Explanation Theorem Used: Pythagorean Theorem: In a right angled triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides. Result used: The Basic Trigonometric Identity: cos ( 2 θ ) = 2 cos 2 θ − 1 . Calculation: Consider the given expression cos ( 2 tan − 1 x ) . cos ( 2 tan − 1 x ) = cos ( 2 θ ) where, θ = tan − 1 x . By Pythagorean Theorem third side of the triangle is as follows, ( Hypotenuse ) 2 = 1 2 + x 2 = 1 + x 2 Hypotenuse = 1 + x 2 Draw a right angle by using tan θ = x 1 as shown in below Figure1, From Figure 1, it is observed that 1 + x 2 represents hypotenuse of the triangle, 1 and x are two legs of the triangle. Now, find the value of cos ( 2 θ ) using the formula stated above

Mylab Math With Pearson Etext -- 18 Week Standalone Access Card -- For Basic Technical Mathematics With Calculus

11th Edition

ISBN: 9780135902912

Author: Allyn J. Washington

Publisher: PEARSON

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Chapter 20.6, Problem 50E

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To find: The value of the expression cos(2tan−1x).

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Chapter 20.6, Problem 49E

Chapter 20.6, Problem 51E

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Construct a know-show table for each statement below that appears to be true.

Problem 3. Pricing a multi-stock option the Margrabe formula The purpose of this problem is to price a swap option in a 2-stock model, similarly as what we did in the example in the lectures. We consider a two-dimensional Brownian motion given by W₁ = (W(¹), W(2)) on a probability space (Q, F,P). Two stock prices are modeled by the following equations: dX = dY₁ = X₁ (rdt+ rdt+0₁dW!) (²)), Y₁ (rdt+dW+0zdW!"), with Xo xo and Yo =yo. This corresponds to the multi-stock model studied in class, but with notation (X+, Y₁) instead of (S(1), S(2)). Given the model above, the measure P is already the risk-neutral measure (Both stocks have rate of return r). We write σ = 0₁+0%. We consider a swap option, which gives you the right, at time T, to exchange one share of X for one share of Y. That is, the option has payoff F=(Yr-XT). (a) We first assume that r = 0 (for questions (a)-(f)). Write an explicit expression for the process Xt. Reminder before proceeding to question (b): Girsanov's theorem…

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…

Chapter 20 Solutions

Mylab Math With Pearson Etext -- 18 Week Standalone Access Card -- For Basic Technical Mathematics With Calculus

Ch. 20.1 - Prob. 1PE Ch. 20.1 - Prob. 2PE Ch. 20.1 - Prob. 3PE Ch. 20.1 - Prob. 1E Ch. 20.1 - Prob. 2E Ch. 20.1 - In Exercises 36, use a calculator to check the...Ch. 20.1 - Prob. 4E Ch. 20.1 - Prob. 5E Ch. 20.1 - Prob. 6E Ch. 20.1 - Prob. 7E

Ch. 20.1 - Prob. 8E Ch. 20.1 - Prob. 9E Ch. 20.1 - Prob. 10E Ch. 20.1 - Prob. 11E Ch. 20.1 - Prob. 12E Ch. 20.1 - Prob. 13E Ch. 20.1 - Prob. 14E Ch. 20.1 - Prob. 15E Ch. 20.1 - Prob. 16E Ch. 20.1 - Prob. 17E Ch. 20.1 - Prob. 18E Ch. 20.1 - In Exercises 19–38, prove the given identities. Ch. 20.1 - Prob. 20E Ch. 20.1 - In Exercises 19–38, prove the given identities. Ch. 20.1 - Prob. 22E Ch. 20.1 - Prob. 23E Ch. 20.1 - Prob. 24E Ch. 20.1 - Prob. 25E Ch. 20.1 - Prob. 26E Ch. 20.1 - Prob. 27E Ch. 20.1 - Prob. 28E Ch. 20.1 - Prob. 29E Ch. 20.1 - Prob. 30E Ch. 20.1 - Prob. 31E Ch. 20.1 - Prob. 32E Ch. 20.1 - Prob. 33E Ch. 20.1 - Prob. 34E Ch. 20.1 - Prob. 35E Ch. 20.1 - Prob. 36E Ch. 20.1 - Prob. 37E Ch. 20.1 - Prob. 38E Ch. 20.1 - Prob. 39E Ch. 20.1 - Prob. 40E Ch. 20.1 - Prob. 41E Ch. 20.1 - Prob. 42E Ch. 20.1 - Prob. 43E Ch. 20.1 - Prob. 44E Ch. 20.1 - Prob. 45E Ch. 20.1 - Prob. 46E Ch. 20.1 - Prob. 47E Ch. 20.1 - Prob. 48E Ch. 20.1 - In Exercises 47–50, for a first-quadrant angle,...Ch. 20.1 - Prob. 50E Ch. 20.1 - Prob. 51E Ch. 20.1 - Prob. 52E Ch. 20.1 - Prob. 53E Ch. 20.1 - Prob. 54E Ch. 20.1 - Prob. 55E Ch. 20.1 - Prob. 56E Ch. 20.1 - Prob. 57E Ch. 20.1 - Prob. 58E Ch. 20.1 - Prob. 59E Ch. 20.1 - Prob. 60E Ch. 20.1 - Prob. 61E Ch. 20.1 - Prob. 62E Ch. 20.1 - Prob. 63E Ch. 20.1 - In Exercise 63–70, solve the given...Ch. 20.1 - Prob. 65E Ch. 20.1 - Prob. 66E Ch. 20.1 - Prob. 67E Ch. 20.1 - Prob. 68E Ch. 20.1 - Prob. 69E Ch. 20.1 - Prob. 70E Ch. 20.1 - Prob. 71E Ch. 20.1 - Prob. 72E Ch. 20.1 - Prob. 73E Ch. 20.1 - Prob. 74E Ch. 20.2 - Prob. 1PE Ch. 20.2 - Prob. 2PE Ch. 20.2 - Prob. 1E Ch. 20.2 - In Exercises 1 and 2, make the given changes in...Ch. 20.2 - Prob. 3E Ch. 20.2 - In Exercises 36, determine the values of the given...Ch. 20.2 - Prob. 5E Ch. 20.2 - Prob. 6E Ch. 20.2 - Prob. 7E Ch. 20.2 - Prob. 8E Ch. 20.2 - Prob. 9E Ch. 20.2 - Prob. 10E Ch. 20.2 - Prob. 11E Ch. 20.2 - In Exercises 1120, simplify the given...Ch. 20.2 - Prob. 13E Ch. 20.2 - Prob. 14E Ch. 20.2 - Prob. 15E Ch. 20.2 - Prob. 16E Ch. 20.2 - Prob. 17E Ch. 20.2 - Prob. 18E Ch. 20.2 - Prob. 19E Ch. 20.2 - Prob. 20E Ch. 20.2 - In Exercises 2124, evaluate each expression by...Ch. 20.2 - In Exercises 21–24, evaluate each expression by...Ch. 20.2 - Prob. 23E Ch. 20.2 - Prob. 24E Ch. 20.2 - Prob. 25E Ch. 20.2 - Prob. 26E Ch. 20.2 - Prob. 27E Ch. 20.2 - Prob. 28E Ch. 20.2 - Prob. 29E Ch. 20.2 - Prob. 30E Ch. 20.2 - Prob. 31E Ch. 20.2 - Prob. 32E Ch. 20.2 - Prob. 33E Ch. 20.2 - Prob. 34E Ch. 20.2 - Prob. 35E Ch. 20.2 - Prob. 36E Ch. 20.2 - Prob. 37E Ch. 20.2 - Prob. 38E Ch. 20.2 - Prob. 39E Ch. 20.2 - Prob. 40E Ch. 20.2 - Prob. 41E Ch. 20.2 - In Exercises 4154, solve the given...Ch. 20.2 - Prob. 43E Ch. 20.2 - Prob. 44E Ch. 20.2 - Prob. 45E Ch. 20.2 - Prob. 46E Ch. 20.2 - Prob. 47E Ch. 20.2 - Prob. 48E Ch. 20.2 - Prob. 49E Ch. 20.2 - Prob. 50E Ch. 20.2 - Prob. 51E Ch. 20.2 - Prob. 52E Ch. 20.2 - Prob. 53E Ch. 20.2 - Prob. 54E Ch. 20.3 - Evaluate cos 90° using values for 45°. Ch. 20.3 - Simplify: Ch. 20.3 - In Exercises 1–4, make the given changes in the...Ch. 20.3 - In Exercises 1–4, make the given changes in the...Ch. 20.3 - In Exercises 1–4, make the given changes in the...Ch. 20.3 - Prob. 4E Ch. 20.3 - Prob. 5E Ch. 20.3 - In Exercises 5–8, determine the values of the...Ch. 20.3 - Prob. 7E Ch. 20.3 - Prob. 8E Ch. 20.3 - Prob. 9E Ch. 20.3 - Prob. 10E Ch. 20.3 - Prob. 11E Ch. 20.3 - In Exercises 9–14, use a calculator to verify the...Ch. 20.3 - Prob. 13E Ch. 20.3 - Prob. 14E Ch. 20.3 - Prob. 15E Ch. 20.3 - Prob. 16E Ch. 20.3 - Prob. 17E Ch. 20.3 - Prob. 18E Ch. 20.3 - Prob. 19E Ch. 20.3 - In Exercises 19–30, simplify the given...Ch. 20.3 - In Exercises 19–30, simplify the given...Ch. 20.3 - Prob. 22E Ch. 20.3 - Prob. 23E Ch. 20.3 - Prob. 24E Ch. 20.3 - Prob. 25E Ch. 20.3 - Prob. 26E Ch. 20.3 - Prob. 27E Ch. 20.3 - Prob. 28E Ch. 20.3 - Prob. 29E Ch. 20.3 - Prob. 30E Ch. 20.3 - Prob. 31E Ch. 20.3 - Prob. 32E Ch. 20.3 - Prob. 33E Ch. 20.3 - Prob. 34E Ch. 20.3 - 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Prob. 12E Ch. 20.5 - Prob. 13E Ch. 20.5 - Prob. 14E Ch. 20.5 - Prob. 15E Ch. 20.5 - In Exercises 5–20, solve the given trigonometric...Ch. 20.5 - Prob. 17E Ch. 20.5 - Prob. 18E Ch. 20.5 - Prob. 19E Ch. 20.5 - Prob. 20E Ch. 20.5 - Prob. 21E Ch. 20.5 - Prob. 22E Ch. 20.5 - Prob. 23E Ch. 20.5 - Prob. 24E Ch. 20.5 - Prob. 25E Ch. 20.5 - Prob. 26E Ch. 20.5 - Prob. 27E Ch. 20.5 - Prob. 28E Ch. 20.5 - Prob. 29E Ch. 20.5 - Prob. 30E Ch. 20.5 - Prob. 31E Ch. 20.5 - Prob. 32E Ch. 20.5 - Prob. 33E Ch. 20.5 - Prob. 34E Ch. 20.5 - Prob. 35E Ch. 20.5 - Prob. 36E Ch. 20.5 - Prob. 37E Ch. 20.5 - Prob. 38E Ch. 20.5 - Prob. 39E Ch. 20.5 - Prob. 40E Ch. 20.5 - Prob. 41E Ch. 20.5 - Prob. 42E Ch. 20.5 - Prob. 43E Ch. 20.5 - Prob. 44E Ch. 20.5 - Prob. 45E Ch. 20.5 - Prob. 46E Ch. 20.5 - Prob. 47E Ch. 20.5 - Prob. 48E Ch. 20.5 - Prob. 49E Ch. 20.5 - Prob. 50E Ch. 20.5 - Prob. 51E Ch. 20.5 - Prob. 52E Ch. 20.5 - Prob. 53E Ch. 20.5 - Prob. 54E Ch. 20.5 - Prob. 55E Ch. 20.5 - Prob. 56E Ch. 20.5 - Prob. 57E Ch. 20.5 - Prob. 58E Ch. 20.5 - Prob. 59E Ch. 20.5 - Prob. 60E Ch. 20.5 - Prob. 61E Ch. 20.5 - Prob. 62E Ch. 20.6 - Prob. 1PE Ch. 20.6 - Prob. 2PE Ch. 20.6 - Prob. 1E Ch. 20.6 - Prob. 2E Ch. 20.6 - Prob. 3E Ch. 20.6 - Prob. 4E Ch. 20.6 - Prob. 5E Ch. 20.6 - Prob. 6E Ch. 20.6 - Prob. 7E Ch. 20.6 - Prob. 8E Ch. 20.6 - Prob. 9E Ch. 20.6 - Prob. 10E Ch. 20.6 - Prob. 11E Ch. 20.6 - In Exercises 11—28, evaluate exactly the given...Ch. 20.6 - In Exercises 11—28, evaluate exactly the given...Ch. 20.6 - In Exercises 11—28, evaluate exactly the given...Ch. 20.6 - Prob. 15E Ch. 20.6 - Prob. 16E Ch. 20.6 - Prob. 17E Ch. 20.6 - In Exercises 11—28, evaluate exactly the given...Ch. 20.6 - Prob. 19E Ch. 20.6 - Prob. 20E Ch. 20.6 - Prob. 21E Ch. 20.6 - Prob. 22E Ch. 20.6 - Prob. 23E Ch. 20.6 - Prob. 24E Ch. 20.6 - Prob. 25E Ch. 20.6 - Prob. 26E Ch. 20.6 - Prob. 27E Ch. 20.6 - Prob. 28E Ch. 20.6 - Prob. 29E Ch. 20.6 - Prob. 30E Ch. 20.6 - Prob. 31E Ch. 20.6 - Prob. 32E Ch. 20.6 - In Exercises 29—36, use a calculator to evaluate...Ch. 20.6 - In Exercises 29—36, use a calculator to evaluate...Ch. 20.6 - Prob. 35E Ch. 20.6 - In Exercises 29—36, use a calculator to evaluate...Ch. 20.6 - Prob. 37E Ch. 20.6 - Prob. 38E Ch. 20.6 - Prob. 39E Ch. 20.6 - Prob. 40E Ch. 20.6 - Prob. 41E Ch. 20.6 - Prob. 42E Ch. 20.6 - Prob. 43E Ch. 20.6 - Prob. 44E Ch. 20.6 - Prob. 45E Ch. 20.6 - Prob. 46E Ch. 20.6 - Prob. 47E Ch. 20.6 - Prob. 48E Ch. 20.6 - Prob. 49E Ch. 20.6 - Prob. 50E Ch. 20.6 - In Exercises 51–56, solve the given problems with...Ch. 20.6 - Prob. 52E Ch. 20.6 - Prob. 53E Ch. 20.6 - Prob. 54E Ch. 20.6 - Prob. 55E Ch. 20.6 - Prob. 56E Ch. 20.6 - Prob. 57E Ch. 20.6 - Prob. 58E Ch. 20.6 - Prob. 59E Ch. 20.6 - Prob. 60E Ch. 20.6 - Prob. 61E Ch. 20.6 - Prob. 62E Ch. 20.6 - Prob. 63E Ch. 20.6 - Prob. 64E Ch. 20.6 - Prob. 65E Ch. 20.6 - Prob. 66E Ch. 20.6 - Prob. 67E Ch. 20.6 - Prob. 68E Ch. 20.6 - Prob. 69E Ch. 20.6 - Prob. 70E Ch. 20.6 - Prob. 71E Ch. 20.6 - Prob. 72E Ch. 20.6 - Prob. 73E Ch. 20.6 - Prob. 74E Ch. 20.6 - Prob. 75E Ch. 20.6 - Prob. 76E Ch. 20 - Prob. 1RE Ch. 20 - Prob. 2RE Ch. 20 - Prob. 3RE Ch. 20 - Prob. 4RE Ch. 20 - Prob. 5RE Ch. 20 - Prob. 6RE Ch. 20 - Prob. 7RE Ch. 20 - Prob. 8RE Ch. 20 - Prob. 9RE Ch. 20 - Prob. 10RE Ch. 20 - Prob. 11RE Ch. 20 - Prob. 12RE Ch. 20 - Prob. 13RE Ch. 20 - Prob. 14RE Ch. 20 - In Exercises 15–22, simplify the given expressions...Ch. 20 - In Exercises 15–22, simplify the given expressions...Ch. 20 - Prob. 17RE Ch. 20 - Prob. 18RE Ch. 20 - Prob. 19RE Ch. 20 - Prob. 20RE Ch. 20 - Prob. 21RE Ch. 20 - Prob. 22RE Ch. 20 - Prob. 23RE Ch. 20 - Prob. 24RE Ch. 20 - Prob. 25RE Ch. 20 - Prob. 26RE Ch. 20 - Prob. 27RE Ch. 20 - Prob. 28RE Ch. 20 - Prob. 29RE Ch. 20 - Prob. 30RE Ch. 20 - Prob. 31RE Ch. 20 - Prob. 32RE Ch. 20 - Prob. 33RE Ch. 20 - Prob. 34RE Ch. 20 - Prob. 35RE Ch. 20 - Prob. 36RE Ch. 20 - Prob. 37RE Ch. 20 - Prob. 38RE Ch. 20 - Prob. 39RE Ch. 20 - Prob. 40RE Ch. 20 - Prob. 41RE Ch. 20 - Prob. 42RE Ch. 20 - Prob. 43RE Ch. 20 - Prob. 44RE Ch. 20 - Prob. 45RE Ch. 20 - Prob. 46RE Ch. 20 - 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The value of the expression cos ( 2 tan − 1 x ) .

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To find: The value of the expression cos(2tan−1x).

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Chapter 20 Solutions