# Call Option and Price

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ACT 4000, FINAL EXAMINATION

ADVANCED ACTUARIAL TOPICS

APRIL 24, 2007

9:00AM - 11:OOAM

University Centre RM 210- 224 (Seats 266- 304)

Instructor: Hal W. Pedersen

You have 120 minutes to complete this examination. When the invigilator instructs you to stop writing you must do so immediately. If you do not abide by this instruction you will be penalised.

Each question is worth 10 points. If the question has multiple parts, the parts are equally weighted unless indicated to the contrary: Provide sufficient reasoning to back up your answer but do not write more than necessary.

This examination consists of 12 questions. Answer each question on a separate page of the exam book. Write your name and student number on each exam book that you use to answer the questions. Good luck!

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Suppose call and put prices are given by

Strike

Call premium

Put premium

80

22

4

100

9

21

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Find the convexity violations.

(1.-)

What spread would you use to effect arbitrage?

105

5

24.80

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I A New York finn is offering a new financial instrument called a "happy calL" It has a payoff function at time T equal to max(.5S, S - K), where S is the price of a stock and K is a fixed strike price. You always get something with a happy call. Let P be the price of the stock at time t

0 and let C, and C2 be the prices of ordinary calIs

with strike prices K and 2K, respectively. The fair price of the happy call is of the fonn

=

CH

Find the constants ex, fJ, and y.

y

~~:

= exP + fJC. + yC2.

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You are interested in stock that will either gain 30% this

year or lose 20% this year. The one-year annual effective rate of interest is 10%. The stock is currently selling for $10.

(1) (4 points) Compute the price of a European call option on this stock with a strike price of $11.50 which expires at the end of the year. (2) (4 points) Compute the hedge portfolio (i.e. the amount of stock and one-year bonds to hold that replicate the option's payoffs) for this European call option.

(1) (2 points) Consider a European put option on this stock with a strike price of $11.50 which expires at the end of the year. ft is possible to structure just the right amount of European call options on this stock with a strike price of $11.50 expiring at the end of the year, together with one-year bonds, and shares of the stock so as to replicate the payoffs from the put option. Determine how many shares of stock, how many call options, and how many one-year bonds are needed to replicate the payoffs from the put option and use this to price the put option.

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A non-dividend-paying stock has a current price of 800p. In any unit of time (t, t + 1) the price of the stock either increases by 25% or decreases by 20%. £1 held in cash between times t and t + 1 receives interest to become £1.04 at time t + 1. The stock price after t time units is denoted by St.

(i)

Calculate the risk-neutral probability measure for the model.

(ii)

Calculate the price (at t = 0) of a derivative contract written on the stock with expiry date t = 2 which pays 1,000p if and only if S2 is not 800p (and otherwise pays 0).

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For a stock index, S

n

= 3.

=

jO.11'f-IO.LeJ

$100, a

=

=

30%, r

5%,

(5

=

3%, and T

=

3. Let

[n refers to the number of binomial periods]

( I)

What is the price of a European put option with a strike of $95?

( 1-)

What is the price of

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S

=

$40, a

=

30%,

all

r

American

call option with a strike of $95?

=

8%, and () =

t-

O.

T (.~ t-

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Suppose you sell a 40-strike put with 91 days to expiration.

( ,)

What is delta?

If the option is on 100 shares, what investment is required for a...

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