{"id":108115,"date":"2021-10-22T10:05:00","date_gmt":"2021-10-22T14:05:00","guid":{"rendered":"https:\/\/ibkrcampus.com\/?p=108115"},"modified":"2022-11-21T09:48:39","modified_gmt":"2022-11-21T14:48:39","slug":"constrained-dynamic-futures-portfolios-with-stochastic-basis","status":"publish","type":"post","link":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/constrained-dynamic-futures-portfolios-with-stochastic-basis\/","title":{"rendered":"Constrained Dynamic Futures Portfolios with Stochastic Basis"},"content":{"rendered":"\n<h2 class=\"wp-block-heading\" id=\"h-overview\"><strong>Overview<\/strong><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">The futures market offers trading instruments on a very diverse range of underlying assets, including precious metals, agricultural commodities, oil and gas, equity, interest rates, volatility, cryptocurrencies, and more. As a consequence, professional futures portfolio managers, commonly known as Commodity Trading Advisors (CTAs), now have access to various assets not only within the same asset class (e.g. gold, silver, and copper), but also across asset classes (e.g. precious metals and volatility). <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Futures portfolios with multiple underlying assets give rise to some new challenges. With more assets and trading instruments, the dimension of the portfolio optimization problem is significantly increased. Furthermore, one must address the dependency structure among the underlying assets and their futures, even if the underlying assets are not traded.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This calls for a stochastic model that can capture the correlation among all the futures and underlying assets while maintaining analytical tractability, numerical efficiency, and interpretability. Motivated by these observations, we introduce in this paper an alternative way to model the <em>joint<\/em> <em>price dynamics <\/em>of the underlying assets and associated futures.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Methodology<\/strong><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">The key idea is as follows. In practice, market frictions and inefficiencies may render the futures price different from the spot price prior to maturity. For each futures contract, the spread between the two prices is called the basis. By no-arbitrage theory, futures prices must converge to the spot price at expiry, so the basis process is expected to converge to zero as the associated futures contract expires as well. This price behavior leads us to (i) express each futures price process through the associated basis, and (ii) model the random basis using a Brownian bridge. For multiple futures contracts, we consider a multidimensional Brownian bridge, where each component converges to zero at the respective maturity.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In our new <a href=\"https:\/\/papers.ssrn.com\/sol3\/papers.cfm?abstract_id=3942191\">article<\/a>, <em>Constrained Dynamic Futures Portfolios with Stochastic Basis<\/em> (see also References below), we analyze the problem of dynamically trading futures with different underlying assets under the stochastic basis model driven by a multidimensional Brownian bridge.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Another key element of our utility maximization problem is the incorporation of portfolio constraints on the futures positions. Our general portfolio setup captures the dollar neutral and market neutral constraints, which are widely used in industry. The optimal strategies for both unconstrained and constrained cases are derived. This is achieved by solving the associated Hamilton-Jacobi-Bellman (HJB) equations. Moreover, we show that the original nonlinear HJB equations are reduced to a system of linear ODEs that can be solved instantly, which in turn generate the optimal futures positions.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Numerical<\/strong> <strong>Example<\/strong><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">In the paper, we present a numerical example to illustrate the results. We consider a market with two different assets and three futures. The trading horizon is set to be strictly less than the futures maturities.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In Figure 1, we display the simulated paths of the underlying asset price, futures price, and corresponding log-bases. In Figure 2, we show the terminal wealth distribution, based on Monte Carlo simulation of the stochastic basis model. In this example, we observe that the portfolio constraint can effectively reduce the tails (and standard deviation) of the wealth distribution.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"589\" height=\"776\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2021\/10\/dynamic-futures-stochastic-basis.png\" alt=\"Constrained Dynamic Futures Portfolios with Stochastic Basis\" class=\"wp-image-108122 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/10\/dynamic-futures-stochastic-basis.png 589w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/10\/dynamic-futures-stochastic-basis-300x395.png 300w\" data-sizes=\"(max-width: 589px) 100vw, 589px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 589px; aspect-ratio: 589\/776;\" \/><\/figure>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><em>Fig. 1: Simulated paths of the underlying asset price, futures price, and corresponding log-bases. Source: Chen et. al (2021) (see References below)<\/em><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"624\" height=\"203\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2021\/10\/dynamic-futures-stochastic-basis-monte-carlo.png\" alt=\"\" class=\"wp-image-108128 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/10\/dynamic-futures-stochastic-basis-monte-carlo.png 624w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/10\/dynamic-futures-stochastic-basis-monte-carlo-300x98.png 300w\" data-sizes=\"(max-width: 624px) 100vw, 624px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 624px; aspect-ratio: 624\/203;\" \/><\/figure>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><em>Fig. 2: The distributions of terminal wealth (based on Monte Carlo simulation) for the unconstrained and constrained portfolios. Source: Chen et. al (2021) (see References below)<\/em><\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>References:<\/strong><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Xiaodong Chen, Tim Leung, and Yang Zhou (2021), Constrained Dynamic Futures Portfolios with Stochastic Basis [<a href=\"https:\/\/papers.ssrn.com\/sol3\/papers.cfm?abstract_id=3942191\">pdf<\/a>], to appear, <strong>Annals of Finance<\/strong><\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Related Articles:<\/strong><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">T. Leung and B. Angoshtari (2021), Optimal Trading of a Basket of Futures Contracts [<a href=\"https:\/\/papers.ssrn.com\/sol3\/papers.cfm?abstract_id=3467897\">pdf<\/a>], Vol.16, pages 253\u2013280, <strong>Annals of Finance<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">T. Leung and Y. Zhou (2021), Optimal Dynamic Futures Portfolio Under a Multifactor Gaussian Framework [<a href=\"https:\/\/papers.ssrn.com\/sol3\/papers.cfm?abstract_id=3905578\">pdf<\/a>], Vol. 24, No. 5, p.2150028,<strong> International Journal of Theoretical &amp; Applied Finance<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"<p>The futures market offers trading instruments on a very diverse range of underlying assets, including precious metals, agricultural commodities, oil and gas, equity, interest rates, volatility, cryptocurrencies, and more.<\/p>\n","protected":false},"author":189,"featured_media":108139,"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":[10536,10537,4922,10340,10538,10539,9780],"contributors-categories":[13668],"class_list":["post-108115","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-applied-finance","tag-constrained-dynamic-futures-portfolios","tag-econometrics","tag-hamilton-jacobi-bellman-hjb-equations","tag-monte-carlo-simulation","tag-multidimensional-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 v27.8) - https:\/\/yoast.com\/product\/yoast-seo-premium-wordpress\/ -->\n<title>Constrained Dynamic Futures Portfolios with Stochastic Basis<\/title>\n<meta name=\"description\" content=\"The futures market offers trading instruments on a very diverse range of underlying assets, including precious metals, agricultural commodities, oil and...\" \/>\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\/108115\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Constrained Dynamic Futures Portfolios with Stochastic Basis | IBKR Quant Blog\" \/>\n<meta property=\"og:description\" content=\"The futures market offers trading instruments on a very diverse range of underlying assets, including precious metals, agricultural commodities, oil and gas, equity, interest rates, volatility, cryptocurrencies, and more.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/constrained-dynamic-futures-portfolios-with-stochastic-basis\/\" \/>\n<meta property=\"og:site_name\" content=\"IBKR Campus US\" \/>\n<meta property=\"article:published_time\" content=\"2021-10-22T14:05:00+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2022-11-21T14:48:39+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/10\/quant-futures.jpg\" \/>\n\t<meta property=\"og:image:width\" content=\"1000\" \/>\n\t<meta property=\"og:image:height\" content=\"563\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/jpeg\" \/>\n<meta name=\"author\" content=\"Tim Leung\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"Tim Leung\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"4 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\n\t    \"@context\": \"https:\\\/\\\/schema.org\",\n\t    \"@graph\": [\n\t        {\n\t            \"@type\": \"NewsArticle\",\n\t            \"@id\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/ibkr-quant-news\\\/constrained-dynamic-futures-portfolios-with-stochastic-basis\\\/#article\",\n\t            \"isPartOf\": {\n\t                \"@id\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/ibkr-quant-news\\\/constrained-dynamic-futures-portfolios-with-stochastic-basis\\\/\"\n\t            },\n\t            \"author\": {\n\t                \"name\": \"Tim Leung\",\n\t                \"@id\": \"https:\\\/\\\/ibkrcampus.com\\\/campus\\\/#\\\/schema\\\/person\\\/63efb8a271eea609385bade5cbb12b7b\"\n\t            },\n\t            \"headline\": \"Constrained Dynamic Futures Portfolios with Stochastic Basis\",\n\t            \"datePublished\": \"2021-10-22T14:05:00+00:00\",\n\t            \"dateModified\": \"2022-11-21T14:48:39+00:00\",\n\t            \"mainEntityOfPage\": {\n\t                \"@id\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/ibkr-quant-news\\\/constrained-dynamic-futures-portfolios-with-stochastic-basis\\\/\"\n\t            },\n\t            \"wordCount\": 629,\n\t            \"publisher\": {\n\t                \"@id\": \"https:\\\/\\\/ibkrcampus.com\\\/campus\\\/#organization\"\n\t            },\n\t            \"image\": {\n\t                \"@id\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/ibkr-quant-news\\\/constrained-dynamic-futures-portfolios-with-stochastic-basis\\\/#primaryimage\"\n\t            },\n\t            \"thumbnailUrl\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/wp-content\\\/uploads\\\/sites\\\/2\\\/2021\\\/10\\\/quant-futures.jpg\",\n\t            \"keywords\": [\n\t                \"Applied Finance\",\n\t                \"Constrained Dynamic Futures Portfolios\",\n\t                \"Econometrics\",\n\t                \"Hamilton-Jacobi-Bellman (HJB) equations\",\n\t                \"Monte Carlo simulation\",\n\t                \"multidimensional Brownian bridge\",\n\t                \"Stochastic Basis\"\n\t            ],\n\t            \"articleSection\": [\n\t                \"Data Science\",\n\t                \"Quant\",\n\t                \"Quant Development\",\n\t                \"Quant North America\",\n\t                \"Quant Regions\"\n\t            ],\n\t            \"inLanguage\": \"en-US\"\n\t        },\n\t        {\n\t            \"@type\": \"WebPage\",\n\t            \"@id\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/ibkr-quant-news\\\/constrained-dynamic-futures-portfolios-with-stochastic-basis\\\/\",\n\t            \"url\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/ibkr-quant-news\\\/constrained-dynamic-futures-portfolios-with-stochastic-basis\\\/\",\n\t            \"name\": \"Constrained Dynamic Futures Portfolios with Stochastic Basis | IBKR Quant Blog\",\n\t            \"isPartOf\": {\n\t                \"@id\": \"https:\\\/\\\/ibkrcampus.com\\\/campus\\\/#website\"\n\t            },\n\t            \"primaryImageOfPage\": {\n\t                \"@id\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/ibkr-quant-news\\\/constrained-dynamic-futures-portfolios-with-stochastic-basis\\\/#primaryimage\"\n\t            },\n\t            \"image\": {\n\t                \"@id\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/ibkr-quant-news\\\/constrained-dynamic-futures-portfolios-with-stochastic-basis\\\/#primaryimage\"\n\t            },\n\t            \"thumbnailUrl\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/wp-content\\\/uploads\\\/sites\\\/2\\\/2021\\\/10\\\/quant-futures.jpg\",\n\t            \"datePublished\": \"2021-10-22T14:05:00+00:00\",\n\t            \"dateModified\": \"2022-11-21T14:48:39+00:00\",\n\t            \"description\": \"The futures market offers trading instruments on a very diverse range of underlying assets, including precious metals, agricultural commodities, oil and gas, equity, interest rates, volatility, cryptocurrencies, and more.\",\n\t            \"inLanguage\": \"en-US\",\n\t            \"potentialAction\": [\n\t                {\n\t                    \"@type\": \"ReadAction\",\n\t                    \"target\": [\n\t                        \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/ibkr-quant-news\\\/constrained-dynamic-futures-portfolios-with-stochastic-basis\\\/\"\n\t                    ]\n\t                }\n\t            ]\n\t        },\n\t        {\n\t            \"@type\": \"ImageObject\",\n\t            \"inLanguage\": \"en-US\",\n\t            \"@id\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/ibkr-quant-news\\\/constrained-dynamic-futures-portfolios-with-stochastic-basis\\\/#primaryimage\",\n\t            \"url\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/wp-content\\\/uploads\\\/sites\\\/2\\\/2021\\\/10\\\/quant-futures.jpg\",\n\t            \"contentUrl\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/wp-content\\\/uploads\\\/sites\\\/2\\\/2021\\\/10\\\/quant-futures.jpg\",\n\t            \"width\": 1000,\n\t            \"height\": 563,\n\t            \"caption\": \"Quant\"\n\t        },\n\t        {\n\t            \"@type\": \"WebSite\",\n\t            \"@id\": \"https:\\\/\\\/ibkrcampus.com\\\/campus\\\/#website\",\n\t            \"url\": \"https:\\\/\\\/ibkrcampus.com\\\/campus\\\/\",\n\t            \"name\": \"IBKR Campus US\",\n\t            \"description\": \"Financial Education from Interactive Brokers\",\n\t            \"publisher\": {\n\t                \"@id\": \"https:\\\/\\\/ibkrcampus.com\\\/campus\\\/#organization\"\n\t            },\n\t            \"potentialAction\": [\n\t                {\n\t                    \"@type\": \"SearchAction\",\n\t                    \"target\": {\n\t                        \"@type\": \"EntryPoint\",\n\t                        \"urlTemplate\": \"https:\\\/\\\/ibkrcampus.com\\\/campus\\\/?s={search_term_string}\"\n\t                    },\n\t                    \"query-input\": {\n\t                        \"@type\": \"PropertyValueSpecification\",\n\t                        \"valueRequired\": true,\n\t                        \"valueName\": \"search_term_string\"\n\t                    }\n\t                }\n\t            ],\n\t            \"inLanguage\": \"en-US\"\n\t        },\n\t        {\n\t            \"@type\": \"Organization\",\n\t            \"@id\": \"https:\\\/\\\/ibkrcampus.com\\\/campus\\\/#organization\",\n\t            \"name\": \"Interactive Brokers\",\n\t            \"alternateName\": \"IBKR\",\n\t            \"url\": \"https:\\\/\\\/ibkrcampus.com\\\/campus\\\/\",\n\t            \"logo\": {\n\t                \"@type\": \"ImageObject\",\n\t                \"inLanguage\": \"en-US\",\n\t                \"@id\": \"https:\\\/\\\/ibkrcampus.com\\\/campus\\\/#\\\/schema\\\/logo\\\/image\\\/\",\n\t                \"url\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/wp-content\\\/uploads\\\/sites\\\/2\\\/2024\\\/05\\\/ibkr-campus-logo.jpg\",\n\t                \"contentUrl\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/wp-content\\\/uploads\\\/sites\\\/2\\\/2024\\\/05\\\/ibkr-campus-logo.jpg\",\n\t                \"width\": 669,\n\t                \"height\": 669,\n\t                \"caption\": \"Interactive Brokers\"\n\t            },\n\t            \"image\": {\n\t                \"@id\": \"https:\\\/\\\/ibkrcampus.com\\\/campus\\\/#\\\/schema\\\/logo\\\/image\\\/\"\n\t            },\n\t            \"publishingPrinciples\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/about-ibkr-campus\\\/\",\n\t            \"ethicsPolicy\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/cyber-security-notice\\\/\"\n\t        },\n\t        {\n\t            \"@type\": \"Person\",\n\t            \"@id\": \"https:\\\/\\\/ibkrcampus.com\\\/campus\\\/#\\\/schema\\\/person\\\/63efb8a271eea609385bade5cbb12b7b\",\n\t            \"name\": \"Tim Leung\",\n\t            \"description\": \"Prof. Tim Leung is the Director, Computational Finance &amp; Risk Management (CFRM) Program https:\\\/\\\/faculty.washington.edu\\\/timleung\\\/\",\n\t            \"sameAs\": [\n\t                \"https:\\\/\\\/faculty.washington.edu\\\/timleung\\\/\"\n\t            ],\n\t            \"url\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/author\\\/timleung\\\/\"\n\t        }\n\t    ]\n\t}<\/script>\n<!-- \/ Yoast SEO Premium plugin. -->","yoast_head_json":{"title":"Constrained Dynamic Futures Portfolios with Stochastic Basis","description":"The futures market offers trading instruments on a very diverse range of underlying assets, including precious metals, agricultural commodities, oil and...","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/www.interactivebrokers.com\/campus\/wp-json\/wp\/v2\/posts\/108115\/","og_locale":"en_US","og_type":"article","og_title":"Constrained Dynamic Futures Portfolios with Stochastic Basis | IBKR Quant Blog","og_description":"The futures market offers trading instruments on a very diverse range of underlying assets, including precious metals, agricultural commodities, oil and gas, equity, interest rates, volatility, cryptocurrencies, and more.","og_url":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/constrained-dynamic-futures-portfolios-with-stochastic-basis\/","og_site_name":"IBKR Campus US","article_published_time":"2021-10-22T14:05:00+00:00","article_modified_time":"2022-11-21T14:48:39+00:00","og_image":[{"width":1000,"height":563,"url":"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/10\/quant-futures.jpg","type":"image\/jpeg"}],"author":"Tim Leung","twitter_card":"summary_large_image","twitter_misc":{"Written by":"Tim Leung","Est. reading time":"4 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"NewsArticle","@id":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/constrained-dynamic-futures-portfolios-with-stochastic-basis\/#article","isPartOf":{"@id":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/constrained-dynamic-futures-portfolios-with-stochastic-basis\/"},"author":{"name":"Tim Leung","@id":"https:\/\/ibkrcampus.com\/campus\/#\/schema\/person\/63efb8a271eea609385bade5cbb12b7b"},"headline":"Constrained Dynamic Futures Portfolios with Stochastic Basis","datePublished":"2021-10-22T14:05:00+00:00","dateModified":"2022-11-21T14:48:39+00:00","mainEntityOfPage":{"@id":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/constrained-dynamic-futures-portfolios-with-stochastic-basis\/"},"wordCount":629,"publisher":{"@id":"https:\/\/ibkrcampus.com\/campus\/#organization"},"image":{"@id":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/constrained-dynamic-futures-portfolios-with-stochastic-basis\/#primaryimage"},"thumbnailUrl":"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/10\/quant-futures.jpg","keywords":["Applied Finance","Constrained Dynamic Futures Portfolios","Econometrics","Hamilton-Jacobi-Bellman (HJB) equations","Monte Carlo simulation","multidimensional Brownian bridge","Stochastic Basis"],"articleSection":["Data Science","Quant","Quant Development","Quant North America","Quant Regions"],"inLanguage":"en-US"},{"@type":"WebPage","@id":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/constrained-dynamic-futures-portfolios-with-stochastic-basis\/","url":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/constrained-dynamic-futures-portfolios-with-stochastic-basis\/","name":"Constrained Dynamic Futures Portfolios with Stochastic Basis | IBKR Quant Blog","isPartOf":{"@id":"https:\/\/ibkrcampus.com\/campus\/#website"},"primaryImageOfPage":{"@id":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/constrained-dynamic-futures-portfolios-with-stochastic-basis\/#primaryimage"},"image":{"@id":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/constrained-dynamic-futures-portfolios-with-stochastic-basis\/#primaryimage"},"thumbnailUrl":"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/10\/quant-futures.jpg","datePublished":"2021-10-22T14:05:00+00:00","dateModified":"2022-11-21T14:48:39+00:00","description":"The futures market offers trading instruments on a very diverse range of underlying assets, including precious metals, agricultural commodities, oil and gas, equity, interest rates, volatility, cryptocurrencies, and more.","inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/constrained-dynamic-futures-portfolios-with-stochastic-basis\/"]}]},{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/constrained-dynamic-futures-portfolios-with-stochastic-basis\/#primaryimage","url":"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/10\/quant-futures.jpg","contentUrl":"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/10\/quant-futures.jpg","width":1000,"height":563,"caption":"Quant"},{"@type":"WebSite","@id":"https:\/\/ibkrcampus.com\/campus\/#website","url":"https:\/\/ibkrcampus.com\/campus\/","name":"IBKR Campus US","description":"Financial Education from Interactive Brokers","publisher":{"@id":"https:\/\/ibkrcampus.com\/campus\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/ibkrcampus.com\/campus\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"en-US"},{"@type":"Organization","@id":"https:\/\/ibkrcampus.com\/campus\/#organization","name":"Interactive Brokers","alternateName":"IBKR","url":"https:\/\/ibkrcampus.com\/campus\/","logo":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/ibkrcampus.com\/campus\/#\/schema\/logo\/image\/","url":"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/05\/ibkr-campus-logo.jpg","contentUrl":"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/05\/ibkr-campus-logo.jpg","width":669,"height":669,"caption":"Interactive Brokers"},"image":{"@id":"https:\/\/ibkrcampus.com\/campus\/#\/schema\/logo\/image\/"},"publishingPrinciples":"https:\/\/www.interactivebrokers.com\/campus\/about-ibkr-campus\/","ethicsPolicy":"https:\/\/www.interactivebrokers.com\/campus\/cyber-security-notice\/"},{"@type":"Person","@id":"https:\/\/ibkrcampus.com\/campus\/#\/schema\/person\/63efb8a271eea609385bade5cbb12b7b","name":"Tim Leung","description":"Prof. Tim Leung is the Director, Computational Finance &amp; Risk Management (CFRM) Program https:\/\/faculty.washington.edu\/timleung\/","sameAs":["https:\/\/faculty.washington.edu\/timleung\/"],"url":"https:\/\/www.interactivebrokers.com\/campus\/author\/timleung\/"}]}},"jetpack_featured_media_url":"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/10\/quant-futures.jpg","_links":{"self":[{"href":"https:\/\/ibkrcampus.com\/campus\/wp-json\/wp\/v2\/posts\/108115","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ibkrcampus.com\/campus\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/ibkrcampus.com\/campus\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/ibkrcampus.com\/campus\/wp-json\/wp\/v2\/users\/189"}],"replies":[{"embeddable":true,"href":"https:\/\/ibkrcampus.com\/campus\/wp-json\/wp\/v2\/comments?post=108115"}],"version-history":[{"count":0,"href":"https:\/\/ibkrcampus.com\/campus\/wp-json\/wp\/v2\/posts\/108115\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/ibkrcampus.com\/campus\/wp-json\/wp\/v2\/media\/108139"}],"wp:attachment":[{"href":"https:\/\/ibkrcampus.com\/campus\/wp-json\/wp\/v2\/media?parent=108115"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/ibkrcampus.com\/campus\/wp-json\/wp\/v2\/categories?post=108115"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/ibkrcampus.com\/campus\/wp-json\/wp\/v2\/tags?post=108115"},{"taxonomy":"contributors-categories","embeddable":true,"href":"https:\/\/ibkrcampus.com\/campus\/wp-json\/wp\/v2\/contributors-categories?post=108115"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}