DOI

10.17077/etd.tkj2s3ik

Document Type

Dissertation

Date of Degree

Fall 2016

Degree Name

PhD (Doctor of Philosophy)

Degree In

Applied Mathematical and Computational Sciences

First Advisor

Jorgensen, Palle

First Committee Member

Li, Tong

Second Committee Member

Strohmer, Gerhard

Third Committee Member

Stroyan, Keith

Fourth Committee Member

Ayati, Bruce

Abstract

Derivative pricing, model calibration, and sensitivity analysis are the three main problems in financial modeling. The purpose of this study is to present an algorithm to improve the pricing process, the calibration process, and the sensitivity analysis of the double Heston model, in the sense of accuracy and efficiency. Using the optimized caching technique, our study reduces the pricing computation time by about 15%. Another contribution of this thesis is: a novel application of the Automatic Differentiation (AD) algorithms in order to achieve a more stable, more accurate, and faster sensitivity analysis for the double Heston model (compared to the classical finite difference methods). This thesis also presents a novel hybrid model by combing the heuristic method Differentiation Evolution, and the gradient method Levenberg--Marquardt algorithm. Our new hybrid model significantly accelerates the calibration process.

Public Abstract

Financial institutions, like banks and insurance companies, are responsible for carefully managing the huge amount of assets collected from different sources (e.g. investors and the government). Although different financial institutions have various investing strategies, they are all facing the risk coming from many unpredictable future events, like the financial crisis. The financial derivatives, serve as insurance for financial institutions, and provide the practical tools to hedge many main kinds of risks. Therefore, the financial derivatives should not be free (just like you should pay the health insurance before having it). To price the financial derivative is not easy, and thus the stochastic calculus as the ideal tool comes in. The double Heston model is one of those well–known practical pricing models which computes the price of many derivatives including European options. This thesis implements the double Heston model in an optimized way which reduces the pricing time by about 15%. The sensitivity analysis of the pricing model is also very important since it provides us the window to mathematically observe the risk. Another contribution of this thesis is: a novel application of the automatic algorithmic differentiation algorithm in order to achieve a more stable, more accurate, and faster sensitivity analysis for the double Heston model (compared to the classical finite difference methods). Lastly, our study presents a novel hybrid calibration method, which mixes the Differentiation Evolution algorithm and the Levenberg–Marquardt algorithm. Our new hybrid model significantly accelerates the calibration process.

Keywords

automatic differentiation, calibration, derivative pricing, gradient-based method, heuristic optimization method, stochastic volatility

Pages

viii, 110 pages

Bibliography

Includes bibliographical references (pages 108-110).

Copyright

Copyright © 2016 Ze Zhao

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