Black Scholes PDE in forward log space - Quantitative Finance Stack Exchange

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post." aria-label="Timeline"><svg aria-hidden="true" class="mln2 mr0 svg-icon iconHistory" width="19" height="18" viewbox="0 0 19 18"><path d="M3 9a8 8 0 1 1 3.73 6.77L8.2 14.3A6 6 0 1 0 5 9l3.01-.01-4 4-4-4h3L3 9Zm7-4h1.01L11 9.36l3.22 2.1-.6.93L10 10V5Z"/></svg></a> </div> </div> <div class="postcell post-layout--right"> <span class="d-none">$\begingroup$</span> <div class="s-prose js-post-body" itemprop="text"> <p>In BS world, we have the stock process in log space <span class="math-container">$dS_t=(r-\frac{1}{2}\sigma^2)dt+\sigma dW$</span>. Let's say we want to price <span class="math-container">$f(t,x)=\mathbb{E}_{t,x}[h(S(T)]$</span>. Using Feynman-kac, we get <span class="math-container">\begin{equation} \frac{\partial f}{\partial t} + (r-\frac{1}{2}\sigma^2)\frac{\partial f}{\partial x}+\frac{1}{2} \sigma^2 \frac{\partial^2 f}{\partial x^2}-rV=0 \end{equation}</span></p> <p>On the other hand, if we consider the forward process (again in log space) <span class="math-container">$F_t=S_t+r(T-t)$</span>, we have the forward process <span class="math-container">$dF_t=-\frac{1}{2}\sigma^2 dt+\sigma dW$</span> and the price becomes <span class="math-container">$f(t,y)=\mathbb{E}_{t,y}[h(F(T)]$</span>. Using F-K again, we get <span class="math-container">\begin{equation} \frac{\partial f}{\partial t} - \frac{1}{2}\sigma^2\frac{\partial f}{\partial y}+\frac{1}{2} \sigma^2 \frac{\partial^2 f}{\partial y^2}-rV=0 \end{equation}</span></p> <p>Somehow I fail to transform the first PDE to the second by change of variable directly from <span class="math-container">$S_t$</span> to <span class="math-container">$F_t$</span>. Since <span class="math-container">$y=x+r(T-t)$</span>, by chain rule, <span class="math-container">$\frac{\partial f}{\partial x}=\frac{\partial f}{\partial x}\frac{\partial y}{\partial x}=\frac{\partial f}{\partial y}$</span>, i.e., the first order is the same and so as the second order. So I end up with <span class="math-container">\begin{equation} \frac{\partial f}{\partial t} +(r-\frac{1}{2}\sigma^2)\frac{\partial f}{\partial y}+\frac{1}{2} \sigma^2 \frac{\partial^2 f}{\partial y^2}-rV=0 \end{equation}</span> which is obviously wrong and I couldn't figure out why.</p> </div> <div class="mt24 mb12"> <div class="post-taglist d-flex gs4 gsy fd-column"> <div class="d-flex ps-relative fw-wrap"> <a href="/web/20220202032205/https://quant.stackexchange.com/questions/tagged/black-scholes" class="post-tag" title="show questions tagged 'black-scholes'" rel="tag">black-scholes</a> <a href="/web/20220202032205/https://quant.stackexchange.com/questions/tagged/pde" class="post-tag" title="show questions tagged 'pde'" rel="tag">pde</a> <a href="/web/20220202032205/https://quant.stackexchange.com/questions/tagged/feynman-kac" class="post-tag" title="show questions tagged 'feynman-kac'" rel="tag">feynman-kac</a> </div> </div> </div> <div class="mb0 "> <div class="mt16 d-flex gs8 gsy fw-wrap jc-end ai-start pt4 mb16"> <div class="flex--item mr16 fl1 w96"> <div class="js-post-menu pt2" data-post-id="69708"> <div class="d-flex gs8 s-anchors s-anchors__muted fw-wrap"> <div class="flex--item"> <a href="/web/20220202032205/https://quant.stackexchange.com/q/69708" rel="nofollow" itemprop="url" class="js-share-link js-gps-track" title="Short permalink to this question" data-gps-track="post.click({ item: 2, priv: 0, post_type: 1 })" data-controller="se-share-sheet" data-se-share-sheet-title="Share a link to this question" data-se-share-sheet-subtitle="" data-se-share-sheet-post-type="question" data-se-share-sheet-social="facebook twitter " data-se-share-sheet-location="1" data-se-share-sheet-license-url="https%3a%2f%2fcreativecommons.org%2flicenses%2fby-sa%2f4.0%2f" data-se-share-sheet-license-name="CC BY-SA 4.0" data-s-popover-placement="bottom-start">Share</a> </div> <div class="flex--item"> <a href="/web/20220202032205/https://quant.stackexchange.com/posts/69708/edit" class="js-suggest-edit-post js-gps-track" data-gps-track="post.click({ item: 6, priv: 0, post_type: 1 })" title="">Improve this question</a> </div> <div class="flex--item"> <button type="button" id="btnFollowPost-69708" class="s-btn s-btn__link js-follow-post js-follow-question js-gps-track" data-gps-track="post.click({ item: 14, priv: 0, post_type: 1 })" data-controller="s-tooltip " data-s-tooltip-placement="bottom" data-s-popover-placement="bottom" aria-controls="" title="Follow this question to receive notifications"> Follow </button> </div> </div> <div class="js-menu-popup-container"></div> </div> </div> <div class="post-signature flex--item"> <div class="user-info "> <div class="user-action-time"> <a href="/web/20220202032205/https://quant.stackexchange.com/posts/69708/revisions" title="show all edits to this post" class="js-gps-track" data-gps-track="post.click({ item: 4, priv: 0, post_type: 1 })">edited <span title="2022-02-01 04:01:30Z" class="relativetime">23 hours ago</span></a> </div> <div class="user-gravatar32"> </div> <div class="user-details" itemprop="author" itemscope itemtype="http://schema.org/Person"> <span class="d-none" itemprop="name">J. Lin</span> <div class="-flair"> </div> </div> </div> </div> <div class="post-signature owner flex--item"> <div class="user-info "> <div class="user-action-time"> asked <span title="2022-02-01 03:47:31Z" class="relativetime">23 hours ago</span> </div> <div class="user-gravatar32"> <a href="/web/20220202032205/https://quant.stackexchange.com/users/39681/j-lin"><div class="gravatar-wrapper-32"><img src="https://web.archive.org/web/20220202032205im_/https://www.gravatar.com/avatar/9f98902a9cb799d9fc920c827387ebe8?s=64&d=identicon&r=PG&f=1" alt="" width="32" height="32" class="bar-sm"></div></a> </div> <div class="user-details" itemprop="author" itemscope itemtype="http://schema.org/Person"> <a href="/web/20220202032205/https://quant.stackexchange.com/users/39681/j-lin">J. Lin</a><span class="d-none" itemprop="name">J. Lin</span> <div class="-flair"> <span class="reputation-score" title="reputation score " dir="ltr">25</span><span title="4 bronze badges" aria-hidden="true"><span class="badge3"></span><span class="badgecount">4</span></span><span class="v-visible-sr">4 bronze badges</span> </div> </div> </div> </div> </div> </div> <span class="d-none">$\endgroup$</span> </div> <span class="d-none" itemprop="commentCount"></span> <div class="post-layout--right js-post-comments-component"> <div id="comments-69708" class="comments js-comments-container bt bc-black-075 mt12 dno" data-post-id="69708" data-min-length="15"> <ul class="comments-list js-comments-list" data-remaining-comments-count="0" data-canpost="false" data-cansee="true" data-comments-unavailable="false" data-addlink-disabled="true"> </ul> </div> <div id="comments-link-69708" data-rep="50" data-anon="true"> <a class="js-add-link comments-link disabled-link" title="Use comments to ask for more information or suggest improvements. 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So first of all <span class="math-container">$$ f(t,x) := E_t (h(S_T)) $$</span> and <span class="math-container">$$ g(t, y) := E_t (h(F_T)). $$</span> Since <span class="math-container">$S_T = F_T$</span>, by no arbitrage we must have <span class="math-container">$$ g(t,y) = f(t,x) = f(t, y - r(T-t)). $$</span></p> <p>This means that <span class="math-container">$$ \frac{\partial g}{\partial t} = \frac{\partial f}{\partial t} + r \frac{\partial f}{\partial x}. $$</span> As you've already pointed out <span class="math-container">$$ \frac{\partial g}{\partial y} = \frac{\partial f}{\partial x}. $$</span> Using this and the PDE satisfied by <span class="math-container">$f$</span> you will then obtain the following PDE for <span class="math-container">$g$</span>: <span class="math-container">$$ \frac{\partial g}{\partial t} -\frac12 \sigma^2 \left( \frac{\partial g}{\partial y} - \frac{\partial^2 g}{\partial y^2}\right) = rg $$</span></p> </div> <div class="mt24"> <div class="d-flex 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href="/web/20220202032205/https://quant.stackexchange.com/users/34971/frido-rolloos"><div class="gravatar-wrapper-32"><img src="https://web.archive.org/web/20220202032205im_/https://i.stack.imgur.com/q0wFx.jpg?s=64&g=1" alt="" width="32" height="32" class="bar-sm"></div></a> </div> <div class="user-details" itemprop="author" itemscope itemtype="http://schema.org/Person"> <a href="/web/20220202032205/https://quant.stackexchange.com/users/34971/frido-rolloos">Frido Rolloos</a><span class="d-none" itemprop="name">Frido Rolloos</span> <div class="-flair"> <span class="reputation-score" title="reputation score " dir="ltr">4,261</span><span title="1 gold badge" aria-hidden="true"><span class="badge1"></span><span class="badgecount">1</span></span><span class="v-visible-sr">1 gold badge</span><span title="9 silver badges" aria-hidden="true"><span class="badge2"></span><span class="badgecount">9</span></span><span class="v-visible-sr">9 silver badges</span><span title="28 bronze badges" aria-hidden="true"><span class="badge3"></span><span class="badgecount">28</span></span><span class="v-visible-sr">28 bronze badges</span> </div> </div> </div> </div> </div> </div> <span class="d-none">$\endgroup$</span> </div> <span class="d-none" itemprop="commentCount">1</span> <div class="post-layout--right js-post-comments-component"> <div id="comments-69713" class="comments js-comments-container bt bc-black-075 mt12 " data-post-id="69713" data-min-length="15"> <ul class="comments-list js-comments-list" data-remaining-comments-count="0" data-canpost="false" data-cansee="true" data-comments-unavailable="false" data-addlink-disabled="true"> <li id="comment-101542" class="comment js-comment " data-comment-id="101542" data-comment-owner-id="39681" data-comment-score="0"> <div class="js-comment-actions comment-actions"> <div class="comment-score js-comment-edit-hide"> </div> </div> <div class="comment-text js-comment-text-and-form"> <div class="comment-body js-comment-edit-hide"> <span class="d-none">$\begingroup$</span> <span class="comment-copy">Indeed, it's about notation. Thanks a lot!</span> <span class="d-none">$\endgroup$</span> <div class="d-inline-flex ai-center"> – <a href="/web/20220202032205/https://quant.stackexchange.com/users/39681/j-lin" title="25 reputation" class="comment-user owner">J. Lin</a> </div> <span class="comment-date" dir="ltr"><span title="2022-02-01 13:50:13Z, License: CC BY-SA 4.0" class="relativetime-clean">13 hours ago</span></span> </div> </div> </li> </ul> </div> <div id="comments-link-69713" data-rep="50" data-anon="true"> <a class="js-add-link comments-link disabled-link" title="Use comments to ask for more information or suggest improvements. 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