Question

A non – dividend – paying stock with a current price of $52, the strike price is $50, the risk free interest rate is 12% pa, the volatility is 30% pa, and the time to maturity is 3 months?

a) Calculate the price of a call option on this stock

b) What is the price of a put option price on this stock?

c) Is the put-call parity of these options hold?

(4 + 4 + 2 = 10 marks)

Please provide step by step answers, thank you!

Answer 1

Sr.	Factor	Denotation	Value
1	Stock Price	S	$52
2	Strike Price	K	$50
3	Risk free interest rate	r	12% p.a.
4	Volatility	sigma	30% p.a.
5	Time to maturity	t	3 months

The formula for value of a call option C for a non-dividend paying stock of price S and strike price K is given as

C=S∗N(d1)−Ke−rt∗N(d2)

• N(d10: Probability that the stock price will be higher than the strike price on the expiry date i.e. call option is in the money. 0 <= N(d1) <= 1.
• d1=ln(S/K)+(r+σ2/2)tσ√t, where r: risk-free interest rate and t: time to expiration of option expressed in years
• Ke−rt: present value of option strike price
• N(d2): Probability that the option will be exercised. 0<= N(d2) <=1
• d2=d1−σ√t

From the given values, we can determine that

d1=ln(52/50)+(0.12+σ2/2)*3/12/(σ√312)

d1=(0.039221+0.04125)/0.15

d1=0.080471/0.15

d1=0.536471

N(d1) will be computed using interpolation from the values in Z tables (Let me know if you want to understand the interpolation from Z tables)

N(d1)=0.704184

d2=d1−σ∗√t

d2=0.536471−0.15

d2=0.386471

N(d2) will be computed using interpolation from the values in Z tables (Let me know if you want to understand the interpolation from Z tables)

N(d2)=0.650426

Substituting the values in the formula, we get

C=S∗N(d1)−Ke−rt∗N(d2)

C=52∗0.704184−50∗e−0.03∗0.650426

C=36.61755−31.56016

C = 5.057387, call option price = 5.0574

Put price calculation

P = S∗-N(-d1)+Ke−rt∗N(-d2)

we have compute the value of N(-d1) and N(-d2) which is

N(-d1) = 1-N(d1) and N(-d2) = 1-N(-d2)

N(-d1) = 1-0.704184 = 0.295816

N(-d2) = 1-0.650426 = 0.349574

P = 52∗-0.295816+50∗e−0.03∗0.349574

P = -15.3824 + 16.962126

P = 1.5796 , Put price = 1.5796

Answer Part 3

C+K/(1+r)t =S+P, c = call price and P = put price

5.0574 + 50/(1+0.12)^3/12 = 52 + 1.5796

5.0574+ 48.6032 = 53.5796

53.6608 > 53.5796, Put call parity does not hold because LHS is not equal to RHS