Stochastic Calculus for Finance I (2004 – Steven Shreve) | Complete Solutions Manual PDF

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, Contents




1 The Binomial No-Arbitrage Pricing Model ................ ]
1.7 Solutions to Exercises... 00... 0 0c cece cere cee cee cee. 1

2 Probability Theory on Coin Toss Space.................... 13
2.9 Solutions to Exercises... 0,0. ccc ccc cece cee eee cee. 13

3 State Prices 2.0. cee beep. 35
3.7 Solutions to Exercises 00... cec
ceee
cece e eee. 35

4 American Derivative Securities .....................-000... 51
4.9 Solutions to Exercises 0.0... 0... cece cece cece cbc e eee. 51


5 Random Walk ....000 0 0. oceec
eee eee. 69
5.8 Solutions to Exercises 0.0.0... ccc ccc ce ec eee eee ee eee. 69

6 Interest-Rate-Dependent Assets .......
0c. .....
.c cece ca ee. 89
6.9 Solutions to Exercises... 0... cc ccc cece cece cece eee p eee. 89

,The Binomial No-Arbitrage Pricing Model




1.7 Solutions to Exercises


Exercise 1.1. Assume in the one-period binomial market of Section 1.1 that.
both H and T have positive probability of occurring. Show that condition
(1.1.2) precludes arbitrage. In other words, show that if Xg = 0 and

X, = 49S) + (1 + r\{Xo ~ AgSo),

then we cannot have AX strictly positive with positive probability unless X,
is strictly negative with positive probability as well, and this is the case re-
gardiess of the choice of the number do.

Solution When X, = 0, we have

X (Hf) = AoS\(H) = u—({l+r) AgSo,
X1 (Ty = AoSy(T) = (d— (1 +7)) Ap So.
According to (1.1.2), u~-{1+7) is positive and d—(1+7) is negative. Therefore,
the only way X,(H) can be positive is for Ap to be positive, and the only way
X,(T) can be positive is for Ap to be negative, Either one of these portfolio
values is positive and the other negative, so both a positive and a negative
portfolio value at time one have a positive probability of occurring, or else
both X;(#) and X,(T) are zero.

Exercise 1.2. Suppose in the situation of Example 1.1.1 that the option sells
for 1.20 at time zero. Consider an agent who begins with wealth X, = 0
and at time zero buys Ap shares of stock and Jy options. The numbers dp
and Io can be either positive or negative or vero. This leaves the agent witb
a cash position of -44» ~ 1.20Iy. If this is positive, it is invested in the
money market: if it is negative, it represents money borrowed from the money

, 2 1 The Binomial No-Arbitrage Pricing Model

market. At time one, the value of the agent’s portfolio of stock, option and: -
rooney market is
.
X, = oS: + Io(Sy -5)7 - 1 (4Ao + 1.20I).
Assume that both 4 and T have positive probability of occurring. Show that
if there is a positive probability that X, is positive, then there is a positive
probability that X, is negative. In other words, one cannot find an arbitrage
when the time-zero price of the option is 1,20,

Solution. Considering the cases of a head and of a tai] on the first toss, and
utilizing the numbers given in Example 1.1.1, we can write:
o
Ai(H) = 84g + 329 - ~ (449 + 1.201),
oe


X(T) = 24,+0-Io- 7 (4do + 1.205)
Adding these, we get.
X (A) + X41 (T) = 10Ap
+ 829 — 110A — 3m = O,
or, equivalently,
X1(H)} = -X,(T).
In other words, either X,(H) and X,(T) are both zero, or they have opposite
sigos. Taking into account that both p > 0 aud q > 0, we conclude that if
there jis a positive probability that 1 is positive, then there is a positive
probability that X, is negative.

Exercise 1.3. ]n the one-period binomial model of Section 1.1, suppose we
want to determine the price at time zero of the derivative security Vj = 94,
i.e., the derivative security pays off the stock price. (This can he regarded as a
European cal] with strike price K = 0). What is the time-zero price Vp given
by the risk-neutral pricing formula (1.1.10)?

Solution. We have

Vo M(H) + (7)
tf




l
ipSi(H) + GS, (T))
{|




L+r
1
[puso + Gd So}
tf




l+r
up
+ dg
= So I+r
_ u((l1+r)—d) +d(u—(14-7))
~ 8 (14r)(u—a)
= So,