2 research outputs found
Stochastic volatility models: calibration, pricing and hedging
Stochastic volatility models have long provided a popular alternative to the Black-
Scholes-Merton framework. They provide, in a self-consistent way, an explanation
for the presence of implied volatility smiles/skews seen in practice. Incorporating
jumps into the stochastic volatility framework gives further freedom to nancial
mathematicians to t both the short and long end of the implied volatility surface.
We present three stochastic volatility models here - the Heston model, the Bates
model and the SVJJ model. The latter two models incorporate jumps in the stock
price process and, in the case of the SVJJ model, jumps in the volatility process. We
analyse the e ects that the di erent model parameters have on the implied volatility
surface as well as the returns distribution. We also present pricing techniques for
determining vanilla European option prices under the dynamics of the three models.
These include the fast Fourier transform (FFT) framework of Carr and Madan as
well as two Monte Carlo pricing methods. Making use of the FFT pricing framework,
we present calibration techniques for tting the models to option data. Speci cally,
we examine the use of the genetic algorithm, adaptive simulated annealing and a
MATLAB optimisation routine for tting the models to option data via a leastsquares
calibration routine. We favour the genetic algorithm and make use of it in
tting the three models to ALSI and S&P 500 option data. The last section of the
dissertation provides hedging techniques for the models via the calculation of option
price sensitivities. We nd that a delta, vega and gamma hedging scheme provides
the best results for the Heston model. The inclusion of jumps in the stock price and
volatility processes, however, worsens the performance of this scheme. MATLAB
code for some of the routines implemented is provided in the appendix
Change point detection in dynamic Gaussian graphical models: the impact of COVID-19 pandemic on the US stock market
Reliable estimates of volatility and correlation are fundamental in economics
and finance for understanding the impact of macroeconomics events on the market
and guiding future investments and policies. Dependence across financial
returns is likely to be subject to sudden structural changes, especially in
correspondence with major global events, such as the COVID-19 pandemic. In this
work, we are interested in capturing abrupt changes over time in the dependence
across US industry stock portfolios, over a time horizon that covers the
COVID-19 pandemic. The selected stocks give a comprehensive picture of the US
stock market. To this end, we develop a Bayesian multivariate stochastic
volatility model based on a time-varying sequence of graphs capturing the
evolution of the dependence structure. The model builds on the Gaussian
graphical models and the random change points literature. In particular, we
treat the number, the position of change points, and the graphs as object of
posterior inference, allowing for sparsity in graph recovery and change point
detection. The high dimension of the parameter space poses complex
computational challenges. However, the model admits a hidden Markov model
formulation. This leads to the development of an efficient computational
strategy, based on a combination of sequential Monte-Carlo and Markov chain
Monte-Carlo techniques. Model and computational development are widely
applicable, beyond the scope of the application of interest in this work