FIN9797 Options HW 3 Fall 2023

Question 2. (a) Use the Black-Scholes formula to find the current price of a European call option on a stock paying no income with strike 60 and maturity 18 months from now. Assume the current stock price is 50, the lognormal volatility of the stock is σ = 20%, and the constant continuously compounded interest rate is r = 10%.

2. Call Options A. How does the price of a call option respond to the following changes, other things equal? Does the price go up or down? Explain briefly the intuition for your answer. (). Stock price falls. (i). Volatility of stock price rises B. Suppose FlyByNight Corporation (FBN) is selling a one-year European call option that has an exercise price of $32. Assume that FBN's stock is currently selling for $20 and that over the coming year the price will either rise to $81 or fall to $11. Also assume that the one-year rate of interest is 10 percent. What would be the market price for this call option? Please explain carefully,

3.2 Find the current price of a one-year, R110-strike American put option on a non- dividend-paying stock whose current price is S(0) = 100. Assume that the continuously compounded interest rate equals r = 0.06. Use a two-period Binomial tree with u = 1.23, and d = 0.86 to calculate the price VP(0) of the put option.

Suppose a stock is currently trading for $35, and in one period it will either increase to $38 or decrease to $33. If the one-period risk-free rate is 6%, what is the price of a European put option that expires in one period and has an exercise price of $35? $0.51 $2.32 $1.55 $3.00 $0.76

3 Using Black-Scholes find the price of a European call option on a non-dividend paying stock when the stock price is $69, the strike price is 70, the risk-free interest rate is 12% per annum, the volatility is 30% per annum, and the time to maturity is three months? What is the value of a put using theses parameters (use put-call parity)? What happens to the price of the call if volatility is 10% and 50%? Show the prices at these volatilites.

5. A "cash-or-nothing binary stock option" is a European-style option and pays some fixed amount of cash to the holder at maturity T > 0 if the stock price at time T > 0 is above a certain threshold K (also referred to as the strike price). Suppose a stock price is currently at $50. Assume that over each of the next two 3-month periods the stock price will either go up by 6% or go down by 5%. The risk-free rate is 5% p.a. with continuous compounding. Use a two-step binomial tree model to compute the arbitrage-free price of a cash-or-nothing option written on that stock which pays $1 if the stock price in three months is above K = $50.

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only answer b)   Question 2. (a) Use the Black-Scholes formula to find the current price of a European call option on a stock paying no income with strike 60 and maturity 18 months from now. Assume the current stock price is 50, the lognormal volatility of the stock is σ = 20%, and the constant continuously compounded interest rate is r = 10%. (b) Repeat part (a) for a European put with strike 60 and maturity 18 months from now

Question 5: A call option on a stock that expires in a year has a strike price of $99. The current stock price is $100 and the one-year risk free interest rate is 10%. The price of this call is $6. a) Is arbitrage possible? What is the arbitrage position? b) do you het this minimum? Find the minimum arbitrage profit for this strategy. When

Suppose company B stock is last time traded at 68.23$, and our analysis shows that in one year from now, its price would either increase or decrease by 25%. (i.e. d=0.75, u=1.25) Assume that risk-free rate is 2.5% and the company pays no dividend in this time period. 4) What would be the payoff for a long position on European Call Option written on Company B equity, with time to maturity of 1 year, and strike price of 75$? Calculate the payoff for the up and down scenarios

*NoChatGPT answers please   6A) What is the price of a European call option on a non-dividend-paying stock when the stock price is $52, the strike price is $50, the risk-free interest rate is 12% per annum, the volatility is 30% per annum, and the time to maturity is 3 months?   6B) What is the assumption of the Black–Scholes–Merton stock option pricing model about the probability distribution of the stock price in one year? What is the assumption about the probability distribution of the continuously compounded rate of return on the stock during the year?

Question I: Suppose the S&R index is 800, the continuously compounded risk-free rate is 5%, and the dividend yield is 0%. A 1-year 815-strike European call costs $75 and a 1-year 815-strike European put costs $45. Consider the strategy of buying the stock, selling the 815-strike call, and buying the 815-strike put. (a) What is the rate of return on this position held until the expiration of the options? (b) What is the arbitrage implied by your answer to (a)? (c) What difference between the call and put prices would eliminate arbitrage? (d) What difference between the call and put prices eliminates arbitrage for strike prices of $780, $800, $820, and $840?

Question 2. What is the price of a European call option on a non-dividend paying stock when the stock price is $52, the strike price is $50, the risk-free interest rate is 12% per annum, the volatility is 30% per annum, and the time to maturity is 3 months?

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aa.3   Consider a two-period binomial model, where each period is 6 months. Assume the stock price is $50.00, = 0.20, r = 0.06 and the dividend yield = 3.5%. What is the lowest strike price where early exercise would occur with an American put option?

Please advise

1. Consider a 4 month European put on a stock with no dividend the follow- ing parameters: S(0) = 305, K (a) Compute the option's vega (b) If o increases by 0.01, what is the approximate increase in the value of the option? 300, r = 0.08, o = 0.25

In this problem we assume the stock price S(t) follows Geometric Brownian Motion described by the following stochastic differential equation: dS = µSdt + o Sdw, where dw is the standard Wiener process and u = 0.13 and o = current stock price is $100 and the stock pays no dividends. 0.20 are constants. The Consider an at-the-money European call option on this stock with 1 year to expiration. What is the most likely value of the option at expiration? Please round your numerical answer to 2 decimal places.

i need the answer quickly

QUESTION: 2. What is the fair value for a six-month European call option with a strike price of $135 over a stock which is trading at $138.15 and has a volatility of 42.5% when the risk free rate is 1.85% using the two step binomial tree? a) What is the delta of this option? b) What is the probability of an up movement in this stock? c) What is the probability of a down movement in this stock? d) What is the proportional move up for this stock e) What is the proportional move down for this stock f) What would be the value of the put option with the same strike price?

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please give me the correct answer fully  DO NOT GIVE ME THE WRONG ANSWER ANSWER EACH COLUMN

Question 10. The prices of European call and put options on a non-dividend paying stock with an expiration date in 12 months and a strike price of $120 are $20 and $5, respectively. The current stock price is $130. What is the implied risk-free rate?

Mf2. Assume a one-period binomial model in which the initial stock price is S = 60 and in each period the stock price can go either up by a factor of u = 7 3 or down by a factor of d = 2 3 . Assume that the simple interest rate over one time period is r = 1 3 .   (a) Determine the fair price of the European put option with strike K = 60.   (b) Assume that instead of the price determined in part (a), the European call option is trading at 11. Prove that there is an arbitrage and explain how the arbitrage can be achieved

6. A 4-month European call option on a dividend paying stock is selling for C = $4. So = $54, K = $50. A dividend of $0.90 is expected in 2 months. r = .12. Assume continuous discounting. Are there opportunities for arbitrage? Explain and show these opportunities.

Suppose that an American put option with a strike price of $155.5 and maturity of 12.0 months costs $11.0. The underlying stock price equals 143. The continuously compounded risk-free rate is 6.5 percent per year. What is the potential arbitrage profit from buying a put option on one share of stock? O 12.401 1.5 no arbitrage profit available 11.943 1.6783

4. (15 pts) The current price of a stock is $50 and we assume it can be modeled by geometric Brownian motion with o = .15. If the interest rate is 5% and we want to sell an option to buy the stock for $55 in 2 years, what should be the initial price of the option for there not to be an arbitrage opportunity?

Without using the Black‐Scholes model, compute the price of a European put option on a non‐dividend‐paying stock with the strike price is $70 when the stock price is $73, the risk‐free interest rate is 10% pa, the volatility is 40% pa, and the time to maturity is 6 months?