
We consider the one-period model studied in class as an example. Namely, we assume
that the current stock price is S0 = 10. At time T, the stock has either moved up to
St = 12 (with probability p = 0.6) or down towards St = 8 (with probability 1−p = 0.4).
We consider a call option on this stock with maturity T and strike price K = 10. The
interest rate on the money market is zero.
As in class, we assume that you, as a customer, are willing to buy the call option on
100 shares of stock for $120. The investor, who sold you the option, can adopt one of the
following strategies:
Strategy 1: (seen in class) Buy 50 shares of stock and borrow $380.
Strategy 2: Buy 55 shares of stock and borrow $430.
Strategy 3: Buy 60 shares of stock and borrow $480.
Strategy 4: Buy 40 shares of stock and borrow $280.
(a) For each of strategies 2-4, describe the value of the investor’s portfolio at time 0,
and at time T for each possible movement of the stock.
(b) For each of strategies 2-4, does the investor have an arbitrage ? Reminder: an
arbitrage is an opportunity to obtain a positive profit without risk.
(c) If you were the investor, which strategy would you adopt and why ? (This is an
open question without a right or wrong answer. Feel free to elaborate.)

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- We consider a one-period market with the following properties: the current stock priceis S0 = 4. At time T = 1 year, the stock has either moved up to S1 = 8 (with probability0.7) or down towards S1 = 2 (with probability 0.3). We consider a call option on thisstock with maturity T = 1 and strike price K = 5. The interest rate on the money marketis 25% yearly.(a) Find the replicating portfolio (φ, ψ) corresponding to this call option.(b) Find the risk-neutral (no-arbitrage) price of this call option.(c) We now consider a put option with maturity T = 1 and strike price K = 3 onthe same market. Find the risk-neutral price of this put option. Reminder: A putoption gives you the right to sell the stock for the strike price K.1(d) An investor with initial capital X0 = 0 wants to invest on this market. He buysα shares of the stock (or sells them if α is negative) and buys β call options (orsells them is β is negative). He invests the cash balance on the money market (orborrows if the amount is…arrow_forwardQ3. Suppose that the market default rate for bonds is given by 0.01, i.e., the probability the market believes that the company may not be able to pay the owner of the bonds is 0.01. Now a credit default swap (CDS) is sold at fair price $ 0.01 per unit. Ackman expects that the true default rate is 0.05, not 0.01. Ackman bought 67 billion units of CDS at the price $ 0.01 per unit. a. Suppose that Ackeman's expectation about the default rate is not correct. I.e., the true default rate for bonds is given by 0.01. Find the cost of the purchase. Find the expected payoff. Find the profit. b. Suppose that Ackeman's expectation is correct. Find the cost of the purchase. Find the expected payoff. Find the profit.arrow_forwardYou are given the following excerpt from a life table, including mortality rates and expected value of a 1 unit whole life insurance payable at end of year of death for (x) using an interest rate of 4%: 9x Ax 65 0.012731 9,120.85 0.488213 65 66 0.013698 9,004.73 0.501394 66 67 0.014785 8,881.39 0.514803 67 68 0.016008 8,750.07 0.528423 68 69 0.017379 8,610.00 0.542232 69 70 0.018915 8,460.37 0.556209 70 71 0.020630 8,300.34 0.570330 71 72 0.022545 8,129.11 0.584573 72 73 0.024682 7,945.84 0.598913 73 74 0.027066 7,749.72 0.613326 74 75 0.029730 7,539.96 0.627784 75 You are also given that during the 2 year select period the mortality rates are a fraction of the normal mortality according to the following table: 91x]+1 3 2 4 9x+1 Using survival probabilities from the select life model described above, find the value of a 5,000 5-year deferred life policy payable at the end of year of death for a policy holder [65] (select age 65). 5,000 5|A[65):arrow_forward
- PLEASE HELPI WITH ALL OF THEMI Exercise 5.20 Consider the following portfolio of annuities-due currently being paid from the assets of a pension fund. Age Number of annuitants 60 40 70 30 80 10 Each annuity has an annual payment of $10000. The lives are assumed to be independent. Assume mortality follows the Standard Ultimate Life Table, with interest at 5% per year. Calculate (a) the expected present value of the total outgo on annuities, (b) the standard deviation of the present value of the total outgo on annu- ities, and (c) the 95th percentile of the distribution of the present value of the total outgo on annuities, using a Normal approximation.arrow_forwardIf customer interarrival times are exponentially distributed with rate 5 customers per hour, then O There is a 8.2% chance that a customer arrives within 5 minutes. There is a 81.8% chance that a customer arrives within 30 minutes. None of the answers are correct. There is a 34.1% chance that a customer arrives within 10 minutes. O There is a 71.3% chance that a customer arrives within 15 minutes.arrow_forwardGabrielle has purchased the 2-year extended warranty from a retailer to cover the value of hers new cellphone in case if it gets damaged or becomes inoperable for the price of $25. Gabrielle's cellphone is worth $1400 and the probability that it gets damaged or becomes inoperable during the length of the extended warranty is estimated to be 4%. Let XX be the retailer's profit from selling the extended warranty. Answer the following questions: 1. Create the probability distribution table for XX : XX outcome profit xx ,$ P(X=x)P(X=x) the cellphone gets damaged or becomes inoperable no claim filed 2. Use the probability distribution table to find the following: E[X]=μX=E[X]=μX= dollars. (Round the answer to 1 decimal place.) SD[X]=σX=SD[X]=σX= dollars. (Round the answer to 1 decimal place.)arrow_forward
- The accompanying data represent the annual rates of return of two companies' stock for the past 12 years. Complete parts (a) through (k). Year Rate of Return of Company 1 Rate of Return of Company 21996 0.203 0.3981997 0.310 0.5101998 0.267 0.4101999 0.195 0.4362000 -0.101 -0.0602001 -0.130 -0.1512002 -0.234 -0.3572003 0.264 0.3282004 0.090 0.2072005 0.030 -0.0142006 0.128 0.0932007 -0.035 0.027 (j) Plot residuals against the rate of return of Company 1. Does the residual plot confirm that the relation between the rate of return of Company 1 and Company 2 is linear? Yes or No? (k) Are there any years where the rate of return of Company 2 was unusual? Yes or No?arrow_forwardAfter an automobile is 1 year old, its rate of depreciation at any time is proportional to its value at that time. If an automobile was purchased on March 1, 2022, and its values on March 1, 2023 and March 1, 2024, were $7000 and $5800 respectively, what is its expected value on March 1, 2028?arrow_forwardD3arrow_forward
- 15. Consider the extract from a life table: X 1x 70 150 71 125 72 100 73 75 74 50 75 25 Calculate e70, the curtate expectation of life at age 70. A. 2.0 B. 2.5 C. 3.0 D. 3.5 E.4.0arrow_forwardHi! I was working on the question below: The Capital Asset Pricing Model (CAPM) is a financial model that assumes returns on a portfolio are normally distributed. Suppose a portfolio has an average annual return of 14.7% (i.e. an average gain of 14.7%) with a standard deviation of 33%. A return of 0% means the value of the portfolio doesn’t change, a negative return means that the portfolio loses money, and a positive return means that the portfolio gains money. And question (a) looks like: What percent of years does this portfolio lose money, i.e. have a return less than 0%? I got a z-score of -0.4455, which corresponds to the p value of 0.3264 on the z-table; I don't understand why the correct answer should be 0.3280 as said by one of the solutions, and I cannot locate such a number on the z-table. Thank you so much!arrow_forwardA consumer is contemplating the purchase of a new smart phone. A consumer magazine reports data on the major brands.Brand A has lifetime (TA),which is exponentially distributed with m=0.2;and Brand B has lifetime (TB),which is exponentially distributed with m = 0.1 (The unit of time is one year) a. Find the expected lifetimes for A and B. If a consumer must choose between the two on the basis of maximizing expected lifetime, which on should be chosed? b.Find the probability that A's lifetime exceeds its expected value.Do the same for B. What do you conclude? c.Suppose one consumer purchases Brand A, and another purchases Brand B.Find the mean and variance of 1) the average lifetime of the two devices and 2) the difference between their lifetimes.(Hint:You must use the rules about means and variances of linear transformations discussed in Chapter 7. Making Hard Decision with decisiontools,3rd edition)arrow_forward
- Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning

