Mathematical Modeling in Finance with Stochastic Processes
6.2. ITO’S FORMULAˆ 201 6.2 Itˆo’s Formula Rating Mathematically Mature: may contain mathematics beyond calculus with proofs. Se ...
202 CHAPTER 6. STOCHASTIC CALCULUS chain rule, and Taylor polynomials and Taylor series that enable us to calculate with funct ...
6.2. ITO’S FORMULAˆ 203 Wiener process. Therefore, the expected value, or mean, of the summation will be zero: E[Y(t)] =E [∫t 0 ...
204 CHAPTER 6. STOCHASTIC CALCULUS Theorem 26 (Itˆo’s formula). IfY(t)is scaled Wiener process with drift, satisfyingdY =r dt+σ ...
6.2. ITO’S FORMULAˆ 205 Guessing Processes from SDEs with Itˆo’s Formula One of the key needs we will have is to go in the oppos ...
206 CHAPTER 6. STOCHASTIC CALCULUS Sources This discussion is adapted from Financial Calculus: An introduction to derivative pri ...
6.3. PROPERTIES OF GEOMETRIC BROWNIAN MOTION 207 Outside Readings and Links: 1. 2. 3. 4. 6.3 Properties of Geometric Brownian Mo ...
208 CHAPTER 6. STOCHASTIC CALCULUS Vocabulary Geometric Brownian Motionis the continuous time stochastic pro- cessz 0 exp(μt+σW ...
6.3. PROPERTIES OF GEOMETRIC BROWNIAN MOTION 209 Figure 6.1: The p.d.f. for a lognormal random variable Now differentiating with ...
210 CHAPTER 6. STOCHASTIC CALCULUS Calculation of the Variance We can calculate the variance of Geometric Brownian Motion by usi ...
6.3. PROPERTIES OF GEOMETRIC BROWNIAN MOTION 211 At each time the Geometric Brownian Motion has lognormal distribution with para ...
212 CHAPTER 6. STOCHASTIC CALCULUS Quadratic Variation of Geometric Brownian Motion The quadratic variation of Geometric Brownia ...
6.3. PROPERTIES OF GEOMETRIC BROWNIAN MOTION 213 What is the probability that Geometric Brownian Motion with param- etersμ= 0 a ...
214 CHAPTER 6. STOCHASTIC CALCULUS ...
Chapter 7 The Black-Scholes Model 7.1 Derivation of the Black-Scholes Equation Rating Mathematically Mature: may contain mathema ...
216 CHAPTER 7. THE BLACK-SCHOLES MODEL Vocabulary Abackward parabolic PDEis a partial differential equation of the formVt+DVxx+ ...
7.4 Derivation of the Black-Scholes Equation best predictor of the market value of a stock is the current price. We will make th ...
218 CHAPTER 7. THE BLACK-SCHOLES MODEL Derivation of the Black-Scholes equation We consider a simple derivative instrument, an o ...
7.1. DERIVATION OF THE BLACK-SCHOLES EQUATION 219 The difference in value between the two portfolios changes by δ(V −Π) = (Vt−ψ( ...
220 CHAPTER 7. THE BLACK-SCHOLES MODEL Cancel theδtterms, and recall thatV −Π =V −φ(t)S−ψ(t)B(t), and φ(t) =VS, so that on the l ...
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