This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 ....
View
Full Document
 Spring '08
 Staff
 Probability theory

Click to edit the document details