International Journal of Theoretical and Applied Finance

A growing body of literature suggests that heavy tailed distributions represent an adequate model for the observations of log returns of stocks. Motivated by these findings, here, we develop a discrete time framework for pricing of European options. Probability density functions of log returns for different periods are conveniently taken to be convolutions of the Student’s $t$-distribution with three degrees of freedom. The supports of these distributions are truncated in order to obtain finite values for the options. Within this framework, options with different strikes and maturities for one stock rely on a single parameter — the standard deviation of the Student’s $t$-distribution for unit period. We provide a study which shows that the distribution support width has weak influence on the option prices for certain range of values of the width. It is furthermore shown that such family of truncated distributions approximately satisfies the no-arbitrage principle and the put-call parity. The relevance of the pricing procedure is empirically verified by obtaining remarkably good match of the numerically computed values by our scheme to real market data.

We develop and study stability properties of a hybrid approximation of functionals of the Bates jump model with stochastic interest rate that uses a tree method in the direction of the volatility and the interest rate and a finite-difference approach in order to handle the underlying asset price process. We also propose hybrid simulations for the model, following a binomial tree in the direction of both the volatility and the interest rate, and a space-continuous approximation for the underlying asset price process coming from a Euler–Maruyama type scheme. We test our numerical schemes by computing European and American option prices.

We propose a new nonparametric technique to estimate the call function based on the superhedging principle. This approach requires minimal assumptions on absence of arbitrage and other market imperfections. The estimates so obtained are then combined with SNP estimates of the actual density of market returns. This permits to investigate the time behavior of the relative distance between the two densities obtained. Our empirical findings suggest that the more the two densities differ, the shorter is time to maturity, suggesting a major role of uncertainty over shorter than longer horizons.

We present an expansion for portfolio optimization in the presence of small, instantaneous, quadratic transaction costs. Specifically, the magnitude of transaction costs has a coefficient that is of the order $𝜖$ small, which leads to the optimization problem having an asymptotically-singular Hamilton–Jacobi–Bellman equation whose solution can be expanded in powers of $\sqrt{𝜖}$. In this paper, we derive explicit formulae for the first two terms of this expansion. Analysis and simulation are provided to show the behavior of this approximating solution.

Given an investment universe, we consider the vector $\rho (w)$ of correlations of all assets to a portfolio with weights $w$. This vector offers a representation equivalent to $w$ and leads to the notion of $\rho$-presentative portfolio, that has a positive correlation, or exposure, to all assets. This class encompasses well-known portfolios, and complements the notion of representative portfolio, that has positive amounts invested in all assets (e.g. the market-cap index). We then introduce the concept of maximally $\rho$-presentative portfolios, that maximize under no particular constraint an aggregate exposure $f(\rho (w))$ to all assets, as measured by some symmetric, increasing and concave real-valued function $f$. A basic characterization is established and it is shown that these portfolios are long-only, diversified and form a finite union of polytopes that satisfies a local regularity condition with respect to changes of the covariance matrix of the assets. Despite its small size, this set encompasses many well-known and possibly constrained long-only portfolios, bringing them together in a common framework. This also allowed us characterizing explicitly the impact of maximum weight constraints on the minimum variance portfolio. Finally, several theoretical and numerical applications illustrate our results.

We study the Markowitz portfolio selection problem with unknown drift vector in the multi-dimensional framework. The prior belief on the uncertain expected rate of return is modeled by an arbitrary probability law, and a Bayesian approach from filtering theory is used to learn the posterior distribution about the drift given the observed market data of the assets. The Bayesian Markowitz problem is then embedded into an auxiliary standard control problem that we characterize by a dynamic programming method and prove the existence and uniqueness of a smooth solution to the related semi-linear partial differential equation (PDE). The optimal Markowitz portfolio strategy is explicitly computed in the case of a Gaussian prior distribution. Finally, we measure the quantitative impact of learning, updating the strategy from observed data, compared to nonlearning, using a constant drift in an uncertain context, and analyze the sensitivity of the value of information with respect to various relevant parameters of our model.

We present a methodology to identify change-points in financial markets where the governing regime shifts from a constant rate-of-return, i.e. normal growth, to a superexponential growth described by a power-law hazard rate. The latter regime corresponds, in our view, to financial bubbles driven by herding behavior of market participants. Assuming that the time series of log-price returns of a financial index can be modeled by arithmetic Brownian motion, with an additional jump process with power-law hazard function to approximate the superexponential growth, we derive a threshold value of the hazard-function control parameter, allowing us to decide in which regime the market is more likely to be at any given time. An analysis of the Standard & Poors 500 index over the last 60 years provides evidence that the methodology has merit in identifying when a period of herding behavior begins, and, perhaps more importantly, when it ends.