Chapter 1
Stochastic Calculus for Finance I: The
Binomial Asset Pricing Model
1.1 The Binomial No-Arbitrage Pricing Model
1.1.
Proof. If we get the up sate, then X
1
= X
1
(H) = ∆
0
uS
0
+ (1 + r)(X
0
− ∆
0
S
0
); if we get the down state,
then X
1
= X
1
(T ) = ∆
0
dS
0
+ (1 + r)(X
0
−∆
0
S
0
). If X
1
has a positive probability of being strictly positive,
then we must either have X
1
(H) > 0 or X
1
(T ) > 0.
(i) If X
1
(H) > 0, then ∆
0
uS
0
+ (1 + r)(X
0
− ∆
0
S
0
) > 0. Plug in X
0
= 0, we get u∆
0
> (1 + r)∆
0
.
By condition d < 1 + r < u, we conclude ∆
0
> 0. In this case, X
1
(T ) = ∆
0
dS
0
+ (1 + r)(X
0
− ∆
0
S
0
) =
∆
0
S
0
[d −(1 + r)] < 0.
(ii) If X
1
(T ) > 0, then we can similarly deduce ∆
0
< 0 and hence X
1
(H) < 0.
So we cannot have X
1
strictly positive with positive probability unless X
1
is strictly negative with positive
probability as well, regardless the choice of the number ∆
0
.
Remark: Here the condition X
0
= 0 is not essential, as far as a property definition of arbitrage for
arbitrary X
0
can be given. Indeed, for the one-period binomial model, we can define arbitrage as a trading
strategy such that P (X
1
≥ X
0
(1 + r)) = 1 and P (X
1
> X
0
(1 + r)) > 0. First, this is a generalization of the
case X
0
= 0; second, it is “proper” because it is comparing the result of an arbitrary investment involving
money and stock markets with that of a safe investment involving only money market. This can also be seen
by regarding X
0
as borrowed from money market account. Then at time 1, we have to pay back X
0
(1 + r)
to the money market account. In summary, arbitrage is a trading strategy that beats “safe” investment.
Accordingly, we revise the proof of Exercise 1.1. as follows. If X
1
has a positive probability of being
strictly larger than X
0
(1 + r), the either X
1
(H) > X
0
(1 + r) or X
1
(T ) > X
0
(1 + r). The first case yields
∆
0
S
0
(u −1 −r) > 0, i.e. ∆
0
> 0. So X
1
(T ) = (1 + r)X
0
+ ∆
0
S
0
(d −1 −r) < (1+r)X
0
. The second case can
be similarly analyzed. Hence we cannot have X
1
strictly greater than X
0
(1 + r) with positive probability
unless X
1
is strictly smaller than X
0
(1 + r) with positive probability as well.
Finally, we comment that the above formulation of arbitrage is equivalent to the one in the textbook.
For details, see Shreve [7], Exercise 5.7.
1.2.
Proof. X
1
(u) = ∆
0
×8 + Γ
0
×3 −
5
4
(4∆
0
+ 1.20Γ
0
) = 3∆
0
+ 1.5Γ
0
, and X
1
(d) = ∆
0
×2 −
5
4
(4∆
0
+ 1.20Γ
0
) =
−3∆
0
−1.5Γ
0
. That is, X
1
(u) = −X
1
(d). So if there is a positive probability that X
1
is positive, then there
is a positive probability that X
1
is negative.
Remark: Note the above relation X
1
(u) = −X
1
(d) is not a coincidence. In general, let V
1
denote the
payoff of the derivative security at time 1. Suppose
¯
X
0
and
¯
∆
0
are chosen in such a way that V
1
can be
replicated: (1 + r)(
¯
X
0
−
¯
∆
0
S
0
) +
¯
∆
0
S
1
= V
1
. Using the notation of the problem, suppose an agent begins
3
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