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Mid-term Financial Derivatives Date:
27
th
July 2007
Total = 40
marks
Time =
1 hour 30 minutes
Note:
This is an open book / open notes / open formula sheet exam – students are allowed to bring in the textbook (Hull), notes, the formula sheet and the normal distribution table. But no exchange of notes / books /formula sheets / tables between the students are allowed during the exam 1.
What happens to the following: a.
A look back call as we increase the frequency with which we observe the asset price in calculating the minimum b.
A down and out call as we increase the frequency with which we observe the asset price in determining whether the barrier has been crossed
[3 + 3 = 6 marks]
2.
Arithmetic average rate options were assumed to be newly issued, and there was no historical average to deal with. Show that no generality was lost in doing so.
[5 marks]
3.
Assume Binomial Option Pricing Model. Show that a contingent claim that pays $1 when the stock prices reaches (S u
i
d
n-I
) and $0 otherwise can be replicated by a portfolio of calls.
[5 marks]
4.
Show that “At – the – Money” options have the maximum time value
[5 marks]
5.
At what stock price is the theta of a European Call the smallest?
[5 marks]
6.
Let {W(t); t ≥ 0} be a Brownian motion. Verify whether 2)(
2
t W
is a martingale or not.
[5 marks]
7.
Consider a 9-month call option with an interest rate of 8% and n = 3 (i.e. there are nodes at the 3 month time point, 6 months time point and the final one at the 9 month time point). The current volatility is 10% but the volatility changes between each time point at the rate of 7%. Let the current spot price be $160 and the strike price is $176. The option is of the American type. What is the premium that is to be paid for this option? Also, find the premium to be paid if the option was a put option with all the other information remaining the same.
[5 + 4 = 9 marks]
Solutions
Solution to question 1:
a.
As we increase the frequency, we observe a more extreme minimum, which increases the value of a look back call. b.
As we increase the frequency with which the asset price is observed, the asset price becomes more likely to hit the barrier and the value of a down and out call goes down.
Solution to question 2:
Let the historical average from m prices be A as of time zero. The terminal payoff for a call is then
−+−−++= −+++
∑∑
==
0,11max0,1max
00
nmmA X nmS X nmS mA
niinii
−+−+++−++++=
∑
=
0,1111max11
0
nmmA X nnmnS nmn
nii
So it becomes 11
+++
nmn
options with strike price
−+−+++
111
nmmA X nnm
.
Solution to question 3:
Consider the butterfly spread with strike prices X
L
, X
M
and X
H
such that
1111
−−+−
−−+−−
<<=<<
ini H iniini M ini Lini
d Su X d Su
d Su X d Su X d Su
with 2 X
M
– X
H
– X
L
= 0. This portfolio pays off
Lini
X d Su
−
−
dollars when the stock price reaches
ini
d Su
−
. Furthermore, its payoff is zero if the stock price finishes at other prices.
Solution to question 4:
We have to prove that the strike price X that maximizes the option’s time value is the current stock price S. Note that time value is defined as V = C – max(S – X, 0). Now,
( )
S X if X C X V and S X if X C X V T x N e
X C
rT
>>+∂∂=∂∂><∂∂=∂∂⇒−=∂∂
−
010
σ
Thus, the time value is maximized at S.
Solution to question 5:
It is as
t r
XeS
)2(
2
σ
+
=
To derive it, note that
''''''
''''''
)(2)()(
)(2)()(
x xrSN t x xSN x N
xt x N rXe
t x xSN x N
S
rt
−+−=−−+−=∂Θ∂
−
σ σ σ
The last equality takes advantage of the fact that
''''
)()(
x xSN xt x N Xe
rt
=−
−
σ
And the Black Scholes formula for European call. With N
’’
(x) = -x N
’
(x) and x
’
= 1/(S
σ√
t), it is not hard to see that the particle derivative is 0 if and only iff -
σ
2
+ (x
σ
/
√
t) – 2r = 0. From here the result follows easily.
Solution to question 6:
22)(0),(|
2)(2)(0),(|
2)(0),(|
2)(
22222
st sW
suuW
sW t W
E suuW
sW E suuW
t W E
−+=≤≤−+
≤≤=≤≤
Thus it is not a martingale.

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