diff --git a/docs/um1.html b/docs/um1.html
index 1e75089..e2cf78c 100644
--- a/docs/um1.html
+++ b/docs/um1.html
@@ -5,7 +5,7 @@
   <meta name="generator" content="pandoc" />
   <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" />
   <meta name="author" content="Keith A. Lewis" />
-  <meta name="dcterms.date" content="2025-02-12" />
+  <meta name="dcterms.date" content="2025-02-15" />
   <title>Simple Unified Model</title>
   <style>
     code{white-space: pre-wrap;}
@@ -66,7 +66,7 @@
 <header id="title-block-header">
 <h1 class="title">Simple Unified Model</h1>
 <p class="author">Keith A. Lewis</p>
-<p class="date">February 12, 2025</p>
+<p class="date">February 15, 2025</p>
 <div class="abstract">
 <div class="abstract-title">Abstract</div>
 A mathematically rigorous framework for valuing, hedging, and managing
diff --git a/docs/ums.html b/docs/ums.html
index 57faf73..59ac2d1 100644
--- a/docs/ums.html
+++ b/docs/ums.html
@@ -4,8 +4,8 @@
   <meta charset="utf-8" />
   <meta name="generator" content="pandoc" />
   <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" />
-  <meta name="dcterms.date" content="2025-01-26" />
-  <title>A Unified Model of Derivative Securities</title>
+  <meta name="dcterms.date" content="2025-02-15" />
+  <title>Simple Unified Model of Derivative Securities</title>
   <style>
     code{white-space: pre-wrap;}
     span.smallcaps{font-variant: small-caps;}
@@ -44,84 +44,151 @@
 </head>
 <body>
 <header id="title-block-header">
-<h1 class="title">A Unified Model of Derivative Securities</h1>
-<p class="date">January 26, 2025</p>
+<h1 class="title">Simple Unified Model of Derivative Securities</h1>
+<p class="date">February 15, 2025</p>
 </header>
-<nav id="TOC" role="doc-toc">
-<ul>
-<li><a href="#appendix" id="toc-appendix">Appendix</a></li>
-</ul>
-</nav>
-<p><span class="math inline">T</span> — totally ordered set of trading
-<em>times</em>.</p>
+<!--
+:!pandoc -t html5 -s --katex=https://cdn.jsdelivr.net/npm/katex@0.12.0/dist/ --css math.css ums.md -o ums.html
+:!pandoc -V fontsize=12pt ums.md -o ums.pdf
+-->
+<p><span class="math inline">T</span> — totally ordered set of
+<em>trading times</em>.</p>
 <p><span class="math inline">I</span> — market <em>instruments</em>.</p>
-<p><span class="math inline">\Omega</span> — all possible
+<p><span class="math inline">\Omega</span> — possible
 <em>outcomes</em>.</p>
-<p><span class="math inline">\mathcal{A}_t</span> — a partition of <span
-class="math inline">\Omega</span> indicating the <em>information</em>
-available at time <span class="math inline">t\in T</span>.</p>
+<p><span class="math inline">(\mathcal{A}_t)_{t\in T}</span> —
+<em>partitions</em><a href="#fn1" class="footnote-ref" id="fnref1"
+role="doc-noteref"><sup>1</sup></a> of <span
+class="math inline">\Omega</span> indicating the information available
+at time <span class="math inline">t\in T</span>.</p>
 <p><span
 class="math inline">X_t\colon\mathcal{A}_t\to\boldsymbol{R}^I</span> —
-<em>prices</em> at time <span class="math inline">t</span> of market
-instruments.</p>
+bounded <em>prices</em><a href="#fn2" class="footnote-ref" id="fnref2"
+role="doc-noteref"><sup>2</sup></a> at times <span
+class="math inline">t\in T</span> of market instruments.</p>
 <p><span
 class="math inline">C_t\colon\mathcal{A}_t\to\boldsymbol{R}^I</span> —
-<em>cash flows</em>, usually 0, at time <span
-class="math inline">t</span> of market instruments.</p>
-<p><span class="math inline">(\tau_j, \Gamma_j)</span> — a finite
-<em>trading strategy</em> of strictly increasing stopping times <span
-class="math inline">\tau_j</span> and corresponding trades <span
+bounded <em>cash flows</em> at times <span class="math inline">t\in
+T</span> of market instruments.</p>
+<p>E.g., coupons, dividends, and futures margin adjustments are cash
+flows.</p>
+<p><span class="math inline">(\tau_j, \Gamma_j)</span> — <em>trading
+strategy</em> of strictly increasing stopping times <span
+class="math inline">\tau_j</span> and shares <span
 class="math inline">\Gamma_j\colon\mathcal{A}_{\tau_j}\to\boldsymbol{R}^I</span>
-with <span class="math inline">\sum_j \Gamma_j = 0</span>.</p>
-<p><span class="math inline">\Delta_t = \sum_{\tau_j &lt; t} \Gamma_j =
-\sum_{s &lt; t} \Gamma_s</span> where <span class="math inline">\Gamma_s
-= \Gamma_j 1(\tau_j = s)</span> — the <em>position</em> resulting from
-trading.</p>
+purchased at <span class="math inline">\tau_j</span>.</p>
+<p>The <em>position</em> resulting from a trading strategy is <span
+class="math inline">\Delta_t = \sum_{\tau_j &lt; t} \Gamma_j = \sum_{s
+&lt; t} \Gamma_s</span> where <span class="math inline">\Gamma_s =
+\Gamma_j 1(\tau_j = s)</span>. Note the strict inequality. It takes time
+for a trade to settle and become a part of the position.</p>
 <p><span class="math inline">V_t = (\Delta_t + \Gamma_t)\cdot X_t</span>
-— the <em>value</em>, or mark-to-market, of the strategy.</p>
+— the <em>value</em>, or <em>mark-to-market</em>, of the strategy.</p>
 <p><span class="math inline">A_t = \Delta_t\cdot C_t - \Gamma_t\cdot
-X_t</span> — the amount showing up in the trading <em>account</em>.</p>
+X_t</span> — the <em>amount</em> showing up in the trading account.</p>
 <p>Arbitrage exists if there is a trading strategy with <span
-class="math inline">A_{\tau_0} &gt; 0</span> and <span
+class="math inline">A_{\tau_0} &gt; 0</span>, <span
 class="math inline">A_t \ge0</span>, <span class="math inline">t &gt;
-\tau_0</span>.</p>
-<p><strong>Fundamental Theorem of Asset Pricing</strong>. There is no
-arbitrage if there exist positive measures <span
+\tau_0</span>, and <span class="math inline">\sum_j \Gamma_j = 0</span>.
+You make money on the first trade and never lose money until the
+position is closed.</p>
+<p><strong>Theorem</strong> (Fundamental Theorem of Asset Pricing)
+<em>There is no arbitrage if there exist <em>deflators</em>, positive
+finitely additive measures<a href="#fn3" class="footnote-ref"
+id="fnref3" role="doc-noteref"><sup>3</sup></a> <span
 class="math inline">D_t</span> on <span
 class="math inline">\mathcal{A}_t</span>, <span class="math inline">t\in
-T</span>, with <span class="math display">
+T</span>, with</em> <span class="math display">
     X_t D_t = (X_u D_u + \sum_{t &lt; s \le u} C_s
 D_s)|_{\mathcal{A}_t}, t\le u.
-</span></p>
-<p><strong>Lemma</strong>. With the above notation <span
+</span> (Note <span class="math inline">Y = E[X|\mathcal{A}]</span> if
+and only if <span class="math inline">Y(P|_\mathcal{A}) =
+(XP)|_\mathcal{A}</span> when <span class="math inline">P</span> is a
+positive measure with mass 1.)</p>
+<p><strong>Claim</strong>. <em>If <span class="math inline">M_t =
+M_u|_{\mathcal{A}_t}</span>, <span class="math inline">t\le u</span>, is
+a <span class="math inline">\boldsymbol{R}^I</span>-valued</em>
+martingale measure <em>and <span class="math inline">D_t\in
+ba(A_t)</span> are positive measures then <span class="math inline">{X_t
+D_t = M_t - \sum_{s\le t} C_s D_s}</span> is arbitrage-free</em>.</p>
+<p><strong>Example</strong>. (Black-Scholes/Merton) <span
+class="math inline">M_t = (r, se^{\sigma B_t - \sigma^2t/2})P</span>,
+<span class="math inline">D_t = e^{-\rho t}P</span> where <span
+class="math inline">P</span> is Wiener measure.</p>
+<!--
+__Example__. (LIBOR Market Model) _If repurchase agreements are available with price 1 at $t$
+and pay $e^{f_t\,dt}$ at time $t$ then the_ stochastic discount _$D_t = e^{-\int_0^t} f_s\,ds}$ is a deflator_.
+-->
+<p><strong>Lemma</strong>. <em>With the above notation</em> <span
 class="math display">
     V_t D_t = (V_u D_u + \sum_{t &lt; s \le u} A_s
 D_s)|_{\mathcal{A}_t}, t\le u.
 </span></p>
-<p>Trading strategies create synthetic instruments where price
-corresponds to value and cash flow corresponds to account.</p>
-<p>Every arbitrage free model has the form <span class="math display">
-    X_t D_t = M_t - \sum_{s\le t} C_s D_s
-</span> where <span class="math inline">M_t =
-M_u|_{\mathcal{A}_t}</span> is a <em>martingale measure</em>.</p>
-<section id="appendix" class="level2">
-<h2>Appendix</h2>
-<p><span class="math inline">X,C\colon\sum_{j=0}^n B(\mathcal{A}_j,
-\boldsymbol{R}^I)</span></p>
-<p><span class="math inline">A\colon\sum_{j=0}^n B(\mathcal{A}_j,
-\boldsymbol{R}^I)\to\sum_{j=0}^n B(\mathcal{A}_j)</span> where <span
-class="math inline">A(\oplus \Gamma_j) = \oplus \Delta_j\cdot C_j -
-\Gamma_j\cdot X_j</span>.</p>
-<p><span class="math inline">\mathcal{G}_0 = \{\oplus \Gamma_j\mid
-\sum_j \Gamma_j = 0\}</span>.</p>
-<p><span class="math inline">\mathcal{P} = \{\oplus A_j\mid A_0 &gt; 0,
-A_j\ge 0\}</span>.</p>
-<p>Arbitrage if there exists <span
-class="math inline">\Gamma\in\mathcal{G}_0</span> with <span
-class="math inline">A(\Gamma)\in\mathcal{P}</span>.</p>
-<p><span class="math inline">A^*\colon \sum_{j=0}^n
-ba(\mathcal{A}_j)\to\sum_{j=0}^n ba(\mathcal{A}_j,
-\boldsymbol{R}^I)</span></p>
-</section>
+<p><strong>Trading strategies create synthetic instruments where price
+corresponds to value and cash flow corresponds to account.</strong></p>
+<p>A (cash settled) derivative contract is specified by stopping times
+<span class="math inline">{\hat{\tau}_j}</span> and cash flows <span
+class="math inline">\hat{A}_j</span>. If there exists a trading strategy
+<span class="math inline">(\tau_j,\Gamma_j)</span> with <span
+class="math inline">{\sum_j \Gamma_j = 0}</span>, <span
+class="math inline">{A_{\hat{\tau}_j} = \hat{A}_j}</span> and <span
+class="math inline">{A_t = 0}</span> (self-financing) otherwise, then a
+perfect hedge exists<a href="#fn4" class="footnote-ref" id="fnref4"
+role="doc-noteref"><sup>4</sup></a>. The value of the derivative is
+determined by <span class="math display">
+    V_t D_t = (\sum_{\hat{\tau}_j &gt; t} \hat{A}_j
+D_{\hat{\tau}_j})|_{\mathcal{A}_t}.
+</span> Note the right hand side is determined by the contract
+specifications and <span class="math inline">(D_t)</span>. Assuming
+<span class="math inline">\tau_0 = 0</span>, <span
+class="math inline">V_0 = \Gamma_0\cdot X_0</span> so the initial hedge
+<span class="math inline">\Gamma_0</span> is the Fréchet derivative
+<span class="math inline">D_{X_0}V_0</span> with respect to <span
+class="math inline">X_0</span>. Since <span class="math inline">V_t =
+(\Gamma_t + \Delta_t)\cdot X_t</span> we have <span
+class="math inline">\Gamma_t = D_{X_t}V_t - \Delta_t</span>. Note <span
+class="math inline">\Delta_t</span> is settled prior to time <span
+class="math inline">t</span>. This does not specify the trading times
+<span class="math inline">\tau_j &gt; 0</span><a href="#fn5"
+class="footnote-ref" id="fnref5"
+role="doc-noteref"><sup>5</sup></a>.</p>
+<aside id="footnotes" class="footnotes footnotes-end-of-document"
+role="doc-endnotes">
+<hr />
+<ol>
+<li id="fn1"><p>A partition of <span class="math inline">\Omega</span>
+is a collection of pairwise disjoint sets with union <span
+class="math inline">\Omega</span>. If <span
+class="math inline">\mathcal{A}</span> is a finite algebra of sets on
+<span class="math inline">\Omega</span> then the atoms of <span
+class="math inline">\mathcal{A}</span> form a partition of <span
+class="math inline">\Omega</span>. Partial information is knowing which
+atom <span class="math inline">\omega\in\Omega</span> belongs to. A
+function <span class="math inline">X\colon\Omega\to\boldsymbol{R}</span>
+is <span class="math inline">\mathcal{A}</span>-measurable if and only
+if it is constant on atoms so <span class="math inline">X</span>
+<em>is</em> a function on the atoms of <span
+class="math inline">\mathcal{A}</span>.<a href="#fnref1"
+class="footnote-back" role="doc-backlink">↩︎</a></p></li>
+<li id="fn2"><p>Prices <em>are</em> bounded. There is a finite amount of
+money in the world. Likewise for the number of shares it is possible to
+trade.<a href="#fnref2" class="footnote-back"
+role="doc-backlink">↩︎</a></p></li>
+<li id="fn3"><p>The dual of bounded functions <span
+class="math inline">B(\Omega)^* \cong ba(\Omega)</span> is the space of
+finitely additive measures on <span class="math inline">\Omega</span>.
+<span class="math inline">L\in B(\Omega)^*</span> corresponds to the
+measure <span class="math inline">\lambda(E) = L1_E</span>. If <span
+class="math inline">P</span> is a positive measure with mass 1 then
+<span class="math inline">Y = E[X|\mathcal{A}]</span> if and only if
+<span class="math inline">Y(P|_\mathcal{A}) =
+(XP)|_\mathcal{A}</span>.<a href="#fnref3" class="footnote-back"
+role="doc-backlink">↩︎</a></p></li>
+<li id="fn4"><p>A perfect hedge never exists.<a href="#fnref4"
+class="footnote-back" role="doc-backlink">↩︎</a></p></li>
+<li id="fn5"><p>Continuous time trading is impossible.<a href="#fnref5"
+class="footnote-back" role="doc-backlink">↩︎</a></p></li>
+</ol>
+</aside>
 </body>
 </html>
diff --git a/docs/vs.html b/docs/vs.html
index 99dc002..2777a59 100644
--- a/docs/vs.html
+++ b/docs/vs.html
@@ -5,7 +5,7 @@
   <meta name="generator" content="pandoc" />
   <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" />
   <meta name="author" content="Keith A. Lewis" />
-  <meta name="dcterms.date" content="2025-02-12" />
+  <meta name="dcterms.date" content="2025-02-15" />
   <title>Vector Space</title>
   <style>
     code{white-space: pre-wrap;}
@@ -48,7 +48,7 @@
 <header id="title-block-header">
 <h1 class="title">Vector Space</h1>
 <p class="author">Keith A. Lewis</p>
-<p class="date">February 12, 2025</p>
+<p class="date">February 15, 2025</p>
 <div class="abstract">
 <div class="abstract-title">Abstract</div>
 A mathematical sweet spot
@@ -65,6 +65,10 @@ <h1 class="title">Vector Space</h1>
 <li><a href="#axioms" id="toc-axioms">Axioms</a></li>
 <li><a href="#standard-and-dual-basis"
 id="toc-standard-and-dual-basis">Standard and Dual Basis</a></li>
+</ul></li>
+<li><a href="#linear-operator" id="toc-linear-operator">Linear
+Operator</a>
+<ul>
 <li><a href="#tensor" id="toc-tensor">Tensor</a></li>
 <li><a href="#inner-product" id="toc-inner-product">Inner
 Product</a></li>
@@ -99,15 +103,16 @@ <h2><span class="math inline">\bm{{{R}}}^n</span></h2>
 \{1,\ldots,n\}</span> then <span class="math inline">i\in
 \bm{{n}}</span> is a shorter notation for <span class="math inline">1\le
 i\le n</span>. If <span class="math inline">A</span> and <span
-class="math inline">B</span> are sets then the <em>set exponential</em>
+class="math inline">B</span> are sets, the <em>set exponential</em>
 <span class="math inline">B^A = \{f\colon A\to B\}</span> is the set of
 all functions from <span class="math inline">A</span> to <span
 class="math inline">B</span>. We identify <span
 class="math inline">\bm{{{R}}}^n</span> with <span
-class="math inline">\bm{{{R}}}^{\bm{{n}}}</span> by <span
-class="math inline">x = (x_1,\ldots,x_n)\in\bm{{{R}}}^n</span> with the
-functon <span class="math inline">x\colon\bm{{n}}\to\bm{{{R}}}</span>
-defined by <span class="math inline">x(i) = x_i</span>, <span
+class="math inline">\bm{{{R}}}^{\bm{{n}}}</span> where <span
+class="math inline">x = (x_1,\ldots,x_n)\in\bm{{{R}}}^n</span>
+corresponds to the functon <span
+class="math inline">x\colon\bm{{n}}\to\bm{{{R}}}</span> defined by <span
+class="math inline">x(i) = x_i</span>, <span
 class="math inline">i\in\bm{{n}}</span>. For any set <span
 class="math inline">I</span>, <span
 class="math inline">\bm{{{R}}}^I</span> is a vector space.</p>
@@ -124,26 +129,26 @@ <h3>Scalar Multiplication and Vector Addition</h3>
 <h3>Axioms</h3>
 <p>For any set <span class="math inline">I</span>, <span
 class="math inline">a,b\in\bm{{{R}}}</span>, and <span
-class="math inline">x, y, z\in\bm{{{R}}}^I</span></p>
-<p><strong>Exercise</strong>. <em>Show <span class="math inline">x + (y
-+ z) = (x + y) + z</span></em>.</p>
-<p><strong>Exercise</strong>. <em>Show <span class="math inline">x + y =
-y + x</span></em>.</p>
-<p><strong>Exercise</strong>. <em>Show <span
-class="math inline">\bm{{0}}+ x = x</span> where <span
-class="math inline">\bm{{0}}(i) = 0</span>, <span
+class="math inline">x, y, z\in\bm{{{R}}}^I</span> show the
+following:</p>
+<p><strong>Exercise</strong>. <em><span class="math inline">x + (y + z)
+= (x + y) + z</span></em>.</p>
+<p><strong>Exercise</strong>. <em><span class="math inline">x + y = y +
+x</span></em>.</p>
+<p><strong>Exercise</strong>. <em><span class="math inline">\bm{{0}}+ x
+= x</span> where <span class="math inline">\bm{{0}}(i) = 0</span>, <span
 class="math inline">i\in I</span></em>.</p>
-<p><strong>Exercise</strong>. <em>Show <span class="math inline">x +
-(-x) = \bm{{0}}</span> where <span class="math inline">(-x)(i) =
+<p><strong>Exercise</strong>. <em><span class="math inline">x + (-x) =
+\bm{{0}}</span> where <span class="math inline">(-x)(i) =
 -(x(i))</span>, for <span class="math inline">i\in I</span></em>.</p>
-<p><strong>Exercise</strong>. <em>Show <span class="math inline">a(bx) =
+<p><strong>Exercise</strong>. <em><span class="math inline">a(bx) =
 (ab)x</span></em>.</p>
-<p><strong>Exercise</strong>. <em>Show <span class="math inline">1x =
+<p><strong>Exercise</strong>. <em><span class="math inline">1x =
 x</span></em>.</p>
-<p><strong>Exercise</strong>. <em>Show <span class="math inline">a(x +
-y) = ax + ay</span></em>.</p>
-<p><strong>Exercise</strong>. <em>Show <span class="math inline">(a +
-b)x = ax + bx</span></em>.</p>
+<p><strong>Exercise</strong>. <em><span class="math inline">a(x + y) =
+ax + ay</span></em>.</p>
+<p><strong>Exercise</strong>. <em><span class="math inline">(a + b)x =
+ax + bx</span></em>.</p>
 <p><em>Hint</em>: Use the properties of real numbers.</p>
 <p>The exercises are the axioms for an <em>abstract vector space</em>
 with scalar multiplication <span class="math inline">{\bm{{{R}}}\times
@@ -152,8 +157,11 @@ <h3>Axioms</h3>
 V}</span> where <span class="math inline">{(x,y)\mapsto x + y}</span>.
 The first four axioms show vector addition is an abelian (commutative)
 group. The last two axioms are the <em>distributive laws</em> connecting
-scalar multiplication and vector addition. Proofs involving only the
-abstract axioms are considered more elegant.</p>
+scalar multiplication and vector addition. Every abstract vector space
+can be represented by <span class="math inline">\bm{{{R}}}^I</span> for
+some set <span class="math inline">I</span>. Two vector spaces</p>
+<p>Proofs involving only the abstract axioms are considered more
+elegant.</p>
 <p>If <span class="math inline">x\in\bm{{{R}}}^I</span> then <span
 class="math inline">(0x)(i) = 0(x(i)) = 0</span> and <span
 class="math inline">(1x)(i) = 1(x(i)) = x(i)</span> for all <span
@@ -195,54 +203,69 @@ <h3>Axioms</h3>
 \end{aligned}
 </span>
 </details>
-<p>If <span class="math inline">n = 1</span> we can identify <span
-class="math inline">\bm{{{R}}}</span> with <span
-class="math inline">\bm{{{R}}}^1</span> by <span
-class="math inline">x\in\bm{{{R}}}</span> corresponds to <span
-class="math inline">(x)\in\bm{{{R}}}^1</span>. We can also identify
-every <span class="math inline">y\in\bm{{{R}}}</span> with the linear
-function from <span class="math inline">\bm{{{R}}}</span> to <span
-class="math inline">\bm{{{R}}}</span> of multiplication by <span
-class="math inline">y</span> denoted by <span class="math inline">y^*x =
-xy</span>, for <span class="math inline">x\in\bm{{{R}}}</span>. These
-trivial definitions are the foundation of a systematic notation for
-higher dimensional calculations. There is no need for “row” and “column”
+<!--
+If $n = 1$ we can identify $\RR$ with $\RR^1$ by $x\in\RR$ corresponds to $(x)\in\RR^1$.
+We can also identify every $y\in\RR$ with the linear function from $\RR$ to $\RR$
+of multiplication by $y$ denoted by $y^*x = xy$, for $x\in\RR$.
+These trivial definitions are the foundation of a systematic notation for
+higher dimensional calculations. There is no need for "row" and "column"
 vectors, the dual of a vector space and composition of linear functions
-generalizes the Einstein summation convention and various “tensor”
-notations invented by less brilliant people. Standard mathematical
-notation suffices.</p>
+generalizes the Einstein summation convention and various "tensor"
+notations invented by less brilliant people. Standard mathematical notation suffices.
+-->
 </section>
 <section id="standard-and-dual-basis" class="level3">
 <h3>Standard and Dual Basis</h3>
 <p>The <em>standard basis</em> of <span
 class="math inline">\bm{{{R}}}^n</span> is <span
 class="math inline">{e_i\in\bm{{{R}}}^n}</span>, <span
-class="math inline">{i\in n}</span>, where <span
+class="math inline">{i\in\bm{{n}}}</span>, where <span
 class="math inline">{e_i = (0,\ldots,1,\ldots,0)}</span> with all
 elements <span class="math inline">0</span> except for a <span
 class="math inline">1</span> in the <span
 class="math inline">i</span>-th position. It is plausible that <span
 class="math inline">{x = (x_1,\ldots,x_n) = x_1 e_1 + \cdots + x_n
 e_n}</span> for <span class="math inline">{x\in\bm{{{R}}}^n}</span>, but
-you should always be suspicious of arguments involving dots.</p>
+you should always be wary of definitions involving dots.</p>
+<p>More generally, the standard basis of <span
+class="math inline">\bm{{{R}}}^I</span> is <span
+class="math inline">{e_i\in\bm{{{R}}}^I}</span>, <span
+class="math inline">{i\in I}</span>, where <span
+class="math inline">{e_i(j) = \delta_{ij}}</span> for <span
+class="math inline">j\in I</span>, where $_{ij} is the <em>Kronecker
+delta</em> defined by <span class="math inline">{δ_{ij} = 1}</span> if
+<span class="math inline">{i=j}</span> and <span
+class="math inline">{δ_{ij} = 0}</span> if <span
+class="math inline">{i\not=j}</span>.</p>
+<p><strong>Exercise</strong>. <em>If <span class="math inline">I</span>
+is finite, show <span class="math inline">x = \sum_{i\in I}
+x(i)e_i</span> for <span
+class="math inline">x\in\bm{{{R}}}^I</span></em>.</p>
+<p><em>Hint</em>: Consider <span class="math inline">x(j)</span>, <span
+class="math inline">j\in I</span>.</p>
 <p>Define the <em>dual basis</em> by <span
-class="math inline">e^*_j\colon\bm{{{R}}}^n\to\bm{{{R}}}</span> by <span
-class="math inline">{e^*_j((x_1,\ldots,x_n)) = x_j}</span>, <span
-class="math inline">{j\in n}</span>. Clearly <span
-class="math inline">e^*_j(e_i) = \delta_{ij}</span>, wWhere <span
-class="math inline">δ_{ij} = 1</span> if <span
-class="math inline">i=j</span> and <span class="math inline">δ_{ij} =
-0</span> if <span class="math inline">i\not= j</span> is the
-<em>Kronecker delta</em>.</p>
-<p>The standard and dual bases allows us to identify <span
-class="math inline">\bm{{{R}}}^n</span> with its <em>dual</em> <span
-class="math inline">(\bm{{{R}}}^n)^*</span>, the linear functionals from
-<span class="math inline">\bm{{{R}}}^n\to\bm{{{R}}}</span>. If <span
-class="math inline">{x = \sum_i x_i e_i\in\bm{{{R}}}^n}</span> define
-<span class="math inline">{x^* = \sum_j x_j
-e_j^*\in(\bm{{{R}}}^n)^*}</span>.</p>
-<p><strong>Exercise</strong>. <em>Show <span class="math inline">y^*x =
-\langle x,y^*\rangle = \sum_i x_i y_i</span></em>.</p>
+class="math inline">e^*_j\colon\bm{{{R}}}^I\to\bm{{{R}}}</span> by <span
+class="math inline">{e^*_j(x) = x(j)}</span>, <span
+class="math inline">{j\in I}</span>. The standard and dual bases allows
+us to identify <span class="math inline">\bm{{{R}}}^I</span> with its
+<em>dual</em> <span class="math inline">(\bm{{{R}}}^I)^*</span>, the
+linear functionals from <span
+class="math inline">\bm{{{R}}}^I\to\bm{{{R}}}</span>. The <em>dual
+pairing</em> of <span class="math inline">x\in\bm{{{R}}}^I</span> and
+<span class="math inline">{y\in\bm{{{R}}}^I}</span> is defined by
+y^<em>(x) = x, y^</em>= _{iI} x(i)y(i)$.</p>
+<p>If <span class="math inline">{x,y\in\bm{{{R}}}^I}</span> the
+<em>inner product</em>, or <em>dot product</em> is <span
+class="math inline">{(x,y) = x\cdot y = \sum_{i\in I}x_i y_i}</span>.
+This is similar, but different from, the dual pairing is a function
+<span
+class="math inline">\langle\cdot,\cdot\rangle\colon\bm{{{R}}}^I\times(\bm{{{R}}}^I)^*\to\bm{{{R}}}</span>
+while the dot product is a function <span
+class="math inline">(\cdot,\cdot)\colon\bm{{{R}}}^I\times\bm{{{R}}}^I\to\bm{{{R}}}</span>.</p>
+</section>
+</section>
+<section id="linear-operator" class="level2">
+<h2>Linear Operator</h2>
 <p>A function <span
 class="math inline">T\colon\bm{{{R}}}^n\to\bm{{{R}}}^m</span> is a
 <em>linear operator</em> if <span class="math inline">{T(ax + y) = aTx +
@@ -272,7 +295,6 @@ <h3>Standard and Dual Basis</h3>
 class="math inline">i\in n</span>, <span class="math inline">k\in
 l</span></em>.</p>
 <p>Matrix multiplication is just composition of linear operators.</p>
-</section>
 <section id="tensor" class="level3">
 <h3>Tensor</h3>
 <p>Write <span class="math inline">\bm{{{R}}}^{-n}</span> for <span
diff --git a/ums.md b/ums.md
index b17d880..e2a52b8 100644
--- a/ums.md
+++ b/ums.md
@@ -1,66 +1,100 @@
+<!--
+:!pandoc -t html5 -s --katex=https://cdn.jsdelivr.net/npm/katex@0.12.0/dist/ --css math.css ums.md -o ums.html
+:!pandoc -V fontsize=12pt ums.md -o ums.pdf
+-->
 ---
-title: A Unified Model of Derivative Securities
+title: Simple Unified Model of Derivative Securities
 classoption: fleqn
 ...
 \newcommand\RR{\boldsymbol{R}}
 \renewcommand\AA{\mathcal{A}}
 
-$T$ — totally ordered set of trading _times_.
+$T$ — totally ordered set of _trading times_.
 
 $I$ — market _instruments_.
 
-$\Omega$ — all possible _outcomes_.
+$\Omega$ — possible _outcomes_.
 
-$\AA_t$ — a partition of $\Omega$ indicating the _information_ available at time $t\in T$.
+$(\AA_t)_{t\in T}$ — _partitions_[^1] of $\Omega$ indicating the information available at time $t\in T$.
 
-$X_t\colon\AA_t\to\RR^I$ — _prices_ at time $t$ of market instruments.
+$X_t\colon\AA_t\to\RR^I$ — bounded _prices_[^2] at times $t\in T$ of market instruments.
 
-$C_t\colon\AA_t\to\RR^I$ — _cash flows_, usually 0, at time $t$ of market instruments.
+$C_t\colon\AA_t\to\RR^I$ — bounded _cash flows_ at times $t\in T$ of market instruments.
 
-$(\tau_j, \Gamma_j)$ — a finite _trading strategy_ of strictly increasing stopping
-times $\tau_j$ and corresponding trades $\Gamma_j\colon\AA_{\tau_j}\to\RR^I$
-with $\sum_j \Gamma_j = 0$.
+E.g., coupons, dividends, and futures margin adjustments are cash flows.
 
+$(\tau_j, \Gamma_j)$ — _trading strategy_ of strictly increasing stopping
+times $\tau_j$ and shares $\Gamma_j\colon\AA_{\tau_j}\to\RR^I$ purchased at $\tau_j$.
+
+The _position_ resulting from a trading strategy is
 $\Delta_t = \sum_{\tau_j < t} \Gamma_j = \sum_{s < t} \Gamma_s$ where
-$\Gamma_s = \Gamma_j 1(\tau_j = s)$ — the _position_ resulting from trading.
+$\Gamma_s = \Gamma_j 1(\tau_j = s)$. Note the strict inequality.
+It takes time for a trade to settle and become a part of the position.
 
-$V_t = (\Delta_t + \Gamma_t)\cdot X_t$ — the _value_, or mark-to-market, of the strategy.
+$V_t = (\Delta_t + \Gamma_t)\cdot X_t$ — the _value_, or _mark-to-market_, of the strategy.
 
-$A_t = \Delta_t\cdot C_t - \Gamma_t\cdot X_t$ — the amount showing up in the trading _account_.
+$A_t = \Delta_t\cdot C_t - \Gamma_t\cdot X_t$ — the _amount_ showing up in the trading account.
 
-Arbitrage exists if there is a trading strategy with $A_{\tau_0} > 0$ and $A_t \ge0$, $t > \tau_0$.
+Arbitrage exists if there is a trading strategy with $A_{\tau_0} > 0$, $A_t \ge0$, $t > \tau_0$, and
+$\sum_j \Gamma_j = 0$. You make money on the first trade and never lose money until
+the position is closed.
 
-__Fundamental Theorem of Asset Pricing__. There is no arbitrage if there exist positive
-measures $D_t$ on $\AA_t$, $t\in T$, with
+__Theorem__ (Fundamental Theorem of Asset Pricing) _There is no arbitrage if there exist _deflators_, positive
+finitely additive measures[^3] $D_t$ on $\AA_t$, $t\in T$, with_
 $$
 	X_t D_t = (X_u D_u + \sum_{t < s \le u} C_s D_s)|_{\AA_t}, t\le u.
 $$
+(Note $Y = E[X|\AA]$ if and only if $Y(P|_\AA) = (XP)|_\AA$ when $P$ is a positive measure with mass 1.)
+
+__Claim__. _If $M_t = M_u|_{\AA_t}$, $t\le u$, is a $\RR^I$-valued_ martingale measure _and
+$D_t\in ba(A_t)$ are positive measures then
+${X_t D_t = M_t - \sum_{s\le t} C_s D_s}$ is arbitrage-free_.
+
+__Example__. (Black-Scholes/Merton) $M_t = (r, se^{\sigma B_t - \sigma^2t/2})P$, $D_t = e^{-\rho t}P$ where
+$P$ is Wiener measure.
 
-__Lemma__. With the above notation
+<!--
+__Example__. (LIBOR Market Model) _If repurchase agreements are available with price 1 at $t$
+and pay $e^{f_t\,dt}$ at time $t$ then the_ stochastic discount _$D_t = e^{-\int_0^t} f_s\,ds}$ is a deflator_.
+-->
+
+__Lemma__. _With the above notation_
 $$
 	V_t D_t = (V_u D_u + \sum_{t < s \le u} A_s D_s)|_{\AA_t}, t\le u.
 $$
 
-Trading strategies create synthetic instruments where price corresponds
-to value and cash flow corresponds to account.
+**Trading strategies create synthetic instruments where price corresponds
+to value and cash flow corresponds to account.**
 
-Every arbitrage free model has the form
+A (cash settled) derivative contract is specified by stopping times ${\hat{\tau}_j}$ and cash
+flows $\hat{A}_j$.  If there exists a trading strategy
+$(\tau_j,\Gamma_j)$ with ${\sum_j \Gamma_j = 0}$, ${A_{\hat{\tau}_j} = \hat{A}_j}$
+and ${A_t = 0}$ (self-financing) otherwise, then a perfect hedge exists[^4].
+The value of the derivative is determined by
 $$
-	X_t D_t = M_t - \sum_{s\le t} C_s D_s
+	V_t D_t = (\sum_{\hat{\tau}_j > t} \hat{A}_j D_{\hat{\tau}_j})|_{\AA_t}.
 $$
-where $M_t = M_u|_{\AA_t}$ is a _martingale measure_.
-
-## Appendix
-
-$X,C\colon\sum_{j=0}^n B(\AA_j, \RR^I)$
-
-$A\colon\sum_{j=0}^n B(\AA_j, \RR^I)\to\sum_{j=0}^n B(\AA_j)$
-where $A(\oplus \Gamma_j) = \oplus \Delta_j\cdot C_j - \Gamma_j\cdot X_j$.
-
-$\mathcal{G}_0 = \{\oplus \Gamma_j\mid \sum_j \Gamma_j = 0\}$.
-
-$\mathcal{P} = \{\oplus A_j\mid A_0 > 0, A_j\ge 0\}$.
-
-Arbitrage if there exists $\Gamma\in\mathcal{G}_0$ with $A(\Gamma)\in\mathcal{P}$.
-
-$A^*\colon \sum_{j=0}^n ba(\AA_j)\to\sum_{j=0}^n ba(\AA_j, \RR^I)$
+Note the right hand side is determined by the contract specifications and $(D_t)$.
+Assuming $\tau_0 = 0$, $V_0 = \Gamma_0\cdot X_0$ so the initial hedge $\Gamma_0$ is the Fréchet 
+derivative $D_{X_0}V_0$ with respect to $X_0$.
+Since $V_t = (\Gamma_t + \Delta_t)\cdot X_t$ we have
+$\Gamma_t = D_{X_t}V_t - \Delta_t$. Note $\Delta_t$ is settled prior to time $t$.
+This does not specify the trading times $\tau_j > 0$[^5].
+
+[^1]: A partition of $\Omega$ is a collection of pairwise disjoint sets with union $\Omega$.
+If $\AA$ is a finite algebra of sets on $\Omega$ then the atoms of $\AA$ form a partition
+of $\Omega$. Partial information is knowing which atom $\omega\in\Omega$ belongs to.
+A function $X\colon\Omega\to\RR$ is $\AA$-measurable if and only if it is constant on atoms
+so $X$ _is_ a function on the atoms of $\AA$.
+
+[^2]: Prices _are_ bounded. There is a finite amount of money in the world.
+Likewise for the number of shares it is possible to trade.
+
+[^3]: The dual of bounded functions $B(\Omega)^* \cong ba(\Omega)$ is the space
+of finitely additive measures on $\Omega$. $L\in B(\Omega)^*$ corresponds to
+the measure $\lambda(E) = L1_E$. If $P$ is a positive measure with mass 1 then
+$Y = E[X|\AA]$ if and only if $Y(P|_\AA) = (XP)|_\AA$.
+
+[^4]: A perfect hedge never exists.
+
+[^5]: Continuous time trading is impossible.
diff --git a/vs.md b/vs.md
index 1b22e31..ba55ba7 100644
--- a/vs.md
+++ b/vs.md
@@ -50,10 +50,10 @@ ${\RR^n = \{(x_1,\ldots,x_n)\mid x_i\in\RR, 1\le i\le n\}}$
 be the cartesian product of $n\in\NN$ copies of the real numbers.
 If bold $\bs{n} = \{1,\ldots,n\}$
 then $i\in \bs{n}$ is a shorter notation for $1\le i\le n$.
-If $A$ and $B$ are sets then the _set exponential_ $B^A = \{f\colon A\to B\}$
+If $A$ and $B$ are sets, the _set exponential_ $B^A = \{f\colon A\to B\}$
 is the set of all functions from $A$ to $B$. 
-We identify $\RR^n$ with $\RR^{\bs{n}}$ by $x = (x_1,\ldots,x_n)\in\RR^n$
-with the functon $x\colon\bs{n}\to\RR$ defined by $x(i) = x_i$, $i\in\bs{n}$.
+We identify $\RR^n$ with $\RR^{\bs{n}}$ where $x = (x_1,\ldots,x_n)\in\RR^n$
+corresponds to the functon $x\colon\bs{n}\to\RR$ defined by $x(i) = x_i$, $i\in\bs{n}$.
 For any set $I$, $\RR^I$ is a vector space.
 
 ### Scalar Multiplication and Vector Addition
@@ -63,23 +63,23 @@ ${(ax)(i) = a(x(i))}$ and ${(x + y)(i) = x(i) + y(i)}$, $a\in\RR$, $x,y\in\RR^I$
 
 ### Axioms
 
-For any set $I$, $a,b\in\RR$, and $x, y, z\in\RR^I$
+For any set $I$, $a,b\in\RR$, and $x, y, z\in\RR^I$ show the following:
 
-__Exercise__. _Show $x + (y + z) = (x + y) + z$_.
+__Exercise__. _$x + (y + z) = (x + y) + z$_.
 
-__Exercise__. _Show $x + y = y + x$_.
+__Exercise__. _$x + y = y + x$_.
 
-__Exercise__. _Show $\zero + x = x$ where $\zero(i) = 0$, $i\in I$_.
+__Exercise__. _$\zero + x = x$ where $\zero(i) = 0$, $i\in I$_.
 
-__Exercise__. _Show $x + (-x) = \zero$ where $(-x)(i) = -(x(i))$, for $i\in I$_.
+__Exercise__. _$x + (-x) = \zero$ where $(-x)(i) = -(x(i))$, for $i\in I$_.
 
-__Exercise__. _Show $a(bx) = (ab)x$_.
+__Exercise__. _$a(bx) = (ab)x$_.
 
-__Exercise__. _Show $1x = x$_.
+__Exercise__. _$1x = x$_.
 
-__Exercise__. _Show $a(x + y) = ax + ay$_.
+__Exercise__. _$a(x + y) = ax + ay$_.
 
-__Exercise__. _Show $(a + b)x = ax + bx$_.
+__Exercise__. _$(a + b)x = ax + bx$_.
 
 _Hint_: Use the properties of real numbers.
 
@@ -88,7 +88,9 @@ multiplication ${\RR\times V\to V}$ where ${(a,x)\mapsto ax = xa}$ and
 binary addition ${V\times V\to V}$ where ${(x,y)\mapsto x + y}$. The
 first four axioms show vector addition is an abelian (commutative)
 group. The last two axioms are the _distributive laws_ connecting scalar
-multiplication and vector addition.
+multiplication and vector addition. Every abstract vector space
+can be represented by $\RR^I$ for some set $I$. Two vector spaces
+
 Proofs involving only the abstract axioms are considered more elegant.
 
 If $x\in\RR^I$ then $(0x)(i) = 0(x(i)) = 0$ and $(1x)(i) = 1(x(i)) = x(i)$ for all $i\in I$ so
@@ -122,6 +124,7 @@ $$
 </details>
 
 
+<!--
 If $n = 1$ we can identify $\RR$ with $\RR^1$ by $x\in\RR$ corresponds to $(x)\in\RR^1$.
 We can also identify every $y\in\RR$ with the linear function from $\RR$ to $\RR$
 of multiplication by $y$ denoted by $y^*x = xy$, for $x\in\RR$.
@@ -130,25 +133,38 @@ higher dimensional calculations. There is no need for "row" and "column"
 vectors, the dual of a vector space and composition of linear functions
 generalizes the Einstein summation convention and various "tensor"
 notations invented by less brilliant people. Standard mathematical notation suffices.
+-->
 
 ### Standard and Dual Basis
 
-The _standard basis_ of $\RR^n$ is ${e_i\in\RR^n}$, ${i\in n}$,
+The _standard basis_ of $\RR^n$ is ${e_i\in\RR^n}$, ${i\in\bs{n}}$,
 where ${e_i = (0,\ldots,1,\ldots,0)}$ with all elements $0$ except for a
 $1$ in the $i$-th position.  It is plausible that ${x = (x_1,\ldots,x_n)
 = x_1 e_1 + \cdots + x_n e_n}$ for ${x\in\RR^n}$, but you should always
-be suspicious of arguments involving dots.
+be wary of definitions involving dots.
+
+More generally, the standard basis of $\RR^I$ is ${e_i\in\RR^I}$, ${i\in
+I}$, where ${e_i(j) = \delta_{ij}}$ for $j\in I$, where $\delta_{ij} is
+the _Kronecker delta_ defined by ${δ_{ij} = 1}$ if ${i=j}$ and
+${δ_{ij} = 0}$ if ${i\not=j}$.
+
+__Exercise__. _If $I$ is finite, show $x = \sum_{i\in I} x(i)e_i$ for $x\in\RR^I$_.
+
+_Hint_: Consider $x(j)$, $j\in I$.
 
-Define the _dual basis_ by
-$e^*_j\colon\RR^n\to\RR$ by ${e^*_j((x_1,\ldots,x_n)) = x_j}$, ${j\in n}$.
-Clearly $e^*_j(e_i) = \delta_{ij}$, wWhere $δ_{ij} = 1$
-if $i=j$ and $δ_{ij} = 0$ if $i\not= j$ is the _Kronecker delta_.
+Define the _dual basis_ by $e^*_j\colon\RR^I\to\RR$ by ${e^*_j(x) = x(j)}$, ${j\in I}$.
+The standard and dual bases allows us to identify $\RR^I$ with its _dual_
+$(\RR^I)^*$, the linear functionals from $\RR^I\to\RR$.
+The _dual pairing_ of $x\in\RR^I$ and ${y\in\RR^I}$ is defined by
+y^*(x) = \langle x, y^*\rangle = \sum_{i\in I} x(i)y(i)$.
 
-The standard and dual bases allows us to identify $\RR^n$ with its _dual_
-$(\RR^n)^*$, the linear functionals from $\RR^n\to\RR$.
-If ${x = \sum_i x_i e_i\in\RR^n}$ define ${x^* = \sum_j x_j e_j^*\in(\RR^n)^*}$.
+If ${x,y\in\RR^I}$ the _inner product_, or _dot product_ is
+${(x,y) = x\cdot y = \sum_{i\in I}x_i y_i}$. This is similar, but different
+from, the dual pairing is a
+function $\langle\cdot,\cdot\rangle\colon\RR^I\times(\RR^I)^*\to\RR$ while
+the dot product is a function $(\cdot,\cdot)\colon\RR^I\times\RR^I\to\RR$.
 
-__Exercise__. _Show $y^*x = \langle x,y^*\rangle = \sum_i x_i y_i$_.
+## Linear Operator
 
 A function $T\colon\RR^n\to\RR^m$ is a _linear operator_ if ${T(ax + y) = aTx + y}$, ${a\in\RR}$, ${x,y\in\RR^n}$.
 
