BUS137 Options Pricing Practice Problems

BUS 137Options 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?