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<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>
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<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
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<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>
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<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/[email protected]/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>
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