Numerical Algorithm For Multi-Parameter Calibration Prblems Based On Heston Model

Since the advent of the Black-Scholes option pricing model,the calibration of the volatility parameters in the option pricing model has been a research hotspot.The Heston model improves the Black-Scholes model based on the characteristics of implied volatility in the market.Due to the non-linearity of Heston option pricing model and the complexity of the data,the volatility parameter calibration problem is an ”ill-posed” inverse problem,which is challenging for our study.This dissertation discusses the multi-parameter calibration problem based on the Heston model of stochastic volatility option pricing,presents the mathematical description of the multi-parameter calibration problem,and proposes a corresponding calibration algorithm.This dissertation consists of six chapters:Chapter 1 introduces the research background and significance of parameter calibration problem based on option pricing models,the recent research progress on option pricing models and model parameter calibration problems at home and abroad.It also briefly describes some preliminary knowledge about option pricing models and the main research contents and resultes of this dissertation.Chapter 2 introduces the Heston stochastic volatility model(direct problem)including its derivation and solution process.Aiming at the difficulty of calculating numerical integration,an effective numerical method is proposed based on the Gauss-Legendre quadrature and GPU parallel computing technology in this dissertation.Chapter 3 discusses the multi-parameter calibration problem(inverse problem)based on the Heston model.Motivated by Tikhonov regularization,the error norms ASPE and RSPE in two regularization functionals are chosen.In order to unify the dimension,the dimensionless treatment is carried out on the regularization terms.The L-Curve method and Morozov's discrepancy principle are appied respectively to select satisfied regularization parameters.The analytic gradient of the regularized functional is derived,and then substituted into the trust region-interior point optimization algorithm.These aspects contribute the multi-parameter calibration algorithm.Chapter 4 conducts numerical experiments on noisy option price data to verify the effectiveness of the multi-parameter calibration numerical algorithm proposed in Chapter 3.Chapter 5 further validates the practicality of the parameter calibration algorithm based on the option market data of the US S & P500,and shows that proposed numerical algorithm can be applied to make predictions with high accuracy.Chapter 6 summarizes the main content and innovations of this dissertation,and points out somen valuable research topics in the future.