Hi can you please help with this ASAP? I would immensely appreciate it. I will attach the file itself also.

For each of the following security combinations draw a graph to show what the payoff would be when the option expires as a function of the stock price at expiration. For parts (a) through (c), assume that the strike price is $40

buy a share and sell a call option on the share.

sell a put and buy a Treasury bill with face value $50 maturing on the exercise date.

buy a call option and a put option on the share.

buy a put option with a strike price of $50 and sell a put option with a strike price of $30.

On November 1, 2005, Home Depot stock closed at 40. You forecast that six months from that date Home Depot stock price will be either $55 or $30. If the stock price rises to $55, then on November 1, 2006 the price will be either $70 or $45. If the stock price falls to $30, then on November 1, 2006 the price will be either $45 or $20. Assume that the six month risk-free rate of return is 2%.

Use the Single-period BOPM to determine the value of a European call option on Home Depot with the exercise price of $40 that expires on May 1, 2006.

Use the Two-period BOPM to determine the value of a European call option on Home Depot with the exercise price of $40 that expires on November 1, 2006. Comparing the two prices, what do you observe?

Assume that today is 1-Nov-05. Determine as many points as possible in today's yield curve (or term structure of interest rates), given that you know that there are three risk-free bonds with the following characteristics:

A zero-coupon bond (pure discount) that matures on 1-May-07 is selling today at 98:02.

A 6% bond (that is, pays semi-annually 3% of its face value as coupon) that matures on 1-May-07 is selling today at 106:29. The coupon is paid out twice a year on 1-May 1^st and November 1st.

An 8% bond (face value $100) that matures on 1-May-06 is selling today at $102.4.

Suppose, there is a 5% bond (face value $100) that will mature on 1-Nov-2006, find the price for this bond. The coupon is paid out twice a year on May 1st and November 1^st.

Assume today is 1-Nov-05. A risk-free zero-coupon (pure discount) bond maturing on 1-Nov-06 is selling today at 94:00. The forward rate for the period 1-NOv-06 to 1-May-07 is 7%. Compute the price of a pure discount (zero-coupon) bond maturing on 1-May-07?

1. For each of the following security combinations draw a graph to show what the payoff would be when the option expires as a function of the stock price at expiration. For parts (a) through (c), assume that the strike price is $40 (a) (b) buy a share and sell a call option on the share. sell a put and buy a Treasury bill with face value $50 maturing on the exercise date. buy a call option and a put option on the share. buy a put option with a strike price of $50 and sell a put option with a strike price of $30. (c) (d) 2. On November 1, 2005, Home Depot stock closed at 40. You forecast that six months from that date Home Depot stock price will be either $55 or $30. If the stock price rises to $55, then on November 1, 2006 the price will be either $70 or $45. If the stock price falls to $30, then on November 1, 2006 the price will be either $45 or $20. Assume that the six month risk-free rate of return is 2%. a. Use the Single-period BOPM to determine the value of a European call option on Home Depot with the exercise price of $40 that expires on May 1, 2006. b. Use the Two-period BOPM to determine the value of a European call option on Home Depot with the exercise price of $40 that expires on November 1, 2006. Comparing the two prices, what do you observe? 3. Assume that today is 1-Nov-05. Determine as many points as possible in today's yield curve (or term structure of interest rates), given that you know that there are three riskfree bonds with the following characteristics: (1) A zero-coupon bond (pure discount) that matures on 1-May-07 is selling today at 98:02. (2) A 6% bond (that is, pays semi-annually 3% of its face value as coupon) that matures on 1-May-07 is selling today at 106:29. The coupon is paid out twice a year on 1-May 1st and November 1st. (3) An 8% bond (face value $100) that matures on 1-May-06 is selling today at $102.4. Suppose, there is a 5% bond (face value $100) that will mature on 1-Nov-2006, find the price for this bond. The coupon is paid out twice a year on May 1st and November 1st. 4. Assume today is 1-Nov-05. A risk-free zero-coupon (pure discount) bond maturing on 1-Nov-06 is selling today at 94:00. The forward rate for the period 1-NOv-06 to 1May-07 is 7%. Compute the price of a pure discount (zero-coupon) bond maturing on 1-May-07? This sheet answers question 1a Payoff for buying a share = S Payoff for selling a call = -max(S-K,0) Inputs Stock price at expiration, S_T $30 $31 $32 $33 $34 $35 $36 $37 $38 $39 $40 $41 $42 $43 $44 $45 $46 $47 $48 $49 $50 Position Strike Long n/a Short $40.00 long s $30 $31 $32 $33 $34 $35 $36 $37 $38 $39 $40 $41 $42 $43 $44 $45 $46 $47 $48 $49 $50 Payoff short c $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 -$1 -$2 -$3 -$4 -$5 -$6 -$7 -$8 -$9 -$10 Total $30 $31 $32 $33 $34 $35 $36 $37 $38 $39 $40 $40 $40 $40 $40 $40 $40 $40 $40 $40 $40 Payoff Diagram $ 60 lon gs sho rt c $ 50 Payoff Instrument Share Call $ 40 $ 30 $ 20 $ 10 $0 $ 30 -$ 10 -$ 20 $ 32 $ 34 $ 36 $ 38 $ 40 Stock Price $ 42 $ 44 $ 46 $ 48 $ 50 This sheet answers question 1b Payoff for selling put = -max(K-S,0) Payoff for buying T-Bill = $50 Inputs Stock price at expiration, S_T $30 $31 $32 $33 $34 $35 $36 $37 $38 $39 $40 $41 $42 $43 $44 $45 $46 $47 $48 $49 $50 Position Strike Short $40.00 Long n/a short p -$10 -$9 -$8 -$7 -$6 -$5 -$4 -$3 -$2 -$1 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 Payoff T-bill $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 Total $40 $41 $42 $43 $44 $45 $46 $47 $48 $49 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 Payoff Diagram $ 60 sho rt p Tbill Total $ 50 Payoff Instrument Put Treasury bill $ 40 $ 30 $ 20 $ 10 $0 $ 30 -$ 10 -$ 20 $ 32 $ 34 $ 36 $ 38 $ 40 Stock Price $ 42 $ 44 $ 46 $ 48 $ 50 This sheet answers question 1c Payoff for buying call = max(S-K,0) Payoff for buying put = max(K-S,0) Inputs Stock price at expiration, S_T $30 $31 $32 $33 $34 $35 $36 $37 $38 $39 $40 $41 $42 $43 $44 $45 $46 $47 $48 $49 $50 Position Strike Long $40.00 Long $40.00 long c $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $1 $2 $3 $4 $5 $6 $7 $8 $9 $10 Payoff long p $10 $9 $8 $7 $6 $5 $4 $3 $2 $1 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 Total $10 $9 $8 $7 $6 $5 $4 $3 $2 $1 $0 $1 $2 $3 $4 $5 $6 $7 $8 $9 $10 Payoff Diagram $ 12 lon gc lon gp Total $ 10 Payoff Instrument Call (c ) Put (p ) $8 $6 $4 $2 $0 $ 30 $ 32 $ 34 $ 36 Stock Price $ 38 $ 40 $ 42 $ 44 $ 46 $ 48 $ 50 This sheet answers question 1d Payoff for buying put = max(K-S,0) Payoff for selling put = -max(K-S,0) Inputs Stock price at expiration, S_T $20 $21 $22 $23 $24 $25 $26 $27 $28 $29 $30 $31 $32 $33 $34 $35 $36 $37 $38 $39 $40 $41 $42 $43 $44 $45 $46 $47 $48 $49 $50 $51 $52 $53 $54 $55 Position Strike Long $50.00 Short $30.00 Payoff long p1 short p2 $30 -$10 $29 -$9 $28 -$8 $27 -$7 $26 -$6 $25 -$5 $24 -$4 $23 -$3 $22 -$2 $21 -$1 $20 $0 $19 $0 $18 $0 $17 $0 $16 $0 $15 $0 $14 $0 $13 $0 $12 $0 $11 $0 $10 $0 $9 $0 $8 $0 $7 $0 $6 $0 $5 $0 $4 $0 $3 $0 $2 $0 $1 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 Total $20 $20 $20 $20 $20 $20 $20 $20 $20 $20 $20 $19 $18 $17 $16 $15 $14 $13 $12 $11 $10 $9 $8 $7 $6 $5 $4 $3 $2 $1 $0 $0 $0 $0 $0 $0 Payoff Diagram $ 35 lon g p1 sho rt p2 $ 30 $ 25 Payoff Instrument Put (p1) Put (p2) $ 20 $ 15 $ 10 $5 $0 $,20 -$ 5 $,25 $,30 $,35 $,40 -$ 10 -$ 15 Stock Price $,45 $,50 $,55 This sheet answers question 1a Payoff for buying a share = S Payoff for selling a call = -max(S-K,0) Inputs Stock price at expiration, S_T $30 $31 $32 $33 $34 $35 $36 $37 $38 $39 $40 $41 $42 $43 $44 $45 $46 $47 $48 $49 $50 Position Strike Long n/a Short $40.00 long s $30 $31 $32 $33 $34 $35 $36 $37 $38 $39 $40 $41 $42 $43 $44 $45 $46 $47 $48 $49 $50 Payoff short c $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 -$1 -$2 -$3 -$4 -$5 -$6 -$7 -$8 -$9 -$10 Total $30 $31 $32 $33 $34 $35 $36 $37 $38 $39 $40 $40 $40 $40 $40 $40 $40 $40 $40 $40 $40 Payoff Diagram $ 60 lon gs sho rt c $ 50 Payoff Instrument Share Call $ 40 $ 30 $ 20 $ 10 $0 $ 30 -$ 10 -$ 20 $ 32 $ 34 $ 36 $ 38 $ 40 Stock Price $ 42 $ 44 $ 46 $ 48 $ 50 This sheet answers question 1b Payoff for selling put = -max(K-S,0) Payoff for buying T-Bill = $50 Inputs Stock price at expiration, S_T $30 $31 $32 $33 $34 $35 $36 $37 $38 $39 $40 $41 $42 $43 $44 $45 $46 $47 $48 $49 $50 Position Strike Short $40.00 Long n/a short p -$10 -$9 -$8 -$7 -$6 -$5 -$4 -$3 -$2 -$1 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 Payoff T-bill $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 Total $40 $41 $42 $43 $44 $45 $46 $47 $48 $49 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 Payoff Diagram $ 60 sho rt p Tbill Total $ 50 Payoff Instrument Put Treasury bill $ 40 $ 30 $ 20 $ 10 $0 $ 30 -$ 10 -$ 20 $ 32 $ 34 $ 36 $ 38 $ 40 Stock Price $ 42 $ 44 $ 46 $ 48 $ 50 This sheet answers question 1c Payoff for buying call = max(S-K,0) Payoff for buying put = max(K-S,0) Inputs Stock price at expiration, S_T $30 $31 $32 $33 $34 $35 $36 $37 $38 $39 $40 $41 $42 $43 $44 $45 $46 $47 $48 $49 $50 Position Strike Long $40.00 Long $40.00 long c $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $1 $2 $3 $4 $5 $6 $7 $8 $9 $10 Payoff long p $10 $9 $8 $7 $6 $5 $4 $3 $2 $1 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 Total $10 $9 $8 $7 $6 $5 $4 $3 $2 $1 $0 $1 $2 $3 $4 $5 $6 $7 $8 $9 $10 Payoff Diagram $ 12 lon gc lon gp Total $ 10 Payoff Instrument Call (c ) Put (p ) $8 $6 $4 $2 $0 $ 30 $ 32 $ 34 $ 36 Stock Price $ 38 $ 40 $ 42 $ 44 $ 46 $ 48 $ 50 This sheet answers question 1d Payoff for buying put = max(K-S,0) Payoff for selling put = -max(K-S,0) Inputs Stock price at expiration, S_T $20 $21 $22 $23 $24 $25 $26 $27 $28 $29 $30 $31 $32 $33 $34 $35 $36 $37 $38 $39 $40 $41 $42 $43 $44 $45 $46 $47 $48 $49 $50 $51 $52 $53 $54 $55 Position Strike Long $50.00 Short $30.00 Payoff long p1 short p2 $30 -$10 $29 -$9 $28 -$8 $27 -$7 $26 -$6 $25 -$5 $24 -$4 $23 -$3 $22 -$2 $21 -$1 $20 $0 $19 $0 $18 $0 $17 $0 $16 $0 $15 $0 $14 $0 $13 $0 $12 $0 $11 $0 $10 $0 $9 $0 $8 $0 $7 $0 $6 $0 $5 $0 $4 $0 $3 $0 $2 $0 $1 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 Total $20 $20 $20 $20 $20 $20 $20 $20 $20 $20 $20 $19 $18 $17 $16 $15 $14 $13 $12 $11 $10 $9 $8 $7 $6 $5 $4 $3 $2 $1 $0 $0 $0 $0 $0 $0 Payoff Diagram $ 35 lon g p1 sho rt p2 $ 30 $ 25 Payoff Instrument Put (p1) Put (p2) $ 20 $ 15 $ 10 $5 $0 $,20 -$ 5 $,25 $,30 $,35 $,40 -$ 10 -$ 15 Stock Price $,45 $,50 $,55 This sheet answers question 2. Inputs Interest Stock price u1 d1 u2 d2 u3 d3 Exercise price q1 q2 q3 2% six-month risk free rate. $40.00 November 1, 2005. 1.38 period 1 up factor. 0.75 period 1 down factor. 1.27 period 2 up factor if share price increases in period 1. 0.82 period 2 down factor if share price increases in period 1. 1.50 period 2 up factor if share price decreases in period 1. 0.67 period 2 down factor if share price decreases in period 1. $40.00 0.43 risk-neutral probability over the first period. 0.44 risk-neutral probability over the second period, if the share price increases in the first period. 0.42 risk-neutral probability over the second period, if the share price decreases in the first period. Stock price lattice Period Stock price 0 1 2 $40.00 $55.00 $30.00 $70.00 $45.00 $20.00 a) Single period BOPM Period Call 0 1 $6.35 $15.00 $0.00 Payoff in period 1 = max(S-K,0). Call premium in period 0 = (q1*$15+(1-q1)*0)/(1+interest) b) Two-period BOPM Period 0 1 2 $30.00 $15.78 $5.00 Call $7.84 $5.00 $2.08 $0.00 Payoff in period 2 = max(S-K,0). Call premium in period 0 = (q1*$15.78+(1-q1)*2.08)/(1+interest) Value of call in period 1 if share price is 55 = (q2*$30.00+(1-q2)*5.00)/(1+interest) Value of call in period 1 if share price is 30 = (q3*$5.00+(1-q3)*0)/(1+interest) Comparing the two option premiums The call option with a longer time to maturity costs more. This is intuitive since a longer time to maturity gives the share price greater scope to increase, thereby resulting in a higher call payoff. This corresponds to a higher call premium given that losses on a call option are capped. This sheet answers question 1a Payoff for buying a share = S Payoff for selling a call = -max(S-K,0) Inputs Stock price at expiration, S_T $30 $31 $32 $33 $34 $35 $36 $37 $38 $39 $40 $41 $42 $43 $44 $45 $46 $47 $48 $49 $50 Position Strike Long n/a Short $40.00 long s $30 $31 $32 $33 $34 $35 $36 $37 $38 $39 $40 $41 $42 $43 $44 $45 $46 $47 $48 $49 $50 Payoff short c $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 -$1 -$2 -$3 -$4 -$5 -$6 -$7 -$8 -$9 -$10 Total $30 $31 $32 $33 $34 $35 $36 $37 $38 $39 $40 $40 $40 $40 $40 $40 $40 $40 $40 $40 $40 Payoff Diagram $ 60 lon gs sho rt c $ 50 Payoff Instrument Share Call $ 40 $ 30 $ 20 $ 10 $0 $ 30 -$ 10 -$ 20 $ 32 $ 34 $ 36 $ 38 $ 40 Stock Price $ 42 $ 44 $ 46 $ 48 $ 50 This sheet answers question 1b Payoff for selling put = -max(K-S,0) Payoff for buying T-Bill = $50 Inputs Stock price at expiration, S_T $30 $31 $32 $33 $34 $35 $36 $37 $38 $39 $40 $41 $42 $43 $44 $45 $46 $47 $48 $49 $50 Position Strike Short $40.00 Long n/a short p -$10 -$9 -$8 -$7 -$6 -$5 -$4 -$3 -$2 -$1 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 Payoff T-bill $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 Total $40 $41 $42 $43 $44 $45 $46 $47 $48 $49 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 Payoff Diagram $ 60 sho rt p Tbill Total $ 50 Payoff Instrument Put Treasury bill $ 40 $ 30 $ 20 $ 10 $0 $ 30 -$ 10 -$ 20 $ 32 $ 34 $ 36 $ 38 $ 40 Stock Price $ 42 $ 44 $ 46 $ 48 $ 50 This sheet answers question 1c Payoff for buying call = max(S-K,0) Payoff for buying put = max(K-S,0) Inputs Stock price at expiration, S_T $30 $31 $32 $33 $34 $35 $36 $37 $38 $39 $40 $41 $42 $43 $44 $45 $46 $47 $48 $49 $50 Position Strike Long $40.00 Long $40.00 long c $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $1 $2 $3 $4 $5 $6 $7 $8 $9 $10 Payoff long p $10 $9 $8 $7 $6 $5 $4 $3 $2 $1 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 Total $10 $9 $8 $7 $6 $5 $4 $3 $2 $1 $0 $1 $2 $3 $4 $5 $6 $7 $8 $9 $10 Payoff Diagram $ 12 lon gc lon gp Total $ 10 Payoff Instrument Call (c ) Put (p ) $8 $6 $4 $2 $0 $ 30 $ 32 $ 34 $ 36 Stock Price $ 38 $ 40 $ 42 $ 44 $ 46 $ 48 $ 50 This sheet answers question 1d Payoff for buying put = max(K-S,0) Payoff for selling put = -max(K-S,0) Inputs Stock price at expiration, S_T $20 $21 $22 $23 $24 $25 $26 $27 $28 $29 $30 $31 $32 $33 $34 $35 $36 $37 $38 $39 $40 $41 $42 $43 $44 $45 $46 $47 $48 $49 $50 $51 $52 $53 $54 $55 Position Strike Long $50.00 Short $30.00 Payoff long p1 short p2 $30 -$10 $29 -$9 $28 -$8 $27 -$7 $26 -$6 $25 -$5 $24 -$4 $23 -$3 $22 -$2 $21 -$1 $20 $0 $19 $0 $18 $0 $17 $0 $16 $0 $15 $0 $14 $0 $13 $0 $12 $0 $11 $0 $10 $0 $9 $0 $8 $0 $7 $0 $6 $0 $5 $0 $4 $0 $3 $0 $2 $0 $1 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 Total $20 $20 $20 $20 $20 $20 $20 $20 $20 $20 $20 $19 $18 $17 $16 $15 $14 $13 $12 $11 $10 $9 $8 $7 $6 $5 $4 $3 $2 $1 $0 $0 $0 $0 $0 $0 Payoff Diagram $ 35 lon g p1 sho rt p2 $ 30 $ 25 Payoff Instrument Put (p1) Put (p2) $ 20 $ 15 $ 10 $5 $0 $,20 -$ 5 $,25 $,30 $,35 $,40 -$ 10 -$ 15 Stock Price $,45 $,50 $,55 This sheet answers question 2. Inputs Interest Stock price u1 d1 u2 d2 u3 d3 Exercise price q1 q2 q3 2% six-month risk free rate. $40.00 November 1, 2005. 1.38 period 1 up factor. 0.75 period 1 down factor. 1.27 period 2 up factor if share price increases in period 1. 0.82 period 2 down factor if share price increases in period 1. 1.50 period 2 up factor if share price decreases in period 1. 0.67 period 2 down factor if share price decreases in period 1. $40.00 0.43 risk-neutral probability over the first period. 0.44 risk-neutral probability over the second period, if the share price increases in the first period. 0.42 risk-neutral probability over the second period, if the share price decreases in the first period. Stock price lattice Period Stock price 0 1 2 $40.00 $55.00 $30.00 $70.00 $45.00 $20.00 a) Single period BOPM Period Call 0 1 $6.35 $15.00 $0.00 Payoff in period 1 = max(S-K,0). Call premium in period 0 = (q1*$15+(1-q1)*0)/(1+interest) b) Two-period BOPM Period 0 1 2 $30.00 $15.78 $5.00 Call $7.84 $5.00 $2.08 $0.00 Payoff in period 2 = max(S-K,0). Call premium in period 0 = (q1*$15.78+(1-q1)*2.08)/(1+interest) Value of call in period 1 if share price is 55 = (q2*$30.00+(1-q2)*5.00)/(1+interest) Value of call in period 1 if share price is 30 = (q3*$5.00+(1-q3)*0)/(1+interest) Comparing the two option premiums The call option with a longer time to maturity costs more. This is intuitive since a longer time to maturity gives the share price greater scope to increase, thereby resulting in a higher call payoff. This corresponds to a higher call premium given that losses on a call option are capped. This sheet answers question 4 Today Maturity Bond price Forward maturity date Forward rate Bond par value 1-Nov-05 1-Nov-06 $94.00 1-May-07 7% I assumed that the forward rate provided is a continuously compounded annual rate. Note, the compounding frequ $100.00 Price of Zero Coupon Bond (ZCB) maturing in 1-May-2007 Price $90.77 =Par value x exp(-forward rate x 0.5) x (price of bond maturing on 1 Nov 06)/Par value. Note - The discount factor over the period 1 Nov 05 to 1 Nov 06 = (price of bond maturing on 1 Nov 06)/Par value. - The discount factor over the period 1 Nov 06 to 1 May 07 = exp(-forward rate x 0.5) equency is not stated in the question. This sheet answers question 1a Payoff for buying a share = S Payoff for selling a call = -max(S-K,0) Inputs Stock price at expiration, S_T $30 $31 $32 $33 $34 $35 $36 $37 $38 $39 $40 $41 $42 $43 $44 $45 $46 $47 $48 $49 $50 Position Strike Long n/a Short $40.00 long s $30 $31 $32 $33 $34 $35 $36 $37 $38 $39 $40 $41 $42 $43 $44 $45 $46 $47 $48 $49 $50 Payoff short c $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 -$1 -$2 -$3 -$4 -$5 -$6 -$7 -$8 -$9 -$10 Total $30 $31 $32 $33 $34 $35 $36 $37 $38 $39 $40 $40 $40 $40 $40 $40 $40 $40 $40 $40 $40 Payoff Diagram $ 60 lon gs sho rt c $ 50 Payoff Instrument Share Call $ 40 $ 30 $ 20 $ 10 $0 $ 30 -$ 10 -$ 20 $ 32 $ 34 $ 36 $ 38 $ 40 Stock Price $ 42 $ 44 $ 46 $ 48 $ 50 This sheet answers question 1b Payoff for selling put = -max(K-S,0) Payoff for buying T-Bill = $50 Inputs Stock price at expiration, S_T $30 $31 $32 $33 $34 $35 $36 $37 $38 $39 $40 $41 $42 $43 $44 $45 $46 $47 $48 $49 $50 Position Strike Short $40.00 Long n/a short p -$10 -$9 -$8 -$7 -$6 -$5 -$4 -$3 -$2 -$1 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 Payoff T-bill $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 Total $40 $41 $42 $43 $44 $45 $46 $47 $48 $49 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 Payoff Diagram $ 60 sho rt p Tbill Total $ 50 Payoff Instrument Put Treasury bill $ 40 $ 30 $ 20 $ 10 $0 $ 30 -$ 10 -$ 20 $ 32 $ 34 $ 36 $ 38 $ 40 Stock Price $ 42 $ 44 $ 46 $ 48 $ 50 This sheet answers question 1c Payoff for buying call = max(S-K,0) Payoff for buying put = max(K-S,0) Inputs Stock price at expiration, S_T $30 $31 $32 $33 $34 $35 $36 $37 $38 $39 $40 $41 $42 $43 $44 $45 $46 $47 $48 $49 $50 Position Strike Long $40.00 Long $40.00 long c $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $1 $2 $3 $4 $5 $6 $7 $8 $9 $10 Payoff long p $10 $9 $8 $7 $6 $5 $4 $3 $2 $1 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 Total $10 $9 $8 $7 $6 $5 $4 $3 $2 $1 $0 $1 $2 $3 $4 $5 $6 $7 $8 $9 $10 Payoff Diagram $ 12 lon gc lon gp Total $ 10 Payoff Instrument Call (c ) Put (p ) $8 $6 $4 $2 $0 $ 30 $ 32 $ 34 $ 36 Stock Price $ 38 $ 40 $ 42 $ 44 $ 46 $ 48 $ 50 This sheet answers question 1d Payoff for buying put = max(K-S,0) Payoff for selling put = -max(K-S,0) Inputs Stock price at expiration, S_T $20 $21 $22 $23 $24 $25 $26 $27 $28 $29 $30 $31 $32 $33 $34 $35 $36 $37 $38 $39 $40 $41 $42 $43 $44 $45 $46 $47 $48 $49 $50 $51 $52 $53 $54 $55 Position Strike Long $50.00 Short $30.00 Payoff long p1 short p2 $30 -$10 $29 -$9 $28 -$8 $27 -$7 $26 -$6 $25 -$5 $24 -$4 $23 -$3 $22 -$2 $21 -$1 $20 $0 $19 $0 $18 $0 $17 $0 $16 $0 $15 $0 $14 $0 $13 $0 $12 $0 $11 $0 $10 $0 $9 $0 $8 $0 $7 $0 $6 $0 $5 $0 $4 $0 $3 $0 $2 $0 $1 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 Total $20 $20 $20 $20 $20 $20 $20 $20 $20 $20 $20 $19 $18 $17 $16 $15 $14 $13 $12 $11 $10 $9 $8 $7 $6 $5 $4 $3 $2 $1 $0 $0 $0 $0 $0 $0 Payoff Diagram $ 35 lon g p1 sho rt p2 $ 30 $ 25 Payoff Instrument Put (p1) Put (p2) $ 20 $ 15 $ 10 $5 $0 $,20 -$ 5 $,25 $,30 $,35 $,40 -$ 10 -$ 15 Stock Price $,45 $,50 $,55 This sheet answers question 2. Inputs Interest Stock price u1 d1 u2 d2 u3 d3 Exercise price q1 q2 q3 2% six-month risk free rate. $40.00 November 1, 2005. 1.38 period 1 up factor. 0.75 period 1 down factor. 1.27 period 2 up factor if share price increases in period 1. 0.82 period 2 down factor if share price increases in period 1. 1.50 period 2 up factor if share price decreases in period 1. 0.67 period 2 down factor if share price decreases in period 1. $40.00 0.43 risk-neutral probability over the first period. 0.44 risk-neutral probability over the second period, if the share price increases in the first period. 0.42 risk-neutral probability over the second period, if the share price decreases in the first period. Stock price lattice Period Stock price 0 1 2 $40.00 $55.00 $30.00 $70.00 $45.00 $20.00 a) Single period BOPM Period Call 0 1 $6.35 $15.00 $0.00 Payoff in period 1 = max(S-K,0). Call premium in period 0 = (q1*$15+(1-q1)*0)/(1+interest) b) Two-period BOPM Period 0 1 2 $30.00 $15.78 $5.00 Call $7.84 $5.00 $2.08 $0.00 Payoff in period 2 = max(S-K,0). Call premium in period 0 = (q1*$15.78+(1-q1)*2.08)/(1+interest) Value of call in period 1 if share price is 55 = (q2*$30.00+(1-q2)*5.00)/(1+interest) Value of call in period 1 if share price is 30 = (q3*$5.00+(1-q3)*0)/(1+interest) Comparing the two option premiums The call option with a longer time to maturity costs more. This is intuitive since a longer time to maturity gives the share price greater scope to increase, thereby resulting in a higher call payoff. This corresponds to a higher call premium given that losses on a call option are capped. This sheet answers question 3 In what follows, yields are calculated on a continuously compounded basis. (1) A zero-coupon bond (pure discount) that matures on 1-May-07 is selling today at 98:02. Bond price Time to maturity Par value 1.5 year Yield 98.06 1.50 years 100.00 maturity proceeds. 1.30% continuous compounded yearly rate. Note Yield = Log(maturity proceeds/bond price) / Time to maturity. (2) An 8% bond (face value $100) that matures on 1-May-06 is selling today at $102.4. Bond price Time to maturity Par value Coupon rate Coupon frequency Coupon amount Maturity proceeds 0.5 year Yield 102.40 I assume that the bond is quoted in decimals, unlike other bonds in this project. 0.50 years 100.00 8% 2 per year. 4.00 104.00 3.10% continuous compounded yearly rate. Note Yield = Log(maturity proceeds/bond price) / Time to maturity. This sheet answers question 4 Today Maturity Bond price Forward maturity date Forward rate Bond par value 1-Nov-05 1-Nov-06 $94.00 1-May-07 7% I assumed that the forward rate provided is a continuously compounded annual rate. Note, the compounding frequ $100.00 Price of Zero Coupon Bond (ZCB) maturing in 1-May-2007 Price $90.77 =Par value x exp(-forward rate x 0.5) x (price of bond maturing on 1 Nov 06)/Par value. Note - The discount factor over the period 1 Nov 05 to 1 Nov 06 = (price of bond maturing on 1 Nov 06)/Par value. - The discount factor over the period 1 Nov 06 to 1 May 07 = exp(-forward rate x 0.5) equency is not stated in the question. This sheet answers question 1a Payoff for buying a share = S Payoff for selling a call = -max(S-K,0) Inputs Stock price at expiration, S_T $30 $31 $32 $33 $34 $35 $36 $37 $38 $39 $40 $41 $42 $43 $44 $45 $46 $47 $48 $49 $50 Position Strike Long n/a Short $40.00 long s $30 $31 $32 $33 $34 $35 $36 $37 $38 $39 $40 $41 $42 $43 $44 $45 $46 $47 $48 $49 $50 Payoff short c $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 -$1 -$2 -$3 -$4 -$5 -$6 -$7 -$8 -$9 -$10 Total $30 $31 $32 $33 $34 $35 $36 $37 $38 $39 $40 $40 $40 $40 $40 $40 $40 $40 $40 $40 $40 Payoff Diagram $ 60 lon gs sho rt c $ 50 Payoff Instrument Share Call $ 40 $ 30 $ 20 $ 10 $0 $ 30 -$ 10 -$ 20 $ 32 $ 34 $ 36 $ 38 $ 40 Stock Price $ 42 $ 44 $ 46 $ 48 $ 50 This sheet answers question 1b Payoff for selling put = -max(K-S,0) Payoff for buying T-Bill = $50 Inputs Stock price at expiration, S_T $30 $31 $32 $33 $34 $35 $36 $37 $38 $39 $40 $41 $42 $43 $44 $45 $46 $47 $48 $49 $50 Position Strike Short $40.00 Long n/a short p -$10 -$9 -$8 -$7 -$6 -$5 -$4 -$3 -$2 -$1 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 Payoff T-bill $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 Total $40 $41 $42 $43 $44 $45 $46 $47 $48 $49 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 Payoff Diagram $ 60 sho rt p Tbill Total $ 50 Payoff Instrument Put Treasury bill $ 40 $ 30 $ 20 $ 10 $0 $ 30 -$ 10 -$ 20 $ 32 $ 34 $ 36 $ 38 $ 40 Stock Price $ 42 $ 44 $ 46 $ 48 $ 50 This sheet answers question 1c Payoff for buying call = max(S-K,0) Payoff for buying put = max(K-S,0) Inputs Stock price at expiration, S_T $30 $31 $32 $33 $34 $35 $36 $37 $38 $39 $40 $41 $42 $43 $44 $45 $46 $47 $48 $49 $50 Position Strike Long $40.00 Long $40.00 long c $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $1 $2 $3 $4 $5 $6 $7 $8 $9 $10 Payoff long p $10 $9 $8 $7 $6 $5 $4 $3 $2 $1 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 Total $10 $9 $8 $7 $6 $5 $4 $3 $2 $1 $0 $1 $2 $3 $4 $5 $6 $7 $8 $9 $10 Payoff Diagram $ 12 lon gc lon gp Total $ 10 Payoff Instrument Call (c ) Put (p ) $8 $6 $4 $2 $0 $ 30 $ 32 $ 34 $ 36 Stock Price $ 38 $ 40 $ 42 $ 44 $ 46 $ 48 $ 50 This sheet answers question 1d Payoff for buying put = max(K-S,0) Payoff for selling put = -max(K-S,0) Inputs Stock price at expiration, S_T $20 $21 $22 $23 $24 $25 $26 $27 $28 $29 $30 $31 $32 $33 $34 $35 $36 $37 $38 $39 $40 $41 $42 $43 $44 $45 $46 $47 $48 $49 $50 $51 $52 $53 $54 $55 Position Strike Long $50.00 Short $30.00 Payoff long p1 short p2 $30 -$10 $29 -$9 $28 -$8 $27 -$7 $26 -$6 $25 -$5 $24 -$4 $23 -$3 $22 -$2 $21 -$1 $20 $0 $19 $0 $18 $0 $17 $0 $16 $0 $15 $0 $14 $0 $13 $0 $12 $0 $11 $0 $10 $0 $9 $0 $8 $0 $7 $0 $6 $0 $5 $0 $4 $0 $3 $0 $2 $0 $1 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 $0 Total $20 $20 $20 $20 $20 $20 $20 $20 $20 $20 $20 $19 $18 $17 $16 $15 $14 $13 $12 $11 $10 $9 $8 $7 $6 $5 $4 $3 $2 $1 $0 $0 $0 $0 $0 $0 Payoff Diagram $ 35 lon g p1 sho rt p2 $ 30 $ 25 Payoff Instrument Put (p1) Put (p2) $ 20 $ 15 $ 10 $5 $0 $,20 -$ 5 $,25 $,30 $,35 $,40 -$ 10 -$ 15 Stock Price $,45 $,50 $,55 This sheet answers question 2. Inputs Interest Stock price u1 d1 u2 d2 u3 d3 Exercise price q1 q2 q3 2% six-month risk free rate. $40.00 November 1, 2005. 1.38 period 1 up factor. 0.75 period 1 down factor. 1.27 period 2 up factor if share price increases in period 1. 0.82 period 2 down factor if share price increases in period 1. 1.50 period 2 up factor if share price decreases in period 1. 0.67 period 2 down factor if share price decreases in period 1. $40.00 0.43 risk-neutral probability over the first period. 0.44 risk-neutral probability over the second period, if the share price increases in the first period. 0.42 risk-neutral probability over the second period, if the share price decreases in the first period. Stock price lattice Period Stock price 0 1 2 $40.00 $55.00 $30.00 $70.00 $45.00 $20.00 a) Single period BOPM Period Call 0 1 $6.35 $15.00 $0.00 Payoff in period 1 = max(S-K,0). Call premium in period 0 = (q1*$15+(1-q1)*0)/(1+interest) b) Two-period BOPM Period 0 1 2 $30.00 $15.78 $5.00 Call $7.84 $5.00 $2.08 $0.00 Payoff in period 2 = max(S-K,0). Call premium in period 0 = (q1*$15.78+(1-q1)*2.08)/(1+interest) Value of call in period 1 if share price is 55 = (q2*$30.00+(1-q2)*5.00)/(1+interest) Value of call in period 1 if share price is 30 = (q3*$5.00+(1-q3)*0)/(1+interest) Comparing the two option premiums The call option with a longer time to maturity costs more. This is intuitive since a longer time to maturity gives the share price greater scope to increase, thereby resulting in a higher call payoff. This corresponds to a higher call premium given that losses on a call option are capped. This sheet answers question 3 In what follows, yields are calculated on a continuously compounded basis. (1) A zero-coupon bond (pure discount) that matures on 1-May-07 is selling today at 98:02. Bond price Time to maturity Par value 1.5 year Yield 98.06 1.50 years 100.00 maturity proceeds. 1.30% continuous compounded yearly rate. Note Yield = Log(maturity proceeds/bond price) / Time to maturity. (2) An 8% bond (face value $100) that matures on 1-May-06 is selling today at $102.4. Bond price Time to maturity Par value Coupon rate Coupon frequency Coupon amount Maturity proceeds 0.5 year Yield 102.40 I assume that the bond is quoted in decimals, unlike other bonds in this project. 0.50 years 100.00 8% 2 per year. 4.00 104.00 3.10% continuous compounded yearly rate. Note Yield = Log(maturity proceeds/bond price) / Time to maturity. (3) A 6% bond that matures on 1-May-07 is selling today at 106:29. The coupon is paid out twice a year on 1-May 1 st and November 1st. Bond price Time to maturity Par value Coupon rate Coupon frequency Coupon amount Maturity proceeds 106.906250 1.50 years 100.00 6% 2 per year. 3.00 103.00 Pricing the bond Discount Time rate 0.5 3.10% 1 1.75% 1.5 1.30% Bond Price 1 year Yield Discount Present Cashflow Factor Value 3.00 0.984615 2.95 3.00 0.982677 2.95 103.00 0.980625 101.00 106.91 1.75% continuous compounded yearly rate. Note The 1-year yield value in cell C43, is derived from using Excel's goal-seek function, which is run as follows: Set cell F45 to value $106.90625 (which is the current bond price) by changing cell C43. Price of 5% bond (face value $100) that will mature on 1-Nov-2006 Par value Coupon rate Coupon frequency Coupon amount Maturity proceeds 100.00 5% 2 per year. 2.50 102.50 Pricing the bond Discount Time rate 0.5 3.10% 1 1.75% Bond Price Discount Present Cashflow Factor Value 2.50 0.984615 2.46 102.50 0.982677 100.72 103.19 This sheet answers question 4 Today Maturity Bond price Forward maturity date Forward rate Bond par value 1-Nov-05 1-Nov-06 $94.00 1-May-07 7% I assumed that the forward rate provided is a continuously compounded annual rate. Note, the compounding frequ $100.00 Price of Zero Coupon Bond (ZCB) maturing in 1-May-2007 Price $90.77 =Par value x exp(-forward rate x 0.5) x (price of bond maturing on 1 Nov 06)/Par value. Note - The discount factor over the period 1 Nov 05 to 1 Nov 06 = (price of bond maturing on 1 Nov 06)/Par value. - The discount factor over the period 1 Nov 06 to 1 May 07 = exp(-forward rate x 0.5) equency is not stated in the