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Table 1 David Michael Q1A call option expiring in 2 months has a market price of $10.75. The current stock price is $70, the strike price is $60, and the risk-free rate is 4% per annum. Calculate the implied volatility. Current Value of stock (S) $70.00 Exercise Price of Option (X) $60 Number of periods to Exercise 2 months (in years(T)) 0.166666667 yr Risk-Free Interest Rate per annum (rf) 4% Volatilityi i.e. Standard Deviation-SD per annum 23% Goal Seeker d1=(Ln(S/X)+ (rf+0.5*SD^2)*T) /(SD*T^0.5) 1.7785 d2=d1(SD*T^0.5) 1.6856 N(d1) 0.9623 N(d2) 0.9541 Value of 2 month $70 Call Option $10.50 SN(d1)Xexp( rfT)N(d2) 4.Consider a stock index currently standing at 2,100. The dividend yield on the index is 3% per annum and the risk-free rate is 1%. A 3-month European call option on the index with a strike price of 2,000 is trading at $105.91. What is the value of a 3-month European put option with a strike price of 2,000? (Hint: Use put-call parity for index options) Current Stock Price $1,083 Exercise Price of O $1,000 Number of periods 0.5 Risk-Free Interest 4% Dividend Yield per 1.00% Value of 6 month Eu $34.94 Value of 6 month European put option (P )at strike price $1000 is -$62.46 C-S*exp(-q*T) +X*(exp(-rf*T) Q8 Suppose you are creating a butterfly spread using call options with 3 different strike prices. Currently, the call price with strike price of $40 is $21.94, the call with strike price of $50 is $11.24, and the call with strike price of $60 is $6.55. What is the initial cash flow of the butterfly spread strategy? If it's a cash outflow, then answer in a negative number. Option buy -21.94 Option buy -6.55 Option sell 22.58 Initial cash flow -5.91 Q10A call with a strike price of $70 costs $7.7. A put with the same strike price and expiration date costs $4.78. If you create a straddle, what is the initial cash flow? If it's a cash outflow, answer in a negative number. Option buy -70 Option buy -4.78 Initial cash flow -74.78 15 45 3 55 8 Premium paid net 5 BEP price 50 16 Currently, a stock price is $80. It is known that at the end of 4 months it will be either $75 or $90. The risk-free rate is 6% per annum with continuous compounding. What is the value of a 4-month European put option with a strike price of $80? stock 80 Price 1 75 Price 2 90 P= E^.006x(412)7 580)908075 80) 0.3892 Value of Put Option e-(0.06 * 4/12) * (1 0.3892) * [Max($80 - $73, 0)] 4.361 Max 7 Q22 Calculate the price of a 4-month European call option on a dividend-paying stock with a strike price of $30 when the current stock price is $34, the risk-free rate is 6% per annum and the volatility is 40% per annum. A dividend of $1.00 is expected in 2 months. Use Black-Scholes formula. Answer is either: 3.05, 3.65, 4.32, or 5.02 Template - Black-Scholes Option Value Input Data Stock Price now (P) Exercise Price of Option (EX) Number of periods to Exercise in year Compounded Risk-Free Interest Rate ( Standard Deviation (annualized ) 30 34 0.1666666667 6.00% 40.00% Output Data Present Value of Exercise Price (PV( .5 33.6617 0.1633 d1 -0.6236 d2 -0.7869 Delta N(d1) Normal Cumulative Density Function 0.2665 Bank Loan N(d2)*PV(EX) 7.2601 Value of Call 0.7335 Value of Put 4.3952 Answer is either: 3.05, 3.65, 4.32, or 5.02 Q6A stock price is currently $100. Over each of the next two six-month periods it is expected to go up by 14% or down by 6%. The risk-free interest rate is 4%. What is the risk-neutral probability that the stock price will increase each period? (Report in % such as 12.34%.) 0 125.44 0.5 Strike price e^rt 100 112 1.0202013398 0.5 0 0 100.8 0.5 0 100 Option value 4.56 100.80 90.00 0.5 0.50 0 10.00 Option value 9.31 81.00 0.5 19 Consider a 3-month European put option on a non-dividend-paying stock, where the stock price is $50, the strike price is $50, the risk-free rate is 3% per annum. Stock price will either move up by 10% or down by 5%, every month. Price the put with binomial trees. Current stock price ( Option strike price (K Time to expiration (T # of tree steps risk-free rate, r dt (length of 1 step) u $50.00 $50.00 0.25 3 3% 0.0833333333 0.10 d 0.05 p 19.0500625521 t = 0 (current) t = 6 months t = 12 months 18 months $0.05 $0.50 $5.00 $50.00 $0.03 $0.25 $2.50 $0.01 $0.13 $0.01 Table 1 David Michael Q1A call option expiring in 2 months has a market price of $10.75. The current stock price is $70, the strike price is $60, and the risk-free rate is 4% per annum. Calculate the implied volatility. Current Value of stock (S) $70.00 Exercise Price of Option (X) $60 Number of periods to Exercise 2 months (in years(T)) 0.166666667 yr Risk-Free Interest Rate per annum (rf) 4% Volatilityi i.e. Standard Deviation-SD per annum 23% Goal Seeker d1=(Ln(S/X)+ (rf+0.5*SD^2)*T) /(SD*T^0.5) 1.7785 d2=d1(SD*T^0.5) 1.6856 N(d1) 0.9623 N(d2) 0.9541 Value of 2 month $70 Call Option $10.50 SN(d1)Xexp( rfT)N(d2) 4.Consider a stock index currently standing at 2,100. The dividend yield on the index is 3% per annum and the risk-free rate is 1%. A 3-month European call option on the index with a strike price of 2,000 is trading at $105.91. What is the value of a 3-month European put option with a strike price of 2,000? (Hint: Use put-call parity for index options) Current Stock Price $1,083 Exercise Price of O $1,000 Number of periods 0.5 Risk-Free Interest 4% Dividend Yield per 1.00% Value of 6 month Eu $34.94 Value of 6 month European put option (P )at strike price $1000 is -$62.46 C-S*exp(-q*T) +X*(exp(-rf*T) Q8 Suppose you are creating a butterfly spread using call options with 3 different strike prices. Currently, the call price with strike price of $40 is $21.94, the call with strike price of $50 is $11.24, and the call with strike price of $60 is $6.55. What is the initial cash flow of the butterfly spread strategy? If it's a cash outflow, then answer in a negative number. Option buy -21.94 Option buy -6.55 Option sell 22.58 Initial cash flow -5.91 Q10A call with a strike price of $70 costs $7.7. A put with the same strike price and expiration date costs $4.78. If you create a straddle, what is the initial cash flow? If it's a cash outflow, answer in a negative number. Option buy -70 Option buy -4.78 Initial cash flow -74.78 15 45 3 55 8 Premium paid net 5 BEP price 50 16 Currently, a stock price is $80. It is known that at the end of 4 months it will be either $75 or $90. The risk-free rate is 6% per annum with continuous compounding. What is the value of a 4-month European put option with a strike price of $80? stock 80 Price 1 75 Price 2 90 P= E^.006x(412)7 580)908075 80) 0.3892 Value of Put Option e-(0.06 * 4/12) * (1 0.3892) * [Max($80 - $73, 0)] 4.361 Max 7 Q22 Calculate the price of a 4-month European call option on a dividend-paying stock with a strike price of $30 when the current stock price is $34, the risk-free rate is 6% per annum and the volatility is 40% per annum. A dividend of $1.00 is expected in 2 months. Use Black-Scholes formula. Answer is either: 3.05, 3.65, 4.32, or 5.02 Template - Black-Scholes Option Value Input Data Stock Price now (P) Exercise Price of Option (EX) Number of periods to Exercise in year Compounded Risk-Free Interest Rate ( Standard Deviation (annualized ) 30 34 0.1666666667 6.00% 40.00% Output Data Present Value of Exercise Price (PV( .5 33.6617 0.1633 d1 -0.6236 d2 -0.7869 Delta N(d1) Normal Cumulative Density Function 0.2665 Bank Loan N(d2)*PV(EX) 7.2601 Value of Call 0.7335 Value of Put 4.3952 Answer is either: 3.05, 3.65, 4.32, or 5.02 Q6A stock price is currently $100. Over each of the next two six-month periods it is expected to go up by 14% or down by 6%. The risk-free interest rate is 4%. What is the risk-neutral probability that the stock price will increase each period? (Report in % such as 12.34%.) 0 125.44 0.5 Strike price e^rt 100 112 1.0202013398 0.5 0 0 100.8 0.5 0 100 Option value 4.56 100.80 90.00 0.5 0.50 0 10.00 Option value 9.31 81.00 0.5 19 Consider a 3-month European put option on a non-dividend-paying stock, where the stock price is $50, the strike price is $50, the risk-free rate is 3% per annum. Stock price will either move up by 10% or down by 5%, every month. Price the put with binomial trees. Current stock price ( Option strike price (K Time to expiration (T # of tree steps risk-free rate, r dt (length of 1 step) u d p $50.00 $50.00 0.25 3 3% 0.0833333333 1.10 0.95 0.3500208507 t = 0 (current) t = 6 months t = 12 months 18 months $66.55 $60.50 $55.00 $50.00 $57.48 $52.25 $47.50 $49.64 $45.13 $42.87 This sheet answers the questions in sheet 1 A call option expiring in 2 months has a market price of $10.75. The current stock price is $70, the strike price is $60, and the risk-free rate is 4% per annum. Calculate the implied volatility. Inputs Current Value of stock (S) Strike price Time to maturity Risk-Free Interest Rate per annum (rf) Volatility d1 d2 N(d1) N(d2) Call option premium 70 60 0.17 yr 0.04 0.3003379548 This implied volatility is found by using the goal seek function. 1.37 =(LN(S/X)+(rf+0.5*SD^2)*T)/(SD*T^0.5) 1.25 =d1-(SD*T^0.5) 0.92 0.89 10.75 SN(d1)Xexp(rfT)N(d2) I think your error stemmed from running a goal seek for a call option of 10.50 instead of 10.75 as required by the question. Otherwise, all your stated formulas are correct. The image below shows the goal seek inputs Consider a stock index currently standing at 2,100. The dividend yield on the index is 3% per annum and the risk-free rate is 1%. A 3-month European call option on the index with a strike price of 2,000 is trading at $105.91. What is the value of a 3-month European put option with a strike price of 2,000? (Hint: Use put-call parity for index options) Current Stock Price (S) Strike price (X) Time to maturity Risk-Free rate (rf) Dividend Yield per annum (q) Call premium Put premium 2,100 2,000 0.25 1% 3% 105.91 16.61 C-S*exp(-q*T)+X*(exp(-rf*T) I think your error stemmed from using input values that differ from those provided in the question. Otherwise, all your stated formulas are correct. This sheet answers question 6 A stock price is currently $100. Over each of the next two six-month periods it is expected to go up by 14% or down by 6%. The risk-free interest rate is 4%. What is the risk-neutral probability that the stock price will increase each period? (Report in % such as 12.34%.) Inputs risk-free rate, r dt (length of 1 step) u d p 4% ... correct. 0.50 ... correct. 1.14 ...the up factor is 1 + 14%. 0.94 ...the down factor is 1 - 6% 40.10% ... Note, that your risk neutral probability lies between 0 and 1 (reasonability check). p is the risk neutral probability of the share price going up. The question only requires you to calculate p. Thus, there is no need to attempt deriving the share price process. Note that the formula you used to compute p under q23 is what you need to compute the risk neutral probability in q6. This sheet answers question 23 Consider a 3-month European put option on a non-dividend-paying stock, where the stock price is $50, the strike price is $50, the risk-free rate is 3% per annum. Stock price will either move up by 10% or down by 5%, every month. Price the put with binomial trees. Inputs Current stock price (S0) Option strike price (K) Time to expiration (T) # of tree steps risk-free rate, r dt (length of 1 step) u d p Discount factor 50 ... correct. 50 ... correct. 0.25 ... correct. 3 ... correct. 0.03 ... correct. 0.08 ... correct. 1.10 ...the up factor is 1 + 10%. 0.95 ...the down factor is 1 - 5% 0.35 ... Note, that your risk neutral probability now lies between 0 and 1. 0.998 Share Price Process Period Time 0 0 1 0.08 2 0.17 3 0.25 Share 50.00 55.00 47.50 60.50 52.25 45.13 66.55 57.48 49.64 42.87 0 0 1 0.08 2 0.17 3 0.25 0.15 3.16 0.24 4.75 0.36 7.13 You get the same share prices if you change the values of u and d in sheet q23. So, the way you designed your share price process is correct Put Option Process Period Time Put 2.10 You omitted the put option process in your solution. Notes on the put option process The put payoff at the last time period (3) is Max(K-S,0). For t=2, enter the following equation for each node: Value of put = Discount factor * {p* + (1-p) *} Then repeat for t=1. Finally, repeat for t=0. This gives you a put premium of $2.10 now (t=0). Table 1 David Michael Q1A call option expiring in 2 months has a market price of $10.75. The current stock price is $70, the strike price is $60, and the risk-free rate is 4% per annum. Calculate the implied volatility. Current Value of stock (S) $70.00 Exercise Price of Option (X) $60 Number of periods to Exercise 2 months (in years(T)) 0.166666667 yr Risk-Free Interest Rate per annum (rf) 4% Volatilityi i.e. Standard Deviation-SD per annum 23% Goal Seeker d1=(Ln(S/X)+ (rf+0.5*SD^2)*T) /(SD*T^0.5) 1.7785 d2=d1(SD*T^0.5) 1.6856 N(d1) 0.9623 N(d2) 0.9541 Value of 2 month $70 Call Option $10.50 SN(d1)Xexp( rfT)N(d2) 4.Consider a stock index currently standing at 2,100. The dividend yield on the index is 3% per annum and the risk-free rate is 1%. A 3-month European call option on the index with a strike price of 2,000 is trading at $105.91. What is the value of a 3-month European put option with a strike price of 2,000? (Hint: Use put-call parity for index options) Current Stock Price $1,083 Exercise Price of O $1,000 Number of periods 0.5 Risk-Free Interest 4% Dividend Yield per 1.00% Value of 6 month Eu $34.94 Value of 6 month European put option (P )at strike price $1000 is -$62.46 C-S*exp(-q*T) +X*(exp(-rf*T) Q8 Suppose you are creating a butterfly spread using call options with 3 different strike prices. Currently, the call price with strike price of $40 is $21.94, the call with strike price of $50 is $11.24, and the call with strike price of $60 is $6.55. What is the initial cash flow of the butterfly spread strategy? If it's a cash outflow, then answer in a negative number. Option buy -21.94 Option buy -6.55 Option sell 22.58 Initial cash flow -5.91 Q10A call with a strike price of $70 costs $7.7. A put with the same strike price and expiration date costs $4.78. If you create a straddle, what is the initial cash flow? If it's a cash outflow, answer in a negative number. Option buy -70 Option buy -4.78 Initial cash flow -74.78 15 45 3 55 8 Premium paid net 5 BEP price 50 16 Currently, a stock price is $80. It is known that at the end of 4 months it will be either $75 or $90. The risk-free rate is 6% per annum with continuous compounding. What is the value of a 4-month European put option with a strike price of $80? stock 80 Price 1 75 Price 2 90 P= E^.006x(412)7 580)908075 80) 0.3892 Value of Put Option e-(0.06 * 4/12) * (1 0.3892) * [Max($80 - $73, 0)] 4.361 Max 7 Q22 Calculate the price of a 4-month European call option on a dividend-paying stock with a strike price of $30 when the current stock price is $34, the risk-free rate is 6% per annum and the volatility is 40% per annum. A dividend of $1.00 is expected in 2 months. Use Black-Scholes formula. Answer is either: 3.05, 3.65, 4.32, or 5.02 Template - Black-Scholes Option Value Input Data Stock Price now (P) Exercise Price of Option (EX) Number of periods to Exercise in year Compounded Risk-Free Interest Rate ( Standard Deviation (annualized ) 30 34 0.1666666667 6.00% 40.00% Output Data Present Value of Exercise Price (PV( .5 33.6617 0.1633 d1 -0.6236 d2 -0.7869 Delta N(d1) Normal Cumulative Density Function 0.2665 Bank Loan N(d2)*PV(EX) 7.2601 Value of Call 0.7335 Value of Put 4.3952 Answer is either: 3.05, 3.65, 4.32, or 5.02 Q6A stock price is currently $100. Over each of the next two six-month periods it is expected to go up by 14% or down by 6%. The risk-free interest rate is 4%. What is the risk-neutral probability that the stock price will increase each period? (Report in % such as 12.34%.) 0 125.44 0.5 Strike price e^rt 100 112 1.0202013398 0.5 0 0 100.8 0.5 0 100 Option value 4.56 100.80 90.00 0.5 0.50 0 10.00 Option value 9.31 81.00 0.5 19 Consider a 3-month European put option on a non-dividend-paying stock, where the stock price is $50, the strike price is $50, the risk-free rate is 3% per annum. Stock price will either move up by 10% or down by 5%, every month. Price the put with binomial trees. Current stock price ( Option strike price (K Time to expiration (T # of tree steps risk-free rate, r dt (length of 1 step) u d p $50.00 $50.00 0.25 3 3% 0.0833333333 1.10 0.95 0.3500208507 t = 0 (current) t = 6 months t = 12 months 18 months $66.55 $60.50 $55.00 $50.00 $57.48 $52.25 $47.50 $49.64 $45.13 $42.87 This sheet answers the questions in sheet 1 A call option expiring in 2 months has a market price of $10.75. The current stock price is $70, the strike price is $60, and the risk-free rate is 4% per annum. Calculate the implied volatility. Inputs Current Value of stock (S) Strike price Time to maturity Risk-Free Interest Rate per annum (rf) Volatility 70 60 0.17 yr 0.04 0.3003379548 This implied volatility is found by using the goal seek function. d1 d2 N(d1) N(d2) 1.37 =(LN(S/X)+(rf+0.5*SD^2)*T)/(SD*T^0.5) 1.25 =d1-(SD*T^0.5) 0.92 0.89 Call option premium 10.75 SN(d1)Xexp(rfT)N(d2) I think your error stemmed from running a goal seek for a call option of 10.50 instead of 10.75 as required by the question. Otherwise, all your stated formulas are correct. The image below shows the goal seek inputs Consider a stock index currently standing at 2,100. The dividend yield on the index is 3% per annum and the risk-free rate is 1%. A 3-month European call option on the index with a strike price of 2,000 is trading at $105.91. What is the value of a 3-month European put option with a strike price of 2,000? (Hint: Use put-call parity for index options) Current Stock Price (S) Strike price (X) Time to maturity Risk-Free rate (rf) Dividend Yield per annum (q) Call premium 2,100 2,000 0.25 1% 3% 105.91 Put premium 16.61 C-S*exp(-q*T)+X*(exp(-rf*T) I think your error stemmed from using input values that differ from those provided in the question. Otherwise, your formulas are correct. Suppose you are creating a butterfly spread using call options with 3 different strike prices. Currently, the call price with strike price of $40 is $21.94, the call with strike price of $50 is $11.24, and the call with strike price of $60 is $6.55. What is the initial cash flow of the butterfly spread strategy? If it's a cash outflow, then answer in a negative number. Let K1, K2 and K3 denote three different strike prices where K1 Buy 1 call option with strike price K1; ---> Buy 1 call option with strike price K3; and ---> Sell 2 call option with strike price K2. From the information provided, we know that: Strike Value Call Number of Position Premium Options K1 $40.00 $21.94 K2 $50.00 $11.24 K3 $60.00 $6.55 Initial cash flow of the butterfly spread strategy Buy Sell Buy 1 2 1 Cash flow -$21.94 $22.48 -$6.55 -$6.01 Your error stemmed from using an incorrect call premium for the sold option. Specifically, you used 11.29 instead of 11.24 in cell B56 (sheet 1). Otherwise, your formulas and portfolio construction are correct. A call with a strike price of $70 costs $7.7. A put with the same strike price and expiration date costs $4.78. If you create a straddle, what is the initial cash flow? If it's a cash outflow, answer in a negative number. A straddle is constructed by buying a call and put with the same strike price. The Corresponding cash flows are shown in the table below. Option Premium Position Call $7.70 Put $4.78 Initial cash flow to create straddle Buy Buy Number of Cash flow Options 1 -$7.70 1 -$4.78 -$12.48 Your error stemmed from using an incorrect call premium. Specifically, you used 70 instead of 7.70 in cell B61 (sheet 1). Otherwise, your formulas and portfolio construction are correct. Currently, a stock price is $80. It is known that at the end of 4 months it will be either $75 or $90. The risk-free rate is 6% per annum with continuous compounding. What is the value of a 4-month European put option with a strike price of $80? In what follows, I use exactly the same approach as the one in sheet "R23" Inputs Current stock price (S0) Option strike price (K) Time to expiration (T) # of tree steps risk-free rate, r dt (length of 1 step) u d p Discount factor 80 80 0.33 1 0.06 0.33 1.13 0.94 0.44 0.980 Share Price Process Period Time 0 0 1 0.33 Share 80.00 90.00 75.00 0 0 1 0.33 Put Option Process Period Time Put 2.74 5.00 Notes on the put option process The put payoff at the last time period (1) is Max(K-S,0). For t=0, enter the following equation for each node: Value of put = Discount factor * {p* + (1-p) *} This gives you the put premium at t=0. Calculate the price of a 4-month European call option on a dividend-paying stock with a strike price of $30 when the current stock price is $34, the risk-free rate is 6% per annum and the volatility is 40% per annum. A dividend of $1.00 is expected in 2 months. Use Black-Scholes formula. Answer is either: 3.05, 3.65, 4.32, or 5.02 Inputs Current Value of stock (S) Strike price Time to maturity 34 30 0.33 years Risk-Free rate (rf) Volatility Dividend Time of dividend PV(Dividend) S* 6% 40% 1 0.17 years 0.99 ...Dividend x exp(-rf*T) 33.01 ...share price adjusted to allow for dividend. d1 d2 N(d1) N(d2) 0.62 =(LN(S*/X)+(rf+0.5*SD^2)*T)/(SD*T^0.5) 0.39 =d1-(SD*T^0.5) 0.73 0.65 Call option premium 5.02 S*N(d1)Xexp({rf}T)N(d2) The BlackScholes formula use is exactly the same as the one used in finding the implied volatility in this worhsheet. The only exception is that we employ S* = S PV(Dividend) instead of S in the Black Scholes formula. This sheet answers question 6 A stock price is currently $100. Over each of the next two six-month periods it is expected to go up by 14% or down by 6%. The risk-free interest rate is 4%. What is the risk-neutral probability that the stock price will increase each period? (Report in % such as 12.34%.) Inputs risk-free rate, r dt (length of 1 step) u d p 4% ... correct. 0.50 ... correct. 1.14 ...the up factor is 1 + 14%. 0.94 ...the down factor is 1 - 6% 40.10% ... Note, that your risk neutral probability lies between 0 and 1 (reasonability check). p is the risk neutral probability of the share price going up. The question only requires you to calculate p. Thus, there is no need to attempt deriving the share price process. Note that the formula you used to compute p under q23 is what you need to compute the risk neutral probability in q6. This sheet answers question 23 Consider a 3-month European put option on a non-dividend-paying stock, where the stock price is $50, the strike price is $50, the risk-free rate is 3% per annum. Stock price will either move up by 10% or down by 5%, every month. Price the put with binomial trees. Inputs Current stock price (S0) Option strike price (K) Time to expiration (T) # of tree steps risk-free rate, r dt (length of 1 step) u d p Discount factor 50 ... correct. 50 ... correct. 0.25 ... correct. 3 ... correct. 0.03 ... correct. 0.08 ... correct. 1.10 ...the up factor is 1 + 10%. 0.95 ...the down factor is 1 - 5% 0.35 ... Note, that your risk neutral probability now lies between 0 and 1. 0.998 Share Price Process Period Time 0 0 1 0.08 2 0.17 3 0.25 Share 50.00 55.00 47.50 60.50 52.25 45.13 66.55 57.48 49.64 42.87 0 0 1 0.08 2 0.17 3 0.25 0.15 3.16 0.24 4.75 0.36 7.13 You get the same share prices if you change the values of u and d in sheet q23. So, the way you designed your share price process is correct Put Option Process Period Time Put 2.10 You omitted the put option process in your solution. Notes on the put option process Price of put The put payoff at the last time period (3) is Max(K-S,0). For t=2, enter the following equation for each node: Value of put = Discount factor * {p* + (1-p) *} Then repeat for t=1. Finally, repeat for t=0. This gives you the put premium at t=0