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2 PSTAT 170 – FINAL EXAM

Problem 1. Ernie has adopted a two-year investment plan in which $10,000 is to

be deposited into his account at time 0. His investment is placed into a stock index

fund that will either gain 25% each year or lose 15% each year. The one-year annual

effective rate of interest is 7.5%. Ernie is concerned about losing money and his investment advisor Bert has offered him the following insurance for protecting the value

of his investment portfolio. The insurance provides for Ernie to receive a minimum

payout at the end of two years which is equal to the value of his initial investment and

in return for this protection, Ernie’s annual investment gain is capped at g% where

g < 25.

Compute a value for g so that Bert neither makes money nor loses money from

providing the insurance to Ernie.

Problem 2. You are given the following data assuming a Black-Scholes model.

• S = $100

• σ = 30%

• r = 0.08

• δ = 0

Suppose you sell a 95-strike put with 3 months to expiration.

(i) (5 points) What is the gamma of this put option?

(ii) (3 points) If the underlying were to decrease by $1 over the next short while,

what effect would that have on this option’s delta?

(iii) (2 points) If all of the data above were the same except that the maturity of the

put option was 30 years, what is the approximate value of the put’s gamma?

Problem 3. Let S = 62, K = 55, r = 0.06, σ = 0.35, δ = 0, T = 1, and pricing is to

be done using the two-period binomial approximation (i.e. n = 2 periods).

A chooser option is an exotic option that lets you choose after a certain period whether

an option is a European call or a European put. Suppose that the time at which you

need to choose is equal to T/2. Note that K is the strike that will apply to the call

or the put depending on what is chosen.

What is the price of this chooser option?

PSTAT 170 – FINAL EXAM 3

Problem 4. Assume the Black-Scholes framework. Consider a one-year at-the-money

European put option on a nondividend-paying stock.

You are given.

(i) The ratio of the put option price to the stock price is less than 5%.

(ii) The Delta (mathematical notation ∆ for this greek) of the put option is s0.4364.

(iii) The continuously compounded risk-free interest rate is 1.2%

Which of the following is closest to the value of σ?

(A) 0.12

(B) 0.14

(C) 0.16

(D) 0.18

(E) 0.20

Problem 5. A stock price index at time t has the distribution according to the

random variable (as per Chapter 18 of McDonald)

St = 1000 × exp

[µ 0.5σ

2

]t + σ

√

tZ

where Z has a standard normal distribution and time is measured in years.

µ = 0.08 and σ = 0.20

(1) Calculate the probability that the stock price decreases by at least 5% at the end

of the next month.

(2) What is the smallest value of σ so that the probability that the stock price declines

by at least 10% over the next day is at least 1%.

4 PSTAT 170 – FINAL EXAM

Problem 6. You are interested in pricing and hedging a European put option on a

stock using a two-period binomial model with notation and set-up as in Chapter 10

of the text.

S0 = 10, u = 1.15, d = 0.92, h = 1, δ = 0, and r = 0

The put option expires at time 2 and has a strike price of 12.

Compute the price of the put option at time 0 and compute the dynamic hedging

strategy needed to replicate the put option payoffs at time 2.

[As you know, dynamic hedging strategy consists of the trading positions in stock

and bond at time 0 and in each of the upstate and downstates at time 1 that are

needed to get the put option payoffs at time 2. In class we used the notation (∆, B),

(∆u, Bu) and (∆d, Bd). So long as you clearly indicate what node the trading strategy

applies to any reasonable notation is acceptable.]

Problem 7. Consider a market maker in a call option written on a stock with the

following information under the Black-Scholes framework (as per Chapter 13 of McDonald).

(i) S0 = 35

(ii) σ = 0.30

(iii) The continuously compounded risk-free interest rate is 6%

(iv) δ = 0

(v) T = 90 days (Call option expires in 90 days, 365 days in a year.)

(vi) The market maker transacts in the option at the Black-Scholes prices.

The market maker sells 100 call options and delta hedges the position at daily frequency over the next two days, closing out her position at the end of the second

day.

If the stock price at the end of the first day is 34.25 and the stock price at the end of

the second day is 35.75, what is the market makers profit or loss?

PSTAT 170 – FINAL EXAM 5

Problem 8. You have been offered an opportunity to take a long or short position

on the following deal.

• In 6 months time the long pays $1026.20 for the asset.

• The long receives the asset from the short exactly 2 months after the $1026.20

payment is made.

• The asset will pay a cash dividend of $10.50 exactly one month after the

payment of $1026.20 is made.

The continuously compounded interest rate is 4.25% (i.e. r = 0.0425) and the current

market price of the asset is $1000.00.

(1) [5 points] If profit is your sole motive, is there an attractive opportunity here?

Explain which side of the transaction you would take (i.e. long or short) and why.

(2) [5 points] Explain how you would hedge out your risk in the transaction you chose

in (1) and compute what your profit is at the time the asset is delivered to the long.

Problem 9. You are interested in pricing European and American put options on a

stock using a four-period binomial model with notation and set-up as in Chapter 10

of the text.

S0 = 30, u = 1.15, d = 0.78, h = 1, δ = 0, and r = 0.04

The European and American put options expire at the end of the 4

th period and each

has a strike price of 25.

As in the text, r as a continuously compounded interest rate.

Compute the price the European put option and the American put option.

Problem 10. Consider the put-call parity relationship for European call and put

options on a stock that pays discrete dividends only. Assume the initial stock price

is S0, the continuously compounded risk-free interest rate is r, the options expire at

time T , the stock price at time T is denoted ST and there are two discrete dividends

paid between time 0 and time T at times t1 and t2, 0 < t1 < t2 < T with amounts

denoted dt1 and dt2

respectively.

Write the put-call parity formula and explain why it is true? (You should provide an

arbitrage argument to justify the formula you provide).

STANDARD NORMAL DISTRIBUTION TABLE

Entries represent Pr(Z ≤ z). The value of z to the first decimal is given in the left column. The second

decimal is given in the top row.

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