Question

1)   Price a call option on Planetary Resources Group stock expires in two periods

With a strike price of $200. The PRG stock is $215 a ton. The price either moves

up with u=1.15, or down with d=1/1.15. The risk-free interest rate is 2%. What is

the value of your option today in period 0?

A)   Recompute the price from above assuming that the risk-free interest rate has

risen to 4%. In one sentence, explain why the price changed the direction it did.

B)   Recompute the price from question 1 assuming that the current price has risen to $230. In one sentence, explain why the price changed the direction it did.

C)   Donets Extraction stock sells for $10. The monthly interest rate is 1%. The

standard deviation of the price of this stock is 100% per year. Use the Black-

Sholes equation to determine the price of a call option with a strike price of 9 that

expires in 6 months? Using Put-Call parity determine the price of a put option on

this stock with a strike of 9 expiring in 6 months.

Answer 1

1)

Note: Assuming that risk free rate of 2% is per period (since not explicitly mentioned whether it's annual). I also assume continuous compounding.

We construct a binomial tree. Formula also shown in picture. Note: S(t+) will be u*S(0); S(t-) will be d*S(0), and in this faishon we can also continue for n=2. {e.g. S(t++) = S0*u*u}

C(t0) = $30.55

(A) In previous Excel Model, we will just replace 'Rf per period' by 0.04 (earlier was 0.02).

C(t0) = $35.61 (as Rf ncreases, time value of money will increase; by using call options investors save more money by not paying for the underlying until later date and earn higher interest meanwhile.)

(B)

In excel model of Q1, we just change S0 to $230 (was 215 earlier).

C(t0) = $43.21 (With Strike of $200, as stock rises from $215 to $230, it becomes more in-the-money, so it becomes more valuable.)

(C) BSM model: we consider time period in terms of years

monthly interest rate = 1%

(1+AnnualRate)=(1+MonthlyRate)¹²

AnnualRate = (1+MonthlyRate)¹² - 1 = 1.01¹²-1 = 12.6825%

. S=10, sigma=1, r=12.6825%, K=9, T=0.5 yrs

Call option = S N(d1) - K e^-rT N(d2)

where d1 = {ln(S/K) + [[r+(sigma²/2)] *T] } / (sigma * squareroot(T))

d2 = d1 - [sigma * sqrt(T)]

N(d1) = NORMDIST(d1,0,1,TRUE) {excel in-built function}

N(d2) = NORMDIST(d2,0,1,TRUE) {excel in-built function}

{We can simply put above formulas in Excel & solve}

By BSM Model, we get Call = 3.3943

Put Call parity will hold good with same maturity & strike Call-Put pair on same underlying.

Call-Put=Stock - PV(K)

Put=Call - Stock + PV(K) = 3.3943 - 10 + 9*exp(-0.126825*0.5) = 1.8413