Stochastic modelling in volatility and its applications in derivatives

Zou, Yihan (2020) Stochastic modelling in volatility and its applications in derivatives. PhD thesis, University of Glasgow.

Abstract

This thesis consists of three articles concentrating on modelling stochastic volatility in commodity as well as equity and applying stochastic volatility models to evaluate financial derivatives and real options. Firstly, we introduce the general background and the incentive of considering stochastic volatility models.

In Chapter 2 we derive tractable analytic solutions for futures and options prices for a linear-quadratic jump-diffusion model with seasonal adjustments in stochastic volatility and convenience yield. We then calibrate our model to data from the fish pool futures market, using the extended Kalman filter and a quasi-maximum likelihood estimator and alternatively using an implied-state quasi-maximum likelihood estimator. We find no statistical evidence of jumps. However, we do find evidence for the positive correlation between salmon spot prices and volatility, seasonality in volatility and convenience yield. In addition we observe a positive relationship between seasonal risk premium and uncertainty within the EU salmon demand. We further show that our model produces option prices that are conform with the observation of implied volatility smiles and skews.

In Chapter 3, we introduce a linear quadratic volatility model with co-jumps and show how to calibrate this model to a rich dataset. We apply general method of moments (GMM) and more specifically match the moments of realized power and multi-power variations, which are obtained from high-frequency stock market data. Our model incorporates two salient features: the setting of simultaneous jumps in both return process and volatility process and the superposition structure of a continuous linear quadratic volatility process and a Lévy-driven Ornstein-Uhlenbeck process. We compare the quality of fit for several mod- els, and show that our model outperforms the conventional jump diffusion or Bates model. Besides that, we find evidence that the jump sizes are not normal distributed and that our model performs best when the distribution of jump-sizes is only specified through certain (co-) moment conditions. A Monte Carlo experiments is employed to confirm this.

Finally, in Chapter 4, we study the optimal stopping problems in the context of American options with stochastic volatility models and the optimal fish harvesting decision with stochastic convenience yield models, in the presence of drift ambiguity. From the perspective of an ambiguity averse agent, we transfer the problem to the solution of a reflected backward stochastic differential equation (RBSDE) and prove the uniform Lipschitz continuity of the generator. We then propose a numerical algorithm with the theory of RBSDEs and a general stratification technique, and an alternative algorithm without using the theory of RBSDEs. We test the accuracy and convergence of the numerical schemes. By comparing to the one dimensional case, we highlight the importance of the dynamic structure of the agent’s worst case belief. Results also show that the numerical RBSDE algorithm with stratification is more efficient when the optimal generator is attainable.

• Item Type: Thesis (PhD)
• Qualification Level: Doctoral
• Subjects: H Social Sciences > HG Finance
• Colleges/Schools: College of Social Sciences > Adam Smith Business School > Economics
• Supervisor's Name: Ewald, Professor Christian and Agarwal, Dr. Ankush
• Date of Award: 2020
• Depositing User: Yihan Zou
• Unique ID: glathesis:2020-81658
• Copyright: Copyright of this thesis is held by the author.
• Date Deposited: 17 Sep 2020 08:42
• Last Modified: 10 Oct 2022 13:48
• Thesis DOI: 10.5525/gla.thesis.81658
• URI: https://theses.gla.ac.uk/id/eprint/81658