To celebrate International Womens' Day, and as part of the #QuantWomen series, we asked Laura Ballotta, Reader in Financial Mathematics as Cass Business School, to discuss her upcoming session at QuantMinds International. Inspired? Read more pieces from leading women working in quantitative finance here.
In this talk, we discuss how the performance of models for equities can be improved in terms of both calibration to option market data and quality of the hedge position. The proposed general model assumes that the risk drivers of equity log-returns and their volatility are captured by means of discrete movements of small size occurring with very high frequency. We note that movements of such small size could potentially be misidentified as diffusion, except for the more realistic levels of skewness and excess kurtosis which they can generate in the underlying distributions over both the short and long period. The resulting distributions prove to be much more agile and flexible in fitting observed volatility surfaces.
In more details, the proposed construction is based on time changed Lévy process for modelling the stochastic joint evolution of stock log-returns and their volatility. The processes are obtained by observing a Lévy process (i.e. a process with independent and stationary increments) on a time scale governed by a so-called business clock; thus uncertainty is subdivided into uncertainty of the time at which the next price change will occur, and uncertainty related to the magnitude of this change. The time change is a convenient tool to equip processes otherwise devoid of dynamics in their variance – such as Lévy processes - with stochastic volatility features. The proposed setting includes risk factors of both diffusive and jump nature, and leverage effects originated by both factors; thus it encompasses a large number of the most commonly used stochastic volatility models, allows for the construction of new potential alternative models, and enables a comparative study of their features in terms of volatility, volatility of volatility and correlation processes. Results point to the attractiveness of models composed exclusively of jumps for both the log-return and the volatility process.
The findings of this study stress the need for hedging and risk management strategies equipped to face not just (rare) crash risk alone, but also and most importantly risks associated with small and intermediate sized jumps. Consequently, we analyse the performance of standard Delta-hedging strategies based on the Delta originating out of our model. Like in the case of the more established Heston model, that turns out to be a particular case of our general setting, an analytical expression for the stock log-returns distribution is not available, this can be anyhow recovered numerically by means of the model characteristic function and well established efficient Fourier inversion techniques. This allows us to recover the model Delta and perform hedging. Indeed the Delta represents the probability (under a suitably specified probability measure) of options being in the money.
We test the proposed model on options on the S&P500 using as relevant base Lévy process the Carr Geman Madan Yor process in order to exploit the very rich structure offered by this parsimonious version of a two-sided tempered stable process. As the S&P500 index can be considered relatively immune to firm/sector specific crashes, it is reasonable to assume that it moves in a 'normal' way on a day-to-day basis most of time, offering therefore an ideal test for flexible distributions originated by highly frequent small discontinuous movements.
The results of our extensive analysis show that in capturing the behaviour of the S&P500 index options, models equipped with jumps outperform those based on pure diffusion. Amongst these models, the ones offering better performance for both in-sample and out-of-sample tests, present high frequency jumps of small magnitude; these results in particular illustrate the role of jumps in financial modelling: although in day-to-day market conditions jumps in the form of sudden large movements (especially drops) are not that frequent, small size jumps with high frequency are the key to generate realistic distributions with levels of skewness and excess kurtosis sufficiently close to reality. Finally, whilst diffusion-only models are unable to offer calibration quality at par with the other models considered in this talk, jump-only models represent valid parsimonious alternatives.