{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/113139"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/113139","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Fractional stochastic volatility models: approximation, calibration and hedging","abstract":"The area of modeling stochastic volatility using continuous time models has a long history and is always an interesting and vibrant area in financial mathematics, where the dynamic of the asset is a diffusion driven by Brownian motion and the dynamic of the volatility is associated with a diffusion driven also by another Brownian motion, instead of a fixed constant as in Black-Scholes model. However recent works have pointed out that there are some observations that the semimartinagales or Markovian models cannot explain, for example volatility persistence or the roughness of the sample paths of volatilities. And that is when the fractional stochastic volatility models are introduced. Our work about fractional stochastic volatility models mainly covers three different directions: hedging, calibration and approximation. In chapter 3, we propose a delta-hedging strategy for a long memory stochastic volatility model (LMSV). This is a model in which the volatility is driven by a fractional Ornstein-Uhlenbeck process with Hurst Index H greater than 1/2. We need to notice that perfect hedging cannot be achieved in non-Markovian fractional volatility framework, and thus we can only study imperfect delta-hedging strategy. We first proved the existence of this strategy by establishing the differentiability of the option price with respect to the underlying asset price. We also compute the so-called hedging bias, i.e. the difference between the Black-Scholes Delta and the LMSV Delta, and we determine when a European-type option is over-hedged or under-hedged. In chapter 4, to address the concern whether roughness can only be observed in the high frequency data, we first propose a new volatility proxy framework using low frequency cumulative option trading entries. Upon that framework we estimate the Hurst Index of the volatility and verify that the volatility exhibits very rough behavior even in the low frequency (daily) settings, which corroborates that roughness should always be taken into consideration when modeling the volatility for option pricing and calibration. In the case of pricing option in the long run we observe that the hurst number varies largely, suggesting the hurst number is a local parameter and is not appropriate to be taken as a constant in the long setting. In chapter 5, we consider the volatility model driven by fractional Ornstein-Unlenbeck process, with Hurst index H less than 1/2. We propose a point process approximation scheme to both the volatility process and the asset price process, and we first prove a Donsker type theorem for the convergence to fractional Ornstein-Uhlenbeck process. With that theorem we prove the weak convergence of the scheme to the log-price process. We also take into account the correlation between the volatility and the stock process, which is also termed as leverage effect. This work will be conducted with respect to Skorohod topology and we will not have interpolating term. The result can be generalized to all point process with jumps consisting of bounded random variables.","abstract_html":"The area of modeling stochastic volatility using continuous time models has a long history and is always an interesting and vibrant area in financial mathematics, where the dynamic of the asset is a diffusion driven by Brownian motion and the dynamic of the volatility is associated with a diffusion driven also by another Brownian motion, instead of a fixed constant as in Black-Scholes model. However recent works have pointed out that there are some observations that the semimartinagales or Markovian models cannot explain, for example volatility persistence or the roughness of the sample paths of volatilities. And that is when the fractional stochastic volatility models are introduced. Our work about fractional stochastic volatility models mainly covers three different directions: hedging, calibration and approximation. In chapter 3, we propose a delta-hedging strategy for a long memory stochastic volatility model (LMSV). This is a model in which the volatility is driven by a fractional Ornstein-Uhlenbeck process with Hurst Index H greater than 1/2. We need to notice that perfect hedging cannot be achieved in non-Markovian fractional volatility framework, and thus we can only study imperfect delta-hedging strategy. We first proved the existence of this strategy by establishing the differentiability of the option price with respect to the underlying asset price. We also compute the so-called hedging bias, i.e. the difference between the Black-Scholes Delta and the LMSV Delta, and we determine when a European-type option is over-hedged or under-hedged. In chapter 4, to address the concern whether roughness can only be observed in the high frequency data, we first propose a new volatility proxy framework using low frequency cumulative option trading entries. Upon that framework we estimate the Hurst Index of the volatility and verify that the volatility exhibits very rough behavior even in the low frequency (daily) settings, which corroborates that roughness should always be taken into consideration when modeling the volatility for option pricing and calibration. In the case of pricing option in the long run we observe that the hurst number varies largely, suggesting the hurst number is a local parameter and is not appropriate to be taken as a constant in the long setting. In chapter 5, we consider the volatility model driven by fractional Ornstein-Unlenbeck process, with Hurst index H less than 1/2. We propose a point process approximation scheme to both the volatility process and the asset price process, and we first prove a Donsker type theorem for the convergence to fractional Ornstein-Uhlenbeck process. With that theorem we prove the weak convergence of the scheme to the log-price process. We also take into account the correlation between the volatility and the stock process, which is also termed as leverage effect. This work will be conducted with respect to Skorohod topology and we will not have interpolating term. The result can be generalized to all point process with jumps consisting of bounded random variables.","abstract_has_math":false,"creators":["Zhao, Qi"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Industrial Engineering","degree_department":null,"school":null,"contributors":["Chronopoulou, Alexandra","Feng, Liming","Sreenivas, Ramavarapu","Sirignano, Justin"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2022,"date_issued":"2022-01-12T22:34:51Z","date_published":"2022-01-12T22:34:51Z","updated_at":"2026-07-22T22:24:53Z","subjects":["Fractional Stochastic Volatility","Hedging","Calibration","Weak Convergence"],"languages":["en"],"rights":["Copyright 2021 Qi Zhao"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/113139","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Chronopoulou, Alexandra","Feng, Liming","Sreenivas, Ramavarapu","Sirignano, Justin"]},{"key":"dc:creator","label":"Author","values":["Zhao, Qi"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2022-01-12T22:34:51Z","2024-01-12T22:35:30Z","2021-07-11","2021-08"]},{"key":"dc:type","label":"Dc Type","values":["text","Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Industrial Engineering"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Fractional Stochastic Volatility","Hedging","Calibration","Weak Convergence"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2021 Qi Zhao"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/113139"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["The area of modeling stochastic volatility using continuous time models has a long history and is always an interesting and vibrant area in financial mathematics, where the dynamic of the asset is a diffusion driven by Brownian motion and the dynamic of the volatility is associated with a diffusion driven also by another Brownian motion, instead of a fixed constant as in Black-Scholes model. However recent works have pointed out that there are some observations that the semimartinagales or Markovian models cannot explain, for example volatility persistence or the roughness of the sample paths of volatilities. And that is when the fractional stochastic volatility models are introduced. Our work about fractional stochastic volatility models mainly covers three different directions: hedging, calibration and approximation. In chapter 3, we propose a delta-hedging strategy for a long memory stochastic volatility model (LMSV). This is a model in which the volatility is driven by a fractional Ornstein-Uhlenbeck process with Hurst Index H greater than 1/2. We need to notice that perfect hedging cannot be achieved in non-Markovian fractional volatility framework, and thus we can only study imperfect delta-hedging strategy. We first proved the existence of this strategy by establishing the differentiability of the option price with respect to the underlying asset price. We also compute the so-called hedging bias, i.e. the difference between the Black-Scholes Delta and the LMSV Delta, and we determine when a European-type option is over-hedged or under-hedged. In chapter 4, to address the concern whether roughness can only be observed in the high frequency data, we first propose a new volatility proxy framework using low frequency cumulative option trading entries. Upon that framework we estimate the Hurst Index of the volatility and verify that the volatility exhibits very rough behavior even in the low frequency (daily) settings, which corroborates that roughness should always be taken into consideration when modeling the volatility for option pricing and calibration. In the case of pricing option in the long run we observe that the hurst number varies largely, suggesting the hurst number is a local parameter and is not appropriate to be taken as a constant in the long setting. In chapter 5, we consider the volatility model driven by fractional Ornstein-Unlenbeck process, with Hurst index H less than 1/2. We propose a point process approximation scheme to both the volatility process and the asset price process, and we first prove a Donsker type theorem for the convergence to fractional Ornstein-Uhlenbeck process. With that theorem we prove the weak convergence of the scheme to the log-price process. We also take into account the correlation between the volatility and the stock process, which is also termed as leverage effect. This work will be conducted with respect to Skorohod topology and we will not have interpolating term. The result can be generalized to all point process with jumps consisting of bounded random variables.","Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2023-08-01","The student, Qi Zhao, accepted the attached license on 2021-07-02 at 22:56.","The student, Qi Zhao, submitted this Dissertation for approval on 2021-07-02 at 22:58.","This Dissertation was approved for publication on 2021-07-11 at 07:39.","DSpace SAF Submission Ingestion Package generated from Vireo submission #16753 on 2022-01-12 at 12:52:52","Made available in DSpace on 2022-01-12T22:34:51Z (GMT). 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However recent works have pointed out that there are some observations that the semimartinagales or Markovian models cannot explain, for example volatility persistence or the roughness of the sample paths of volatilities. And that is when the fractional stochastic volatility models are introduced. Our work about fractional stochastic volatility models mainly covers three different directions: hedging, calibration and approximation. In chapter 3, we propose a delta-hedging strategy for a long memory stochastic volatility model (LMSV). This is a model in which the volatility is driven by a fractional Ornstein-Uhlenbeck process with Hurst Index H greater than 1/2. We need to notice that perfect hedging cannot be achieved in non-Markovian fractional volatility framework, and thus we can only study imperfect delta-hedging strategy. We first proved the existence of this strategy by establishing the differentiability of the option price with respect to the underlying asset price. We also compute the so-called hedging bias, i.e. the difference between the Black-Scholes Delta and the LMSV Delta, and we determine when a European-type option is over-hedged or under-hedged. In chapter 4, to address the concern whether roughness can only be observed in the high frequency data, we first propose a new volatility proxy framework using low frequency cumulative option trading entries. Upon that framework we estimate the Hurst Index of the volatility and verify that the volatility exhibits very rough behavior even in the low frequency (daily) settings, which corroborates that roughness should always be taken into consideration when modeling the volatility for option pricing and calibration. In the case of pricing option in the long run we observe that the hurst number varies largely, suggesting the hurst number is a local parameter and is not appropriate to be taken as a constant in the long setting. In chapter 5, we consider the volatility model driven by fractional Ornstein-Unlenbeck process, with Hurst index H less than 1/2. We propose a point process approximation scheme to both the volatility process and the asset price process, and we first prove a Donsker type theorem for the convergence to fractional Ornstein-Uhlenbeck process. With that theorem we prove the weak convergence of the scheme to the log-price process. We also take into account the correlation between the volatility and the stock process, which is also termed as leverage effect. This work will be conducted with respect to Skorohod topology and we will not have interpolating term. The result can be generalized to all point process with jumps consisting of bounded random variables.","Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2023-08-01","The student, Qi Zhao, accepted the attached license on 2021-07-02 at 22:56.","The student, Qi Zhao, submitted this Dissertation for approval on 2021-07-02 at 22:58.","This Dissertation was approved for publication on 2021-07-11 at 07:39.","DSpace SAF Submission Ingestion Package generated from Vireo submission #16753 on 2022-01-12 at 12:52:52","Made available in DSpace on 2022-01-12T22:34:51Z (GMT). No. of bitstreams: 2 ZHAO-DISSERTATION-2021.pdf: 1458164 bytes, checksum: 244f6db8a585fd60d3f826847563f3ab (MD5) LICENSE.txt: 4204 bytes, checksum: b03da40a724ec38a2a82459af04081e4 (MD5) Previous issue date: 2021-07-11","Embargo set by: Seth Robbins for item 121065 Lift date: 2024-01-12T22:35:30Z Reason: Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system","Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system","U of I Only"],"dc:format":["application/pdf"],"dc:identifier":["http://hdl.handle.net/2142/113139"],"dc:language":["en"],"dc:rights":["Copyright 2021 Qi Zhao"],"dc:subject":["Fractional Stochastic Volatility","Hedging","Calibration","Weak Convergence"],"dc:title":["Fractional stochastic volatility models: approximation, calibration and hedging"],"dc:type":["text","Thesis"],"thesis:degree_discipline":["Industrial Engineering"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:24:53Z"}