Question

Consider the situation where the “Zeus” stock price 3 months from the expiration
of an option is $32, the exercise price of the option is $30, the risk-free rate is 6%
per annum, and the volatility is 25% per annum.

a. Calculate the price of the European Call and European Put respectively.
b. If the quoted price of the call is $2.75, can you argue that the call is
undervalued?
c. If the quoted price of the put is $1.98, can you argue that the put is
overvalued?
d. Show by the means of well-drawn diagrams that “Option-Pricing” is a
ZERO-SUM game. Explain your answers analytically.

Answer 1

a) From the Black Scholes Model, Value of Call option (C) is given by

C=S*N(d1)-K*e^(-r*t) * N(d2)

where S is the current stock price =$32

K is the strike price = $30

r is the risk free rate = 6%=0.06

t is the time till maturity in years = 3/12 = 0.25 and

d1= ( ln(S/K) + (r + s^2/2) *t ) / (s*t^0.5)

where s is the standard deviation or volatility = 0.25

and d2 = ( ln(S/K) + (rd- s^2/2) *t ) / (s*t^0.5)

Putting the values , we get , d1 = 0.6988 and N(d1) = area under normal distribution upto d1 =0.757664

and d2 =   0.5738 and N(d2) = 0.7169512

So, C = $3.0569 or $3.06

Now using put call parity equation

p = C+Ke^(-rt) -S = $0.6103 or $0.61

b) If the quoted price of call option is $2.75, the call option is undervalued as the theoretical price should be $3.06

c) If the quoted price of put option is $1.98, the put option is overvalued as the theoretical price should be $0.61

d) Payoff of a call option buyer is given by = max(St-K,0) where St is the stock price at maturity and K is the strike price. & profit is given by Profit = max(St-K,0) - C where C is the call option premium

The Payoff and profit of the seller of the call option are exactly opposite

Profit = C -max(St-K, 0)

Using the above option values C = $3.06, K = $30 and various possible values of St, the payoff of Seller and Buyer are as shown in the table and graph below

St	Profit of Seller	Profit of Buyer	Total Profit of Buyer and Seller
25	3.06	-3.06	0
26	3.06	-3.06	0
27	3.06	-3.06	0
28	3.06	-3.06	0
29	3.06	-3.06	0
30	3.06	-3.06	0
31	2.06	-2.06	0
32	1.06	-1.06	0
33	0.06	-0.06	0
34	-0.94	0.94	0
35	-1.94	1.94	0

It can be seen clearly that the call option buyer and Seller has exactly negative profits. Sum of profits is 0. So the Call option pricing is a zero sum game,

Similarly the put option pricing is also a zero sum game.

Profit for buyer of put option at expiration = max(K-St,0) -P

The Payoff and profit of the seller of the put option are exactly opposite

Profit = P -max(K-St, 0)

Using the above option values C = $3.06, K = $30 and various possible values of St, the payoff of Seller and Buyer are as shown in the table and graph below

St	Profit of Seller	Profit of Buyer	Total Profit of Buyer and Seller
25	-4.39	4.39	0
26	-3.39	3.39	0
27	-2.39	2.39	0
28	-1.39	1.39	0
29	-0.39	0.39	0
30	0.61	-0.61	0
31	0.61	-0.61	0
32	0.61	-0.61	0
33	0.61	-0.61	0
34	0.61	-0.61	0
35	0.61	-0.61	0

It can be seen clearly that the put option buyer and Seller has exactly negative profits. Sum of profits is 0. So the Put option pricing is a zero sum game,