TD - Binomial model for asset price changes

Question
Consider the simple binomial model for asset price changes:
The asset price at time t = 0 os S + 0 and at time t = 1 it can either be S2
with the probability 0 < p < 1 or S1 < S2 with probability 1 − p. Show that
unless
S1 < S0e
r < S2
where r is the risk free rate, this model is arbitragabile. (You may assume
that short sales are allowed.)
A derivative security, V , is written on this asset. At time t = 0 its value is
V0. At time t = 1 its value can be either V1, if the underlying’s price is S1,
or V2 if the underlying’s price is S2.
(a) Let V
p
0 denote the present value (at time t = 0) of the expected value
of V at time t = 1. Give a formula for V
p
0
in terms of p, r, V1 and V2.
(b) Construct a risk free portfolio containing bot V and S and use an
arbitrage argument to show that this leads to a “fair price” for V0, say
V
∆
0
, in terms of S0, S1, S2, V1 and V2 but not p.
(c) Construct a replicating strategy, in terms of S and cash invested at the
risk-free rate r, which leads to an arbitrage free price for V0, say V
R
0
,
in terms of S0, S1, S2, V1 and V2 but not p.
(d) Deduce from wither (b) or (c) that there is a number q, which may
be regarded as a “risk neutral” probability, associated with the underlying’s price, such that the “fair value” of V0 is the present value (at
t = 0) of the expected value of V at time t = 1.
(e) Assuming that either V2 < V1 or V2 > V1, and that p > q, show that
a trader using either of the prices V
∆
0 = V
T
)
from (b) or (c) would
necessarily be able to arbitrage a trader using the price V
p
0
from (a).