Check the sample mean by using the calculator. Check the sample standard deviation by using the calculator.

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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…

Answer 1

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Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

Chapter 7.4, Problem 14P

(a)

To determine

Check the sample mean by using the calculator.

Check the sample standard deviation by using the calculator.

(b)

To determine

Find the 99% confidence interval for μ1−μ2.

(c)

To determine

Interpret the confidence interval in the context of the problem.

Identify whether the interval consist of numbers that are all positive or all negative or of different signs.

Explain whether professional football players tend to have a higher population mean weight than professional basketball players at 99% level of confidence.

(d)

To determine

Identify and explain the distribution that is used for calculating interval.

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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…

The scores of 8 students on the midterm exam and final exam were as follows. Student Midterm Final Anderson 98 89 Bailey 88 74 Cruz 87 97 DeSana 85 79 Erickson 85 94 Francis 83 71 Gray 74 98 Harris 70 91 Find the value of the (Spearman's) rank correlation coefficient test statistic that would be used to test the claim of no correlation between midterm score and final exam score. Round your answer to 3 places after the decimal point, if necessary. Test statistic: rs =

Answer 2

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Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

Chapter 7.4, Problem 14P

(a)

To determine

Check the sample mean by using the calculator.

Check the sample standard deviation by using the calculator.

(b)

To determine

Find the 99% confidence interval for μ1−μ2.

(c)

To determine

Interpret the confidence interval in the context of the problem.

Identify whether the interval consist of numbers that are all positive or all negative or of different signs.

Explain whether professional football players tend to have a higher population mean weight than professional basketball players at 99% level of confidence.

(d)

To determine

Identify and explain the distribution that is used for calculating interval.

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Answer 3

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Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

Chapter 7.4, Problem 14P

(a)

To determine

Check the sample mean by using the calculator.

Check the sample standard deviation by using the calculator.

(b)

To determine

Find the 99% confidence interval for μ1−μ2.

(c)

To determine

Interpret the confidence interval in the context of the problem.

Identify whether the interval consist of numbers that are all positive or all negative or of different signs.

Explain whether professional football players tend to have a higher population mean weight than professional basketball players at 99% level of confidence.

(d)

To determine

Identify and explain the distribution that is used for calculating interval.

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Answer 4

Question

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

Chapter 7.4, Problem 14P

(a)

To determine

Check the sample mean by using the calculator.

Check the sample standard deviation by using the calculator.

(b)

To determine

Find the 99% confidence interval for μ1−μ2.

(c)

To determine

Interpret the confidence interval in the context of the problem.

Identify whether the interval consist of numbers that are all positive or all negative or of different signs.

Explain whether professional football players tend to have a higher population mean weight than professional basketball players at 99% level of confidence.

(d)

To determine

Identify and explain the distribution that is used for calculating interval.

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Answer 5

Chapter 7.4, Problem 14P

(a)

To determine

Check the sample mean by using the calculator.

Check the sample standard deviation by using the calculator.

(b)

To determine

Find the 99% confidence interval for μ1−μ2.

(c)

To determine

Interpret the confidence interval in the context of the problem.

Identify whether the interval consist of numbers that are all positive or all negative or of different signs.

Explain whether professional football players tend to have a higher population mean weight than professional basketball players at 99% level of confidence.

(d)

To determine

Identify and explain the distribution that is used for calculating interval.

Answer 6

Chapter 7.4, Problem 14P

(a)

To determine

Check the sample mean by using the calculator.

Check the sample standard deviation by using the calculator.

Answer 7

Chapter 7.4, Problem 14P

(a)

To determine

Check the sample mean by using the calculator.

Check the sample standard deviation by using the calculator.

Answer 8

(b)

To determine

Find the 99% confidence interval for μ1−μ2.

Answer 9

(b)

To determine

Find the 99% confidence interval for μ1−μ2.

Answer 10

(c)

To determine

Interpret the confidence interval in the context of the problem.

Identify whether the interval consist of numbers that are all positive or all negative or of different signs.

Explain whether professional football players tend to have a higher population mean weight than professional basketball players at 99% level of confidence.

Answer 11

(c)

To determine

Interpret the confidence interval in the context of the problem.

Identify whether the interval consist of numbers that are all positive or all negative or of different signs.

Explain whether professional football players tend to have a higher population mean weight than professional basketball players at 99% level of confidence.

Answer 12

(d)

To determine

Identify and explain the distribution that is used for calculating interval.

Answer 13

(d)

To determine

Identify and explain the distribution that is used for calculating interval.

Answer 14

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Check the sample mean by using the calculator. Check the sample standard deviation by using the calculator.

Solution Summary: The author explains how to calculate the mean and standard deviation for x_1 using Ti 83 calculator.

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Check the sample mean by using the calculator.

Check the sample standard deviation by using the calculator.

Find the 99% confidence interval for μ1−μ2.

Identify and explain the distribution that is used for calculating interval.

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