PartII_DiscreteTimeModels

# PartII_DiscreteTimeModels - STAT/ACTSC 446/846 Part II...

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STAT/ACTSC 446/846 Part II Discrete Time Financial Models * Chapters 10-11 from the textbook: Derivatives Markets, 2nd Edition, R.L. McDonald * Chapters 5-6 from the textbook Financial Eco- nomics, Panjer et al. 1)Binomial Tree Model i) Introduction ii) Continuous-Time limit iii) Binomial Model Parameters iv) Recursive Valuation v) Trading/ replicating strategies vi) Dividends vii) American Options 2) Discrete Time Securities Market i) Single-Period Model ii) Arbitrage/ State Price Vector iii) Fundamental Theorem Asset Pricing/ Risk Neutral Probability Measures iv) Completeness in one-period model v) Multiperiod model vi) Fundamental Theorem Asset Pricing and Com- pleteness in Multiperiod Model

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STAT/ACTSC 446/846 Part II Discrete Time Financial Models 1)Binomial Tree Model i) Introduction ii) Continuous-Time limit iii) Binomial Model Parameters iv) Recursive Valuation v) Trading/ replicating strategies vi) Dividends vii) American Options
1 The Binomial Option-Pricing Model It represents the simplest and best known example of a multiperiod valuation model. It provides a powerful and flexible tool for option valuation. Assumptions: – there are N time periods with each period of size h so that the time to maturity is T=Nh . – there are two securities: • a bank account B(n), n=0,. .,N , with a constant rate • a risky security S(n), n=0,. .,N , which each period either goes up by the factor u>1 or goes down by the factor d , 0 < d < 1 .

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2 The Binomial Model (continued) For two time periods: After n time periods, there are n+1 possible stock prices The deterministic bank account satisfies where r is the continuously compounded interest rate per year. . ,.., 1 , 0 , ) 0 ( ) ; ( n j d u S j n S j n j = = , ) ( rnh e n B =
3 The Binomial Model (continued) This model is arbitrage free and complete : – the arbitrage-free condition is guaranteed by imposing the condition u > e rh > d Thus, the risk-neutral probability measure Q, Q~P , is defined such that the probability of going up is equal to q=(e rh -d)/(u-d). For this model, the value at time 0 of a European derivative with payoff function g : ))] ( ( [ ) 0 ( N S g e E V rhN Q = ). ) 0 ( ( ) 1 ( 0 j N j j N j N j j N rhN d u S g q q e = =

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4 The Binomial Model (continued) For a European call option with exercise price K and the payoff at time T=Nh equal to the binomial model value at t=0 is where m is the smallest integer such that S(0)u m d N-m >k , and q*=exp(-rh)qu. Similarly, an expression for a European put option can be obtained. Or, one may use put- call parity : ] ) 0 ( [ ) 1 ( ) 0 ( K S d u q q e c j N j j N j N m j j N rhN = = + = ] ) ( [ ) ( K N S N g ), , ; ( ) , ; ( ) 0 ( * q N m Ke q N m S rNh Φ Φ = , ) 1 ( )! ( ! ! ) , ; ( j N j N m j p p j N j N p N m = = Φ . ) 0 ( ) 0 ( ) 0 ( rNh Ke S p c =
5 The Binomial Model in Greater Detail Continuous-time limit of the binomial model – how large N need to be chosen?

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PartII_DiscreteTimeModels - STAT/ACTSC 446/846 Part II...

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