Efficient simulation of clustering jumps with CIR intensity

We introduce a broad family of generalised self-exciting point processes with CIR-type intensities, and develop associated algorithms for their exact simulation. The underlying models are extensions of the classical Hawkes process, which already has numerous applications in modelling the arrival of events with clustering or contagion effect in finance, economics and many other fields. Interestingly, we find that the CIR-type intensity together with its point process can be sequentially decomposed into simple random variables, which immediately leads to a very efficient simulation scheme. Our algorithms are also pretty accurate and flexible. They can be easily extended to further incorporate externally-excited jumps, or, to a multidimensional framework. Some typical numerical examples and comparisons with other well known schemes are reported in detail. In addition, a simple application for modelling a portfolio loss process is presented

A dynamic contagion process for modelling contagion risk in finance and insurance

We introduce a new point process, the dynamic contagion process, by generalising the Hawkes process and Cox process with shot noise intensity. Our process includes both self-excited and externally excited jumps, which could be used to model the dynamics of contagion impact from endogenous and exogenous factors of the underlying system. We systematically analyse the theoretical distributional properties of this new process, based on the piecewise-deterministic Markov process theory developed in Davis (1984), and the extension of the martingale methodology used in Dassios and Embrechts (1989). The analytic expressions of the Laplace transform of the intensity process and probability generating function of the point process are derived. A simulation algorithm is provided for further industrial implementation and statistical analysis. Some extensions of this process and comparison with other similar processes are also investigated. The major object of this study is to produce a general mathematical framework for modelling the dependence structure of arriving events with dynamic contagion, which has the potential to be applicable to a variety of problems in economics, finance and insurance. We apply our research to the default probability of credit risk and ruin probability of risk theory

Stochastic volatility models in financial econometrics: an application to South Africa

Thesis (Ph.D.)--University of the Witwatersrand, Faculty of Commerce, Law & Management, School of Economic and Business Sciences, 2015.The dissertation carries out a study to understand asset price behaviour in South Africa. This is investigated through the application of stochastic volatility models to trace the characteristics of high frequency financial data; daily temperature, exchange rates, interest rates, stock and house prices. Innovation in the derivatives market has seen the introduction of weather derivatives as a risk mitigation tool against adverse weather movements. Chapter Two applies three different time series models of temperature to estimate payoffs to determine which method offers the best hedging strategy in four South African cities. Results from the study suggest that the seasonality GARCH method of estimating payoffs for temperature based weather derivatives offers superior performance compared to the Cumulative Cooling Degree Days (CDD) and the historical method. This suggests that the seasonality GARCH method can be applied in these cities to hedge against adverse temperature movements. In Chapter three we consider the estimation methodology for jump diffusion models and GARCH models. Chapter four investigates volatility on exchange rate data. Use is made of the british pound/south african rand, euro/south african rand and u.s dollar/ south african rand exchange rates. The research introduces a jump diffusion model to trace the behaviour of exchange rate data. Estimation results are able to match the summary statistics in mean, variance, skewness and kurtosis. Results from the model can also explain the volatility smile for short and medium term maturities. A fat tailed GARCH model is introduced to capture the persistence in volatility on exchange rate data. Results from this chapter have an implication for pricing currency options to offer leverage to organisations affected by exchange rate risk. Chapter five extends the analysis to study the behaviour of short term interest rates, making use of the 90 Day Treasury bill (T-Bill) rate. The chapter considers a variant application of the Chan et al. (1992) model for short term interest rates wherein a jump diffusion model is introduced. The results match the summary statistics equivalent suggesting the capability of the model specification. Splitting the estimation period suggests that the jump size is highest post inflation target though with a smaller intensity. However, the 90 Day T-Bill shows higher volatility after inflation targeting though with a lesser intensity. These findings have a bearing on valuation of short term interest derivatives and also investigating multi factor models of interest rates. In chapter six four vi sectors (banking, mining, media and leisure) are considered to explore movements in stock prices. A jump diffusion model is applied to get estimation results. The results confirm related studies that stock prices have incidents of volatility which can be captured by a jump diffusion model. The results also shed light on the importance of portfolio diversification considering the different results across the sectors investigated. The implication also lies in understanding market efficiency. Chapter seven applies the jump diffusion model on house prices to understand more on the drivers of volatility on house prices. The interesting results on this chapter can be summarised as follows; the four different house segments have almost similar jump sizes though the small house price segment has highest intensity. This can point to expectations and volatility from participants in this segment at a higher level than for other segments over different regimes over the study period. The estimated higher moments were not normalised as had happened for the three previous chapters after introducing the jump diffusion model. Results from this chapter have an application to valuing mortgage premium across different house price segments. It is recommended that rigorous research on asset prices using various approaches be considered as it goes a long way in informing policy makers and investors to mitigate risk in an environment of volatile asset prices. With the growing interest in weather derivatives world-wide, there is a need to educate farmers, government entities, potential counter-parties and other organisations affected by weather related risk on the importance of weather derivatives so that a foundation is laid for trading in this special type of insurance

Exact simulation of Ornstein-Uhlenbeck tempered stable processes

There are two types of tempered stable (TS) based Ornstein–Uhlenbeck (OU) processes: (i) the OU-TS process, the OU process driven by a TS subordinator, and (ii) the TS-OU process, the OU process with TS marginal law. They have various applications in financial engineering and econometrics. In the literature, only the second type under the stationary assumption has an exact simulation algorithm. In this paper we develop a unified approach to exactly simulate both types without the stationary assumption. It is mainly based on the distributional decomposition of stochastic processes with the aid of an acceptance–rejection scheme. As the inverse Gaussian distribution is an important special case of TS distribution, we also provide tailored algorithms for the corresponding OU processes. Numerical experiments and tests are reported to demonstrate the accuracy and effectiveness of our algorithms, and some further extensions are also discussed