Simple Formulas to Option Pricing and Hedging in the Black- Scholes Model

For option whose striking price equals the forward price of the underlying asset, the Black-Scholes pricing formula can be approximated in closed-form. A interesting result is that the derived equation is not only very simple in structure but also that it can be immediately inverted to obtain an explicit formula for implied volatility. In this contribution we present and compare the accuracy of three of such approximation formulas. The numerical analysis shows that the first order approximations are close only for small maturities, Polya approximations are remarkably accurate for a very large range of parameters, while logistic values are the most accurate only for extreme maturities.Option pricing; hedging; Taylor, Polya and logistic approximations

An efficient binomial approach to the pricing of options on stocks with cash dividends

In this contribution, we consider options written on stocks which pay cash dividends. Dividend payments have an effect on the value of options: high dividends imply lower call premia and higher put premia. While exact solutions to problems of evaluating both European and American call options and European put options are available in the literature, 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 a shift in the tree; as a result, the tree is no longer reconnecting beyond any dividend date. Methods based on non-recombining trees give consistent results, but they are computationally expensive. We analyze binomial algorithms and performed some empirical experiments.Options on stocks, discrete dividends, binomial lattices

Simulation techniques for generalized Gaussian densities

This contribution deals with 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 also useful in modeling economic and financial data. For values of the shape parameter a within a certain range, the distribution presents heavy tails. In particular, the cases a=1/3 and a=1/2 are considered. For such values of the shape parameter, different simulation methods are assessed.Generalized Gaussian density, heavy tails, transformations of rendom variables, Monte Carlo simulation, Lambert W function

Cumulative prospect theory and second order stochastic dominance criteria: an application to mutual funds performance

In this note using the rules of stochastic dominance of the second order and the recent cumulative prospect theory for classified, according to their performance, a set of common funds. The criteria used are closely linked to the preferences of decision maker and refer to either hypothesis of aversion and of seeking to risk both hypothesis on the sign of derived second of the function which characterizes the losses and gains.

Indici di volatilità

Gli indici di volatilità sono strumenti finanziari innovativi che hanno come scopo principale la misurazione della volatilità implicita dei mercati a breve e medio termine. Il più noto e utilizzato è l’indice americano VIX, che viene divulgato in tempo reale dal CBOE e stima la volatilità a 30 giorni del famoso indice azionario S&P 500. Per il suo calcolo si considerano solo i prezzi di mercato di opzioni call e put out-of-the-money. Il valore dell’indice, pertanto, non solo risulta indipendente da ogni tipo di modello che può essere assunto per descrivere la dinamica dell’attività sottostante, ma consente anche di isolare la volatilità attesa dagli altri fattori che influenzano il prezzo delle opzioni quali i dividendi, i tassi di interesse e il tempo che manca alla scadenza. Il calcolo del VIX è basato su un’approssimazione discreta del valore teorico dei contratti di tipo variance swap e, in quanto tale, è inficiato da diversi errori che comportano delle implicazioni negative su numerosi strumenti finanziari negoziati sia sui mercati ufficiali che sui mercati OTC.Volatility indexes are innovative financial instruments whose main purpose is measuring implied volatility of the markets in the short and medium term. The best known and used one is the American VIX, which is published in real time by the CBOE and estimates the 30 days volatility of the famous S&P 500 stock index. For its calculation, only the market prices of out-of-the-money call and put options are considered. The value of the index, therefore, not only is independent of any type of model that can be used to describe the dynamics of the underlying asset, but also it allows to isolate the volatility effect from other factors influencing the price of options, such as dividends, interest rates and the time to maturity. The VIX calculation is based on a discrete approximation of the theoretical value of the variance swap contracts; such an approximation is affected by several errors that implies negative effects on several financial instruments traded both on official and OTC markets

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

Insurance premium calculation under continuous cumulative prospect theory

We define a premium principle under the continuous cumulative prospect theory which extends the equivalent utility principle. In prospect theory risk attitude and loss aversion are shaped via a value function, whereas a transformation of objective probabilities, which is commonly referred as probability weighting, models probabilistic risk perception. In cumulative prospect theory, probabilities of individual outcomes are replaced by decision weights, which are differences in transformed, through the weighting function, countercumulative probabilities of gains and cumulative probabilities of losses, with outcomes ordered from worst to best. 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 study some properties of the behavioral premium principle. We also assume an alternative framing of the outcomes; then we discuss several applications to the pricing of insurance contracts

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

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