{"id":32409,"date":"2025-02-25T13:00:15","date_gmt":"2025-02-25T13:00:15","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=32409"},"modified":"2025-03-05T21:04:33","modified_gmt":"2025-03-05T21:04:33","slug":"the-one-period-binomial-model-and-the-no-arbitrage-condition","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/en\/the-one-period-binomial-model-and-the-no-arbitrage-condition\/","title":{"rendered":"The One-Period Binomial Model and the No-Arbitrage Condition"},"content":{"rendered":"<style>\np, ul, ol{\ntext-align: justify;\n}\nh1{\ntext-align:center;\ntext-transform: uppercase;\n}\nh2{\ntext-align:center;\ntext-transform: uppercase;\nfont-size:24pt;\n}\nh3 { \n    text-align: center;\n    text-transform: uppercase;\n    font-size: 24px !important;\n}\n<\/style>\n<h1>The One-Period Binomial Model and the No-Arbitrage Condition<\/h1>\n<p style=\"text-align:center;\"><em><b>Abstract:<\/b><br \/>\nImagine a casino where you can bet on a game and, regardless of the outcome, always make money. Sounds too good to be true, right? In financial markets, such opportunities arise due to the possibility of arbitrage; however, they are quickly eliminated by the actions of market participants. In this class, we explore the one-period binomial model and the no-arbitrage condition, analyzing how asset prices, interest rates, and investment strategies eliminate the possibility of risk-free profits. Through detailed examples and rigorous mathematical proof, we will reveal the fundamental principles that underpin financial stability and why detecting an arbitrage opportunity is only the beginning of a much more complex story.<\/em>\n<\/p>\n<p style=\"text-align:center;\"><b>Learning Objectives<\/b><br \/>\nBy the end of this class, the student will be able to:\n<\/p>\n<ol>\n<li><strong>Understand<\/strong> the one-period binomial model and its application in financial asset valuation.<\/li>\n<li><strong>Identify<\/strong> the fundamental elements of the one-period binomial model: underlying asset, up and down factors, and risk-free asset.<\/li>\n<li><strong>Understand<\/strong> the construction and function of a self-financing portfolio in the binomial model.<\/li>\n<li><strong>Understand<\/strong> the no-arbitrage condition in financial markets and how it prevents the possibility of risk-free profits through self-financing portfolios.<\/li>\n<li><strong>Evaluate<\/strong> the existence of arbitrage opportunities in a market by analyzing the no-arbitrage condition.<\/li>\n<li><strong>Analyze<\/strong> how arbitrage affects asset prices and causes market adjustments.<\/li>\n<li><strong>Describe<\/strong> the effect of stock lending rates on arbitrage strategies and the no-arbitrage condition.<\/li>\n<li><strong>Explain<\/strong> through mathematical models how market adjustments occur after the emergence of arbitrage opportunities.<\/li>\n<li><strong>Understand<\/strong> the formal proof of the no-arbitrage condition theorem.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><b><u>TABLE OF CONTENTS<\/u><\/b><br \/>\n<a href=\"#1\">What is the one-period binomial model?<\/a><br \/>\n<a href=\"#2\">How to recognize a market with arbitrage opportunities and its rapid dissolution<\/a><br \/>\n<a href=\"#3\">Proof of the No-Arbitrage Condition Theorem<\/a><br \/>\n<a href=\"#4\">Conclusion<\/a>\n<\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/-oshd8mj6bg?si=8Dgu1tTvP8giiEwd\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe><\/center><br \/>\n<a name=\"1\"><\/a><\/p>\n<h2>What is the One-Period Binomial Model?<\/h2>\n<p>The <strong>one-period binomial model<\/strong> is a mathematical model used in finance to describe the evolution of an asset&#8217;s price in a discrete-time framework. It is called \u00abbinomial\u00bb because, in each time period, the asset price can only move in two possible directions: up or down. This model is widely used in the valuation of financial derivatives, especially options, and serves as the foundation for the multi-period binomial model.<\/p>\n<h3>Elements of the Model<\/h3>\n<p>The one-period binomial model is based on the following fundamental elements:<\/p>\n<ul>\n<li>\n<p><strong>An underlying asset:<\/strong> Represented by its price <span class=\"katex-eq\" data-katex-display=\"false\">S(t)<\/span> at time <span class=\"katex-eq\" data-katex-display=\"false\">t<\/span>. At the initial moment <span class=\"katex-eq\" data-katex-display=\"false\">t=0<\/span>, the asset price is <span class=\"katex-eq\" data-katex-display=\"false\">S(0)<\/span>. At time <span class=\"katex-eq\" data-katex-display=\"false\">t=1<\/span>, its price can move to one of two possible values, denoted as <span class=\"katex-eq\" data-katex-display=\"false\">S(1,\\text{up})<\/span> (price if it goes up) or <span class=\"katex-eq\" data-katex-display=\"false\">S(1,\\text{down})<\/span> (price if it goes down):<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\nS(1) =\n\n\\begin{cases}\n\nS(1,\\text{up}) = S(0)  u, &amp; \\text{with probability } p, \\\\\n\nS(1,\\text{down}) = S(0)  d, &amp; \\text{with probability } 1 - p.\n\n\\end{cases}\n\n<\/span>\n<p>Where the coefficients <span class=\"katex-eq\" data-katex-display=\"false\">u<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">d<\/span> represent the up and down factors of the price, satisfying the relationship:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\lt d \\lt 1 \\lt u<\/span>.<\/p>\n<p>This relationship also ensures that future prices remain strictly positive, as established by the fundamental assumptions of the <strong>simple market model.<\/strong><\/p>\n<\/li>\n<li><strong>Probabilities:<\/strong> It is assumed that the probability of the asset increasing in price is <span class=\"katex-eq\" data-katex-display=\"false\">p<\/span>, and the probability of it decreasing is <span class=\"katex-eq\" data-katex-display=\"false\">1 - p<\/span>, with <span class=\"katex-eq\" data-katex-display=\"false\">0 \\lt p \\lt 1<\/span>. This constraint guarantees that both asset movements are possible and prevents deterministic situations where the price always rises or always falls, which would invalidate the binomial model and create arbitrage opportunities.<\/li>\n<li><strong>A risk-free asset:<\/strong> A bond or financial instrument is introduced, whose value grows predictably with a risk-free interest rate <span class=\"katex-eq\" data-katex-display=\"false\">r<\/span>. Its price in the next period is given by <span class=\"katex-eq\" data-katex-display=\"false\">A(1) = A(0)(1+r)<\/span>.<\/li>\n<\/ul>\n<h3><b>Theorem:<\/b> No-Arbitrage Condition in a One-Period Binomial Model<\/h3>\n<p>Let an asset have an initial price <span class=\"katex-eq\" data-katex-display=\"false\">S(0) \\gt 0<\/span>, and its value at time <span class=\"katex-eq\" data-katex-display=\"false\">t=1<\/span> follows the previously described binomial structure. Suppose there exists a risk-free asset (bond) with a price <span class=\"katex-eq\" data-katex-display=\"false\">A(1) = A(0)(1+r)<\/span>, where <span class=\"katex-eq\" data-katex-display=\"false\">r<\/span> is the risk-free rate. Then, the market is arbitrage-free if and only if the up and down factors satisfy the following condition:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">0 \\lt d \\lt 1 + r \\lt u<\/span>\n<p>In an arbitrage-free market, it is not possible to construct a self-financing portfolio that generates risk-free profits.<\/p>\n<h3>What is a Self-Financing Portfolio?<\/h3>\n<p>A <strong>self-financing portfolio<\/strong> is an investment strategy where no additional capital is required, as any purchase of assets is financed by the sale of others within the same portfolio. In other words, no external funds are injected to implement it.<\/p>\n<p>If a self-financing portfolio can be constructed in a market that guarantees a profit in all possible scenarios, then an arbitrage opportunity exists. The no-arbitrage condition implies that it is not possible to construct such portfolios.<\/p>\n<p>Mathematically, a self-financing portfolio is constructed as follows:<\/p>\n<ul>\n<li><strong>Position in the risky asset:<\/strong> Buy or short-sell <i>x<\/i> units of the asset whose initial price is <i>S(0)<\/i>.<\/li>\n<li><strong>Position in the risk-free asset:<\/strong> Invest or borrow an amount <i>y<\/i> in a bond with price <i>A(0)<\/i> and risk-free rate <i>r<\/i>.<\/li>\n<li><strong>Self-financing condition:<\/strong> The following equation must hold:<\/li>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> V(0) = x S(0) + y A(0) = 0. <\/span>\n<li><strong>Evaluation in the next period:<\/strong> At <i>t = 1<\/i>, the portfolio value is:<\/li>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> V(1) = \\begin{cases} x S(1,\\text{up}) + y A(1), &amp; \\text{if the price goes up}, \\\\ x S(1,\\text{down}) + y A(1), &amp; \\text{if the price goes down}. \\end{cases} <\/span>\n<\/ul>\n<p>If there exists a combination of <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">y<\/span> such that <span class=\"katex-eq\" data-katex-display=\"false\">V(1) \\geq 0<\/span> in both scenarios and <span class=\"katex-eq\" data-katex-display=\"false\">V(1) \\gt 0<\/span> in at least one, an arbitrage opportunity has been found.<\/p>\n<h3>How to Recognize a Market Without Arbitrage Opportunities Using the Theorem<\/h3>\n<p>Suppose an asset has an initial price of <span class=\"katex-eq\" data-katex-display=\"false\">S(0) = 100<\/span> dollars, and in the next period, its price can be:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\nS(1) = \\begin{cases}\n\nS(1,\\text{up}) = S(0) u = 120, &amp; \\text{if the price goes up}, \\\\\n\nS(1,\\text{down}) = S(0) d = 90, &amp; \\text{if the price goes down}.\n\n\\end{cases}\n\n<\/span>\n<p>Meanwhile, a bond grows from <span class=\"katex-eq\" data-katex-display=\"false\">A(0) = 100<\/span> to <span class=\"katex-eq\" data-katex-display=\"false\">A(1) = 105<\/span>, with <span class=\"katex-eq\" data-katex-display=\"false\">r = 5\\%<\/span>. Based on this, we will check for arbitrage by simply verifying the no-arbitrage condition:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">0 \\lt d \\lt 1+r\\lt u<\/span>.<\/p>\n<p>From the given data, we have:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> 0 \\lt 0.9 \\lt 1.05 \\lt 1.2 <\/span>\n<p>Since the inequality holds, it is not possible to construct a self-financing portfolio with risk-free profits, ensuring the consistency of the binomial model.<\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/LcbshxYeYjI?si=uoOXYUtRn31B-KKI\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe><\/center><br \/>\n<a name=\"2\"><\/a><\/p>\n<h2>How to Recognize a Market with Arbitrage Opportunities and Its Rapid Dissolution<\/h2>\n<p>Consider an asset with an initial price of <span class=\"katex-eq\" data-katex-display=\"false\">S(0) = 100<\/span> dollars. In the next period, its price may evolve as follows:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\nS(1) = \\begin{cases}\n\nS(1,\\text{up}) = S(0) u = 105.2, &amp; \\text{if the price goes up}, \\\\\n\nS(1,\\text{down}) = S(0) d = 82, &amp; \\text{if the price goes down}.\n\n\\end{cases}\n\n<\/span>\n<p>The price of the risk-free asset is <span class=\"katex-eq\" data-katex-display=\"false\">A(0) = 100<\/span>, and in the next period, it grows to <span class=\"katex-eq\" data-katex-display=\"false\">A(1) = 107<\/span>, with a risk-free rate of <span class=\"katex-eq\" data-katex-display=\"false\">r = 7\\%<\/span>.<\/p>\n<p>We check the no-arbitrage condition:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> 0 \\lt 0.82 \\lt 1.07 \\not\\lt 1.052 <\/span>\n<p>Since the inequality <span class=\"katex-eq\" data-katex-display=\"false\">1+r \\lt u<\/span> does not hold, arbitrage is possible in this market. To see this, we will construct a self-financing portfolio using the following process:<\/p>\n<ul>\n<li><b>A short sale of one share:<\/b> The risky asset is sold short at <span class=\"katex-eq\" data-katex-display=\"false\">S(0) = 100<\/span>, meaning the investor must borrow one share to sell it in the market.<\/li>\n<li><b>Investment in the risk-free asset:<\/b> The $100 obtained from the short sale is invested in bonds.<\/li>\n<li><b>Repurchase of the share in the next period:<\/b>\n<ul>\n<li>If the price drops to 82, the net profit is <span class=\"katex-eq\" data-katex-display=\"false\">107 - 82 = 25<\/span>.<\/li>\n<li>If the price rises to 105.2, the net profit is <span class=\"katex-eq\" data-katex-display=\"false\">107 - 105.2 = 1.8<\/span>.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p>In both cases, the investor obtains risk-free profits, confirming the existence of arbitrage.<\/p>\n<h3>\ud83d\udccc Market Adjustments to an Arbitrage Strategy<\/h3>\n<p>However, in an efficient market, these opportunities do not persist. As more investors detect this inefficiency, they begin to execute arbitrage strategies through <strong>short selling<\/strong>, leading to several important effects:<\/p>\n<ul>\n<li><b>Increase in the supply of the risky asset:<\/b> Short selling means many investors borrow and sell shares in the market, increasing the supply of available shares. This increase in supply creates <strong>downward pressure on the initial price<\/strong> <span class=\"katex-eq\" data-katex-display=\"false\">S(0)<\/span>.<\/li>\n<li><b>Adjustment in the asset\u2019s future prices:<\/b> Since <span class=\"katex-eq\" data-katex-display=\"false\">S(1, \\text{up}) = S(0) u<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">S(1, \\text{down}) = S(0) d<\/span>, the decline in <span class=\"katex-eq\" data-katex-display=\"false\">S(0)<\/span> leads to a readjustment of the values of <span class=\"katex-eq\" data-katex-display=\"false\">u<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">d<\/span>, affecting their relationship with the risk-free rate <span class=\"katex-eq\" data-katex-display=\"false\">1 + r<\/span>. This tends to restore the no-arbitrage condition.<\/li>\n<li><b>Impact on the bond price:<\/b> As investors use the funds obtained from short selling to invest in bonds, the demand for bonds increases. This generates <strong>an increase in the present price of the bond<\/strong> <span class=\"katex-eq\" data-katex-display=\"false\">A(0)<\/span>. Since the bond\u2019s future value remains <span class=\"katex-eq\" data-katex-display=\"false\">A(1) = 107<\/span>, this <strong>reduces the effective return on investment<\/strong> in bonds, adjusting the risk-free asset&#8217;s yield.<\/li>\n<li><b>Cost of short selling:<\/b> Investors who borrow shares to short sell must pay a <strong>stock lending rate<\/strong> <span class=\"katex-eq\" data-katex-display=\"false\">r_s<\/span>. This rate represents an additional cost, which may reduce the net arbitrage profits.<\/li>\n<\/ul>\n<h3>\ud83d\udccc How Does the Stock Lending Rate Affect Arbitrage?<\/h3>\n<p>If the stock lending rate <span class=\"katex-eq\" data-katex-display=\"false\">r_s<\/span> is high, it can reduce or even eliminate the net profit from arbitrage. The corrected equation for the final value of the arbitrage strategy is:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\nV(1) = A(0)(1 + r - r_s) - S(1)\n\n<\/span>\n<p>Where:<\/p>\n<ul>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">r_s<\/span> is the stock lending rate.<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">A(0)(1+r)<\/span> represents the investment in the bond.<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">S(1)<\/span> is the cost of repurchasing the stock at the end of the period.<\/li>\n<\/ul>\n<p>Incorporating the stock lending rate <span class=\"katex-eq\" data-katex-display=\"false\">r_s<\/span>, the No-Arbitrage condition adjusts as follows:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">0 \\lt d \\lt 1 + r - r_s \\lt u<\/span>\n<p>For this particular case, the values of <span class=\"katex-eq\" data-katex-display=\"false\">r_s<\/span> that satisfy the relation are:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> 0 \\lt 0.82 \\lt 1.07 - r_s \\lt 1.052 <\/span>\n<p>This implies that:<\/p>\n<ul>\n<li><b>If <span class=\"katex-eq\" data-katex-display=\"false\">0 \\leq r_s \\lt 0.018<\/span>:<\/b> The arbitrage opportunity persists, as the profit remains positive in both scenarios.<\/li>\n<li><b>If <span class=\"katex-eq\" data-katex-display=\"false\">0.018 \\leq r_s \\leq 0.25<\/span>:<\/b> Arbitrage disappears, as the stock lending cost balances the equation, eliminating risk-free profits.<\/li>\n<li><b>If <span class=\"katex-eq\" data-katex-display=\"false\">r_s \\gt 0.25<\/span>:<\/b> In this case, no rational investor would carry out the operation, as the borrowing cost exceeds any possible benefit. Since the future value of the portfolio would be negative in all scenarios, a self-financing portfolio in this context is mathematically impossible.<\/li>\n<\/ul>\n<h3>\ud83d\udccc What Happens If Losses Consume the Portfolio? Forced Liquidation and Margin Call<\/h3>\n<p>If the stock lending rate <span class=\"katex-eq\" data-katex-display=\"false\">r_s<\/span> is so high that it guarantees secure losses (<span class=\"katex-eq\" data-katex-display=\"false\">r_s \\gt 0.25<\/span>), the broker automatically intervenes to prevent the investor&#8217;s account from going into a negative balance. This results in a <strong>forced liquidation<\/strong>, also known as a <b>margin call<\/b>.<\/p>\n<h5>\ud83d\udd39 Forced Liquidation Process:<\/h5>\n<ol>\n<li><b>The bond is automatically sold:<\/b>\n<p>The broker liquidates the bond investment <span class=\"katex-eq\" data-katex-display=\"false\">A(0)(1 + r)<\/span> to obtain cash.<\/p>\n<\/li>\n<li><b>Repurchase of the stock to close the short position:<\/b>\n<p>With the available cash, the broker <strong>repurchases the stock<\/strong> at the market price <span class=\"katex-eq\" data-katex-display=\"false\">S(1)<\/span> to return it to the lender.<\/p>\n<\/li>\n<li><b>Debt settlement and position closure:<\/b>\n<p>If the available balance after selling the bond <strong>does not cover the repurchase of the stock<\/strong>, the investor is left with a negative balance, which could lead to legal consequences or require additional fund deposits.<\/p>\n<\/li>\n<li><b>Consolidated loss:<\/b>\n<p>The operation, which was already losing from the beginning, closes with a total loss determined by:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\text{Final Loss} = S(1) - A(0)(1 + r - r_s) <\/span>\n<p>If the <strong>final loss<\/strong> exceeds the available cash in the investor\u2019s account, they lose all their capital and may face a debt with the broker.<\/p>\n<\/li>\n<\/ol>\n<h3>\ud83d\udccc How Is the No-Arbitrage Condition Restored?<\/h3>\n<p>When the stock lending rate <span class=\"katex-eq\" data-katex-display=\"false\">r_s<\/span> is sufficiently low, the arbitrage opportunity remains, encouraging investors to execute short sales in large volumes to secure a risk-free profit.<\/p>\n<p><b>For this analysis, consider that the stock lending rate is <span class=\"katex-eq\" data-katex-display=\"false\">r_s = 0.015<\/span>.<\/b><\/p>\n<p>The high activity driven by this low interest rate causes a market readjustment, which, over time, restores the no-arbitrage condition. In particular, the following effects are observed:<\/p>\n<ul>\n<li><b>Drop in the initial stock price <span class=\"katex-eq\" data-katex-display=\"false\">S(0)<\/span>:<\/b> The high demand for short selling increases the supply of shares in the market, exerting <strong>downward pressure<\/strong> on its initial price. As <span class=\"katex-eq\" data-katex-display=\"false\">S(0)<\/span> decreases, the growth and decline factors <span class=\"katex-eq\" data-katex-display=\"false\">u<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">d<\/span> adjust proportionally, modifying the asset\u2019s future prices and its relationship with the risk-free rate.<\/li>\n<li><b>Increase in the present value of the bond <span class=\"katex-eq\" data-katex-display=\"false\">A(0)<\/span>:<\/b> Investors use the funds obtained from short selling to acquire bonds, causing an increase in their <strong>demand<\/strong>. This raises their present price <span class=\"katex-eq\" data-katex-display=\"false\">A(0)<\/span>, reducing the effective return on bond investments and affecting the perception of the risk-free rate.<\/li>\n<\/ul>\n<p>These combined effects lead to a progressive readjustment of market parameters. The drop in <span class=\"katex-eq\" data-katex-display=\"false\">S(0)<\/span> and the increase in <span class=\"katex-eq\" data-katex-display=\"false\">A(0)<\/span> modify the structure of the coefficients <span class=\"katex-eq\" data-katex-display=\"false\">u<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">d<\/span>, as well as the relationship between the risk-free rate <span class=\"katex-eq\" data-katex-display=\"false\">r<\/span> and the stock lending rate <span class=\"katex-eq\" data-katex-display=\"false\">r_s<\/span>, until the no-arbitrage condition is restored:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">0 \\lt d \\lt 1 + r - r_s \\lt u<\/span>\n<h4>\ud83d\udd39 Modeling the Price Adjustment<\/h4>\n<p>The adjustment process can be modeled using the adjustment coefficients <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span>, which represent the correction factors applied to the present value of bonds and stocks, respectively.<\/p>\n<p>These coefficients modify the current values of assets, adjusting the factors <span class=\"katex-eq\" data-katex-display=\"false\">u<\/span>, <span class=\"katex-eq\" data-katex-display=\"false\">d<\/span>, and <span class=\"katex-eq\" data-katex-display=\"false\">r<\/span> until the no-arbitrage condition is restored. That is, the initial stock price is adjusted from <span class=\"katex-eq\" data-katex-display=\"false\">S(0)<\/span> to <span class=\"katex-eq\" data-katex-display=\"false\">\\beta S(0)<\/span>, while the present value of the bond changes from <span class=\"katex-eq\" data-katex-display=\"false\">A(0)<\/span> to <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha A(0)<\/span>.<\/p>\n<p>As a result, the new values of <span class=\"katex-eq\" data-katex-display=\"false\">u<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">d<\/span> are defined in terms of these adjustment coefficients:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\nu&#039; = \\dfrac{S(1,\\text{up})}{\\beta S(0)}, \\quad d&#039; = \\dfrac{S(1,\\text{down})}{\\beta S(0)}\n\n<\/span>\n<p>Similarly, the new risk-free rate <span class=\"katex-eq\" data-katex-display=\"false\">r&#039;<\/span> is adjusted according to the new present value of the bond:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\nr&#039; + 1 = \\dfrac{A(1)}{\\alpha A(0)}\n\n<\/span>\n<p>This leads to a reformulated no-arbitrage condition:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">0 \\lt \\dfrac{S(1,\\text{down})}{\\beta S(0)} \\lt \\dfrac{A(1)}{\\alpha A(0)} - r_s \\lt \\dfrac{S(1,\\text{up})}{\\beta S(0)} <\/span>\n<p>Solving for the adjustment coefficients, we obtain:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\beta \\gt \\dfrac{A(0)S(1,\\text{down})\\alpha}{S(0)(A(1) - r_s A(0)\\alpha)}\n\n<\/span>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\beta \\lt \\dfrac{A(0)S(1,\\text{up})\\alpha}{S(0)(A(1) - r_s A(0)\\alpha)}\n\n<\/span>\n<p>If we apply the specific values of the problem and consider that stock prices decrease while bond values increase, we obtain:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\n\\beta &amp;\\gt \\dfrac{ 82 \\alpha}{107 - 1.5\\alpha} \\\\ \\\\\n\n\\beta &amp;\\lt \\dfrac{105.2 \\alpha}{107 - 1.5\\alpha } \\\\ \\\\\n\n\\beta &amp;\\lt 1 \\\\ \\\\\n\n\\alpha &amp;\\gt 1\n\n\\end{array}\n\n<\/span>\n<p>The solution to this system is visualized in the darkest region of the following graph:<\/p>\n<p><center><br \/>\n<img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2025\/02\/coef-correccion.jpg\" alt=\"\" width=\"892\" height=\"677\" class=\"aligncenter size-full wp-image-32197 lazyload\" \/><noscript><img decoding=\"async\" src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2025\/02\/coef-correccion.jpg\" alt=\"\" width=\"892\" height=\"677\" class=\"aligncenter size-full wp-image-32197 lazyload\" srcset=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2025\/02\/coef-correccion.jpg 892w, http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2025\/02\/coef-correccion-300x228.jpg 300w, http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2025\/02\/coef-correccion-768x583.jpg 768w\" sizes=\"(max-width: 892px) 100vw, 892px\" \/><\/noscript><br \/>\n<\/center><\/p>\n<p>Therefore, a possible combination of values towards which the market could converge to eliminate the arbitrage opportunity is, for example, <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha=1.05<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">\\beta=0.95<\/span>.<\/p>\n<p>With this, the corrected coefficients are:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\nu^\\prime &amp;= \\dfrac{S(1,\\text{up})}{\\beta S(0)}  = \\dfrac{105.2}{0.95\\cdot 100} \\approx 1.107 \\\\ \\\\\n\nd^\\prime &amp;= \\dfrac{S(1,\\text{down})}{\\beta S(0)}  = \\dfrac{82}{0.95\\cdot 100} \\approx 0.863 \\\\ \\\\\n\nr^\\prime + 1 &amp;= \\dfrac{A(1)}{\\alpha A(0)} = \\dfrac{107}{1.05 \\cdot 100} \\approx 1.019\n\n\\end{array}\n\n<\/span>\n<p>Thus, the no-arbitrage condition is satisfied:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">0 \\lt d^\\prime \\lt 1+r^\\prime - r_s \\lt u^\\prime<\/span>\n<p>Replacing the obtained values:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">0 \\lt 0.863 \\lt 1.019 - 0.015 = 1.004 \\lt 1.107<\/span>\n<p>Additionally, the corrected values of the assets in the present moment can be calculated due to the pressure exerted by investors seeking to take advantage of the arbitrage opportunity:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\nA^\\prime(0) &amp;= \\alpha A(0) = 1.05\\cdot 100 = 105 \\\\ \\\\\n\nS^\\prime(0) &amp;= \\beta S(0) = 0.95\\cdot 100 = 95\n\n\\end{array}\n\n<\/span>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/iBmmjdFzVDI?si=BETKzTcGMB4yiZ8R\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe><\/center><br \/>\n<a name=\"3\"><\/a><\/p>\n<h2>Proof of the No-Arbitrage Condition Theorem<\/h2>\n<p>Up to this point, we have explored the workings of the no-arbitrage condition theorem. Now, we will proceed to develop its proof step by step. To do so, it is useful to identify the signals that indicate the presence of an arbitrage opportunity:<\/p>\n<ul>\n<li>\n<p><b>Relationship between the returns of risky assets and risk-free bonds:<\/b><\/p>\n<p>If the return of the risky asset in its worst-case scenario exceeds the risk-free rate, then it is possible to finance its purchase by borrowing at this rate, ensuring a risk-free profit even in the worst case.<\/p>\n<p>Similarly, if the risk-free rate exceeds the return of the risky asset in its best scenario, then arbitrage can be constructed by short selling the asset and investing in bonds, thereby obtaining a risk-free profit.<\/p>\n<\/li>\n<li>\n<p><b>Relationship between the risk-free rate and the lending rate:<\/b><\/p>\n<p>Complementing the previous point, it is important to distinguish between the borrowing rate <span class=\"katex-eq\" data-katex-display=\"false\">r_s<\/span> and the risk-free rate <span class=\"katex-eq\" data-katex-display=\"false\">r<\/span>, especially when analyzing arbitrage strategies or short selling. In general, the following relationship holds:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">-1\\leq r \\leq r_s<\/span>\n<p>If this relationship does not hold, arbitrage can be obtained by borrowing at the lower rate <span class=\"katex-eq\" data-katex-display=\"false\">r_s<\/span> and investing in bonds at the higher rate <span class=\"katex-eq\" data-katex-display=\"false\">r<\/span>, thereby securing a risk-free profit. If this opportunity existed, investors would exploit it until the market adjusted the rates, eliminating arbitrage. Additionally, lenders usually require a higher rate to compensate for the risk of default.<\/p>\n<p>In simplified financial models, it is often assumed that <span class=\"katex-eq\" data-katex-display=\"false\">r_s = r<\/span>, and in most cases, it is also imposed that <span class=\"katex-eq\" data-katex-display=\"false\">r \\geq 0<\/span> to avoid negative rates, although this is not strictly necessary.<\/p>\n<\/li>\n<li>\n<p><b>Conditions for the existence of arbitrage in a portfolio:<\/b><\/p>\n<p>The value of a portfolio at present time <span class=\"katex-eq\" data-katex-display=\"false\">t=0<\/span> is given by:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">V(0) = xS(0) + y A(0)<\/span>\n<p>where <span class=\"katex-eq\" data-katex-display=\"false\">S(0)<\/span> represents the present value of stocks and <span class=\"katex-eq\" data-katex-display=\"false\">A(0)<\/span> the present value of bonds. In the future time <span class=\"katex-eq\" data-katex-display=\"false\">t=1<\/span>, the value of the portfolio will depend on the evolution of the risky asset:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">V(1) =\n\n\\begin{cases}\n\nx S(0) u + y A(0) (1 + r), &amp;\\text{if the price goes up},\\\\\n\nx S(0) d + y A(0) (1 + r), &amp;\\text{if the price goes down}.\n\n\\end{cases}<\/span>\n<p>An arbitrage opportunity exists if and only if it is possible to construct a portfolio <span class=\"katex-eq\" data-katex-display=\"false\">(x,y)<\/span> that satisfies the following three conditions:<\/p>\n<ol>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">V(0)=0<\/span>, meaning the portfolio is self-financing and requires no initial investment.<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">V(1)\\geq 0 <\/span> in all possible market states, guaranteeing no losses.<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">V(1) \\gt 0<\/span> in at least one of the possible states, ensuring a strictly positive profit.<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<p>To develop this proof, we introduce the following notation convention:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rcl}\n\nV(1,\\omega) &amp;=&amp; xS(1,\\omega) + yA(1).\n\n\\end{array}\n\n<\/span>\n<p>Where <span class=\"katex-eq\" data-katex-display=\"false\">\\omega<\/span> can be <span class=\"katex-eq\" data-katex-display=\"false\">\\text{up}<\/span> or <span class=\"katex-eq\" data-katex-display=\"false\">\\text{down}<\/span>. Additionally, it is necessary to mathematically express the condition that holds when there exists a portfolio <span class=\"katex-eq\" data-katex-display=\"false\">(x,y)<\/span> that exploits an arbitrage opportunity. This is formulated as follows:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{l}\n\nV(0) = 0, \\\\\n\n\\forall \\omega \\quad V(1,\\omega) \\geq 0, \\\\\n\n\\exists \\omega \\quad V(1,\\omega) &gt; 0.\n\n\\end{array}\n\n<\/span>\n<p>With these concepts clear, we can now mathematically and rigorously define the expression that characterizes an arbitrage opportunity:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n\\text{Arbitrage}:= &amp; V(0) = 0 \\wedge  (\\exists xy\\in\\mathbb{R}\\setminus\\{0\\})(\\forall \\omega \\quad V(1,\\omega) \\geq 0) \\wedge \\cdots \\\\\n\n&amp; \\cdots \\wedge  (\\exists xy\\in\\mathbb{R}\\setminus\\{0\\})(\\exists \\omega \\quad V(1,\\omega) \\gt 0) \\\\ \\\\\n\n\\text{No-Arbitrage}:= &amp; \\neg \\text{Arbitrage}\\\\\n\n= &amp; V(0) \\neq 0 \\vee  \\neg(\\exists xy\\in\\mathbb{R}\\setminus\\{0\\})(\\forall \\omega \\quad V(1,\\omega) \\geq 0) \\vee \\cdots \\\\\n\n&amp; \\cdots \\vee  \\neg(\\exists xy\\in\\mathbb{R}\\setminus\\{0\\})(\\exists \\omega \\quad V(1,\\omega) \\gt 0)\n\n\\end{array}<\/span>\n<p>Finally, the set of premises <span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{H}<\/span> upon which the proof is developed is expressed as follows:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rcl}\n\n\\mathcal{H} &amp;=&amp; \\left\\{  \\right. V(0)=xS(0) + yA(0) = 0, \\\\ \\\\\n\n&amp; &amp;V(t,\\omega) = xS(t,\\omega) + yA(t), A(0), S(0) \\gt 0, \\\\ \\\\\n\n&amp; &amp;  S(1) = \\begin{cases} S(1, \\text{up})  = S(0)u &amp; \\text{with probability } p \\\\ S(1,\\text{down})  = S(0)d &amp; \\text{with probability } 1-p \\end{cases},  \\\\ \\\\\n\n&amp; &amp;  0 \\lt d \\lt u , \\left.  A(1) = A(0)(1+r), r\\geq -1 \\right\\}\n\n\\end{array}<\/span>\n<p>This set not only includes the premises of the theorem but also the underlying conditions of the one-period binomial model.<\/p>\n<p>With these principles established, we proceed to mathematically prove the relationship that must hold in an arbitrage-free market.<\/p>\n<h3>Formal Proof of the Theorem:<\/h3>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n(1) &amp; \\mathcal{H} \\models V(0) =xS(0) + yA(0) = 0 &amp; \\text{; Assumption} \\\\\n\n(2) &amp; \\mathcal{H} \\models V(1,\\omega) =xS(1,\\omega) + yA(1) &amp; \\text{; Assumption} \\\\\n\n(3) &amp; \\mathcal{H} \\models A(0) \\gt  0 &amp; \\text{; Assumption} \\\\\n\n(4) &amp; \\mathcal{H} \\models S(0) \\gt  0 &amp; \\text{; Assumption} \\\\\n\n(5) &amp; \\mathcal{H} \\models r \\gt  -1 &amp; \\text{; Assumption} \\\\\n\n(6) &amp; \\mathcal{H} \\models A(1) = (1+r) A(0) &amp; \\text{; Assumption} \\\\\n\n(7) &amp;\\color{red}\\mathcal{H} \\models 0 \\lt d \\lt u \\color{black}&amp; \\text{; Assumption} \\\\ \\\\\n\n(8) &amp; \\mathcal{H} \\models S(1) = \\begin{cases}S(1,\\text{up})=S(0)u &amp; \\text{, with probability } p \\\\ S(1,\\text{down}) = S(0)d &amp; \\text{, with probability } 1-p\\end{cases} &amp; \\text{; Assumption} \\\\ \\\\\n\n(9) &amp; \\mathcal{H} \\models y = \\dfrac{-xS(0)}{A(0)} \\wedge x\\in\\mathbb{R} &amp; \\text{; From (1)} \\\\\n\n(10)&amp; \\mathcal{H} \\models V(1,\\omega) =xS(1,\\omega) - \\dfrac{xS(0)}{A(0)} A(1) &amp; \\text{; From (2,9)} \\\\\n\n(11)&amp; \\mathcal{H} \\models V(1,\\omega) =xS(1,\\omega) - x(1+r)S(0) &amp; \\text{; From (6,10)} \\\\\n\n &amp;\\text{This is the future value of a portfolio financed with a loan} &amp;\\\\\n\n &amp;\\text{at interest rate $r$ to finance a stock purchase.} &amp;\\\\\n\n(12)&amp; \\mathcal{H}\\cup\\{1+r\\leq d\\} \\models 0 \\leq (1+r)S(0) \\leq \\underbrace{S(0) d}_{S(1,\\text{down})} \\lt \\underbrace{S(0) u}_{S(1,\\text{up})} &amp; \\text{; From (4,5,7,8)}\\\\\n\n(13)&amp; \\mathcal{H}\\cup\\{1+r\\leq d\\} \\models x(1+r)S(0) \\leq xS(1,\\omega) \\leftrightarrow x\\gt 0 &amp; \\text{; From (12)}\\\\\n\n(14)&amp; \\mathcal{H}\\cup\\{1+r\\leq d\\} \\models (\\exists xy\\in\\mathbb{R}\\setminus\\{0\\})(\\forall \\omega\\quad V(1,\\omega) \\geq 0) &amp;\\text{; From (2,9,13)}\\\\\n\n(15)&amp; \\mathcal{H}\\cup\\{1+r\\leq d\\} \\models V(1,\\omega) \\gt 0 \\leftrightarrow y \\gt \\dfrac{-xS(1,\\omega)}{A(1)} = \\dfrac{-xS(1,\\omega)}{(1+r)A(0)} &amp; \\text{; From (2,3,6,7.8)}\\\\\n\n(16)&amp;\\mathcal{H}\\cup\\{1+r\\leq d\\} \\models (\\exists xy\\in\\mathbb{R}\\setminus\\{0\\})(\\exists \\omega\\quad V(1,\\omega)\\gt 0) &amp;\\text{; From (14,15)}\\\\\n\n(17)&amp; \\mathcal{H}\\cup\\{1+r\\leq d\\} \\models \\text{Arbitrage} &amp;\\text{; From (1,14,16)}\\\\\n\n(18)&amp; \\color{red}\\mathcal{H}\\cup\\{\\text{No-Arbitrage}\\} \\models d \\lt 1+r\\color{black}&amp; \\text{; Proof by Contradiction (17)}\\\\ \\\\\n\n(19)&amp; \\mathcal{H}\\cup\\{u \\leq 1+r\\} \\models 0 \\lt \\underbrace{S(0)d}_{S(1,\\text{down})} \\lt \\underbrace{S(0)u}_{S(1,\\text{up})} \\leq (1+r)S(0) &amp; \\text{; From (4,5,7,8)}\\\\\n\n(20)&amp; \\mathcal{H}\\cup\\{u \\leq 1+r\\} \\models xS(1,\\omega) \\leq x(1+r)S(0) \\leftrightarrow x\\gt 0  &amp;\\text{; From (19)} \\\\\n\n(21)&amp;\\mathcal{H}\\cup\\{u \\leq 1+r\\} \\models \\tilde{V}(0) = - V(0) = 0 &amp; \\text{; From (1)}\\\\\n\n(22)&amp;\\mathcal{H}\\cup\\{u \\leq 1+r\\} \\models\\tilde{V}(1,\\omega)=-V(1,\\omega) &amp; \\\\\n\n &amp;\\phantom{\\mathcal{H}\\cup\\{u \\leq 1+r\\} \\models\\tilde{V}(1,\\omega)}=-xS(1,\\omega)+x(1+r)S(0) &amp; \\text{;From (11)}\\\\\n\n &amp;\\text{This is the future value of a portfolio financed by short-selling a stock} &amp;\\\\\n\n &amp;\\text{to buy a bond growing at interest rate $r$.} &amp; \\\\\n\n(23)&amp;\\mathcal{H}\\cup\\{u \\leq 1+r\\} \\models (\\exists xy\\in\\mathbb{R}\\setminus\\{0\\})(\\forall \\omega\\quad \\tilde{V}(1,\\omega) \\geq 0) &amp; \\text{; From (2,9,20,22)}\\\\\n\n(24)&amp;\\mathcal{H}\\cup\\{u \\leq 1+r\\} \\models \\tilde{V}(1,\\omega)\\gt 0 \\leftrightarrow y \\lt \\dfrac{-xS(1,\\omega)}{A(1)} = \\dfrac{-xS(1,\\omega)}{(1+r)A(0)} &amp;\\text{; From (2,3,4,6,22)}\\\\\n\n(25)&amp;\\mathcal{H}\\cup\\{u \\leq 1+r\\} \\models(\\exists xy\\in\\mathbb{R}\\setminus\\{0\\})(\\exists \\omega\\quad \\tilde{V}(1,\\omega)\\gt 0) &amp;\\text{; From (23,24)}\\\\\n\n(26)&amp;\\mathcal{H}\\cup\\{u \\leq 1+r\\} \\models \\text{Arbitrage} &amp;\\text{; From (21,23,25)}\\\\\n\n(27)&amp;\\color{red}\\mathcal{H}\\cup\\{\\text{No-Arbitrage}\\} \\models 1+r \\lt u\\color{black}&amp; \\text{; Proof by Contradiction (26)}\\\\\n\n(28) &amp;\\mathcal{H}\\cup\\{\\text{No-Arbitrage}\\} \\models 0\\lt d\\lt1+r\\lt u &amp;\\text{;\\color{red}Conjunction (7,18,27)}\\color{black} \\\\\n\n(29)&amp; \\boxed{\\mathcal{H} \\models\\text{No-Arbitrage}\\rightarrow 0\\lt d\\lt1+r\\lt u}  &amp; \\text{; From (28)}\\\\ \\\\\n\n(37)&amp; \\color{blue}\\mathcal{H} \\models 0\\lt d\\lt 1+r \\lt u \\leftrightarrow \\text{No-Arbitrage}\\color{black}\\quad\\blacksquare &amp; \\text{; From (29)}\n\n\\end{array}<\/span>\n<p><a name=\"4\"><\/a><\/p>\n<h2>Conclusion<\/h2>\n<p>The one-period binomial model and the no-arbitrage condition are fundamental pillars in financial theory, providing a structured framework for asset valuation and market stability. Throughout this article, we have analyzed how arbitrage opportunities, while theoretically attractive, are quickly eliminated by market forces through adjustments in asset prices and interest rates. We have mathematically demonstrated that the relationship between an asset\u2019s growth and decline factors and the risk-free rate is key to ensuring an efficient market free of risk-free profit opportunities. Furthermore, we observed that even when arbitrage opportunities arise, mechanisms such as price pressure, borrowing costs, and the reconfiguration of market parameters inevitably lead to the restoration of equilibrium. With this understanding, it becomes clear that arbitrage is not merely a temporary anomaly but a fundamental element in the dynamics of financial markets that drives their efficiency and mathematical consistency.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The One-Period Binomial Model and the No-Arbitrage Condition Abstract: Imagine a casino where you can bet on a game and, regardless of the outcome, always make money. Sounds too good to be true, right? In financial markets, such opportunities arise due to the possibility of arbitrage; however, they are quickly eliminated by the actions of [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":32196,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":12,"footnotes":""},"categories":[749,903],"tags":[],"class_list":["post-32409","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-economics-and-finance","category-financial-mathematics"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>The One-Period Binomial Model and the No-Arbitrage Condition - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Discover the One-Period Binomial Model: its logic, financial applications, and how it prevents arbitrage with a rigorous mathematical 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