{"id":183452,"date":"2022-12-26T14:00:00","date_gmt":"2022-12-26T19:00:00","guid":{"rendered":"https:\/\/ibkrcampus.com\/traders-insight\/hull-white-2-factor-model-3-simulation\/"},"modified":"2023-02-13T16:23:46","modified_gmt":"2023-02-13T21:23:46","slug":"hull-white-2-factor-model-3-simulation","status":"publish","type":"post","link":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/hull-white-2-factor-model-3-simulation\/","title":{"rendered":"Hull-White 2-factor Model: 3) Simulation"},"content":{"rendered":"\n<p>This post discretizes Hull-White 2-factor model and provide derivations of the simulation equations.<\/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\n\n\n<li><a href=\"https:\/\/kiandlee.blogspot.com\/2021\/06\/hull-white-2-factor-model-2-zero-coupon.html\" rel=\"noreferrer noopener\" target=\"_blank\">Hull-White 2-factor model : 2) Zero coupon bond<\/a><\/li>\n<\/ul>\n\n\n\n<p>We try to price an interest derivatives which have cashflows at times&nbsp;<i>T<sub>1<\/sub>,T<sub>2<\/sub>,&#8230;,T<sub>N<\/sub><\/i>. When we let&nbsp;<i>f(T<sub>j<\/sub>)<\/i>&nbsp;denote a cash flow at time&nbsp;<i>T<sub>j<\/sub><\/i>, the price of this product is<\/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-3-factor-model-sh-fintech-simulation-1.png\" alt=\" class=\" class=\"wp-image-170751 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n<p>This pricing is the risk-neutral pricing and needs cash flows and discount factors from future interest rate simulations.<\/p>\n\n\n\n<p>Since&nbsp;<i>dW<sub>1<\/sub>(t)<\/i>&nbsp;and&nbsp;<i>dW<sub>2<\/sub>(t)<\/i>&nbsp;follow&nbsp;<img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/02\/hull-white-3-factor-model-sh-fintech-simulation-2.png\" alt=\"=\"\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\">&nbsp;independently,&nbsp;<i>dx(t)<\/i>&nbsp;and&nbsp;<i>dy(t)<\/i>&nbsp;can be discretized as follows.<\/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-3-factor-model-sh-fintech-simulation-3.png\" alt=\" class=\" class=\"wp-image-170753 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n<p>Here,&nbsp;<em>\u03f51<\/em>&nbsp;and&nbsp;<em>\u03f52&nbsp;<\/em>are random numbers from the standard normal distribution.<\/p>\n\n\n\n<p>Like HW 1-factor model, the same discretized time axis is used as follows&nbsp; <img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/02\/hull-white-3-factor-model-sh-fintech-simulation-4.png\" alt=\"=\"\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\"><\/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-3-factor-model-sh-fintech-simulation-5.png\" alt=\" class=\" class=\"wp-image-170755 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n<p>For this time axis, stochastic process of discretized&nbsp;<em>x(t)<\/em>&nbsp;and&nbsp;<em>y(t)<\/em>&nbsp;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-3-factor-model-sh-fintech-simulation-6.png\" alt=\" class=\" class=\"wp-image-170756 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n<p>Since&nbsp;<em>x<sub>t0<\/sub><\/em>, <em>x<sub>t1<\/sub><\/em>,&nbsp;<em>x<sub>t2<\/sub><\/em>,&nbsp;<em>x<sub>t3<\/sub><\/em>, &#8230;, and&nbsp;<em>y<sub>t0<\/sub><\/em>, <em>y<sub>t1<\/sub><\/em>,&nbsp;<em>y<sub>t2<\/sub><\/em>,&nbsp;<em>y<sub>t3<\/sub><\/em>, &#8230; are easily obtained from this scenario generating equation, discount factors at time&nbsp;<em>T<sub>j<\/sub><\/em>&nbsp;is<\/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-3-factor-model-sh-fintech-simulation-7.png\" alt=\" class=\" class=\"wp-image-170757 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n<p>Cash flow at time&nbsp;<em>T<sub>j<\/sub><\/em>&nbsp;is<\/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-3-factor-model-sh-fintech-simulation-8.png\" alt=\" class=\" class=\"wp-image-170758 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n<p>Therefore, for discount factors and cash flows from this scenario, the price of interest derivatives&nbsp;<em>P<sub>0<\/sub><\/em>&nbsp;which has cash flows&nbsp;<em>f(T<sub>1<\/sub>),f(T<sub>2<\/sub>),&#8230;,f(T<sub>N<\/sub>)<\/em>&nbsp;at time&nbsp;<em>T<sub>1<\/sub>,T<sub>2<\/sub>,&#8230;,T<sub>N<\/sub><\/em>&nbsp;respectively under the risk-neutral measure is as follows.<\/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-3-factor-model-sh-fintech-simulation-9.png\" alt=\" class=\" class=\"wp-image-170759 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n<p>The present value is the average from iterating this process with a number of scenario. This is the same case with the Hull-White 1-factor model except for correlated related cross terms. Expressions for this kind of terms are presented as product (or multiplication) terms not as square terms.<\/p>\n\n\n\n<p>This manipulation for cross terms also holds for numerical integration and calculation which are also similar to 1-factor case. So we do not repeat these similar works.<\/p>\n\n\n\n<p><em>Visit <a href=\"https:\/\/kiandlee.blogspot.com\/2021\/06\/hull-white-2-factor-model-3-simulation.html\">SH Fintech Modeling<\/a> to learn more about this topic.<\/em><\/p>\n","protected":false},"excerpt":{"rendered":"<p>This post discretizes Hull-White 2-factor model and provide derivations of the simulation equations.<\/p>\n","protected":false},"author":662,"featured_media":183462,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[339,338,350,341,344],"tags":[806,4922,13010,14327],"contributors-categories":[13728],"class_list":{"0":"post-183452","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-data-science","13":"tag-econometrics","14":"tag-hull-white-2-factor-model","15":"tag-stochastic-process","16":"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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