BUS 137 Options Pricing Practice Problems Total of 200 points. 1) (16 points) The stock of ABC, Inc. is currently trading at $94. It pays no dividends and its volatility is 25%. The continuously-compounded risk-free rate is 4%. For a strike price of 90, determine the price of a put option with maturity dates shown in the following table. If the stock pays dividends of 1.5%, use the Black-Scholes method to calculate the price for the same stock and enter the values in the last column of the table. BT = Binomial tree method, and BS = Black-Scholes method Maturity Date One-step BT Two-Step BT BS, No Dividends BS- Dividends 1.5% 1-month 6-month 1-year 15-month 2) (16 points) For the stock in Question 1, fill out the following table corresponding to a call option with a strike price of 100. Maturity Date One-step BT Two-Step BT BS, No Dividends BS- Dividends 1.5% 1-month 6-month 1-year 15-month 3) (24 points) For the stock in Question 1, suppose the maturity date in two months. Determine the price of put and call options with the strike prices shown in the following table using the method indicated. K 85 90 95 100 Two-Step BT Call Put BS, No Dividends Call Put BS, 1.5% Dividends Call Put For problems 4 – 10, please show all your work. If not specified, round up your answer to four decimal places. 4) (25 points) The current price of a certain stock is $70. The stock pays no dividends. In six months, the stock price will either increase to $84.273 or decrease to $61.740. The continuously-compounded risk-free rate is 6%. Consider a 6-month put option with a strike price of $65. Using a one-step BT, determine u, d, Cu, Cd, P* (neutral-risk probability), delta, B and the price of the put option. 5) (25 points) The current price of a stock is $94. The stock pays no dividends. In one year, the stock price will either increase to X or decrease to Y. The continuously-compounded riskfree rate is 6% and the risk-neutral probability is 47.5%. A one year call option with the strike price of 98 is priced at 5.50. Determine X. 6) (25 points) The current price of a stock is $94. The stock pays no dividends. In six months, the stock price will either increase to X or decrease to Y. The continuously-compounded riskfree rate is 6% and the risk-neutral probability is 47.5%. A 6-month put option with the strike price of 90 is priced at 3.07. Determine Y. 7) (25 points) Recall a (long) straddle is made up of one long call and one long put with the same strike price and maturity date. The payoff (not the profit) of a straddle is always positive. Refer to Week 3 slide 6. You can treat a straddle as a call option and price the portfolio as one option without pricing the individual options building it. The current price of a stock is $94. The stock pays no dividends. In three months, the stock price will either increase to 105 or decrease to 80. The continuously-compounded risk-free rate is 6%. Using a one-step BT, determine Cu, Cd, P* (neutral-risk probability), delta, B and the price of the straddle. 8) (12 points) The current price of a stock is $94. The stock pays no dividends. A call option with six months to maturity and strike price of 98 costs 5.50. The option has the following Greeks: Delta = 0.514 Gamma = 0.111 Theta = -0.02 (daily) Estimate the price of the option using the delta-gamma-theta approximation, if one week later, a) the price of the stock is $96 b) the price of the stock is $90 9) (12 points) Same information as the previous question. Use delta approximation to estimate the price of the stock if the price of the call option is 5.00 after two weeks. 10) (20 points) Same information as the previous question. A call option with six months to maturity and strike price of 98 costs 5.50. An investor creates a portfolio consisting of three long calls and two short calls, one long put, and one short put. a) What is the sum of all delta’s of this portfolio if the stock does not pay dividends? b) What is the sum of all delta’s of this portfolio if the stock pays 2% dividends?