14.7. A stock price is currently $40.

Assume that the expected return from the stock is 15% and that its volatility is 25%.

What is the probability distribution for the rate of return (with continuous compounding) earned over a 2-year period?

In this case μ=0.15, σ=0.25

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The expected return is 11.88% per annum and standard deviation
is 17.7%.

14.13. What is the price of a European call option on a non-dividend-paying stock when the stock price is $52, the strike price is $50, the risk-free interest rate is 12% per annum, the volatility is 30% per annum, and the time to maturity is 3 months?

S₀ = 52, K = 50, r = 0.12, σ = 0.3, T = 3/12

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14.16. A call option on a non-dividend-paying stock has a market price of $2+1/2. The stock price is $15, the exercise price is $13, the time to maturity is 3 months, and the risk-free interest rate is 5% per annum. What is the implied volatility?

c = 2.5, S₀ = 15, K = 13, r = 0.05, T = 3/12.

We will use iterative search procedure:

if σ = 0.2 then

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if σ = 0.3 then

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if σ = 0.4 then

[image]

as c = 2.507 > 2.5, we’ll take σ = 0.39 then

[image]

as c = 2.487 < 2.5, we’ll take σ = 0.396 then

[image]

c = 2.499 ≈2.5 then we conclude that implied volatility is about
3.96%. This is made by using trial-and-error method, the
following range was considered from σ=0.39 through σ=0.40

14.26. A stock price is currently $50.

Assume that the expected return from the stock is 18% and its volatility is 30%.

What is the probability distribution for the stock price in 2 years?

Calculate the mean and standard deviation of the distribution.

Determine the 95% confidence interval.

S₀ = 50, K = 13,  r = 0.05, T = 2,  µ = 0.18, σ = 0.3

[image]

The mean of the stock price is:

[image]

Variance of the stock price is:

[image]

Standard deviation of the stock price in 2 years is:

[image]

95% confidence intervals for lnS_T are

[image]

or

[image]

These correspond to 95% confidence limits for S_T of

[image]

14.27. Suppose that observations on a stock price (in dollars) at the end of each of 15 consecutive weeks are as follows: 30.2, 32.0, 31.1, 30.1, 30.2, 30.3, 30.6, 33.0, 32.9, 33.0, 33.5, 33.5, 33.7, 33.5, 33.2.

Estimate the stock price volatility.

What is the standard error of your estimate?

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an estimate of standard deviation of weekly returns is (n = 14):

[image]

The volatility per annum is therefore

[image]

or 20.83%.

The standard error of this estimate is

[image]

14.29. Consider an option on a non-dividend-paying stock when the stock price is $30, the exercise price is $29, the risk-free interest rate is 5% per annum, the volatility is 25% per annum, and the time to maturity is four months.

What is the price of the option if it is a European call?

What is the price of the option if it is an American call?

What is the price of the option if it is a European put?

Verify that put–call parity holds.

S₀ = 30, K = 29,  r = 0.05, T = 4/12,  σ = 0.25

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The European call price is

[image]

The American call price is the same as the European call price.
It is $2.53.

The European put price is
[image]

or $1.05.

Put-call parity states that:

[image]

Knowing that p =1.05, c = 2.53, we compute

[image]

This proves that put-call parity holds.