Introduction to Probability and Statistics
Introduction to Probability and Statistics
14th Edition
ISBN: 9781133103752
Author: Mendenhall, William
Publisher: Cengage Learning
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Chapter 1, Problem 1.64SE

(a)

To determine

To construct the stem and leaf display for the data.

(b)

To determine

To describe the shape of the distribution.

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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

Chapter 1 Solutions

Introduction to Probability and Statistics

Ch. 1.3 - Jeans A manufacturer of jeans has plants in...Ch. 1.3 - Prob. 1.12ECh. 1.3 - Want to Be President? Would you want to be the...Ch. 1.3 - Facebook Fanatics The social networking site...Ch. 1.3 - Back to Work How long does it take you to adjust...Ch. 1.5 - Construct a stem and leaf plot for these 50...Ch. 1.5 - Refer to Exercise 1.16. Construct a relative...Ch. 1.5 - Prob. 1.18ECh. 1.5 - A discrete variable can take on oniy the values 0....Ch. 1.5 - Prob. 1.20ECh. 1.5 - Prob. 1.21ECh. 1.5 - Prob. 1.22ECh. 1.5 - Cheeseburgers Create a dotplot for the number of...Ch. 1.5 - Prob. 1.24ECh. 1.5 - Prob. 1.25ECh. 1.5 - Prob. 1.26ECh. 1.5 - Education Pays Off! Education pays off, according...Ch. 1.5 - Preschool The ages (in months) at which 50...Ch. 1.5 - Prob. 1.29ECh. 1.5 - Prob. 1.30ECh. 1.5 - Prob. 1.31ECh. 1.5 - Prob. 1.32ECh. 1.5 - Prob. 1.33ECh. 1.5 - RBC Counts The red blood cell count of a healthy...Ch. 1.5 - Prob. 1.35ECh. 1.5 - Prob. 1.36ECh. 1.5 - Hazardous Waste How safe is your neighborhood? Are...Ch. 1 - Quantitative or Qualitative? Identify each...Ch. 1 - Symmetric or Skewed? Do you expect the...Ch. 1 - Continuous or Discrete? Identify each variable as...Ch. 1 - Prob. 1.41SECh. 1 - Continuous or Discrete, again Identify each...Ch. 1 - Aqua Running Aqua running has been suggested as a...Ch. 1 - Prob. 1.44SECh. 1 - Ages of Pennies We collected 50 pennies and...Ch. 1 - Prob. 1.46SECh. 1 - Presidential Vetoes Here is a list of the 44...Ch. 1 - Prob. 1.48SECh. 1 - Prob. 1.49SECh. 1 - Prob. 1.50SECh. 1 - Prob. 1.51SECh. 1 - Prob. 1.52SECh. 1 - Prob. 1.53SECh. 1 - Student Heights The self-reported heights of 105...Ch. 1 - Prob. 1.55SECh. 1 - Pulse Rates A group of 50 biomedical students...Ch. 1 - Prob. 1.57SECh. 1 - Prob. 1.58SECh. 1 - Prob. 1.59SECh. 1 - Prob. 1.60SECh. 1 - Prob. 1.61SECh. 1 - Old Faithful The data below are the wailing times...Ch. 1 - Prob. 1.63SECh. 1 - Prob. 1.64SECh. 1 - Prob. 1.65SECh. 1 - The number of Starhucks coffee shops in cities...Ch. 1 - What’s Normal? The 98.6 degree standard for human...
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