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 [...]... 1.2.4 Let τ1 resp.τ2 be stopping times on (Ω, F, P) with respect to the filtrations (Ft )t∈I resp (Gt )t∈I Let Ft Gt = σ(Ft , Gt ) Then τ1 ∧ τ2 : Ω → R (τ1 ∧ τ2 )(ω) = min(τ1 (ω), τ2 (ω)) is a stopping time with respect to the filtration (Ft Gt )t∈I Given a stochastic process and a stopping time we can define a new stochastic process by stopping the old one In case the stopping time is finite, we can define... called finite if τ (Ω) ⊂ I A stopping time is called bounded if there exists T ∗ ∈ I such that P{ω|τ (ω) ≤ T ∗ } = 1 The following exercises leads to many examples of stopping times Exercise 1.2.3 Let (Xt , Ft )t∈I be a continuous stochastic process with 11 values in Rn and let A ⊂ Rn be a closed subset Then τ :Ω → R τ (ω) := inf {t ∈ I|Xt (ω) ∈ A} is a stopping time with respect to the filtration (Ft )t∈I... martingale For stochastic integration a class slightly bigger than martingales will play an important role This class is called local martingales To define it, we first need to define what we mean by a stopping time : Definition 1.2.2 A stopping time with respect to a filtration (Ft )t∈I is an F measurable random variable τ : Ω → I ∪ {∞} such that for all t ∈ I we have τ −1 (It ) ∈ Ft A stopping time is called... )t∈I be a stochastic process and τ a stopping time with respect to (Ft )t∈I Then we define a new stochastic process (Xtτ )t∈I with respect to the same filtration (Ft )t∈I as Xtτ (ω) = Xt (ω) , ∀t ≤ τ (ω) Xτ (ω) (ω) , ∀t > τ (ω) (1.10) If τ is finite, we define a random variable Xτ on (Ω, F, P) as Xτ (ω) := Xτ (ω) (ω) (1.12) Also we can define a new σ-algebra : 12 Definition 1.2.4 Let τ be a stopping time with... assume that V0 (ϕ) = 0 Let ϕ ∈ Φ be strictly self financing and τj′ be a sequence of stopping times as in (2.14) Let τj′′ be another sequence of stopping times as in Definition 1.2.5 such that corresponding to Definition 2.3.1 the stopped ˜ ′′ discounted price processes X τj are P∗ -martingales Let us define new ˜ stopping times τj = τj′ ∧ τj′′ Then X τj are still P∗ - martingales ( use the optional sampling... a short introduction into 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... x1 + x2 + + xm ) 1.4 Black and Scholes’ Financial Market Model In this section we will introduce into the standard Black-Scholes model which describes the motion of a stock price and bond First consider the 0 following situation At time t = 0 you put S0 units of money onto your bank account and the bank has a constant deterministic interest rate r > 0 If after time t > 0 you want your money back, the... the NobelPrize in economics in 1997 for Merton and Scholes ( Black was already dead at this time ) 1 2 Exercise 1.4.1 Let St = S0 · e(b− 2 σ )t+σWt denote the price of a stock in the standard Black-Scholes model Compute the expectation E(St ) and variance var(St ) 19 Chapter 2 Financial Market Theory In this section we will introduce into the theory of financial markets The treatment here is as general... context is clear To establish the connection between martingale measures and arbitrage, we must restrict ourself to a special class of trading strategies which is from the real world financial market point of view very natural Let us assume that at time t0 a trader enters a market ( only consisting of tradeable assets ) and buys assets according to ϕt0 Then the worth of his portfolio at time t0 is Vt0... with respect to the filtration (Ft )t∈I Then Fτ := {A ∈ F|A ∩ τ −1 (It ) ∈ Ft ∀ t ∈ I} (1.14) is called the σ-algebra of events up to time τ This is indeed a σ-algebra The following is a generalization of Theorem 19.3 in the Probability Theory lecture Theorem 1.2.1 Optional Sampling Theorem Let (Xt , Ft )t∈I be a right -continuous martingale and τ1 ,τ2 be bounded stopping times with respect to (Ft )t∈I . 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. : 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. stopping time with respect to the filtration (F t G t ) t∈I . Given a stochastic process and a stopping time we can define a new stochastic process by stopping the old one. In case the stopping time is