lec1n - Foundations of Financial Engineering Professor S...

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Foundations of Financial Engineering Professor S. Kou Department of IEOR, Columbia University Lecture 1.a Introduction 1. What is this course about? 2. Why take this course? 1

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Question 1. What is this course about? Financial engineering uses mathematical and statistical tools to study complicated f nancial markets and products. In other words, we will build mathematical/statistical mod- els. In this course you will learn Basic theory of asset pricing and investment, such as ar- bitrage, option pricing, utility maximization, and CAPM. Applications to f xed income, credit, FX, and commodity markets 2
Two successful examples of FE: Example 1. How to invest? A very di cult question. No answer yet. The answer also depends on who asked the question. More precisely, the investment strategies for individuals and cor- porations are di f erent. However, we do know some facts: 3

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(1) Arbitrage opportunities cannot exist. Cambridge English Dictionary: Arbitrage. noun [U] SPECIALIZED the method on the stock exchange of buying something in one place and selling it in another place at the same time, in order to make a pro f t from the di f erence in price in the two places. More precisely, there cannot be two portfolio strategies with the same initial cost and one has always a better f nal payo f than the other one with probability one. 4
(2) Given the same return, people prefer to have a portfolio with a smaller variance. (3)Demandmustbeequa ltosupp ly . Theseareparto fthe“e cient market hypothesis”. Amazingly, many results can be derived based on only these. 5

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For example: (a) Minimizing the variance, subject to the return being the same, leads to the mean-variance analysis (Markowitz, Nobel prize in 1990). (b) No arbitrage opportunity implies that the corporation dividend policy (i.e. how much a corporation pays dividend to its shareholders) is irrelevant in an ideal market (MM the- orem, Nobel prize in 1990). (c) The fact that demand must be equal to supply and the mean variance analysis lead to the Capital Asset Pricing Model (CAPM, Nobel prize in 1990). 6
How can these simple assumptions lead to such important results? The key is mathematics and statistics, which enables people to build models to reach the furthermost conclusion with only minimum assumptions. 7

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Example 2. Pricing of Complex Financial Products. Derivatives are securities whose values depend on (derived from) the values of the underlying assets. In many cases, the underlying assets are f nancial securities. For example, a payo f from a European call option max(0 ,S ( T ) K ) , and from a European put option max(0 ,K S ( T )) , where K is the strike price. Call and put options are most liquid derivatives traded in market, and their quote can be found, for example, on the web page of yahoo. Show the yahoo web page.
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lec1n - Foundations of Financial Engineering Professor S...

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