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FNCE 30007 DERIVATIVE SECURITIES

Tutorial Questions

The Binomial Model (Part I)

QUESTION 1

Assume you roll a pair of fair dice. Let X be the random variable whose value

represents the sum of numbers rolled. Let Y be the random variable whose

value represents the square of the sum of the numbers rolled.

(i) What is the set of possible values of X ?

(ii) What is the probability of rolling a "seven" ?

(iii) Plot the probability function of X

(iv) What is the expected value of X ?

(v) What is the expected value of Y ?

QUESTION 2

Consider a two period binomial model. Let j be the random variable whose

value measures the number of up movements in the stock price at maturity.

(i) What is the set of possible values of j ?

(ii) Plot the probability function of j assuming the probability of the stock

price increasing each period is 50%

(iii) Repeat parts (i) and (ii) for a three period binomial model.

(A three period model can be considered to be a series of three x one period

models).

QUESTION 3

In the derivation of the one period model it was stated that  implies that  where . Prove this result.

QUESTION 4

Assume:

· current stock price = $20

· stock price changes by +/-10% each 3 months with equal probability

· European put option, strike $21, maturity 3 months

· constant riskfree rate of 12% p.a.

Use a one period model to calculate the current value of the put.

QUESTION 5

Use a two period binomial model to value the option in question 4 assuming the

stock price changes by +/- 5% each 1.5 months with equal probability.

QUESTION 6

Assume the current price of a stock is $100. Each month, the stock price is

expected to increase by 2% (with 40% probability) or fall by 1% (with 60%

probability). Consider a 2 month European call option on the stock with a strike

of $100 and assume the riskfree interest rate is constant at 12% p.a.

(i) Use a two period binomial model to value the option.

(ii) Assume the stock price rises in the first month. What is the value of the

option at that time ?

(iii) Assume the stock price falls in the first month. What is the value of the

option at that time ?

(iv) Use your answers in parts (ii) and (iii) in combination with a one period

binomial model (with  month) to value the option.

(v) Is this a realistic model for movements in the underlying stock price?