{"id":163602,"date":"2022-10-26T11:44:45","date_gmt":"2022-10-26T15:44:45","guid":{"rendered":"https:\/\/ibkrcampus.com\/?p=163602"},"modified":"2022-11-21T09:59:24","modified_gmt":"2022-11-21T14:59:24","slug":"hull-white-2-factor-model-1-introduction","status":"publish","type":"post","link":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/hull-white-2-factor-model-1-introduction\/","title":{"rendered":"Hull-White 2-factor Model: 1) Introduction"},"content":{"rendered":"\n<p>This post introduces Hull-White 2-factor model and derives integrations of some important stochastic process which are ingredients of short rate process.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-introduction\">Introduction<\/h3>\n\n\n\n<p>We are going to derive the Hull-White 2-factor model.<\/p>\n\n\n\n<p>Given money market account<em>&nbsp;<strong>B<sub>t<\/sub><\/strong><\/em>&nbsp;as a numeraire under the Q measure, short rate&nbsp;<strong><em>r(t)<\/em><\/strong>&nbsp;is assumed as follows.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"560\" height=\"146\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2022\/10\/Hull-White-2-factor-SH-Fintech-Modeling-1.png\" alt=\"\" class=\"wp-image-163613 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2022\/10\/Hull-White-2-factor-SH-Fintech-Modeling-1.png 560w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2022\/10\/Hull-White-2-factor-SH-Fintech-Modeling-1-300x78.png 300w\" data-sizes=\"(max-width: 560px) 100vw, 560px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 560px; aspect-ratio: 560\/146;\" \/><\/figure>\n\n\n\n<p>Here&nbsp;<strong><em>a(t)<\/em><\/strong>,&nbsp;<strong><em>b(t)<\/em><\/strong>&nbsp;and&nbsp;<strong><em>\u03c3(t)<\/em><\/strong>,&nbsp;<strong><em>\u03b7(t)<\/em><\/strong>&nbsp;are mean-reversion and volatility parameters for each process respectively.&nbsp;<em><strong>Wx(t)<\/strong><\/em>&nbsp;and&nbsp;<strong><em>Wy(t)<\/em><\/strong>&nbsp;are correlated standard Wiener process and&nbsp;<strong><em>\u03c6(t)<\/em><\/strong>&nbsp;is the deterministic process which is adapted to an initial term structure.<\/p>\n\n\n\n<p>Like 1-factor model,&nbsp;<strong><em>\u03b8(t)<\/em><\/strong>&nbsp;and&nbsp;<em><strong>\u03c6(t)<\/strong><\/em>&nbsp;are reflected in the process of derivation implicitly. Hence our focus is on&nbsp;<em><strong>x(t)<\/strong><\/em>+<strong><em>y(t)x(t)<\/em><\/strong>+<strong><em>y(t)<\/em><\/strong>.<\/p>\n\n\n\n<p>Using&nbsp;<strong><em>W1(t)<\/em><\/strong>&nbsp;and&nbsp;<strong><em>W2(t)<\/em><\/strong>&nbsp;as independent Wiener processes,<strong><em>&nbsp;x(t)<\/em><\/strong>&nbsp;and&nbsp;<strong><em>y(t)<\/em><\/strong>&nbsp;can be rewritten.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"546\" height=\"71\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2022\/10\/Hull-White-2-factor-SH-Fintech-Modeling-2.png\" alt=\"\" class=\"wp-image-163620 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2022\/10\/Hull-White-2-factor-SH-Fintech-Modeling-2.png 546w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2022\/10\/Hull-White-2-factor-SH-Fintech-Modeling-2-300x39.png 300w\" data-sizes=\"(max-width: 546px) 100vw, 546px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 546px; aspect-ratio: 546\/71;\" \/><\/figure>\n\n\n\n<p>For any&nbsp;<strong><em>s(&lt;t)<\/em><\/strong>, we can get the integrated from of&nbsp;<strong><em>r(t)<\/em><\/strong>&nbsp;from&nbsp;<strong><em>dr(t)<\/em><\/strong>&nbsp;as follows.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"543\" height=\"173\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2022\/10\/Hull-White-2-factor-SH-Fintech-Modeling-3.png\" alt=\"\" class=\"wp-image-163626 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2022\/10\/Hull-White-2-factor-SH-Fintech-Modeling-3.png 543w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2022\/10\/Hull-White-2-factor-SH-Fintech-Modeling-3-300x96.png 300w\" data-sizes=\"(max-width: 543px) 100vw, 543px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 543px; aspect-ratio: 543\/173;\" \/><\/figure>\n\n\n\n<p>The derivation of the above equation is skipped because that is the similar logic of the corresponding derivation of HW 1-factor model.<\/p>\n\n\n\n<p>Using these results, we will derive a zero coupon bond price of Hull-White 2-factor model in the next post.<\/p>\n\n\n\n<p><em>For additional insight on this topic visit the <a href=\"https:\/\/kiandlee.blogspot.com\/2021\/06\/hull-white-2-factor-model-1-introduction.html\">SH Fintech Modeling<\/a>&nbsp;Blog.<\/em><\/p>\n","protected":false},"excerpt":{"rendered":"<p>This post introduces Hull-White 2-factor model and derives integrations of some important stochastic process which are ingredients of short rate process.<\/p>\n","protected":false},"author":662,"featured_media":163630,"comment_status":"closed","ping_status":"open","sticky":true,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[339,343,338,350,341,344,342],"tags":[4922,4941,13010],"contributors-categories":[13728],"class_list":{"0":"post-163602","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-data-science","8":"category-programing-languages","9":"category-ibkr-quant-news","10":"category-quant-asia-pacific","11":"category-quant-development","12":"category-quant-regions","13":"category-r-development","14":"tag-econometrics","15":"tag-financial-mathematics","16":"tag-hull-white-2-factor-model","17":"contributors-categories-sh-fintech-modeling"},"pp_statuses_selecting_workflow":false,"pp_workflow_action":"current","pp_status_selection":"publish","acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin 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