Question

1. A stock is currently valued at 100, and next year it will have a value of 150 with probability 0.5 or a value of 70 with probability 0.5. The risk-free interest rate is 5%, and the average return on the market index is 10%. What do you expect the beta of the stock is?

2. In the example above, what are the risk neutral probabilities?

3. Use the risk neutral probabilities to value a call option with exercise price 100, which matures next year.

4. Use put-call parity to value the corresponding put option (i.e. exercise price is also 100, and it matures next year).

Answer 1

Q1) What is the beta of stock?

Expected return = 0.5*150 + 0.5*70

= 110

Return = (110-100)/100 = 10%

Using CAPM

Expected return = Risk free rate + Beta * ( market return - risk free rate)

10% = 5% + Beta * (10% - 5%)

Beta = 1

Q2) Calculating risk neutral probability?

Let u=150, d=70, r=5%, S=100

risk neutral probability = p

S*(1 + r)^T = p*u + (1-p)*d

100 * 1.05 = p*150 + (1-p)*70

p = 0.44

Q3) Valuing Call option which matures

next

year?

C

all option will get exercised if the value of the stock is 150

value of call option today is V(0)=e^-rt*p*(150-100)

V(0) = e^-0.05*1*0.44*50

V(0) = 20.93

Q4) Put Call Parity?

C + PV(X) = P + S

where C = Callprice, PV(X) = Present value of strike price X, P = put price, S = Spot price

C = 20.93

S = 100

PV(X) = 100*e^-rt = 95.12

Solving, we get the value of Put P = 16.05