lec 1 - Lecture 1: The Tree Approach I - Binomial Branch...

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Lecture 1: The Tree Approach I -- Binomial Branch Framework Framework: a risky stock S , a riskless cash bond D , and a derivative F . (Capital letters represent random variables and lower cases represent realizations.) Question: What is the initial price of the derivative? ± The unconscious statistician might pay ( ) 0 (1 ) rt ud f ep f p f −∆ =+ , i.e. the expectation of the future payoffs discounted by the risk-free rate. ± But, this answer is wrong! Why? The relationship between a derivative and its underlying is the same no matter what discount factor we use, as long as we use the consistent probability measure. ± the appropriate discount rate µ and the true probability p () 0 ) t f f p f the appropriate discount rate --- very difficult to determine the true probability p --- often not observable; s 0 s u s d p 1-p t=0 t= t 1 e p 1-p t=0 f 0 ?? f u f d p 1-p t=0 e The stock price starts at 0 s ( 0 S = s 0 ). A short tick later ( t ), the stock price can jump up to s u ( t S = s u ) with probability p and jump down to s d ( t S = s d ) with probability (1-p). t= t t= t The bond price is not random. Let r be the continuously compounding risk-free rate. If the initial price of a cash bond is 0 D =1, then a short tick later, the bond price will grow to t D = 0 D e with probability 1 no matter the stock price goes up or down. A derivative pays off f u if the stock price goes up and f d if the stock price goes down. S D F
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MFIN7003: Mathematical Techniques of Finance I_Dr. Rujing Meng 2 ± the risk-free discount rate r and the risk-neutral probability q () 0 (1 ) rt ud f eq f q f −∆ =+ the risk-free discount rate --- observable; the risk-neutral probability --- easy to calculate. How? No-arbitrage Definition of Arbitrage: The simultaneous buying and selling of a security at two different prices in two different markets, resulting in profits without risk. Perfectly efficient markets present no arbitrage opportunities. Perfectly efficient markets seldom exist, but, arbitrage opportunities are often precluded because of transactions costs. 1 Axiom: There is no arbitrage in financial markets. Based on this principle, we have the following law: The Law of One Price: If two securities have the same future payoffs, then they must have the same price.
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This note was uploaded on 04/20/2010 for the course MATH 456 taught by Professor Blyth during the Spring '10 term at Athens State.

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lec 1 - Lecture 1: The Tree Approach I - Binomial Branch...

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