Problem 4 Consider a stock with current price $60. You are given: • Dividends of $1 each will be paid every three months; the next dividend will be paid in 2 months. • σ = 0.3. ⚫ The continuously compounded risk-free interest rate is 4%. Use the Black-Scholes methodology to price a nine-month at-the-money European put option on the stock.

Consider 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).

Suppose that a stock price is currently 35 dollars, and it is known that four months from now, the price will be either 51 dollars or 29 dollars. Find the value of a European call option on the stock that expires four months from now, and has a strike price of 39 dollars. Assume that no arbitrage opportunities exist and a risk-free interest rate of 10 percent.Answer =dollars.

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.

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 option

Consider a stock with a current price of P $27 Suppose that over the next 6 months the stock price will either go up by a factor of 1.41 or down by a factor of 071. Consider a call option on the stock with a strike price of $25 that expires in 6 months. The nsk-free rate is 6%. (1) Using the binomial model, what are the ending values of the stock price? What are the payoffs of the call option? (2) Suppose you write one call option and buy N shares of stock How many shares must you buy to create a portfolo with a riskless payoff Ge, a hedge portfolio)? What is the payoff of the portfolio? 13)What.is the.present.value of the hedge port- Tolot What &the value of phe calt.option? (4) What s a teplieatirg portfolio What is 2otrage?

Consider a European call option on a non-dividend-paying stock where the stock price is $40, the strike price is $40, the risk-free rate is 4% per annum, the volatility is 30% per annum, and the time to maturity is 6 months. (a) Calculate u, d, and p for a two-step tree. (b) Value the option using a two-step tree.

A European put option on a non-dividend paying stock has strike price of $40 and time to maturity 9 months. Assume the risk-free interest rate is 5% per annum, the volatility is 20% per annum and the current stock price is $38. Using the Black-Scholes model, calculate the price of the European put option.

The current price of a non-dividend paying stock is $30. Use a two -step tree to value a European call option on the stock with a strike price of $32 that expires in 6 months. Each step is 3 months, the risk free rate is 8% per annum with continuous compounding. What is the option price when the volatility is 20%? (Hint: Calculate u and d using the CRR approach.) A. $1.48 B. $1.08 C. $1.68 D. $1.28

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.