Pricing and Hedging Index Options with a Dominant Constituent Stock

In this paper, we examine the pricing and hedging of an index option where one constituents stock plays an overly dominant role in the index. Under a Geometric Brownian Motion assumption we compare the distribution of the relative value of the index if the dominant stock is modeled separately from the rest of the index, or not. The former is equivalent to the relative index value being distributed as the sum of two lognormal random variables and the latter is distributed as a single lognormal random variable. Since these are not equal in distribution, we compare the two models. The validity of this theoretical result is verified against empirical stock market data. We look at two main models representing these cases: first, we use numerical methods to solve the two-dimensional problem directly; second, we make simplifying assumptions to reduce the two-dimensional Black-Scholes problem to a one-dimensional Black-Scholes problem that can be solved analytically. Since the terminal conditions of an option are usually non-smooth the numerical methods are verified by comparison to a Monte Carlo simulated solution. Attributes of the models that we compare are the relative option price differences and expected hedging profits. We compare the models for various volatilities, dominance levels, correlations and risk free rates. This work is significant in options trading because when a stock becomes dominant in its index the distribution of the returns changes. Even if the effect is small, given the millions of dollars exposed to index option trades, it has a material impact

Extracting expectations from currency option prices: a comparison of methods

This paper compares the goodness-of-fit and the stability of six methods used to extract risk-neutral probability density functions from currency option prices. We first compare five existing methods commonly employed to recover risk-neutral density functions from option prices. Specifically, we compare the methods introduced by Shimko (1993), Madan and Milne (1994), Malz (1996), Melick and Thomas (1997) and Bliss and Panigirtzoglou (2002). In addition, we propose a new method based on the piecewise cubic Hermite interpolation of the implied volatility function. We use data on 12 emerging market currencies against the US dollar and find that the piecewise cubic Hermite interpolation method is by far the method with the best accuracy in fitting observed option prices. We also find that there is a relative tradeoff between the goodness-of-fit and the stability of the methods. Thus, methods which have a better accuracy in fitting observed option prices appear to be more sensitive to option pricing errors, while the most stable methods have a fairly disappointing fitting. However, for the first two PDF moments as well as the quartiles of the risk-neutral distributions we find that the estimates do not differ significantly across methods. This suggests that there is a large scope for selection between these methods without essentially sacrificing the accuracy of the analysis. Nonetheless, depending on the particular use of these PDFs, some methods may be more suitable than othersRisk-neutral probability density functions, option pricing, exchange rate expectations

Log-skew-normality of ocean turbulence

The statistics of intermittent ocean turbulence is the key link between physical understanding of turbulence and its global implications. The log-normal distribution is the standard but imperfect assumed distribution for the turbulent kinetic energy dissipation rate. We argue that as turbulence is often generated by multiple changing sources, a log-skew-normal (LSN) distribution is more appropriate. We show the LSN distribution agrees excellently and robustly with observations. The heavy tail of the LSN distribution has important implications for sampling of turbulence in terrestrial and extraterrestrial analogous systems

Density forecasting in financial risk modelling

As a result of an increasingly stringent regulation aimed at monitoring financial risk exposures, nowadays the risk measurement systems play a crucial role in all banks. In this thesis we tackle a variety of problems, related to density forecasting, which are fundamental to market risk managers. The computation of risk measures (e.g. Value-at-Risk) for any portfolio of financial assets requires the generation of density forecasts for the driving risk factors. Appropriate testing procedures must then be identified for an accurate appraisal of these forecasts. We start our research by assessing whether option-implied densities, which constitute the most obvious forecasts of the distribution of the underlying asset at expiry, do actually represent unbiased forecasts. We first extract densities from options on currency and equity index futures, by means of both traditional and original specifications. We then appraise them, via rigorous density forecast evaluation tools, and we find evidence of the presence of biases. In the second part of the thesis, we focus on modelling the dynamics of the volatility curve, in order to measure the vega risk exposure for various delta-hedged option portfolios. We propose to use a linear Kalman filter approach, which gives more precise forecasts of the vega risk exposure than alternative, well-established models. In the third part, we derive a continuous time model for the dynamics of equity index returns from a data set of 5-minute returns. A model inferred from high-frequency typical of risk measures calculations. The last part of our work deals with evaluating density forecasts of the joint distribution of the risk factors. We find that, given certain specifications for the multivariate density forecast, a goodness-of-fit procedure based on the Empirical Characteristic Function displays good statistical properties in detecting misspecifications of different nature in the forecasts

Stochastic Volatility Effects on Correlated Log-Normal Random Variables

The transition density function plays an important role in understanding and explaining the dynamics of the stochastic process. In this paper, we incorporate an ergodic process displaying fast moving fluctuation into constant volatility models to express volatility clustering over time. We obtain an analytic approximation of the transition density function under our stochastic process model. Using perturbation theory based on Lie–Trotter operator splitting method, we compute the leading-order term and the first-order correction term and then present the left and right skew scenarios through numerical study