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Chapter10 Applications of Stochastic

Chapter10 Applications of Stochastic - Chapter 10...

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Chapter 10 Applications of Stochastic Calculus in Finance Since Black and Scholes (1973) and Merton (1973), the idea of using stochastic calculus for modelling prices of risky assets (e.g., share prices of stock, stock indices such as Dow Jones, foreign exchange rates, interest rates, etc.) has been generally accepted. This led to a new branch of applied probability theory, the field of mathematical finance. It is a symbiosis of stochastic modelling, economic reasoning and practical financial engineering. In this chapter, we consider the Black-Scholes model for pricing options (in particular, a European option), using two different approaches (with em- phasis on the latter): (i) PDE approach; (ii) probabilistic approach The basic financial concepts needed are: bond, stock, option, portfo- lio, volatility, trading strategy, hedging, maturity of a contract, self-financing, arbitrage . On the other hand, some terminology in stochastic calculus will be encountered such as: Brownian motion (BM), Geometric BM, Ito’s inte- gral, Stochastic differential equivation (SDE), martingale, equivalent martin- gale measure, etc. The key tools include Ito’s Lemma, Girsanov Theorem (change of measure theorem), Martingale representation theorem, Feymann-Kac theorem 1
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10.1 Introduction to some basic concepts We are given a filtered probability space (Ω , F , {F t , t [0 , T ] } , P ). 1. Risky asset: stock The price S t of a risky asset (called stock ) is supposed to follow a geometric Brownian motion (BM): dS t S t = μdt + σdB t ( P ) ⇐⇒ S t = S 0 e ( μ - 0 . 5 σ 2 ) t + σB t ( P ) (1.1) Remark 10.1 (a) The measure P is also called the market measure. Formally, we have E ˆ dS t S t ! = μdt, and V ar ˆ dS t S t ! = σ 2 V ar ( dB t ) = σ 2 dt. So μ and σ are the mean and standard deviation (or volatility ) of rate of return (or relative return, or log return) per unit of time. (b) In (6.19), the rates of return are assumed to be: (i) constant per unit of time; (ii) independent for different time intervals. How- ever, empirical evidence indicates that, although the rates of return are uncorrelated for different time periods, they are not independent since the absolute values or the squares of rates of returns are corre- lated). Therefore, the geometric BM is only a reasonable, but crude, first approximation to a real price process. Incidentally, in time series, GARCH models are introduced to take care of the correlated volatilities. (c) If σ 2 = 0 , then S t = S 0 e μt . Namely, taking away the noise, stock markets grow exponentially. People in economics believe in expo- nential growth, and empirical evidence generally supports this belief as well. 2. Riskless asset: bond Suppose that there is also a bond ( riskless asset ) market with fixed interest rate, r , t β t = rdt ⇐⇒ β t = β 0 e rt . For simplicity, we might assume that the initial capital β 0 = 1.
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