1. Consider a 4 month European put on a stock with no dividend the follow- ing parameters: S(0) = 305, K = 300, r = 0.08, o = 0.25 (a) Compute the option's vega (b) If o increases by 0.01, what is the approximate increase in the value of the option?
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- 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.Consider an American put option (K=$100) expiring in one year on a stock trading for $84. The return volatility on the stock is 27.3% and the riskless rate is 5%. Find the price of the option using a Binomial Model with two steps. (Respond with two decimal places, such as "-12.34") 26.59 Correct Answer: 18.28Consider a 3-month European call option on a non-dividend-paying stock. The current stock price is $20, the risk-free rate is 6% per annum, and the strike price is $20. Assume a risk-neutral world. You calculate the following values using the Black-Scholes-Merton model: d1 = 0.2000 N(d1) = 0.5793 d2 = 0.1000 N(d2) = 0.5398 a) What is the probability that the call option will be exercised? b) What is the expected stock price at the option’s expiration in 3 months? Assume that all values of the stock price less than $20 are counted as zero. c) What is the expected payoff on the option at expiration (in 3 months)? d) Calculate the PV of the expected payoff from part c).
- 3. Consider a non-dividend paying stock whose initial stock price is 62 and has a log- volatility of σ = 0.20. The interest rate r = 10%, compounded monthly. Consider a 5-month option with a strike price of 60 in which after exactly 3 months the purchaser may declare this option a (European) call or put option. Assume u = 1.05943 and d = = 0.94390 (a) Compute the values of the binomial lattice for 5 1 month period. 0 1 2 3 4 5 62 (b) Compute the appropriate risk-free rate. (c) Find the risk-neutral probability p of going up? (d) Find the values of call option and put option along this lattice: 0 5.85 1 2 3 4 5 call option 0 1 2 3 4 5 1.40 put option3. Consider a European call option on a non-dividend paying stock with exercise price 100 USD and expiration in one month. Interest rate is 1 percent and volatility of the stock is 0.4. Stock price 90 USD. Option price?? Use excel.b) Suppose you are given the following information: Current market price of a share= R200 000 Option’s exercise price = R300 000 Time until the option expires= 5 yrs Risk free rate =4% Standard deviation of the returns on the share= 0.35 Required: i. Calculate d1 and d2 ii. Suppose N(d1) =0.7517 and N(d2) =0.4602; calculate the price of the call option on the share
- 3. Consider a European put option on a non-dividend paying stock with exercise price 100 USD and expiration in one month. Interest rate is 1 percent and volatility of the stock is 0.4. Stock price 90 USD. Option price?? Use excel.Consider a European call option and a European put option that have the same underlying stock, the same strike price K = 40, and the same expiration date 6 months from now. The current stock price is $45. a) Suppose the annualized risk-free rate r = 2%, what is the difference between the call premium and the put premium implied by no-arbitrage? b) Suppose the annualized risk-free borrowing rate = 4%, and the annualized risk-free lending rate = 2%. Find the maximum and minimum difference between the call premium and the put premium, i.e., C − P such that there is no arbitrage opportunities.3. A stock has a 15 percent change of moving either up or down per period and is currently priced at $25. Using a one period binomial model, and assuming that the risk-free rate is 10 percent, complete the following. a. Determine the possible stock prices at the end of the first period. b. Calculate the intrinsic values at expiration of an at-the-money European call option. c. Find the value of the option today. d. Construct a hedge by combining a position in stock with a position in the call. Show that the return on the hedge is the risk-free rate regardless of the outcome, assuming that the call sells for the value you obtained in c. e. Determine the rate of return from a riskless hedge if the call is selling for $3.50 when the hedge is initiated.
- 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 $0.76The following information is given about an Option on a stock: S(0)=$31, X=$34, rf=9%, variance (sigma squared)=20%, T=182.5 days, Dividend $1.75 in 45 days (1) Calculate the price of a European put option using the Black-Scholes pricing model (show all workings including d1, d2, N(d1), N(d2)) (2) Calculate the price of the corresponding European call option (hint: put call parity) (3) Suppose you feel that the put option is overpriced. What strategy should you use to exploit the apparent mispricing? (4) The market has entered a state of significant volatility, and you believe the implied volatility is incorrect. You believe it should be 10% higher. What trade would you undertake to exploit this arbitrageSuppose you wrote an European call option on Stock A with an exercise of $40 and call premium of $3.3. What would be your profit if the stock price at the expiration date is $38?