competition Between Banks with Different Ownership Structure: Implications for Performance, Risk and Welfare

This paper analyses the effects of competition between banks with different ownership structures on financial stability, social welfare, risk-taking incentives and performance. Specifically, we present a model of strategic competition in the retail banking sector where a profit-maximizing bank, i.e., commercial bank competes against a non-for-profit maximizing bank, i.e., stakeholder bank.Our main conclusions are that the presence of stakeholder banks increases systemic financial stability, social welfare, and banking competition. We also show that stakeholder banks are less risk-inclined and obtain a higher market share than commercial banks; and that any bank is less stable and less profitable when competing against a stakeholder bank. Our predictions are validated by the empirical evidence. Thus, our model contributes to the current debate on the future of stakeholder banks suggesting that policy makers worried about systemic financial stability may favor a stakeholder approach in the banking system.