chapter4 - Chapter 4: Black-Scholes Model Chapter 4:...

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Unformatted text preview: Chapter 4: Black-Scholes Model Chapter 4: Black-Scholes Model In this chapter we will discuss Black-Scholes model, one of the most important continuous-time 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 continuous-time securities model is very similar to discrete-time 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 continuous-time. In Section 4.2, we will introduce Ito integral, which naturally corresponds to self-financing trading strategy. Then we will introduce Itos lemma in section 4.3. Itos lemma is a very useful mathematical tool to deal with randomness in continuous-time. In section 4.4 we will develop Black Scholes model and derive Black-Scholes formula for European call/put option. In section 4.5 we will present more discussions such as hedging in Black-Scholes model. Section 1. Brownian Motion We first define a probability space ), , , ( P F where { } : = t F F t is a filtration of - 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 - 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 left-continuous. 4.1. Definition: A Brownian motion over ) , , ( P F 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 Chapter 4: Black-Scholes Model where . , > R The parameter is called the drift coefficient, and the parameter 2 is called the diffusion coefficient. A Brownian motion with 1 , = = is called standard Brownian motion. 4.2. Brownian motion exists! Moreover, there is a version of Brownian motion such that the function ) ( t B t is continuous (for almost all ). (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 , : (-------- ----- &- =- - = =- - =- = = & where the first step comes from Markov property of the Brownian motion, the second...
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chapter4 - Chapter 4: Black-Scholes Model Chapter 4:...

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