homework 2 FNCE 4040

Put call parity

C+ Ke^-rT= p+S0

The price of a non-dividend paying stock is $19 and the price of a three-month European call option on the stock with a strike price of $20 is $1. The risk-free rate is 4% per annum. What is the price of a three-month European put option with a strike price of $20?

* put call parity problem* PCP= C+Ke^(-rT)= P+S0 $1+$20e^(-.04*(3/12))= p+$19 p= $1.80

What is meant by a protective put?

A long position in the stock and a long position in the put

What position in call options is equivalent to a protective put?

A long position in a call option plus a long position in a bond????

Which of the following is not a reason for not exercising an American call option on a non-dividend-paying stock early (yes, there are two "not's" in this question, sorry).

The stock price will surely increase more

A stock price is currently $40. It is known that at the end of one month it will be either $42 or $38. The risk-free interest rate is 8% per annum with continuous compounding. What is the value of a one-month European call option with a strike price of $39?

***binomial Trees** 1) set the equation for delta where you long delta shares and short 1 call option --> if stock rises to 42, value of call will be (42-39) =$3 --> if stock falls to 38, value of call will be $0 Therefore--> 42(Delta)-3= 38(Delta) delta= .75 -->plug delta back in = value of the portfolio= $28.5 2) find the PV of the portfolio(FVe^-rt) --> 28.5e^(-0.08% X 1/12)= $28.31 3)calculate the value of the call option S0 X Delta - F= PV of portfolio $40* .75- F= $28.31 F= $1.69---> value of one month euro call option

A stock price is currently $50. It is known that at the end of six months it will be either $45 or $55. The risk-free interest rate is 10% per annum with continuous compounding. What is the value of a six-month European put option with a strike price of $50?

***binomial Trees** 1) set the equation for delta where you long delta shares and short 1 put option --> if stock rises to 55, value of put will be $0 --> if stock falls to 45, value of put will be (50-45)= $5 Therefore--> 55(Delta)= 45(Delta)-5 delta= -0.5 -->plug delta back in = value of the portfolio= -$27.5 2) find the PV of the portfolio(FVe^-rt) --> -27.5e^(-0.1% X 6/12)= -$26.16 3)calculate the value of the call option S0 X Delta - F= PV of portfolio $50* -.5- F= -$26.16 F= $1.16---> value of 6 month euro put option

(Based on Problem 11.14, Revised for Canvas) The price of a European call that expires in six months and has a strike price of $30 is $2. The underlying stock price is $29, and a dividend of $0.50 is expected in two months and again in five months. Interest rates (all maturities) are 10%. What is the price of a European put option that expires in six months and has a strike price of $30?

*put call parity with dividends* Strike price = $30 underlying stock price = $29 value of one european call option share = $2 maturity = 6 months , 2 months , 5 months Divident =$ 0.50 Risk free rate = 10% Formula of put option price = = Value of one european call option - underlying stock price + ( strike price * er t) + divident amount = 2 - 29 +(30 * e-10 * 6/12) + (0.50 * e-10 * 2/12) + (0.50 * e-10 * 5/12) = 2 - 29 + (30 * 0.9512) + (0.50 * 0.9834) + (0.50 * 0.9591) = - 27 + 28.536 + 0.4917 + 0.4795 = 2.51 Put option price = 2.51

A four-month European call option on a dividend-paying stock is currently selling for $5. The stock price is $64, the strike price is $60, and a dividend of $0.80 is expected in one month. The risk-free interest rate is 12% per annum for all maturities. What opportunities are there for an arbitrageur (solve for minimum arbitrage profits in present value terms)?

Bounds on European call option: S0 > c > S0 − P V (D) − Ke−rT P V (D) = Present Value of Dividends (if any) S0= $64 PV(D)= $.80e^(-.12*(1/12))= $0.792 K=$60 r= 0.12% T= 3/12 $5 != 66-.792-60e^(-0.12*(3/12)) $5!= $5.56 thus, arbitrage opp of 5.56-5= $0.56 short stock at $64, then in 4 months, but a call for price of $60, plus another call option for a dividend paying stock for5$--> net gain on arbitrage will be $5.56-$5

Suppose that put options on a stock with strike prices $30 and $35 cost $4 and $7, respectively. Construct a table that shows the profit and payoff for a bear spread. Note, if your answer for one of the blanks involves the stock price (S T), use the letter S. For example, if your answer is 15 − S T, type "15-S"

bearish= pessimistic profit= payoff-initial investment 1) long/buy put option for 35--> A 2)short/sell put option for 30--> B initial investment= (-7 from long option) + (+4 from short option)= -3 1) if St>=35--> Payoff= A(0 bc after strike price is becomes valueless) + B(0 bc after strike price is becomes valueless)= 0 Profit= 0-3= -3 2) if 30<=St<= 35--> Payoff= A(35-st because when long put it has value before hitting the strike) + B(0 bc after strike price is becomes valueless) Profit= 35-st-3 3) if St<30--> Payoff= A(35-st because when long put it has value before hitting the strike ) + B(St-30 because its below the strike price)-->5 Profit= 5-3= 2

Suppose that put options on a stock with strike prices $30 and $35 cost $4 and $7, respectively. Construct a table that shows the profit and payoff for a bull spread. Note, if your answer for one of the blanks involves the stock price (S T), use the letter S. For example, if your answer is 15 − S T, type "15-S"

bull spread = optimistic--> move from right to left profit= payoff-initial investment 2)long/buy put option for 30(k1)--> A 1) short/sell put option for 35(k2)--> B initial investment= (+ 7 from shorting a put) (-4 for buying a put option)= +3 inflow 1) if St>=35--> Payoff= A(0 bc at K1 or higher option becomes worthless) + B(0 bc at K2 or higher option becomes worthless) Profit= $0 payoff + $3 premium for short put 2) if 30<=St<= 35--> Payoff= A(0 because worthless above K1) + B( S-35 because loss until stock price equals strike or higher) Profit= s-35 loss +3 premium from short put 3) if St<30--> Payoff= A(+30-st because profit until st is K or above ) + B(S-35 because loss until stock price equals strike or higher)--> 30-35= -5 Profit= -5+ 3 premium from short put