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# L2 - Lecture 2 Discrete Model Framework and Binomial Trees...

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Unformatted text preview: Lecture 2 Discrete Model Framework and Binomial Trees 2.1 Basic formulation We start with a simple discrete model framework with several basic elements. 2.1.1 Asset price dynamics • A finite sample space Ω = { ω 1 ,...,ω K } . • A probability measure P on Ω with P ( ω ) > ∀ ω ∈ Ω. • A filtration FF = {F t , t = 0 , 1 ,...,T } with F t- 1 ⊆ F t , t = 1 ,...,T , where F t contains the information about the financial market available to the investors at time t . Usually, t = , 1 ,...,T represent T + 1 trading dates. Since T < ∞ , this is called a finite horizon model or a multiperiod model. • A riskless bank account process B = { B ( t ) , t = 0 , 1 ,...,T } , where B (0) = 1 and B ( t ) > ∀ t . B ( t ) is thought of as the time t value of a money market account when \$1 is deposited at time 0. Hence B ( t ) is nondecreasing in t . Moreover, the quantity r ( t ) = [ B ( t )- B ( t- 1)] /B ( t- 1) is thought of as the interest rate pertaining to the time interval ( t- 1 ,t ]. • N risky security processes S n = { S n ( t ) , t = 0 , 1 ,...,T } , n = 1 ,...,N , where S n ( t ) ≥ 0 is thought of as the time t price of risky security n (e.g. stock or bond). Note that B,S 1 ,...,S N are considered to be stochastic processes, i.e. for each t , B ( t ), S 1 ( t ) ,..., S N ( t ) are all functions of ω . To ease the notation, the dependence on ω is usually not shown unless necessary. Furthermore, B,S 1 ,...,S N are assumed to be adapted to the filtration FF . A stochastic process { X ( t ) } is said to be adapted to the filtration FF if for each t , the random variable X ( t ) is measurable with respect to F t , i.e. the information about X ( t ) is contained in F t . 2.1.2 Trading strategies A trading strategy h = ( h ,h 1 ,...,h N ) is a vector of processes h n = { h n ( t ) , t = 1 ,...,T } , n = , 1 ,...,N . Note that h n (0) is not specified, because for n = 1 ,...,N , h n ( t ) is interpreted as the number of units (e.g. shares of stock) that the investor owns (i.e. carries forward) from time t- 1 to time t , whereas h ( t ) B ( t- 1) represents the amount of money invested in the bank account at time t- 1. A negative value of h n ( t ) corresponds to borrowing money from the bank (when n = 0) or selling short security n (when n = 1 ,...,N ). h is also called a portfolio . A trading strategy is a rule that specifies the investor’s position in each security n at each time t and in each state of the world ω . In general, this rule should allow the investor to choose a position 1 in the securities based on the available information thus far without “looking into the future”. This is done by introducing the concept of predictability ....
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L2 - Lecture 2 Discrete Model Framework and Binomial Trees...

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