Binomial model

There are 6 questions on this homework. 5 problems worth 4 points each, and 1 bonus problem for an
additional 4 points.
Exercise 1. Consider a one step binomial model. The initial stock price is $50. There is a 80% chance the
stock price will rise to $65 and a 20% chance it will fall to $40. The risk free bond gets 2%.
• Price a call option with strike of $50.
• Price a put option with strike of $50.
• Does put-call parity hold? Show whether it does or does not.
• What would the option prices be if the probabilities were 30% chance the stock price rises and 70%
chance it falls?
Exercise 2. (No picking specific numbers) Recall that the hedge ratio for a European call option is given
by
H =
Cu − Cd
uS0 − dS0
What happens to the hedge ratio in the limit as it gets farther in the money and as it gets farther out of
the money, i.e., solve for
lim
S0→∞
H =? lim
S0→0
H =? (1)
(Hint: Note that Cu and Cd are really functions of the underlying stock price S0.)
Exercise 3. (Again no picking numbers) Consider a stock that pays no dividends on which a futures
contract, a call option, and a put option trade. The maturity date for all three contracts is T, the strike
price of both the put and the call is K, and the futures price is F. Prove that if K = F, then the price of
the call option equals the price of the put option.
Exercise 4. Consider a portfolio that consists of the following derivatives: 1) a put option purchased with
strike price x + 10, 2) two put options written (sold) with strike price x + 5, and 3) a put option purchased
with strike price x. The stock price at maturity is ST . What are the payoffs at expiration of this portfolio?
Write the payoffs and draw the payoff graph.
Problem Set 5 Page 1
Economics 4751 Financial Economics Spring 2021
Exercise 5. Consider one stock and a risk free bond that gets rate r = 0. Examine the following payoff graph.
1 2 3
1
2
3
ST
P ayoffT
How many/much of the following do you need to buy for your portfolio to have the payoff graphed above?
Shares of Stock
Cash (risk free bond)
Call options with K = 1
Call options with K = 2
For example if you need four shares of stock and twenty dollars of cash invested in risk free rate write
4, 20, 0, 0, in the lines above. Also, note that it is the amount of cash you invest initially that you should
write on that line, and amounts may be negative.
Draw a graph of the payoffs from each of the four parts of your portfolio listed above.
Exercise 6. (Use Excel or similar) Recall from class that the Black-Scholes formula for a European call
option at current time t with maturity T and strike price K is given by
C(St, t) = N(d1)St − N(d2)Ke−r(T −t)
where
d1 =
ln( St
K
) + (r +
σ
2
2
)(T − t)
σ
√
T − t
d2 = d1 − σ
√
T − t
And N(·) is the standard normal cumulative distribution function (NORM.S.DIST(·,True) in Excel 2010).
From put-call parity we can write that the price of a put option is
P(St, t) = Ke−r
(T − t) − St + C(St, t)
= N(−d2)Ke−r(T −t) − N(−d1)St
We will use Black-Scholes to price put options on the S&P 500 with maturity as close to 1 year as you
can find and different strike prices, and compare them to the actual prices of the options. Go to CBOE’s
website or something similar and get data on prices and “implied volatility” of put options on the S&P 500
with strike prices of 3600, 3800, 4000, 4200, 4400, and 4600 and maturity as close to 1 year as you can find