In the setting of polynomial jump-diffusion dynamics, we provide a formula for computing correlators, namely, cross-moments of the process at different time points along its path. The formula involves only linear combinations of the exponential of the so-called generator matrix, extending the well-known moment formula for polynomial processes. The developed framework allows to replace costly simulations with more accurate estimates, and it may be used for increasing the accuracy in financial pricing, such as for path-dependent options or in a stochastic volatility models context.
{"title":"Correlators of Polynomial Processes","authors":"F. Benth, Silvia Lavagnini","doi":"10.1137/21m141556x","DOIUrl":"https://doi.org/10.1137/21m141556x","url":null,"abstract":"In the setting of polynomial jump-diffusion dynamics, we provide a formula for computing correlators, namely, cross-moments of the process at different time points along its path. The formula involves only linear combinations of the exponential of the so-called generator matrix, extending the well-known moment formula for polynomial processes. The developed framework allows to replace costly simulations with more accurate estimates, and it may be used for increasing the accuracy in financial pricing, such as for path-dependent options or in a stochastic volatility models context.","PeriodicalId":48880,"journal":{"name":"SIAM Journal on Financial Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2019-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"64314814","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Rough volatility models are very appealing because of their remarkable fit of both historical and implied volatilities. However, due to the non-Markovian and non-semimartingale nature of the volatility process, there is no simple way to simulate efficiently such models, which makes risk management of derivatives an intricate task. In this paper, we design tractable multi-factor stochastic volatility models approximating rough volatility models and enjoying a Markovian structure. Furthermore, we apply our procedure to the specific case of the rough Heston model. This in turn enables us to derive a numerical method for solving fractional Riccati equations appearing in the characteristic function of the log-price in this setting.
{"title":"Multifactor Approximation of Rough Volatility Models","authors":"Eduardo Abi Jaber, Omar El Euch","doi":"10.1137/18m1170236","DOIUrl":"https://doi.org/10.1137/18m1170236","url":null,"abstract":"Rough volatility models are very appealing because of their remarkable fit of both historical and implied volatilities. However, due to the non-Markovian and non-semimartingale nature of the volatility process, there is no simple way to simulate efficiently such models, which makes risk management of derivatives an intricate task. In this paper, we design tractable multi-factor stochastic volatility models approximating rough volatility models and enjoying a Markovian structure. Furthermore, we apply our procedure to the specific case of the rough Heston model. This in turn enables us to derive a numerical method for solving fractional Riccati equations appearing in the characteristic function of the log-price in this setting.","PeriodicalId":48880,"journal":{"name":"SIAM Journal on Financial Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76582726","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-10-24DOI: 10.1201/9781315373768-15
G. Campolieti, R. Makarov
{"title":"Interest-Rate Modelling and Derivative Pricing","authors":"G. Campolieti, R. Makarov","doi":"10.1201/9781315373768-15","DOIUrl":"https://doi.org/10.1201/9781315373768-15","url":null,"abstract":"","PeriodicalId":48880,"journal":{"name":"SIAM Journal on Financial Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2018-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76721874","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Mathematics of Compounding","authors":"G. Campolieti, R. Makarov","doi":"10.1201/9781315373768-1","DOIUrl":"https://doi.org/10.1201/9781315373768-1","url":null,"abstract":"","PeriodicalId":48880,"journal":{"name":"SIAM Journal on Financial Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2018-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86875298","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-10-24DOI: 10.1201/9781315373768-18
G. Campolieti, R. Makarov
{"title":"Numerical Applications to Derivative Pricing","authors":"G. Campolieti, R. Makarov","doi":"10.1201/9781315373768-18","DOIUrl":"https://doi.org/10.1201/9781315373768-18","url":null,"abstract":"","PeriodicalId":48880,"journal":{"name":"SIAM Journal on Financial Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2018-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91162017","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-10-24DOI: 10.1201/9781315373768-11
G. Campolieti, R. Makarov
{"title":"Introduction to Continuous-Time Stochastic Calculus","authors":"G. Campolieti, R. Makarov","doi":"10.1201/9781315373768-11","DOIUrl":"https://doi.org/10.1201/9781315373768-11","url":null,"abstract":"","PeriodicalId":48880,"journal":{"name":"SIAM Journal on Financial Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2018-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74484163","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Replication and Pricing in the Binomial Tree Model","authors":"G. Campolieti, R. Makarov","doi":"10.1201/9781315373768-7","DOIUrl":"https://doi.org/10.1201/9781315373768-7","url":null,"abstract":"","PeriodicalId":48880,"journal":{"name":"SIAM Journal on Financial Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2018-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85698594","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-10-24DOI: 10.1201/9781315373768-14
G. Campolieti, R. Makarov
{"title":"American Options","authors":"G. Campolieti, R. Makarov","doi":"10.1201/9781315373768-14","DOIUrl":"https://doi.org/10.1201/9781315373768-14","url":null,"abstract":"","PeriodicalId":48880,"journal":{"name":"SIAM Journal on Financial Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2018-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80392152","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Primer on Pricing Risky Securities","authors":"G. Campolieti, R. Makarov","doi":"10.1201/9781315373768-2","DOIUrl":"https://doi.org/10.1201/9781315373768-2","url":null,"abstract":"","PeriodicalId":48880,"journal":{"name":"SIAM Journal on Financial Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2018-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83452064","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-10-24DOI: 10.1201/9781315373768-16
G. Campolieti, R. Makarov
{"title":"Alternative Models of Asset Price Dynamics","authors":"G. Campolieti, R. Makarov","doi":"10.1201/9781315373768-16","DOIUrl":"https://doi.org/10.1201/9781315373768-16","url":null,"abstract":"","PeriodicalId":48880,"journal":{"name":"SIAM Journal on Financial Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2018-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72904518","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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