Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN: 9781133382119
Author: Swokowski
Publisher: Cengage
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[画像:Problem 1. Multi-stock model We consider a 2-stock model similar to the one studied in class. Namely, we consider = S(1) S(2) = S(1) exp (σ1B(1) + (M1 - 0/1 ) S(2) exp (02B(2) + (H2- M2 where (B(1) ) +20 and (B(2) ) +≥o are two Brownian motions, with t≥0 Cov (B(1), B(2)) = p min{t, s}. " The purpose of this problem is to prove that there indeed exists a 2-dimensional Brownian motion (W+)+20 (W(1), W(2))+20 such that = S(1) S(2) = = S(1) exp (011W(1) + (μ1 - 01/1) t) 롱) S(2) exp (021W (1) + 022W(2) + (112 - 03/01/12) t). where σ11, 21, 22 are constants to be determined (as functions of σ1, σ2, p). Hint: The constants will follow the formulas developed in the lectures. (a) To show existence of (Ŵ+), first write the expression for both W. (1) and W (2) functions of (B(1), B(2)). as (b) Using the formulas obtained in (a), show that the process (WA) is actually a 2- dimensional standard Brownian motion (i.e. show that each component is normal, with mean 0, variance t, and that their covariance is 0, which yields independence).]
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Transcribed Image Text:Problem 1. Multi-stock model We consider a 2-stock model similar to the one studied in class. Namely, we consider = S(1) S(2) = S(1) exp (σ1B(1) + (M1 - 0/1 ) S(2) exp (02B(2) + (H2- M2 where (B(1) ) +20 and (B(2) ) +≥o are two Brownian motions, with t≥0 Cov (B(1), B(2)) = p min{t, s}. " The purpose of this problem is to prove that there indeed exists a 2-dimensional Brownian motion (W+)+20 (W(1), W(2))+20 such that = S(1) S(2) = = S(1) exp (011W(1) + (μ1 - 01/1) t) 롱) S(2) exp (021W (1) + 022W(2) + (112 - 03/01/12) t). where σ11, 21, 22 are constants to be determined (as functions of σ1, σ2, p). Hint: The constants will follow the formulas developed in the lectures. (a) To show existence of (Ŵ+), first write the expression for both W. (1) and W (2) functions of (B(1), B(2)). as (b) Using the formulas obtained in (a), show that the process (WA) is actually a 2- dimensional standard Brownian motion (i.e. show that each component is normal, with mean 0, variance t, and that their covariance is 0, which yields independence).
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