Problem 10. Chapter 8. Our portfolio contains one European put and

Problem 10. Chapter 8.
Our portfolio contains one European put and one American put both at K = 32 (let us call these
respectively H and h), plus one straddle of American options at K = 35, which is nothing but the sum
of an American option and an American call (let us denote these respectively by by P and C). Since all
these are long positions, the price of the Meirkiec is simply the sum of these three derivatives:
Meirkiec = h + H + C + P.
We use the 2-step formula for a binomial tree to determine the price of the European option, and go node
by node to determine the price of the American options.
First, the values of the stock at future times are
Su = 30(1.12) = 33.6
Sd = 30(.92) = 27.6
Suu = 30(1.12)2 = 37.632
Sud = 30(1.12)(0.92) = 30.912
Sdd = 30(0.92)2 = 25.392
In turn, this gives the possible payo↵s for each of the derivatives at maturity
huu = Huu = max{32
37.632, 0} = 0
hud = Hud = max{32
30.912, 0} = 1.088
hdd = Hdd = max{32
25.392, 0} = 6.608
(at maturity a European and a American option are identical)
Puu = max{35
37.632, 0} = 0
Pud = max{35
30.912, 0} = 3.088
Pdd = max{35
25.392, 0} = 9.608
Cuu = max{37.632
35, 0} = 2.632
Cud = max{30.912
35, 0} = 0
Cdd = max{25.392
35, 0} = 0
As for the risk neutral probability, we have
p=
er t d
e(0.04)(1/2) 0.92
=
⇡ 0.501, 1 p ⇡ 0.499
u d
1.12 0.92
e rT = e (0.04)1 ⇡ 0.9607.
We use the 2-step formula to compute the European put h,
h=e
0.04
p2 huu + 2p(1
p)hud + (1
p)2 hdd
= e 0.04 0 + 2(0.501)(0.499)(1.088) + (0.499)2 (6.608)
= 2.10345
As for the American put h, we proceed node by node. First we compute Hu and Hd using the 1-step
formula and the definition of an American option,
Hu = max{e (0.04)/2((0.501)0 + (0.499)(1.088)), 32
= 0.532154
33.6}
Hu = max{e (0.04)/2((0.501)(1.088) + (0.499)(6.608)), 32
= 4.4
27.6}
From here, we compute the price
H = max{e (0.04)/2((0.501)(0.532154) + (0.499)(4.4)), 2}
= 2.413.
Now, an American call without dividends is always the same as a European call, so that the price of C
can be computed as if it was European! We use the 2-step formula and get
C=e
0.04
p2 Cuu + 2p(1
p)Cud + (1
p)2 Cdd
= e 0.04 (0.501)2 (2.632) + 0 + 0
= 0.6347
The American put is computed as the previous American put, and one can see after the standard step by
step computation that
P =5
Adding up the prices, we see that the price of the Meirkiec is
2.1035 + 2.413 + 0.6347 + 5 = 10.15168$.
Problem 12. Chapter 8.
a) By exercising early, one gets K S0 which is positive (since by assumption K > S0 u > S0 , thanks to
u > 1). On the other hand, by exercising later, using risk neutral pricing, one gets
e
rT
(K
S0 erT ) = e
rT
K
S0
Since r > 0 we know that e rT < 1, therefore e rT K S0 < K S0 , so exercising later would always
result in a smaller return. Thus in this case it is preferable to exercise the option early.
b) Since K < S0 d, the strike price is below both future values of the option, therefore it would never be
profitable to exercise in the future (i.e. the payo↵ at all future nodes is = 0). On the other hand, S0 d < S0
(since d < 1) so that also K < S0 , showing that is is also not profitable to exercise the option right now.
In conclusion, in the case K < S0 d the option can never be exercised so its value is 0.
c) The previous two parst show that ✓ > S0 u means one always exercises early and ✓ < S0 d means one
never exercises. Therefore we look for ✓ between S0 d and So u. What we want of ✓ is that
✓
S0 = e
rT
(phu + (1
p)hd )
(where hu , hd denotes the payo↵s at future times). Since ✓ < S0 u, we know that it won’t make sense to
exercise if the stock goes up, so hu = 0, and one gets
✓
Solving for ✓, we get the formula
✓ = S0
S0 = e
✓
u
rT
(1
p)hd
1 de rT
d + 1 ue
rT
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