Martina Nardon

Simulation techniques for generalized Gaussian densities

This contribution deals with the Monte Carlo simulation of generalized Gaussian random variables. Such a parametric family of distributions has been proposed in many applications in science to describe physical phenomena and in engineering, and it seems to be also useful in modelling economic and financial data. For values of the shape parameter α within a certain range, the distribution presents heavy tails. In particular, the cases α=1/3 and α=1/2 are considered. For such values of the shape parameter, different simulation methods are assessed.

A Behavioural Approach to the Pricing of European Options

Empirical studies on quoted options highlight deviations from the theoretical model of Black and Scholes; this is due to different causes, such as assumptions regarding the price dynamics, markets frictions and investors’ attitude toward risk. In this contribution, we focus on this latter issue and study how to value options within the continuous cumulative prospect theory. According to prospect theory, individuals do not always take their decisions consistently with the maximization of expected utility. Decision makers have biased probability estimates; they tend to underweight high probabilities and overweight low probabilities. Risk attitude, loss aversion and subjective probabilities are described by two functions: a value function and a weighting function, respectively. As in Versluis et al. [15], we evaluate European options; we consider the pricing problem both from the writer’s and holder’s perspective, and extend the model to the put option. We also use alternative probability weighting functions.

Binomial algorithms for the evaluation of options on stocks with fixed per share dividends

We consider options written on assets which pay cash dividends. Dividend payments have an effect on the value of options: high dividends imply lower call premia and higher put premia. Recently, Haug et al. [13] derived an integral representation formula that can be considered the exact solution to problems of evaluating both European and American call options and European put options. For American-style put options, early exercise may be optimal at any time prior to expiration, even in the absence of dividends. In this case, numerical techniques, such as lattice approaches, are required. Discrete dividends produce discrete shift in the tree; as a result, the tree is no longer reconnecting beyond any dividend date. While methods based on non-recombining trees give consistent results, they are computationally expensive. In this contribution, we analyse binomial algorithms for the evaluation of options written on stocks which pay discrete dividends and perform some empirical experiments, comparing the results in terms of accuracy and speed.

Simulating a Generalized Gaussian Noise with Shape Parameter 1/2

This contribution deals with Monte Carlo simulation of generalized Gaussian random variables. Such a parametric family of distributions has been proposed for many applications in science and engineering to describe physical phenomena. Its use also seems interesting in modeling economic and financial data. For low values of the shape parameter α, the distribution presents heavy tails. In particular, α = 1/2 is considered and for such a value of the shape parameter, different simulation methods are assessed.

A Note on the Shape of the Probability Weighting Function

The focus of this contribution is on the transformation of objective probability, which in Prospect Theory is commonly referred as probability weighting. Empirical evidence suggests a typical inverse-S shaped function: decision makers tend to overweight small probabilities, and underweight medium and high probabilities; moreover, the probability weighting function is initially concave and then convex. We apply different parametric weighting functions proposed in the literature to the evaluation of derivative contracts and to insurance premium principles.

European option pricing under cumulative prospect theory with constant relative sensitivity probability weighting functions

In this contribution, we evaluate European financial options under continuous cumulative prospect theory. In prospect theory, risk attitude and loss aversion are shaped via a value function, while a probability weighting function models probabilistic risk perception. We focus on investors’ probability risk attitudes, as probability weighting may be one of the possible causes of the differences between empirically observed options prices and theoretical prices obtained with the Black and Scholes formula. We consider alternative probability weighting functions; in particular, we adopt the constant relative sensitivity weighting function, whose parameters have a direct interpretation in terms of curvature and elevation. Curvature models optimism and pessimism when one moves from extreme probabilities, whereas elevation can be interpreted as a measure of relative optimism. We performed a variety of numerical experiments and studied the effects of these features on options prices and implied volatilities.

Covered Call Writing and Framing: A Cumulative Prospect Theory Approach

The covered call writing, which entails selling a call option on one’s underlying stock holdings, is perceived by investors as a strategy with limited risk. It is a very popular strategy used by individual, professional and institutional investors. Previous studies analyze behavioral aspects of the covered call strategy, indicating that hedonic framing and risk aversion may explain the preference of such a strategy with respect to other designs. In this contribution, following this line of research, we extend the analysis and apply Cumulative Prospect Theory in its continuous version to the evaluation of the covered call strategy and study the effects of alternative framing.

Covered call writing in a cumulative prospect theory framework

The covered call writing, which entails selling a call option on one’s underlying stock holdings, is perceived by investors as a strategy with limited risk. It is a very popular strategy used by individual, professional and institutional investors; moreover, the CBOE developed the Buy Write Index (BXM) which tracks the performance of a synthetic covered call strategy on the S&P500 Index. Previous studies analyze behavioral aspects of the covered call strategy, indicating that hedonic framing and risk aversion may explain the preference of such a strategy with respect to other designs. In this contribution, following this line of research, we extend the analysis and apply Cumulative Prospect Theory in its continuous version to the evaluation of the covered call strategy and study the effects of alternative framing.

Probability Weighting Functions

Cumulative prospect theory (CPT) has been proposed as an alternative to expected utility theory to explain irregular behavior by economic agents. CPT comprises two key transformations: one of outcome values and the other of objective probabilities. Risk attitudes are derived from the shapes of these transformations as well as their interaction. The focus of this contribution is on the transformation of objective probability, which is commonly referred as probability weighting function. We review different families of weighting functions proposed in the literature and study their features.

European Option Pricing with Constant Relative Sensitivity Probability Weighting Function

We evaluate European financial options under continuous cumulative prospect theory. Within this framework, it is possible to model investors� attitude toward risk, which may be one of the possible causes of mispricing. We focus on probability risk attitudes and consider alternative probability weighting functions. In particular, curvature of the weighting function models optimism and pessimism when one moves from extreme probabilities, whereas elevation can be interpreted as a measure of relative optimism. The constant relative sensitivity weighting function is the only one, amongst those in the literature, which is able to model separately curvature and elevation. We are interested in studying the effects of both these features on options prices.