Math 4550: Financial Mathematics.
HW #8: Expected Payoff Pricing
Answers
Problem 1:

Use the fact that


1
e
0 S 2 T
e
 x2
dx   and simple substitution to prove that

S
1
(ln( )( r   2 )T )2
S0
2
2 2T
dS  1 .
---------------------------ln(
Answer: Let x 

(ln(

1
e
0 S 2 T
S
1
)  (r   2 )T
S0
2
1
, then dx 
dS . Hence
S 2T
 2T
S
1
) ( r   2 )T ) 2
S0
2
2 2T

dS 


1

e x dx  1
2
Notice how the limits changed, since limS  x   and limS 0 x   .
Problem 2:
Assume there is a company X with S0=$100. Assume r=1%, and σ=35%.
Find the value of a binary option that pays $20 in a year if the stock price is
below $80 then. (for an answer just show the integral (in Maple notation),
and the answer as obtained with Maple)
----------------------------
6.171361206
The price is $6.17
Problem 3:
Assume there is a company Z with S0=$100. Assume r=1%, and σ=25%.
(a) Calculate the price of a 1-year call option on Z with strike K=200.
0.03328636731
(b) Calculate the price of a 1-year put option on Z with strike K=50.
0.01274691145
(c) Suppose that we sell 1 million of these puts and 1 million of these calls.
Draw the profit diagram for this portfolio. Find the maximum profit.
MaxProfit:=1,000,000*(0.03328636731+0.01274691145)= $46,033
Profit Diagram:
(d) Calculate the probability that we make this maximum profit.
0.9942671346
the probability is 99.4%!!
(e) Suppose that against all odds, the stock ends the year at $25. Calculate
our loss.
Loss=25,000,000 – 46,033= $24,953,967
Problem 4:
Keeping the parameters of company Z, find the value of an option that has the following
payoff diagram:
23.80933429
Problem 5:
On March 30, 2015, the share price of Apple stood at $125. A Call-option expiring on
July 17, 2015 (take T=3.5/12) with a strike of $130 cost $5.00. Find the volatility of
Apple implied by this option price (assume that Apple pays a $.47 dividend before
expiry, also assume that the interest rate is .25%)
5.001028751
so the implied volatility is about .2695
Compare this to the historic volatility of Apple based on stock prices from Jan 1 to March
30 (calculated in Excel): .277
Problem 6:
Assume there is a company Y with S0=$100. Assume r=2%, and σ=30%.
Find the value of an exponential option that pays $ 100e
S

100
in a year.
36.9545
Problem 7:
True or False? (If true, give a reason, if False, give a counter example)
1. `When the riskless interest rate is 0%, then the Put option costs exactly as much
as the Call Option , when the strike is `at the money’ (i.e., K=S0)’
2. `For any given (European) option, the chance that one makes money in the end is
50-50’.
Answers: 1. True: Put call parity says S0  P  C  Ke rT . When r  0 and K  S0 this
simplifies to P  C
2 . False. Consider our problem 3d: the chance for this option combination to make
money was 99.5%!