Question

4. Consider a portfolio consisting of 6 stocks and 10 put options on the stocks. The current stock price is So = 100 and the stike price of the options is X = 100. The stock price can take on only two values at maturity T given by Su = 120 and Sd = 90. The risk-free rate is given by 5%. (a) What is the payoff at maturity of the portfolio? (b) Calculate the "Delta" of the portfolio which is defined as the price change of the put over the price change of the stock, or A = P.-PAL (c) How does your answer change in (a) and (b) when considering a put option with strike price X = 110? (d) How many put options with strike price X = 110 are necessary in the portfolio to have a certain payoff at maturity? (e) Find the value of a put option Po with strike price X = 100. (f) Find the value of a call option Co with X = 100 by using the Put-Call-Parity. Verify that your answer by calculating the price Co directly.

please type your answer

Answer 1

1. Pay off of put option is max of (X-S(T),0).

from the given information k is 100 and Spot price at maturity is 120 so S(T) is > X in this case pay off is "0".

for another spot value S(T) is 90 and X is 100 In this case payoff is X-S(T) = 100-90 = 10

Where X is Exercise price and S(T) is the spot price at maturity

2. Delta is

Change in value of option / change in future spot price

from the given data Future spot price FP1= Su = 120

FP2 = Sd =90

Where Pu is FP1- X= 120-100= 20

Pd is FP2-X= 90-100=0

Delta is Pu-Pd/ Su-Sd= 20-0/120-90 =20/30

.67

C) Put option with strike price X= 110

Payoff at S(T) is 120

S(T) is > X . 120>110 so pay off is 0

Payoff at S(T) is 90

In this case S(T) is < X

Payoff is X-S(T)= 110-90=20

Delta @ X= 110

Su & Sd is same as 120& 90

Pu  is FP1 - X= 120-110 = 10

Pd is FP2-X = 90-110 =0

So Delta is 10-0/120-90 = 10/30 = .33

D) No of put options

1/options Delta = 1/.33 = 3 puts per stock .

no of stocks is 6 and total 18 put options for 6 shares held.

E) value of put option:

Option delta x ( amount to be invested at risk free rate- current stock price)

= .67 x ( 90 x e^-rt) = .67 x ( 90 x e^.05xT)

maturity time is not given in the problem