The blank space in the statement “The of the sample mean , x ¯ , is the probability distribution of all possible values of the random variable x ¯ computed from a sample of size n from a population with mean μ and standard deviation σ.”

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Chapter 8.1, Problem 1AYU

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To fill: The blank space in the statement “The of the sample mean, x¯, is the probability distribution of all possible values of the random variable x¯ computed from a sample of size n from a population with mean μ and standard deviation σ.”

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In this problem, we consider a Brownian motion (W+) t≥0. We consider a stock model (St)t>0 given (under the measure P) by d.St 0.03 St dt + 0.2 St dwt, with So 2. We assume that the interest rate is r = 0.06. The purpose of this problem is to price an option on this stock (which we name cubic put). This option is European-type, with maturity 3 months (i.e. T = 0.25 years), and payoff given by F = (8-5)+ (a) Write the Stochastic Differential Equation satisfied by (St) under the risk-neutral measure Q. (You don't need to prove it, simply give the answer.) (b) Give the price of a regular European put on (St) with maturity 3 months and strike K = 2. (c) Let X = S. Find the Stochastic Differential Equation satisfied by the process (Xt) under the measure Q. (d) Find an explicit expression for X₁ = S3 under measure Q. (e) Using the results above, find the price of the cubic put option mentioned above. (f) Is the price in (e) the same as in question (b)? (Explain why.)

Problem 4. Margrabe formula and the Greeks (20 pts) In the homework, we determined the Margrabe formula for the price of an option allowing you to swap an x-stock for a y-stock at time T. For stocks with initial values xo, yo, common volatility σ and correlation p, the formula was given by Fo=yo (d+)-x0Þ(d_), where In (±² Ꭲ d+ õ√T and σ = σ√√√2(1 - p). дго (a) We want to determine a "Greek" for ỡ on the option: find a formula for θα (b) Is дго θα positive or negative? (c) We consider a situation in which the correlation p between the two stocks increases: what can you say about the price Fo? (d) Assume that yo< xo and p = 1. What is the price of the option?

We consider a 4-dimensional stock price model given (under P) by dẴ₁ = µ· Xt dt + йt · ΣdŴt where (W) is an n-dimensional Brownian motion, π = (0.02, 0.01, -0.02, 0.05), 0.2 0 0 0 0.3 0.4 0 0 Σ= -0.1 -4a За 0 0.2 0.4 -0.1 0.2) and a E R. We assume that ☑0 = (1, 1, 1, 1) and that the interest rate on the market is r = 0.02. (a) Give a condition on a that would make stock #3 be the one with largest volatility. (b) Find the diversification coefficient for this portfolio as a function of a. (c) Determine the maximum diversification coefficient d that you could reach by varying the value of a? 2

Answer 2

Question

Chapter 8.1, Problem 1AYU

To determine

To fill: The blank space in the statement “The of the sample mean, x¯, is the probability distribution of all possible values of the random variable x¯ computed from a sample of size n from a population with mean μ and standard deviation σ.”

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Answer 3

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Chapter 8.1, Problem 1AYU

To determine

To fill: The blank space in the statement “The of the sample mean, x¯, is the probability distribution of all possible values of the random variable x¯ computed from a sample of size n from a population with mean μ and standard deviation σ.”

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Answer 4

Question

Chapter 8.1, Problem 1AYU

To determine

To fill: The blank space in the statement “The of the sample mean, x¯, is the probability distribution of all possible values of the random variable x¯ computed from a sample of size n from a population with mean μ and standard deviation σ.”

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Answer 5

Chapter 8.1, Problem 1AYU

To determine

To fill: The blank space in the statement “The of the sample mean, x¯, is the probability distribution of all possible values of the random variable x¯ computed from a sample of size n from a population with mean μ and standard deviation σ.”

Answer 6

Chapter 8.1, Problem 1AYU

To determine

To fill: The blank space in the statement “The of the sample mean, x¯, is the probability distribution of all possible values of the random variable x¯ computed from a sample of size n from a population with mean μ and standard deviation σ.”

Answer 7

Chapter 8.1, Problem 1AYU

To determine

To fill: The blank space in the statement “The of the sample mean, x¯, is the probability distribution of all possible values of the random variable x¯ computed from a sample of size n from a population with mean μ and standard deviation σ.”

Answer 8

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Answer 11

Ch. 8.1 - 1. The ________ ________ of the sample mean, , is...Ch. 8.1 - Prob. 2AYU Ch. 8.1 - 3. The standard deviation of the sampling...Ch. 8.1 - 4. True or False: The distribution of the sample...Ch. 8.1 - Prob. 5AYU Ch. 8.1 - Prob. 6AYU Ch. 8.1 - 7. A simple random sample of size n = 10 is...Ch. 8.1 - 8. A simple random sample of size n = 40 is...Ch. 8.1 - In Problems 9–12, determine and from the given...Ch. 8.1 - Prob. 10AYU

The blank space in the statement “The of the sample mean , x ¯ , is the probability distribution of all possible values of the random variable x ¯ computed from a sample of size n from a population with mean μ and standard deviation σ.”

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To fill: The blank space in the statement “The of the sample mean, x¯, is the probability distribution of all possible values of the random variable x¯ computed from a sample of size n from a population with mean μ and standard deviation σ.”

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Chapter 8 Solutions

The blank space in the statement “The ________ ________ of the sample mean , x ¯ , is the probability distribution of all possible values of the random variable x ¯ computed from a sample of size n from a population with mean μ and standard deviation σ.”

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To fill: The blank space in the statement “The ________ ________ of the sample mean, x¯, is the probability distribution of all possible values of the random variable x¯ computed from a sample of size n from a population with mean μ and standard deviation σ.”

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Chapter 8 Solutions

The blank space in the statement “The of the sample mean , x ¯ , is the probability distribution of all possible values of the random variable x ¯ computed from a sample of size n from a population with mean μ and standard deviation σ.”

To fill: The blank space in the statement “The of the sample mean, x¯, is the probability distribution of all possible values of the random variable x¯ computed from a sample of size n from a population with mean μ and standard deviation σ.”