Problem 1. Use the put-call parity relationship to solve this problem. The stock

Question

Problem 1. Use the put-call parity relationship to solve this problem. The stock does not pay any dividend. a) Show how one can replicate a one-year zero-coupon Treasury bond with a face value of S100 using a share of stock, a put and a call. b) Suppose that the current stock price S S100, the price of a European put on the stock p-$15, and the price of a European call on the stock c = $35 (strike prices of put and call $100). What must be the one-year risk-free interest rate? Show that if the one-year risk-free interest rate is lower than in your answer to part b), there would be an arbitrage opportunity. (Hint: use what you find in part a) and notice that the price of the zero-coupon bond would be too high.) c)

Explanation / Answer

a. put call parity exists when:

a one year zero coupon treasury bond is contructed by buying a stock, buying one put option and selling another call option with same exercise price.

Then:

Lets assume:

example is taken to derive exercise price with a risk free rate of return of 5% with time to expiration of 1 year.

if 200 is invested in a risk free treasury bill for 1 year with 5% rate of interest, it yields 200*5%= 10 in a year.

if put call parity holds good, it should also yield 10 in a year.

Lets take below scenarios:

in every scenario, return on stock is 10 which equals the return given by zero coupon bond or any risk free bond.

b. substituting the given variables in the above equation gives

80+15-35= 100/ (1+r)

60 = 100/(1+r)

1+r= 100/60= 1.66667

r= 66.67%

risk free interest rate= 66.67%

c. zero coupon bond maturing in one year with 100 payoff and 60 price today is having interest rate of 66.67% as per b.

if the interest rate is 50% instead of 66.67% the price of bond will be= 100/ (1+50%)= 66.67 which is higher than price in b. 60

Arbitrage:

Net investment required today to buy stock plus one put option and sell one call option with same exercie price of 100 is= 80+15-35= 60

the amount of 60 borrowed at risk free rate of 50% and repaid on maturity.

and the payoff on stock will be at any scenario:

the payoff is 40 and total investment is worth= 100

now the loan to be repaid at 50% interest rate= 60 * 50% + 60= 90

Net profit due to arbitrage= 100-90= 10

This amount of 10 is earned without any risk and capital investment.

S0+p0 = X/(1+r)T + c0

Related Questions

Navigate

• Browse (All)
• Browse P
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.