PPT 5 - 5. Binomial Tree Model Reading: Chapter 12, Chapter...

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5. Binomial Tree Model Reading: Chapter 12, Chapter 11 of Luenberger 1
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5.1 A Toy Model That Could Later Be Made Very Realistic Suppose a stock is currently at price S. In a later time t, it could move up and down, but only to one of the two known states: We do not know the probability p that the stock will go up, or ( 1-p ) that it will go down. So the stock price at t is random . Is there a way to calculate p? How do we value a call option C at t=0 ? 2 or , 0. Su Sd u d 
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How to make the toy model more realistic (Chapter 11; will not cover in detail here) Make t very small, and attach many t s one after the other. Small fixed bistate changes in each t can add to yield any value of change in a stock price at larger time intervals. We can match the historical volatility of this stock with the volatility of the binomial tree model to yield 3 and tt u e d e  
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We Will Show: The Probability of an Up Move (in a Risk-Neutral World (No Arbitrage)) is: contract futures a for 1 rate free - risk foreign the is here currency w a for index the on yield dividend the e index wher stock a for stock paying d nondividen a for a r e a q e a e a d u d a p f t r r t q r t r f ) ( ) ( 4
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Risk-Neutral Pricing of an Option The current price of a Call option (or any other derivative option) is: 5 [ (1 ) ] Same as the expected value of the option price at the next time step, converted to present value. rt ud C pC p C e 
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0 0 S C stock price today call price today 6 Stock Price = $22 Option Price = $1 Stock Price = $18 Option Price = $0 Stock price = $20 Option Price=? 1 1 S C stock price at maturity (3 mos) call price at maturity (3 mos) 5.2 Constructing a Deterministic Portfolio 3-Month Call Option with strike K = 21
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The Riskless Hedge 0 0 0 P S C   7 Consider the Portfolio: long Δ shares, short 1 call option 22Δ – 1 18Δ Portfolio is riskless when 22 – 1 = 18Δ or Δ = 0.25 1 1 1 P S C Value today Value in 3 months is random depending on stock price is random 1 P 1 S
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5.3 Valuing the Portfolio Assumed risk-free rate: 12% Riskless portfolio is long .25 shares and short a call option The value of the portfolio in 3 months (maturity) is The value of the portfolio today is 1 1 1 .25 22 1 4.50 P S C     8 .12 .25 .12 .25 01 4.5 4.367 P P e e
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This note was uploaded on 03/23/2012 for the course AMATH 541 taught by Professor Kk.t during the Winter '11 term at University of Washington.

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PPT 5 - 5. Binomial Tree Model Reading: Chapter 12, Chapter...

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