{"id":89275,"date":"2021-05-25T16:15:00","date_gmt":"2021-05-25T20:15:00","guid":{"rendered":"https:\/\/ibkrcampus.com\/?p=89275"},"modified":"2022-11-21T09:47:33","modified_gmt":"2022-11-21T14:47:33","slug":"a-multidimensional-scaled-brownian-bridge-model-for-stochastic-basis","status":"publish","type":"post","link":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/a-multidimensional-scaled-brownian-bridge-model-for-stochastic-basis\/","title":{"rendered":"A Multidimensional Scaled Brownian Bridge Model for Stochastic Basis"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Futures are standardized bilateral contracts of agreement to buy or sell an asset at a pre-determined price at a pre-specified time in the future. The underlying assets can be physical commodities, market indexes, or financial instruments. The Chicago Mercantile Exchange (CME), which is the world\u2019s largest futures exchange, averages well over 15&nbsp;million futures contracts traded per day.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In the futures market, it is commonly observed that the futures prices may deviate from the spot price of the underlying asset.&nbsp;The differential between\u202ffutures&nbsp;and&nbsp;spot\u202fprices, called the&nbsp;<em>basis<\/em>, can be positive or negative but are expected to converge to zero or near-zero at the expiration of the futures contract.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For each underlying asset, there are multiple futures with different maturities (ranging from 1 month to over a year). And for each futures contract, there is one basis process. Therefore, when we consider all different spot assets and their associated futures, there are a high number of basis processes.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Moreover, these stochastic processes are clearly\u202f<strong>dependent<\/strong>. <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">First,\u00a0different\u00a0underlying\u00a0assets, such as silver and gold, can\u00a0be\u00a0correlated. Second, the futures\u00a0contracts\u00a0written on the same underlying asset are clearly driven by a common source of randomness, among other factors. For anyone trading futures\u00a0&#8212;\u00a0on the same underlying or different assets &#8212;\u00a0it is crucial to understand the dependence structure among these processes.\u00a0<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Multidimensional Scaled Brownian Bridge<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This motivates us to develop a novel model to capture the joint dynamics of stochastic basis from different underlying&nbsp;assets&nbsp;and different futures&nbsp;contracts. Once this model is built,&nbsp;one can&nbsp;apply it to dynamic futures trading, as studied in this&nbsp;paper:&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">B.\u00a0Angoshtari\u00a0\u00a0and\u00a0T. Leung,\u202f<em>Optimal Trading of a Basket of Futures Contracts\u00a0<\/em>[<a href=\"https:\/\/www.google.com\/url?q=https%3A%2F%2Fpapers.ssrn.com%2Fsol3%2Fpapers.cfm%3Fabstract_id%3D3467897&amp;sa=D&amp;sntz=1&amp;usg=AFQjCNEaXz7YASPCmsnHQ0UT827TPPNbDA\" target=\"_blank\" rel=\"noreferrer noopener\">pdf<\/a>; <a href=\"https:\/\/www.google.com\/url?q=https%3A%2F%2Fdoi.org%2F10.1007%2Fs10436-019-00357-w&amp;sa=D&amp;sntz=1&amp;usg=AFQjCNGeWFV4cxyza8Vkhi6EFC6HCBfUsg\" target=\"_blank\" rel=\"noreferrer noopener\">link<\/a>],\u202f<strong>Annals of Finance<\/strong>, 2020\u00a0<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The&nbsp;<strong>Multidimensional Scaled Brownian Bridge (MSBB)<\/strong>\u202fis a continuous-time stochastic model described by the following multidimensional stochastic differential equation (SDE):&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" width=\"1006\" height=\"112\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2021\/05\/image-119.png\" alt=\"\" class=\"wp-image-89304 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/05\/image-119.png 1006w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/05\/image-119-700x78.png 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/05\/image-119-300x33.png 300w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/05\/image-119-768x86.png 768w\" data-sizes=\"(max-width: 1006px) 100vw, 1006px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 1006px; aspect-ratio: 1006\/112;\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">where<\/p>\n\n\n\n<figure class=\"wp-block-image img-twothird\"><img decoding=\"async\" width=\"492\" height=\"190\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2021\/05\/image-120.png\" alt=\"\" class=\"wp-image-89306 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/05\/image-120.png 492w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/05\/image-120-300x116.png 300w\" data-sizes=\"(max-width: 492px) 100vw, 492px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 492px; aspect-ratio: 492\/190;\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image img-twothird\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2021\/05\/image-121.png\" alt=\"\" class=\"wp-image-89307 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\"><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">and\u202f<strong>W<\/strong>\u202fconsists of Brownian motions.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In fact,\u202f<strong>Z<\/strong>\u202fis a N-dimensional process where each component is a 1-dimensional\u202f<a href=\"https:\/\/medium.com\/@timleungresearch\/a-stopped-scaled-brownian-bridge-model-for-basis-trading-6b1af54cee7f\" target=\"_blank\" rel=\"noreferrer noopener\">scaled Brownian bridge<\/a>:&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2021\/05\/image-122.png\" alt=\"\" class=\"wp-image-89310 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">The stochastic differential equation for a 1-d scaled Brownian bridge. Note that the scaled Brownian bridges Z\u1d62 are correlated.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The SDE for the multidimensional scaled Brownian bridge has a unique&nbsp;solution&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2021\/05\/image-123-1100x246.png\" alt=\"\" class=\"wp-image-89313 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 1100px; aspect-ratio: 1100\/246;\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Here, we used the shorthand notation for the diagonal matrix:&nbsp;diag&nbsp;(a\u1d62 )&nbsp;=&nbsp;diag(a\u2081&nbsp;,&nbsp;. . . ,&nbsp;a<sub>N<\/sub>).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The&nbsp;<strong>mean function&nbsp;<\/strong>of\u202f<strong>Z<\/strong>\u202fis given&nbsp;by&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2021\/05\/image-124-1100x136.png\" alt=\"\" class=\"wp-image-89315 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 1100px; aspect-ratio: 1100\/136;\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">where<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2021\/05\/image-125-1100x255.png\" alt=\"\" class=\"wp-image-89317 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 1100px; aspect-ratio: 1100\/255;\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">\u03ba\u1d62 is the scaling parameter.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">And the\u202f<strong>covariance function<\/strong>\u202fis&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2021\/05\/image-126-1100x136.png\" alt=\"\" class=\"wp-image-89319 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 1100px; aspect-ratio: 1100\/136;\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Figure 1 illustrates the simulated paths of\u202f<strong>Z<\/strong>,\u202f<strong>S<\/strong>, and\u202f<strong>F<\/strong>\u202ffor two pairs of futures and underlying assets (i.e.&nbsp;N = 2). Here, each Z is the log basis (i.e.&nbsp;log(F\/S)) for the corresponding asset and futures contract. The plots for (Z<sub>t<\/sub>,1) and (Z<sub>t<\/sub>,2) also show the 95% confidence intervals of the log-bases.&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2021\/05\/image-127-1100x575.png\" alt=\"\" class=\"wp-image-89320 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 1100px; aspect-ratio: 1100\/575;\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Figure 1: Simulated sample paths of S and F for two pairs of futures and spot (i.e.&nbsp;N = 2), and the corresponding 2-dimensional basis processes Z.&nbsp;Source:&nbsp;Angoshtari&nbsp;and Leung (2020), available at\u202f<a href=\"https:\/\/arxiv.org\/pdf\/1910.04943.pdf\" target=\"_blank\" rel=\"noreferrer noopener\">https:\/\/arxiv.org\/pdf\/1910.04943.pdf<\/a>&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">These plots showcase two characteristics of the log-bases.&nbsp;First, they are\u202f<strong>mean-reverting<\/strong>\u202fin that any deviation from their mean is corrected. Second, they partially converge to zero at the end of the trading horizon (T = 0.25) as evident by narrowing of the confidence intervals. Indeed, (Z<sub>t<\/sub>,1)&nbsp;and (Z<sub>t<\/sub>,2) are Brownian bridges that converge to zero at T\u2081&nbsp;= 0.27 and T\u2082&nbsp;= 0.254, respectively. This convergence is not realized since trading stops at T = 0.25 in this&nbsp;particular example.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Since,&nbsp;Z<em><sub>t<\/sub><\/em>|Z\u2080&nbsp;is a multivariate normal random variable, this means&nbsp;that&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" width=\"1100\" height=\"90\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2021\/05\/image-128-1100x90.png\" alt=\"\" class=\"wp-image-89325 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/05\/image-128-1100x90.png 1100w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/05\/image-128-700x57.png 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/05\/image-128-300x25.png 300w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/05\/image-128-768x63.png 768w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/05\/image-128.png 1150w\" data-sizes=\"(max-width: 1100px) 100vw, 1100px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 1100px; aspect-ratio: 1100\/90;\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">has a\u202f<strong>chi-squared distribution<\/strong>\u202fwith\u202f<em>N<\/em>\u202fdegrees of freedom. This relationship is utilized for obtaining the 95% confidence regions of&nbsp;Z<em><sub>t<\/sub><\/em>&nbsp;represented by dashed blue ellipses. These plots also illustrate partial convergence of log-basis at the end of time horizon.&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2021\/05\/image-129-1100x340.png\" alt=\"\" class=\"wp-image-89326 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 1100px; aspect-ratio: 1100\/340;\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Simulated values of (Z\u2081, Z\u2082\u00a0)\u00a0based on 400 Simulated paths observed at three different times: t = 0.05 (left), t = 0.15 (middle), and t = 0.25 (right). The dotted lines represent the border of the 95% confidence region for (Z\u2081, Z\u2082), conditional on Z\u2080\u00a0= (0,0). Source:\u00a0Angoshtari\u00a0and Leung (2020), available at\u202f<a href=\"https:\/\/arxiv.org\/pdf\/1910.04943.pdf\" target=\"_blank\" rel=\"noreferrer noopener\">https:\/\/arxiv.org\/pdf\/1910.04943.pdf<\/a>\u00a0<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Simulation<\/strong>&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The SDE solution for the multidimensional scaled Brownian bridge&nbsp;actually lends&nbsp;itself to a simulation algorithm. We refer to the paper for details, but the main idea is to discretize the time horizon in&nbsp;<em>M<\/em>&nbsp;time steps, simulate independent Gaussian random variables, and put them in the right places as follows:&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2021\/05\/image-130-1100x688.png\" alt=\"\" class=\"wp-image-89329 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 1100px; aspect-ratio: 1100\/688;\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Source:\u00a0Angoshtari\u00a0and Leung (2020), available at <a href=\"https:\/\/arxiv.org\/pdf\/1910.04943.pdf\" target=\"_blank\" rel=\"noreferrer noopener\">https:\/\/arxiv.org\/pdf\/1910.04943.pdf<\/a>\u00a0<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For any notations not explained herein, please refer to the papers&nbsp;below.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Takeaways&nbsp;<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For trading systems that involve multiple futures and assets, it is imperative to properly capture the dependence among price&nbsp;processes.\u202f<strong>MSBB\u202f<\/strong>described herein is designed for modeling the joint dynamics among futures and their spot assets. Through the solution to the stochastic differential question, this model is straightforward to simulate. With simulated sample paths, one can test the performance of trading strategies.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For research discussions, reach out to\u202fProf.&nbsp;Leung\u202f<a href=\"https:\/\/www.linkedin.com\/in\/timstleung\/\" target=\"_blank\" rel=\"noreferrer noopener\"><strong>here<\/strong><\/a>.\u202f&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Reference<\/strong><strong>s<\/strong>&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">B.\u00a0Angoshtari\u00a0\u00a0and\u00a0T. Leung,\u202f<em>Optimal Trading of a Basket of Futures Contracts<\/em><strong>\u202f<\/strong>[<a href=\"https:\/\/www.google.com\/url?q=https%3A%2F%2Fpapers.ssrn.com%2Fsol3%2Fpapers.cfm%3Fabstract_id%3D3467897&amp;sa=D&amp;sntz=1&amp;usg=AFQjCNEaXz7YASPCmsnHQ0UT827TPPNbDA\" target=\"_blank\" rel=\"noreferrer noopener\">pdf<\/a>; <a href=\"https:\/\/www.google.com\/url?q=https%3A%2F%2Fdoi.org%2F10.1007%2Fs10436-019-00357-w&amp;sa=D&amp;sntz=1&amp;usg=AFQjCNGeWFV4cxyza8Vkhi6EFC6HCBfUsg\" target=\"_blank\" rel=\"noreferrer noopener\">link<\/a>],\u202f<strong>Annals of Finance<\/strong>, 2020\u00a0<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">B.\u00a0Angoshtari\u00a0 and T. Leung,\u00a0Optimal Dynamic Basis Trading [<a href=\"https:\/\/link.springer.com\/article\/10.1007\/s10436-019-00348-x\" target=\"_blank\" rel=\"noreferrer noopener\">link<\/a>; <a href=\"https:\/\/www.google.com\/url?q=https%3A%2F%2Frdcu.be%2FbGlF5&amp;sa=D&amp;sntz=1&amp;usg=AFQjCNFYVi1e0vy3I35HFsNIPK6hsZM7iA\" target=\"_blank\" rel=\"noreferrer noopener\">read online<\/a>],\u202f<strong>Annals of Finance<\/strong>, Vol.15, Issue 3, pp. 307\u2013335, 2019\u00a0\u00a0\u00a0<\/p>\n","protected":false},"excerpt":{"rendered":"<p>For trading systems that involve multiple futures and assets, it is imperative to properly capture the dependence among price processes.\u202fMSBB\u202fdescribed herein is designed for modeling the joint dynamics among futures and their spot assets.<\/p>\n","protected":false},"author":189,"featured_media":89349,"comment_status":"closed","ping_status":"open","sticky":true,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":"","jetpack_post_was_ever_published":false},"categories":[339,338,341,352,344],"tags":[9783,9781,9782,9779,9778,9780],"contributors-categories":[13668],"class_list":["post-89275","post","type-post","status-publish","format-standard","has-post-thumbnail","category-data-science","category-ibkr-quant-news","category-quant-development","category-quant-north-america","category-quant-regions","tag-chi-squared-distribution","tag-covariance-function","tag-mean-reverting","tag-multidimensional-scaled-brownian-bridge-model","tag-scaled-brownian-bridge","tag-stochastic-basis","contributors-categories-computational-finance-risk-management-university-of-washington"],"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 v28.0) - https:\/\/yoast.com\/product\/yoast-seo-premium-wordpress\/ -->\n<title>A Multidimensional Scaled Brownian Bridge Model for Stochastic Basis<\/title>\n<meta name=\"description\" content=\"For trading systems that involve multiple futures and assets, it is imperative to properly capture the dependence among price...\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.interactivebrokers.com\/campus\/wp-json\/wp\/v2\/posts\/89275\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"A Multidimensional Scaled Brownian Bridge Model for Stochastic Basis | IBKR Quant Blog\" \/>\n<meta property=\"og:description\" content=\"For trading systems that involve multiple futures and assets, it is imperative to 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