FIN 4021 Fall 2015
Homework Assignment #3
Due: Friday, December 4th
Submit one copy of your solutions per group. You may email this to me anytime on the due
date. Include the names of all your group members and your group’s number. Give your file a
title that is very easy for me to identify.
I suggest you use something like
FIN_4021_HW_3_Group_XX.
1. Black-Scholes-Merton Option Pricing: Use EXCEL to answer this question. Suppose that
Syzsmanski & Co. (SYZ) stock is trading at $69.81. The annualized standard deviation of
daily returns for SYZ is 71.65%. The T-Bill rate is 4.95%.
a. Call Price without Dividends: Assume that SYZ pays no dividends. What is the BSM
price of a call with an exercise price of $70 that matures in 146 days (assume that a
year is 252 days)?
Note: Use the NORMSDIST function in EXCEL to find N(d1) and N(d2).
b. Call Price with Dividends: Assume that SYZ pays dividends at annual rate of 1%.
Now, what is the BSM price of a call with an exercise price of $70 that matures in
146 days (assume that a year is 252 days)?
c. Put Price without Dividends: Assume that SYZ pays no dividends. What is the BSM
price of a put with an exercise price of $70 that matures in 146 days (assume that a
year is 252 days)?
d. Put Price with Dividends: Assume that SYZ pays dividends at annual rate of 1%.
Now, what is the BSM price of a put with an exercise price of $70 that matures in
146 days (assume that a year is 252 days)?
2. Using EXCEL to compute the BSM model ‘Greeks’: Suppose that Ono & Co. (ONO) stock
does not pay dividends. The annualized standard deviation of daily returns for ONO is
20%. The T-Bill rate is 5%. In this problem we will analyze puts and calls with a strike
price of $100 and 6 months to maturity.
a. Delta: In the BMS model, the sensitivity of an option’s price to a change in the
underlying stock price is called Delta. The Delta for puts and calls is given by:
∆ = ( 1 )
and
∆ = −(−1 )
On the same graph, plot the Put and Call Deltas for ONO for stock for prices ranging
from $75 to $125.
Note 1: You can use the NORMSDIST function in EXCEL to find N(d1) and N(-d1).
Note 2: The Call Delta ranges from 0 to 1. The Put Delta ranges from -1 to 0.
b. Gamma: As in the binomial model, delta in the BSM model changes with the value
of the underlying asset. The sensitivity of delta to changes in the underlying stock
price is called Gamma. The Gamma for puts and calls is given by:
1
Γ = Γ =
1
√ √2
−(1 )2
2
Plot the Call Gamma for ONO for stock for prices ranging from $75 to $125.
Note: Gamma is always positive and is maximized near the money.
c. Theta: Theta measures the impact of time on option values. For reasons you will
see in this calculation, it often called time decay. The Theta for puts and calls is
given by:
Θ = −
Θ = −
1
2√ √2
1
2√ √2
−(1 )2
2
−(−1 )2
2
− − ( 2 )
+ − (−2 )
On the same graph, plot the Call and Put Theta for ONO for stock prices ranging
from $75 to $125.
Note: the Put Theta will plot a constant distance above the Call Theta. Also, the Call
Theta will always be negative (time decay always reduces call values); but, the Put
Theta should be positive for low stock prices (deep in-the-money). At these low
prices, the passage of time actually increases the value of the put because we are
coming closer to receiving the strike price (in this range the time value of money
dominates the insurance value of the option).
d. Vega: The BSM model assumes the volatility of the underlying asset is constant. In
reality, of course, this isn’t true. Traders compute the sensitivity of BSM model
prices to changes in volatility. The result is called Vega. The Vega for puts and calls
is given by:
= =
√
√2
−(1 )2
2
Plot the Call Vega for ONO for stock prices ranging from $75 to $125.
Note: Vega is always positive and is maximized near the money.
3. Implied Volatility: Suppose the S&P 500 (SPX) is trading at 1,512.84. SPX has a dividend
yield of 1.62%. The T-Bill rate is 4.62%. We will evaluate options with a maturity of 61
days (out of 252 trading days).
Consider the following observed market prices for puts and calls:
Exercise Price
Call Price
Put Price
1,450
1,500
1,520
1,540
1,600
$105.60
$21.70
$63.40
$32.40
$45.50
$39.50
$40.00
$42.00
$11.50
$76.50
Compute the implied volatility for each option. Plot the implied volatility for each
option on the same graph where the x-axis is exercise price and the y-axis is volatility.
Connect the put volatility estimates (so that they trace out a smile, smirk, etc.). Do the
same for the call volatility estimates.
Note: the put and call volatility estimates will not generally be the same even at the
same strike price.
Solution strategy: Compute the difference between the BSM model price and the
observed market price for each exercise price. Use a dummy value of σ to compute the
initial BSM model price. You can use the EXCEL SOLVER tool to find the implied volatility
by trial & error. Instruct SOLVER to set the difference between the BSM model price
and the observed market price to zero (within a very small distance) by changing the
volatility.
