week5 - Momentum and Technical Analysis.ppt

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1、12.1Introduction to Binomial TreesChapter 12-continued12.2GeneralizationA derivative lasts for time T and is dependent on a stockSu uSd dS12.3Generalization(continued)lConsider the portfolio that is long D shares and short 1 derivativelThe portfolio is riskless when SuD u=Sd D d orSuD uSdD d12.4Gene

2、ralization(continued)lValue of the portfolio at time T is Su D ulValue of the portfolio today is (Su D u)erTlAnother expression for the portfolio value today is S D flHence =S D (Su D u)erT 12.5Generalization(continued)lSubstituting for D we obtain =p u+(1 p)d erTwhere 12.6Risk-Neutral Valuationl=p

3、u+(1 p)d e-rTlThe variables p and(1 p)can be interpreted as the risk-neutral probabilities of up and down movementslThe value of a derivative is its expected payoff in a risk-neutral world discounted at the risk-free rateSu uSd dSp(1 p)12.7Original Example RevisitedlSince p is a risk-neutral probabi

4、lity20e0.12 0.25=22p+18(1 p);p=0.6523lAlternatively,we can use the formulaSu=22 u=1Sd=18 d=0S p(1 p)12.8Valuing the Option The value of the option is e0.120.25 0.65231+0.34770 =0.633Su=22 u=1Sd=18 d=0S0.65230.347712.9A Two-Step ExamplelEach time step is 3 monthslK=21,r=12%20221824.219.816.212.10Valu

5、ing a Call OptionlValue at node B=e0.120.25(0.65233.2+0.34770)=2.0257lValue at node A=e0.120.25(0.65232.0257+0.34770)=1.2823201.2823221824.23.219.80.016.20.02.02570.0ABCDEF12.11A Put Option Example;K=52K=52,Dt=1yrr=5%504.1923604072048432201.41479.4636ABCDEF12.12What Happens When an Option is America

6、n 505.0894604072048432201.414712.0ABCDEF12.13DeltalDelta(D)is the ratio of the change in the price of a stock option to the change in the price of the underlying stocklThe value of D varies from node to node12.14Choosing u and dOne way of matching the volatility is to set where s is the volatility a

7、nd Dt is the length of the time step.This is the approach used by Cox,Ross,and Rubinstein12.15The Probability of an Up Move for CRR modelBinomial Trees in PracticeChapter 18Stock Prices on the Tree(Figure 18.2,page 393)S0u 2 S0u 4 S0d 2 S0d 4 S0 S0u S0d S0 S0 S0u 2 S0d 2 S0u 3 S0u S0d S0d 3 17Backwa

8、rds InductionlWe know the value of the option at the final nodeslWe work back through the tree using risk-neutral valuation to calculate the value of the option at each node,testing for early exercise when appropriate18Example:Put OptionS0=50;K=50;r=10%;s=40%;T=5 months=0.4167;Dt=1 month=0.0833The p

9、arameters imply u=1.1224;d=0.8909;a=1.0084;p=0.507319Example(continued)Figure 18.3,page 39520Calculation of DeltaDelta is calculated from the nodes at time Dt 21Trees and Dividend YieldslWhen a stock price pays continuous dividends at rate q we construct the tree in the same way but set a=e(r q)DtlF

10、or options on stock indices,q equals the dividend yield on the indexlFor options on a foreign currency,q equals the foreign risk-free ratelFor options on futures contracts q=r22Binomial Tree for Dividend Paying StocklProcedure:lDraw the tree for the stock price less the present value of the dividendslCreate a new tree by adding the present value of the dividends at each nodelThis ensures that the tree recombines and makes assumptions similar to those when the Black-Scholes model is used23