derivatives assigment 2

7/30/2019 derivatives assigment 2

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F714

Assignment 2

Instructions

This assignment can be done individually or with one partner, and is due in class onThursday December 1. Please use 4 decimal points in your answers. All interest rates herein are expressed with continuous compounding. Please use

continuous compounding in your answers.

Please ensure that your answers are as complete as possible. For numerical questions,simply calculating the numbers out will not be sufficient. Please also provide mewith the reasons for your calculations. Correct thinking processes are much moreimportant than correct numbers. Therefore, regardless of whether or not yournumbers are correct, no marks will be given if (a) no explanation of your calculationsis provided; or (b) the explanation provided is incorrect. On the other hand, even if

your numbers are incorrect, partial marks will be given if your thinking process iscorrect.

Please note that marks can only be given based on what you write. Therefore, Irequest that you attempt to convey your ideas to me as clearly as possible. I will notmake a guess as to what you intend to mean if you do not make it obvious and clear.

When your answers involve taking positions in securities or lending/borrowing,please mention all the relevant details such as the timing of the transactions, the sideof the contracts (e.g., long or short forward), the length of the contracts (e.g., 6-monthfutures), the exercise prices (in case of options) and the interest rates (in case oflending or borrowing).

Please be very careful with the logic of your solutions. For example, if the price of anasset at some future point in time is random, you cannot assume that you know thatprice. I consider this kind of mistake to be fundamental, and so anyone who makes itwill get zero point for that question.

The majority of the questions in this assignment are taken (or are adapted) fromquestions from past exams. Therefore, it is important that you try to do every

question on your own (even though you're working with a partner). Please take

this opportunity to familiarize yourself with what the exams will be like and,

more importantly, to develop your thinking skill.


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1) (3 points) Lalaa Corp. wants to invest its cash surplus of $1m. in a GIC (i.e.,certificate of deposit) for a period of 6 months. The best GIC rate that it could find is5.5% p.a. continuous compounding. [Note: This is the rate that Lalaa can lend itsmoney. It is not its borrowing rate. Nor is it anybody else's lending or borrowingrate.] Mr. Dipsey, the companys treasurer, has asked you to explore the possibility

of creating a synthetic lending transaction by a portfolio of options and stocks. Youobserved the following information:

Security Price6-month European call whose exercise price = $32 $2.106-month European put whose exercise price = $32 $2.92the underlying (non-dividend-paying) stock $30

Based on the above observations, will Lalaa be better off with the GIC or thesynthetic lending? If the synthetic lending is better, please also show how it can becreated.

2) Consider a stock whose current price is $100 and volatility is 20% p.a. The stock isexpected to pay a dividend of $2 in one months time. The risk-free rate is 5% p.a.continuously compounding.

(a)(3 points) Using a two-period binomial tree, what is the value of a 2-monthAmerican call whose exercise price is $101?

(b)(3 points) Using a two-period binomial tree, what is the value of a 2-monthAmerican put whose exercise price is $101?

3) (4 points) An Average Strike call is a type of Asian option that pays offmax{0, STS

avg) at maturity where S

avgis the average of the stock prices from the

start of the call to its maturity. For example, consider the following path for stockprices:

Day Price

0 Start of option $1001 $1102 $1213 $133.1

In this case, the average stock price = (100 + 110 + 121 + 133.1)/4 = $116.025.[Note that although there are 4 days and 3 periods, the average is taken from thestart date (i.e. over the 4 prices).]

Consider an Average Strike call with a maturity of 3 months (and can be exercisedonly at maturity). The underlying stock is currently traded at $100, while the risk-free rate is 5% p.a. Use a 3-period Binomial tree to price this call, given that youhave determined that u = 1.1 andd= 0.9. Also, at time 0, what is the (risk-neutral)probability that this call will end up in the money?


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Note that although the tree for the stock price recombines, the calls payoff dependson the path that the stock price has taken. For example, the payoff under the pathuud is no longer the same as under the path udu, since the average prices under thetwo paths are not the same. Nevertheless, the probabilities that the two paths will

occur are the same.

4) (4 points) In its most simple form, a convertible bond is similar to a fixed-rate bondwith a provision that at maturity, the holders can choose to either receive the principalback, or convert the principal repayment into shares of the issuing company at a pre-determined conversion rate.

Suppose your company wants to raise funds by issuing this most simple form ofconvertible bonds. The bond will have a maturity of 3 years and will be issued at par(i.e., you sell the bond at its face value, $1,000). Coupons will be paid every year. At

maturity, each bond allows its holder to convert the principal repayment into 20shares of your companys stock. The stock is currently traded at $40/share. Itsvolatility is 25% per annum. The current risk-free rate is 4% for all terms. Yourcompanys credit rating is currently BBB. Typically, BBB-rated firms have to pay3% p.a. more than the risk-free rate when they borrow. That is, if your companywanted to issue a straight bond (i.e., one with no conversion provision) now, it wouldhave to pay 7% p.a. in coupons in order for the bond to be priced at par.

You were asked to set the coupon rate for the bond. Under a 3-period binomialframework, what coupon rate will you set? Note that according to bond convention,the coupon rate that you set will be in annual compounding because the bond in thiscase pays coupons every year. For example, if you set a coupon rate of 5% p.a., itmeans that every year the holder will receive $50.

[Hint: The buyers of your convertible bonds have an option. They pay for this optionby having to accept a lower coupon rate on the bond.]

5) Remember the friend who offered you a bet on Research in Motion (RIM) stock inthe midterm exam? Shes back with a different offer. She still believes that the priceof RIM, now at $20 per share, is still too high. She is prepared to back up herconviction by offering you the following bet. If in exactly one year from now RIMstock price is higher than $20/share, she will give you one hundred times the amountof the difference (e.g., suppose the price in one year is $23 per share, you will get$300). On the other hand, if the price in one year ends up below $20/share, you willhave to give her only seventy times the amount of the difference (e.g., suppose theprice in one year is $18 per share, you will have to give her $140). Suppose that theexpected return on RIM stock is 9% p.a., and the volatility is 30% p.a. Suppose alsothat the current risk-free rate is 3% p.a. and Research in Motion does not plan to paydividends in the next year. What is the value of this bet to you?


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6) An investment bank recently introduced a new security called "SuperBear Principal-Protected Note". This security has a 2-year term, and will make money for investorsif the markets drop. That is, its return is based on the percentage decline, if any, inthe S&P 500 index over its life. Specifically, the payoffs are as follows:

-

If after 2 years, the S&P 500 index ends up below the current level, then therate of return on this note will be positive and twice as much as the percentagedecline in the index. For example, suppose the index drops by 10% over the2-year term. The rate of return on this note will then be +20%.

- If after 2 years, the S&P 500 index ends up at the current level or higher, therate of return on this note will be zero, and the investors will get theirprincipal (i.e., $100 per unit) back.

Suppose that the S&P 500 index follows a geometric Brownian motion with anexpected return of 9% p.a. and a standard deviation of 30% p.a. Suppose also that thedividend yield on the index is zero (i.e., no dividend), and that the risk-free rate is 4%

p.a. for all terms.

(a)(3 points) What should be the fair price of this note?(b)(2 points) What is the real-world probability that the rate of return on the note will

be positive but not more than 20%?

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derivatives assigment 2