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Unformatted text preview: Chapter 4: BlackScholes Model Chapter 4: BlackScholes Model In this chapter we will discuss BlackScholes model, one of the most important continuoustime financial model. Black Scholes (1973) first developed this model and the whole idea of pricing and hedging financial products. Since then it has been widely used in corporate finance, financial management, and many other practices. The idea of studying continuoustime securities model is very similar to discretetime securities model. However, the mathematical facts are more involved in the continuous time model. In section 3.1, we will first introduce Brownian motion. This is an important mathematical concept to model randomness (uncertainty) in continuoustime. In Section 4.2, we will introduce Ito integral, which naturally corresponds to selffinancing trading strategy. Then we will introduce Ito’s lemma in section 4.3. Ito’s lemma is a very useful mathematical tool to deal with randomness in continuoustime. In section 4.4 we will develop Black Scholes model and derive BlackScholes’ formula for European call/put option. In section 4.5 we will present more discussions such as hedging in BlackScholes model. Section 1. Brownian Motion We first define a probability space ), , , ( P F Ω where { } : ≥ = t F F t is a filtration of  σ algebras such that: ) ( , ) 3 ( , ). 2 ( , ). 1 ( F A B P F B A s F F t s F F F s s t s t s ∈ & = ∈ ⊆ ∀ = < ∀ ⊆ ⊆ > & Where (1) is the increasing property, (2) is “ right continuous ”, (3) is the complete property. Since we use those  σ algebras to explain the information set, all possible information s t F t > , is equivalent to the information . s F However, because of new information at time s, the filtration is not “leftcontinuous”. 4.1. Definition: A Brownian motion over ) , , ( P F Ω is a (stochastic) process { } : ≥ = t B B t , which is adapted with respect to the filtration F, and satisfying (i), = B (ii). For every , 1 k t t t < < < ≤ ¡ the random variables (rv) 1 tk k t t B B are independent, (iii). t s s t s t N B B s t < ∀ ≈ )), ( ), ( ( 2 σ μ Chapter 4: BlackScholes Model where . , > ∈ σ μ R The parameter μ is called the drift coefficient, and the parameter 2 σ is called the diffusion coefficient. A Brownian motion with 1 , = = σ μ is called standard Brownian motion. 4.2. Brownian motion exists! Moreover, there is a version of Brownian motion such that the function ) ( ω t B t → is continuous (for almost all ). Ω ∈ ω (This is a deep fact). For this course we make use of this fact and other properties of the Brownian motions as follows. 4.3. For , 1 1 n n t t t t < < < < ≤ & we have dx e t t b b B B P b B b b B B P n i b B b B P n n n n tn n tn tn n i n t t x b b n n n n t t n t n n t t i t n t ) ( 2 1 1 1 1 1 2 1 1 1 1 ) ( 2 1 ) ( )  ( ) 1 , , 1 , : ( ∞ & = ≤ = = ≤ = = = ≤ π & where the first step comes from “Markov property of the Brownian motion”, the second...
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 Winter '09
 Adam
 Normal Distribution, Pricing, Black–Scholes, Geometric Brownian motion

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