Mathematical Finance pptx

Electronic copy of this paper is available at: http://ssrn.com/abstract=976593 Mathematical Finance Introduction to continuous time Financial Market models Dr. Christian-Oliver Ewald School of Economics and Finance University of St.Andrews Electronic copy of this paper is available at: http://ssrn.com/abstract=976593 Abstract These are my Lecture Notes for a course in Continuous Time Finance which I taught in the Summer term 2003 at the University of Kaiser- slautern. I am aware that the notes are not yet free of error and the manuscrip needs further improvement. I am happy about any com- ment on the notes. Please send your comments via e-mail to ce16@st- andrews.ac.uk. Working Version March 27, 2007 1 Contents 1 Stochastic Processes in Continuous Time 5 1.1 Filtrations and Stochastic Processes . . . . . . . . . . . . 5 1.2 Special Classes of Stochastic Processes . . . . . . . . . . . 10 1.3 Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . 15 1.4 Black and Scholes’ Financial Market Model . . . . . . . . 17 2 Financial Market Theory 20 2.1 Financial Markets . . . . . . . . . . . . . . . . . . . . . . . 20 2.2 Arbitrage . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.3 Martingale Measures . . . . . . . . . . . . . . . . . . . . . 25 2.4 Options and Contingent Claims . . . . . . . . . . . . . . . 34 2.5 Hedging and Completeness . . . . . . . . . . . . . . . . . 36 2.6 Pricing of Contingent Claims . . . . . . . . . . . . . . . . 38 2.7 The Black-Scholes Formula . . . . . . . . . . . . . . . . . 42 2.8 Why is the Black-Scholes model not good enough ? . . . . 46 3 Stochastic Integration 48 3.1 Semi-martingales . . . . . . . . . . . . . . . . . . . . . . . 48 3.2 The stochastic Integral . . . . . . . . . . . . . . . . . . . . 55 3.3 Quadratic Variation of a Semi-martingale . . . . . . . . . 65 3.4 The Ito Formula . . . . . . . . . . . . . . . . . . . . . . . . 71 3.5 The Girsanov Theorem . . . . . . . . . . . . . . . . . . . . 76 3.6 The Stochastic Integral for predictable Processes . . . . . 81 3.7 The Martingale Representation Theorem . . . . . . . . . 84 1 4 Explicit Financial Market Models 85 4.1 The generalized Black Scholes Model . . . . . . . . . . . . 85 4.2 A simple stochastic Volatility Model . . . . . . . . . . . . 93 4.3 Stochastic Volatility Model . . . . . . . . . . . . . . . . . . 95 4.4 The Poisson Market Model . . . . . . . . . . . . . . . . . . 100 5 Portfolio Optimization 105 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 105 5.2 The Martingale Method . . . . . . . . . . . . . . . . . . . 108 5.3 The stochastic Control Approach . . . . . . . . . . . . . . 119 2 Introduction Mathematical Finance is the mathematical theory of financial markets. It tries to develop theoretical models, that can be used by “prac tition- ers” to evaluate certain data from “real” financial markets. A model cannot be “right” or wrong, it can only be good or bad ( for practical use ). Even “bad” models can be “good” for theoretical insight. Content of the lecture : Introduction to continuous time financial market models. We will give precise mathematical definitions, what we do understand under a financial market, until this let us think of a financial market as some place where people can buy or sell financial derivatives. During the lecture we will give various examples for financial deriva- tives. The following definition has been taken from [Hull] : A financial derivative is a financial contract, whose value at expire is determined by the prices of the underlying financial assets ( here we mean Stocks and Bonds ). We will treat options, futures, forwards, bonds etc. It is not necessary to have financial background. 3 During the course we will work with methods from Probability theory, Stochastic Analysis and Partial Differ- ential Equations. The Stochastic Analysis and Partial Differential Equations methods are part of the course, the Probability Theory methods should be known from courses like Probability Theory and Prama Stochastik. 4 Chapter 1 Stochastic Processes in Continuous Time Given the present, the price S t of a certain stock at some future time t is not known. We cannot look into the future. Hence we consider this price as a random variable. In fact we have a whole family of random variables S t , for every future time t. Let’s assume, that the random variables S t are defined on a complete probability space (Ω, F, P), now it is time 0 , 0 ≤ t < ∞ and the σ-algebra F contains all possible infor- mation. Choosing sub σ-algebras F t ⊂ F containing all the information up to time t, it is natura l to assume that S t is F t measurable, that is the stock price S t at time t only depends on the past, not on the future. We say that (S t ) t∈[0,∞) is F t adapted and that S t is a stochastic pro- cess. Throughout this chapter we assume that (Ω, F, P) is a complete probability space. If X is a topological space, then we think of X as a measurable space with its associated Borel σ-algebra which we denote as B(X). 1.1 Filtrations and Stochastic Processes Let us denote with I any subset of R. Definition 1.1.1. A family (F t ) t∈I of sub σ-algebras of F such that F s ⊂ 5 F t whenever s < t is called a filtration of F. Definition 1.1.2. A family (X t , F t ) t∈I consisting of F t -measurable R n - valued random variables X t on (Ω, F, P) and a Filtration (F t ) t∈I is called an n-dimensional stochastic process. The case where I = N corresponds to stochastic processes in discrete time ( see Probability Theory, chapter 19 ). Since this section is devoted to stochastic processes in continuous time, from now on we think of I as a connected subinterval of R ≥0 . Often we just speak of the stochastic process X t , if the reference to the filtration (F t ) t∈I is clear. Also we say that (X t ) t∈I is (F t ) t∈I adapted. If no filtration is given, we mean the stochastic process (X t , F t ) t∈I where F t = F X t = σ(X s |s ∈ I, 0 ≤ s ≤ t) is the σ-algebra generated by the random variables X s up to time t. Given a stochastic process (X t , F t ) t∈I , we can consider it as function of two variables X : Ω × I → R n , (ω, t) → X t (ω). On Ω × I we have the product σ-algebra F ⊗B(I) and for I t := {s ∈ I|s ≤ t} (1.1) we have the product σ-algebras F t ⊗ B(I t ). Definition 1.1.3. The stochastic process (X t , F t ) t∈I is called measur- able if the associated map X : Ω×I → R n from (1.1) is (F⊗B(I))/B(R n ) measurable. It is called progressively measurable, if for all t ∈ I the restriction of X to Ω × I t is (F t ⊗ B(I t )/B(R n ) measurable. In this course we will only consider measurable processes. So from now on, if we speak of a stochastic process, we mean a measurable 6 stochastic process. Working with stochastic processes the following space is of fundamen- tal importance : (R n ) I := Map(I, R n ) = {ω : I → R n } (1.2) i.e. the maps from I to R n . For any t ∈ I we have the so called evalua- tion map ev t : (R n ) I → R n ω → ω(t) Definition 1.1.4. The σ-algebra on (R n ) I σ cyl := σ(ev s |s ∈ I) generated by the evaluation maps is called the σ-algebra of Borel cylinder sets. F t := σ cyl,t := σ(ev s |s ∈ I, s ≤ t) defines a filtration of σ cyl . Whenever we consider (R n ) I as a measur- able space, we consider it together with th is σ-algebra and this filtra- tion. The space (R n ) I has some important subspaces : C(I, R n ) := {ω : I → R n | ω is continuous } (1.3) C + (I, R n ) := {ω : I → R n | ω is right-continuous } (1.4) C − (I, R n ) := {ω : I → R n | ω is left-continuous } (1.5) 7 [...]... Brownian motion but we won’t give a proof for its existence There are many nice proofs available in the literature, but everyone of them gets technical at a certain point So as in most courses about Mathematical Finance, we will keep the proof of existence for a special course in stochastic analysis Definition 1.3.1 Let (Wt , Ft )t∈[0,∞) be an R-valued continuous stochastic process on (Ω, F, P) Then (Wt . http://ssrn.com/abstract=976593 Mathematical Finance Introduction to continuous time Financial Market models Dr. Christian-Oliver Ewald School of Economics and Finance University. Control Approach . . . . . . . . . . . . . . 119 2 Introduction Mathematical Finance is the mathematical theory of financial markets. It tries to develop theoretical