Paolo Pianca

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.

Option pricing bounds with standard risk aversion preferences

Abstract For a theoretical valuation of a financial option, various models have been proposed that require specific hypotheses regarding both the stochastic process driving the price behaviour of the underlying security and market efficiency. When some of these assumptions are removed, we obtain an uncertainty interval for the option price. Up to now, the most restrictive intervals for option prices have been obtained using the decreasing absolute risk aversion (DARA) rule in a state-preference approach. Precautionary saving entails the concept of prudence; in particular, decreasing absolute prudence is a necessary and sufficient condition that guarantees that the saving of wealthier people is less sensitive to the risk associated to future incomes. If this condition is coupled with the DARA assumption we obtain standard risk aversion (SRA), which guarantees on the one hand that introducing a zero-mean background risk to wealth makes people less willing to accept another independent risk and on the other hand that an increase in the risk of the returns distribution of an asset reduces the demand for this asset. The main idea of this contribution is to apply decreasing absolute prudence and SRA rules in a state-preference context in order to obtain efficient bounds for the value of European-style options portfolio strategies. Lower and upper bounds for the options portfolio value are obtained by solving non-linear optimization problems. The numerical experiments carried out show the efficiency of the technique proposed.

On the relative efficiency of nth order and DARA stochastic dominance rules

It is known that third order stochastic dominance implies DARA dominance while no implications exist between higher orders and DARA dominance. A recent contribution points out that, with regard to the problem of determining lower and upper bounds for the price of a financial option, the DARA rule turns out to improve the stochastic dominance criteria of any order. In this paper the relative efficiency of the ordinary stochastic dominance and DARA criteria for alternatives with discrete distributions are compared, in order to see if the better performance of DARA criterion is also suitable for other practical applications. Moreover, the operational use of the stochastic dominance techniques for financial choices is deepened.

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 more informative estimation procedure for the parameters of a diffusion process

The estimation procedures for the parameters of a diffusion process with constant coefficients have mainly focused on volatility. Nevertheless, even if the knowledge of the volatility alone suffices to compute the Black and Scholes option prices, other financial application models assume that the price dynamics follows a log-normal process and requires the knowledge of both parameters. On the other hand, while the usual ML estimator of volatility gives satisfactory results, the estimation of drift is much less accurate; moreover, the drift-estimated value highly depends on the phases of the business cycle included in the sample data. This contribution explicitly imposes a risk aversion or risk neutral assumption into the ML estimation procedure and makes a constrained maximization of the sample likelihood function. The aim is twofold: to obtain estimated values which are consistent with a widely accepted assumption and use the risk aversion constraint in order to improve the accuracy of the estimates.

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.