An analysis between implied and realised volatility in the Greek Derivatives Market

In this article, we examine the relationship between implied and realised volatility in the Greek derivative market. We examine the differences between realised volatility and implied volatility of call and put options for at-the-money index options with a two-month expiration period. The findings provide evidence that implied volatility is not an efficient estimate of realised volatility. Implied volatility creates overpricing, for both call and put options, in the Greek market. This is an indication of inefficiency for the market. In addition, we find evidence that realised volatility ‘Granger causes’ implied volatility for call options, and implied volatility of call options ‘Granger causes’, the implied volatility of put option

Adiabaticity Conditions for Volatility Smile in Black-Scholes Pricing Model

Our derivation of the distribution function for future returns is based on the risk neutral approach which gives a functional dependence for the European call (put) option price, C(K), given the strike price, K, and the distribution function of the returns. We derive this distribution function using for C(K) a Black-Scholes (BS) expression with volatility in the form of a volatility smile. We show that this approach based on a volatility smile leads to relative minima for the distribution function ("bad" probabilities) never observed in real data and, in the worst cases, negative probabilities. We show that these undesirable effects can be eliminated by requiring "adiabatic" conditions on the volatility smile

Robust Estimation of Shape-Constrained State Price Density Surfaces

Given a theoretical pricing model, an implied volatility can be extracted from an option’s market price. Given a set of options with the same maturity and a range of strike prices, it is possible to extract (an approximation to) the entire risk-neutral probability density without having to assume a theoretical pricing model. There are a variety of related methods to do this, but all are subject to certain problems, including the fact that the data never exist to allow full estimation of the tails. Some methods produce improper densities with negative portions. In this article, Ludwig introduces a neural network approach to extract risk-neutral densities from option prices, imposing only a small number of constraints, such as probabilities must be nonnegative and an option’s price must be above intrinsic value. The resulting densities are smooth and sensible, even for days that other approaches find extremely difficult to handle

Sequential Quadratic Programming Method for Volatility Estimation in Option Pricing

Our goal is to identify the volatility function in Dupires equation from given option prices. Following an optimal control approach in a Lagrangian framework, a globalized sequential quadratic programming (SQP) algorithm combined with a primal-dual active set strategy is proposed. Existence of local optimal solutions and of Lagrange multipliers is shown. Furthermore, a sufficient second-order optimality condition is proved. Finally, some numerical results are presented underlining the good properties of the numerical scheme