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Unformatted text preview: Notes for multiperiod discretetime models The results for the multiperiod model are very similar to the ones for the singleperiod model, but we need to modify several definitions to take into account the fact that we are looking now at dynamic processes. For instance, instead of just talking about random variables, we need to talk about stochastic processes ; instead of having stateprice vectors, we have stateprice processes ; the riskneutral property of a probability measure now needs to be stated in terms of martingales , which in turn require the concept of adapted processes and conditional expectation . In addition, an important new feature is that we must now specify how information is gathered by investors. This handout is meant to supplement the slides and allow you to have more time to listen in class while we discuss these more advanced concepts. Notation used and assumption about S 1 : we write vector S ( k ) = ( S 1 ( k ) ,S 2 ( k ) ,...,S N ( k )) to denote the vector of random variables representing the price of each of the N securities at time k , and S j ( k,ω ) is the price of security j at time k when the underlying state is ω . The collection of random variables S j = { S j ( k ) ,k = 0 , 1 ,...,T } is called a stochastic process . For each ω ∈ Ω, the function k → S j ( k,ω ) is called a sample path of the stochastic process S j . Similarly to the oneperiod model, we write S ( k, Ω) to denote the matrix that contains on its i th row the prices S 1 ( k,ω i ) ,...,S N ( k,ω i ) of the N securities at time k for the i th state of the world ω i . We assume that S 1 is a bank account with S 1 (0 ,ω ) = 1 for all ω , and therefore S 1 ( k,ω ) is a nondecreasing function of k . The random variables i k = S 1 ( k + 1) S 1 ( k ) 1 ≥ , k = 0 , 1 ,...,T are the oneperiod interest rates or short rates . When i k ( ω ) = i k for all k and ω , then the interest rate process { i k ,k = 0 , 1 ,...,T } is deterministic. Assumptions about information gathered by investors : • investors observe all prices, and remember all past and present prices. • investors start with full knowledge of Ω, P (the “realworld” probability measure) and S j ( k,ω ) for j = 1 ,...,N,k = 0 ,...,T . 1 • S j (0 ,ω ) is constant over ω (same initial price in each state of the world) • For each security, there is a unique sample path for each ω ∈ Ω, so that at time T the investors know which state ω has occurred. • At k < T , no additional information is given to the investors (cannot guess the future)....
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 Spring '09
 Adam
 Probability theory, probability measure, Ψ, riskneutral probability measure, sj

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