Question

A stock price is currently $60. Assume that the expected return from the stock is 15% and its volatility is 25% per annum.

What is the probability distribution of the stock price, ST, in six months?

Φ ( μ, σ2 ) = Φ ( , )

Calculate a 95% (with 1.96 standard deviation) confidence interval of the stock price, ST, in six months.

What is the probability that a six-month European call option on the stock with an exercise price of $70 will be exercised?

Answer 1

Given that Stock Price, S0 = $60, Time, T (in years) = 0.5

Expected return, μ = 0.15 and volatility, σ = 25%

We use the probability distribution of the stock price in two years using lognormal distribution is given by:-

lnST=Φ(lnS0+(μ−σ2/2)×T, σ2×T)

lnST=Φ(ln60+(0.15−0.252/2)×0.5,0.252×0.5)

lnST=Φ(4.14, .182)

Therefore, the probability of the stock price in 6 months is

lnST=Φ(4.14, 0.182)

The mean of the stock price, E(ST) is given by:-

E(ST)=S0er×T

E(ST)=60 * e0.15×0.5

E(ST)=60*e0.075

E(ST) = $64.67

The standard deviation of the stock price, σST is given by:-

σST=S0erT* Square root of (eσ2*T-1)

σST=60*e0.15×0.5 *square root of (e.25^2*0.5-1)

σST = $11.51

Therefore, the mean of the stock price is $64.67 and the standard deviation of the stock price is $11.51.

95% confidence interval for lnST are:-

We use normal tables to find the critical value at α/2 = 0.05/2 = 0.025 is 1.96

4.14 ± 1.96 x 0.18

3.79, 4.49

Now, the corresponding 95% confidence interval for ST are

e3.79 and e4.49

44.26 and 89.12