{"categories":["Risk Management"],"contentHtml":"<h4>Lecture Notes</h4><h5>Project</h5><p>On problem 2, either you optimize for risk-free or risky assets. The question becomes unconstrained optimization.</p><h5>Final</h5><p>Either we have the same thing as Exam 1, released on 13 or 19? Bonus question on Project 2, but requires reading. Stochastic Interest Rate Models (Vasicek)</p><p>Yields are going up, because people believe that Trump will create inflationary pressure.</p><p>Last video in Modules is worth watching!</p><h5>Forward Contract</h5><p>How do you price a forward contract? The underlying asset is unknown. Both parties have some views on how this process will evolve, but the price must be fair.</p><p>He has to pay <math><msub><mi>𝐹</mi><mi>𝑡</mi></msub><mo form=\"prefix\" stretchy=\"false\">(</mo><mi>𝑇</mi><mo stretchy=\"false\">)</mo></math>.</p><p>And he can sell it for the value of the underlying <math><msub><mi>𝑆</mi><mi>𝑡</mi></msub></math>.</p><p>We can simulate each time, where one party ins and one loses. How do you determine the actual price.</p><p>By expectation, they should both be better off.</p><p><math><mi>𝔼</mi><mo form=\"prefix\" stretchy=\"false\">[</mo><msub><mi>𝑉</mi><mi>𝐴</mi></msub><mo form=\"postfix\" stretchy=\"false\">]</mo><mo>=</mo><mi>𝔼</mi><mo form=\"prefix\" stretchy=\"false\">[</mo><msub><mi>𝑉</mi><mi>𝐵</mi></msub><mo stretchy=\"false\">]</mo></math></p><h3>Week 11</h3><h4>Lecture Notes</h4><p>We’re going to continue derivatives, uninteresting because I’m taking 620. Then we will start linear risk, more interesting.</p><h5>Risk Neutral Valuation</h5><p>A financial engineer is someone who doesn’t know finance or engineering.</p><p>I can bundle two products together and have an outcome, so the price of the combination should be the outcome. A derivative has some underlying value. The forward contract has the value of the stock price with the risk-free price increase of the interest rate.</p><p>The futures of EUR/USD tell us what people think the exchange rate will be at that time. The <em>expected value</em>.</p><p>We need to ensure that there is no arbitrage in this process.</p><p>The intuition of risk-neutral is that you will replace the <math><mi>𝜇</mi></math> in the GBM with the risk-free rate, because that’s the expected return. This keeps things consistent with no-arbitrage pricing.</p><p>Why do dividends matter when it comes to the forward contract?</p><p>You can buy the asset today and start getting dividends, or you can get the forward contract that will give you the asset in the future.</p><p>The higher the dividend, the more I’m missing. So if the yield of my dividend is q, then the future price should be lowered to</p><p><math><msub><mi>𝐹</mi><mi>𝑡</mi></msub><mo form=\"prefix\" stretchy=\"false\">(</mo><mi>𝑇</mi><mo form=\"postfix\" stretchy=\"false\">)</mo><mo>=</mo><msub><mi>𝑆</mi><mi>𝑡</mi></msub><msup><mi>𝑒</mi><mrow><mo stretchy=\"false\">(</mo><mi>𝑟</mi><mo lspace=\"0em\" rspace=\"0em\">−</mo><mi>𝑞</mi><mo form=\"postfix\" stretchy=\"false\">)</mo><mi>𝜏</mi></mrow></msup></math></p><p>What do we do if the forward price is high? Remember, that’s how you acquire the asset through the contract, and then the <math><msub><mi>𝑆</mi><mi>𝑡</mi></msub></math> formula is how you acquire it through the underlying. You want to short the contract and buy the underlying.</p><h5>Example Forward Dividend</h5><p>We have</p><p><math><msub><mi>𝑆</mi><mn>0</mn></msub><mo>=</mo><mn>990</mn></math></p><p><math><mi>𝐹</mi><mo>=</mo><mn>1000</mn></math></p><p><math><mi>𝑞</mi><mo>=</mo><mn>0.02</mn></math></p><p><math><mi>𝑟</mi><mo>=</mo><mn>0.04</mn></math></p><p><math><mi>𝜏</mi><mo>=</mo><mn>0.25</mn></math></p><p>The fair price for <math><mi>𝐹</mi></math> is $995, so there is a $5 mispricing per contract.</p><h5>Interest Rate Parity</h5><p>When you’re thinking of buying a stock that pays dividend, the same argument is used for foreign exchange. If I’m buying euros, the exchange rate should take into account the risk-free interest rate for euros, which is different from dollars.</p><p>Let’s say I pay $106 to get €100. There is a foreign interest rate that the price will increase at, different from the USD risk-free interest rate.</p><p>This is a similar formula as for dividends</p><p><math><msub><mi>𝐹</mi><mi>𝑡</mi></msub><mo form=\"prefix\" stretchy=\"false\">(</mo><mi>𝜏</mi><mo form=\"postfix\" stretchy=\"false\">)</mo><mo>=</mo><msub><mi>𝑆</mi><mi>𝑡</mi></msub><msup><mi>𝑒</mi><mrow><mo stretchy=\"false\">(</mo><mi>𝑟</mi><mo lspace=\"0em\" rspace=\"0em\">−</mo><msup><mi>𝑟</mi><mo>∗</mo></msup><mo form=\"postfix\" stretchy=\"false\">)</mo><mi>𝜏</mi></mrow></msup></math></p><p>where r* is the foreign interest rate.</p><p>Let’s say you need to deliver ₤1 million six months from now, when the GBP/USD rate is 1.308, and USD risk-free rate 2%. You might think that the forward price should be</p><p><math><msub><mi>𝐹</mi><mn>0</mn></msub><mo form=\"prefix\" stretchy=\"false\">(</mo><mn>0.5</mn><mo form=\"postfix\" stretchy=\"false\">)</mo><mo>=</mo><mn>1.308</mn><msup><mi>𝑒</mi><mrow><mn>0.02</mn><mo lspace=\"0em\" rspace=\"0em\">×</mo><mn>0.5</mn></mrow></msup><mo>≈</mo><mn>1.321</mn></math>.</p><p>But actually this forward contract is <em>higher</em> than the fair price because it doesn’t take into account the foreign interest rate. If this was the true price, you would have arbitrage.</p><p>In order to exploit this, I would borrow $1.308 today and buy €</p><p>If the market becomes stronger, than there’s inflation, which will cause the interest rates to go up.</p><h5>Futures</h5><p>Standardized, traded on exchanges.</p><p>They are marked to market, which means that every day someone is going to bother you to ask you what the price is.</p><p>You can also buy these on margin, so the trade will clear. An initial margin must be posted before you initiative a position. The clearinghouses will ensure that if the person I’m trading with disappears, they will find someone else to take his place and give me my money.</p><p>There’s a joke in Hull, for trading cattle. There was a new intern who didn’t close the position and ended up double short.</p>","contentSourceFormat":"typst","contentTypst":"=== Lecture Notes\n<lecture-notes>\n==== Project\n<project>\nOn problem 2, either you optimize for risk-free or risky assets. The question becomes unconstrained optimization.\n\n==== Final\n<final>\nEither we have the same thing as Exam 1, released on 13 or 19? Bonus question on Project 2, but requires reading. Stochastic Interest Rate Models (Vasicek)\n\nYields are going up, because people believe that Trump will create inflationary pressure.\n\nLast video in Modules is worth watching!\n\n==== Forward Contract\n<forward-contract>\nHow do you price a forward contract? The underlying asset is unknown. Both parties have some views on how this process will evolve, but the price must be fair.\n\nHe has to pay $F_t\\(T\\)$.\n\nAnd he can sell it for the value of the underlying $S_t$.\n\nWe can simulate each time, where one party ins and one loses. How do you determine the actual price.\n\nBy expectation, they should both be better off.\n\n$bb(E)\\[V_A\\]= bb(E)\\[V_B\\]$\n\n== Week 11\n<week-11>\n=== Lecture Notes\n<lecture-notes-1>\nWe're going to continue derivatives, uninteresting because I'm taking 620. Then we will start linear risk, more interesting.\n\n==== Risk Neutral Valuation\n<risk-neutral-valuation>\nA financial engineer is someone who doesn't know finance or engineering.\n\nI can bundle two products together and have an outcome, so the price of the combination should be the outcome. A derivative has some underlying value. The forward contract has the value of the stock price with the risk-free price increase of the interest rate.\n\nThe futures of EUR/USD tell us what people think the exchange rate will be at that time. The #emph[expected value].\n\nWe need to ensure that there is no arbitrage in this process.\n\nThe intuition of risk-neutral is that you will replace the $mu$ in the GBM with the risk-free rate, because that's the expected return. This keeps things consistent with no-arbitrage pricing.\n\nWhy do dividends matter when it comes to the forward contract?\n\nYou can buy the asset today and start getting dividends, or you can get the forward contract that will give you the asset in the future.\n\nThe higher the dividend, the more I'm missing. So if the yield of my dividend is q, then the future price should be lowered to\n\n$F_t\\(T\\)= S_t e^(\\(r - q\\)tau)$\n\nWhat do we do if the forward price is high? Remember, that's how you acquire the asset through the contract, and then the $S_t$ formula is how you acquire it through the underlying. You want to short the contract and buy the underlying.\n\n==== Example Forward Dividend\n<example-forward-dividend>\nWe have\n\n$S_0 = 990$\n\n$F = 1000$\n\n$q = 0.02$\n\n$r = 0.04$\n\n$tau = 0.25$\n\nThe fair price for $F$ is \\$995, so there is a \\$5 mispricing per contract.\n\n==== Interest Rate Parity\n<interest-rate-parity>\nWhen you're thinking of buying a stock that pays dividend, the same argument is used for foreign exchange. If I'm buying euros, the exchange rate should take into account the risk-free interest rate for euros, which is different from dollars.\n\nLet's say I pay \\$106 to get €100. There is a foreign interest rate that the price will increase at, different from the USD risk-free interest rate.\n\nThis is a similar formula as for dividends\n\n$F_t\\(tau\\)= S_t e^(\\(r - r^(*)\\)tau)$\n\nwhere r\\* is the foreign interest rate.\n\nLet's say you need to deliver ₤1 million six months from now, when the GBP/USD rate is 1.308, and USD risk-free rate 2%. You might think that the forward price should be\n\n$F_0\\(0.5\\)= 1.308 e^(0.02 times 0.5) approx 1.321$.\n\nBut actually this forward contract is #emph[higher] than the fair price because it doesn't take into account the foreign interest rate. If this was the true price, you would have arbitrage.\n\nIn order to exploit this, I would borrow \\$1.308 today and buy €\n\nIf the market becomes stronger, than there's inflation, which will cause the interest rates to go up.\n\n==== Futures\n<futures>\nStandardized, traded on exchanges.\n\nThey are marked to market, which means that every day someone is going to bother you to ask you what the price is.\n\nYou can also buy these on margin, so the trade will clear. An initial margin must be posted before you initiative a position. The clearinghouses will ensure that if the person I'm trading with disappears, they will find someone else to take his place and give me my money.\n\nThere's a joke in Hull, for trading cattle. There was a new intern who didn't close the position and ended up double short.","dataUrl":"https://sharifhsn.dev/api/posts/forward-contracts.json","date":"2024-11-07","datePublished":"2024-11-07","description":"On problem 2, either you optimize for risk-free or risky assets. The question becomes unconstrained optimization. Either we have the same thing as Exam 1, released on 13 or 19?…","site":"https://sharifhsn.dev","slug":"forward-contracts","source":"FE-535 | Risk Management","sourceUrl":null,"tags":["Risk Management","Forward Contracts","Futures"],"title":"Forward Contracts","url":"https://sharifhsn.dev/blog/forward-contracts/","version":"1","wordCount":769}