{"id":183119,"date":"2022-12-14T15:25:00","date_gmt":"2022-12-14T20:25:00","guid":{"rendered":"https:\/\/ibkrcampus.com\/traders-insight\/hull-white-2-factor-model-2-zero-coupon-bond\/"},"modified":"2023-02-13T16:22:08","modified_gmt":"2023-02-13T21:22:08","slug":"hull-white-2-factor-model-2-zero-coupon-bond","status":"publish","type":"post","link":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/hull-white-2-factor-model-2-zero-coupon-bond\/","title":{"rendered":"Hull-White 2-factor Model: 2) Zero Coupon Bond"},"content":{"rendered":"\n<p>This post derives the expression of zero coupon bond price of Hull-White 2-factor model.<\/p>\n\n\n\n<p>Earlier posts on Hull-White 2-factor model<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/kiandlee.blogspot.com\/2021\/06\/hull-white-2-factor-model-1-introduction.html\" rel=\"noreferrer noopener\" target=\"_blank\">Hull-White 2-factor model : 1) Introduction<\/a><\/li>\n<\/ul>\n\n\n\n<p>Let&nbsp;<em>P(t,T)<\/em>&nbsp;denotes the price of zero-coupon bond with maturity&nbsp;<em>T<\/em>&nbsp;at time&nbsp;<em>t<\/em>. Assuming&nbsp;<em>F<sub>t<\/sub><\/em>&nbsp;as the information generated by&nbsp;<em>x(t)<\/em>&nbsp;and&nbsp;<em>y(t)<\/em>&nbsp;up to time&nbsp;<em>t<\/em>,&nbsp;<em>P(t,T)<\/em>&nbsp;have the following form.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/02\/hull-white-2-factor-model-sh-fintech-1.png\" alt=\" class=\" class=\"wp-image-169976 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n<p>To solve for&nbsp;<em>P(t,T)<\/em>, we need to know the implementable expression for <img decoding=\"async\" class=\"wp-image-169990 lazyload\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/02\/hull-white-2-factor-model-sh-fintech-9.png\" alt=\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\"> &nbsp;except for&nbsp;<em>\u03c6(u)<\/em>&nbsp;because&nbsp;<em>\u03c6(u)<\/em>&nbsp;is not stochastic but deterministic process. Integrating&nbsp;<em>x(u)+y(u)<\/em>&nbsp;from time&nbsp;<em>t<\/em>&nbsp;to&nbsp;<em>T<\/em>, we can get the following result.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/02\/hull-white-2-factor-model-sh-fintech-2.png\" alt=\" class=\" class=\"wp-image-169977 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n<p>Here,&nbsp;<img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/02\/hull-white-2-factor-model-sh-fintech-10.png\" alt=\"=\"\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\">&nbsp;and&nbsp;<img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/02\/hull-white-2-factor-model-sh-fintech-11.png\" alt=\"=\"\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\">. The derivation above is of the same logic for the case of Hull-White 1-factor model.<\/p>\n\n\n\n<p><em>V(t,T)<\/em>&nbsp;is also given using&nbsp;<strong>It\u00f4 isometry<\/strong>&nbsp;like Hull-White 1-factor model.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/02\/hull-white-2-factor-model-sh-fintech-3.png\" alt=\" class=\" class=\"wp-image-169978 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n<p>Similar to HW 1-factor model, we can find that&nbsp;<img decoding=\"async\" class=\"wp-image-169997 lazyload\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/02\/hull-white-2-factor-model-sh-fintech-12.png\" alt=\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\">&nbsp;follows the normal distribution of a mean&nbsp;<em>x(t)B1(t,T)+y(t)B2(t,T)<\/em>&nbsp;and a variance&nbsp;<em>V(t,T)<\/em>.<\/p>\n\n\n\n<p>Using the fact that&nbsp;<img decoding=\"async\" class=\"wp-image-170000 lazyload\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/02\/hull-white-2-factor-model-sh-fintech-13.png\" alt=\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\">&nbsp;with a normally distributed random variable&nbsp;<em>Y<\/em>&nbsp;which has a mean&nbsp;<em>\u03bc<\/em>&nbsp;and a variance&nbsp;<em>\u03c3<sup>2<\/sup><\/em>, the price of zero-coupon bond becomes<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/02\/hull-white-2-factor-model-sh-fintech-4.png\" alt=\" class=\" class=\"wp-image-169979 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n<p>It is argued that Hull-White model is consistent to the no-arbitrage assumption (perfect fit) if market discount factor&nbsp;<em>P(0,T)<\/em>&nbsp;satisfies the following condition.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/02\/hull-white-2-factor-model-sh-fintech-5.png\" alt=\" class=\" class=\"wp-image-169980 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n<p>Using the above relationship,<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/02\/hull-white-2-factor-model-sh-fintech-6.png\" alt=\" class=\" class=\"wp-image-169981 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n<p>As the above expression for function&nbsp;<em>\u03c6(.)<\/em>&nbsp;holds, the price of zero-coupon bond has the following form.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/02\/hull-white-2-factor-model-sh-fintech-7.png\" alt=\" class=\" class=\"wp-image-169982 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n<p>Substituting&nbsp;<em>V(t,T)<\/em>&nbsp;into the equation, the price of zero coupon bond&nbsp;<em>P(t,T)<\/em>&nbsp;is reformulated as<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/02\/hull-white-2-factor-model-sh-fintech-8.png\" alt=\" class=\" class=\"wp-image-169985 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n<p>From this post, we know that most of derivation regarding HW 2-factor model is similar to the HW 1-factor model.<\/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-2-zero-coupon.html\">SH Fintech Modeling Blog<\/a>. <\/em><\/p>\n","protected":false},"excerpt":{"rendered":"<p>This post derives the expression of zero coupon bond price of Hull-White 2-factor model.<\/p>\n","protected":false},"author":662,"featured_media":183133,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[339,338,350,341,344],"tags":[4922,13010,14279],"contributors-categories":[13728],"class_list":{"0":"post-183119","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-data-science","8":"category-ibkr-quant-news","9":"category-quant-asia-pacific","10":"category-quant-development","11":"category-quant-regions","12":"tag-econometrics","13":"tag-hull-white-2-factor-model","14":"tag-zero-coupon-bond","15":"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 v26.9 (Yoast SEO v27.3) - 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