X4Binomial_model_II

X4Binomial_model_II - Binomial Option Pricing II Chapters...

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Slide 11-1 Binomial Option Pricing: II Chapters 10 & 11
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Slide 11-2 Review: Risk neutral option pricing • Method: – use risk-neutral discount rate to compute an option price as a discounted expected payoff, – where the expectation is based on the following risk- neutral probability of an up-move: • Why does this work? p e d u d r h * = ( ) δ To replicate an option, we need positions in the under- lying asset and the riskfree asset. To value an option, we must use a discount rate that depends on the structure of the replication portfolio. The option value does not depend on the expected return of the underlying asset.
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Slide 11-3 Review: binomial trees • Binomial tree: If one step in the tree corresponds to a period of length h , the up- and down factors are: where is the annual volatility of the continuously compounded stock return. • Why this model? d = e ( r δ ) h σ h u = e ( r ) h + h Continuously compounded stock returns are additive. We assume that these returns are idependently and identically normally distributed. The variance of the return over a period h is 2 h . The stock price S t+h = S t e r t,t+h
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Slide 11-4 This class • We will see many numerical example. • Start with the example used in the last class. – Extension: 2 periods • American options – Extension: early exercise • Options on other assets – Extension: binomial trees on forward prices
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Slide 11-5 Example: pricing an European call with two years to maturity and K=$40 Current stock price = $41, riskfree rate r = 8% p.a., dividend yield δ = 0%, stock return vola = 30% p.a. u=e 0.08+0.3 =1.462 d=e 0.08-0.3 =0.803 S u =u x $41=$59.954 S d =d x $41=$32.903 S uu =u 2 x $41=$87.669 S ud =S du =u x d x $41=$48.114 S dd =d 2 x $41=$26.405
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Slide 11-6 Recursive analysis (1) To price an option with two binomial periods, we work backwards through the tree. Year 2, Stock Price=$87.669 : Since we are at expiration, the option value is max (0, S K ) = $47.669. Year 2, Stock Price=$48.114 : Similarly, the option value is $8.114. Year 2, Stock Price=$26.405 : Since the option is out-of-the- money, its value is 0.
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Slide 11-7 Recursive analysis (2) Δ S uu e δ h +Be rh =$47.669 S ud e h +Be rh =$8.114 Δ S u +B
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Slide 11-8 Pricing the call option Year 1, Stock Price=$59.954: At this node, we compute the option value using equation (10.3). where u = 1.462 and d = 0.803 (as before). Year 1, Stock Price=$32.903: Again using equation (10.3), the option value is $3.187. Year 0, Stock Price = $41: Similarly, the option value is computed to be $10.737. 029 . 23 $ 803 . 0 462 . 1 e 462 . 1 114 . 8 $ 803 . 0 462 . 1 803 . 0 e 669 . 47 $ e 08 . 0 08 . 0 08
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This note was uploaded on 10/25/2009 for the course 15 15.402 taught by Professor Bergman during the Fall '09 term at MIT.

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X4Binomial_model_II - Binomial Option Pricing II Chapters...

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