Option prices in stochastic volatility models

Abstract : We study option pricing problems in stochastic volatility models. In the first part of this thesis we focus on American options in the Heston model. We first give an analytical characterization of the value function of an American option as the unique solution of the associated (degenerate) parabolic obstacle problem. Our approach is based on variational inequalities in suitable weighted Sobolev spaces and extends recent results of Daskalopoulos and Feehan (2011, 2016) and Feehan and Pop (2015). We also investigate the properties of the American value function. In particular, we prove that, under suitable assumptions on the payoff, the value function is nondecreasing with respect to the volatility variable. Then, we focus on an American put option and we extend some results which are well known in the Black and Scholes world. In particular, we prove the strict convexity of the value function in the continuation region, some properties of the free boundary function, the Early Exercise Price formula and a weak form of the smooth fit principle. This is done mostly by using probabilistic techniques.In the second part we deal with the numerical computation of European and American option prices in jump-diffusion stochastic volatility models. We first focus on the Bates-Hull-White model, i.e. the Bates model with a stochastic interest rate. We consider a backward hybrid algorithm which uses a Markov chain approximation (in particular, a “multiple jumps” tree) in the direction of the volatility and the interest rate and a (deterministic) finite-difference approach in order to handle the underlying asset price process. Moreover, we provide a simulation scheme to be used for Monte Carlo evaluations. Numerical results show the reliability and the efficiency of the proposed methods.Finally, we analyze the rate of convergence of the hybrid algorithm applied to general jump-diffusion models. We study first order weak convergence of Markov chains to diffusions under quite general assumptions. Then, we prove the convergence of the algorithm, by studying the stability and the consistency of the hybrid scheme, in a sense that allows us to exploit the probabilistic features of the Markov chain approximation
Document type :
Theses
Complete list of metadatas

Cited literature [20 references]  Display  Hide  Download

https://hal.archives-ouvertes.fr/tel-01961071
Contributor : Abes Star <>
Submitted on : Thursday, May 23, 2019 - 2:24:07 PM
Last modification on : Friday, May 24, 2019 - 1:13:39 AM

File

TH2018PESC1132.pdf
Files produced by the author(s)

Identifiers

  • HAL Id : tel-01961071, version 2

Collections

Citation

Giulia Terenzi. Option prices in stochastic volatility models. Statistics [math.ST]. Université Paris-Est; Università degli studi di Roma "Tor Vergata", 2018. English. ⟨NNT : 2018PESC1132⟩. ⟨tel-01961071v2⟩

Share

Metrics

Record views

52

Files downloads

17