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ACTL3004: Weeks 78
Financial Economics for Insurance and
Superannuation: Weeks 78
Discrete Time Derivative Valuation
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ACTL3004: Weeks 78
Binomial Lattice Model: European Option Valuation
Introduction
Binomial Lattice Model: European Option Valuation:
Introduction
Consider an investment world where we can only invest in two ±nancial instruments:
1.
risky stock/share which pays no dividends
We shall denote by
S
(
t
)
the price of this stock at time
t
where
t
=
0
,
1
,
2
, ...
S
(
t
)
is
random.
2.
riskfree zerocoupon bond/cash account
We shall denote by
B
t
the value of this cash account at time
t
per unit invested at time 0.
Assume
r
is the riskfree rate compounded continuously so that
B
t
=
e
rt
.
In addition, we shall assume:
1.
We can hold arbitrarily large amounts (positive or negative) of stocks or cash.
2.
The securities market is arbitragefree.
3.
There are no trading costs, i.e. market is also considered frictionless.
4.
There are no minimum/maximum units of trading.
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ACTL3004: Weeks 78
Binomial Lattice Model: European Option Valuation
Principle of No Arbitrage
Principle of No Arbitrage
◮
Arbitrage
means a riskfree trading pro±t.
◮
We say that an
arbitrage
opportunity exists in the capital/securities
market if either:
1.
an investor is able to make a deal that would give him or her an
immediate pro±t, with no risk of future loss; or
2.
an investor is able to make a deal that has zero initial outlay (or
cost), no risk of future loss, and a nonzero probability of a future
pro±t.
◮
In the capital market, the principle of ”no arbitrage” is generally
assumed. No arbitrage opportunities must exist in the market.
◮
The ”Law of One Price” states that in a ”no arbitrage” situation, any
two securities or combination of securities that give exactly the same
payments must have the same price.
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ACTL3004: Weeks 78
Binomial Lattice Model: European Option Valuation
Binomial Branch Model
Binomial Branch Model
Consider one time tick  start at time 0, and one tick (of length
δ
t
) later we
arrive at time 1
The stock is assumed to either go up to
s
3
or down to
s
2
. The probability of
the stock rising is p.
The bond starts off as 1 and rises to
e
r
δ
t
, where
r
is the risk free rate
Suppose we want to price a derivative that pays
f
3
at node 3 or
f
2
at node 2.
Eg for a forward
f
3
=
s
3
−
K
f
2
=
s
2
−
K
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View Full DocumentACTL3004: Weeks 78
Binomial Lattice Model: European Option Valuation
Binomial Branch Model
Stock and Bond Strategy
We know from forward pricing that pure expected value pricing, ie
E
P
b
e

r
δ
t
f
(
S
(
1
))
B
is not good enough.
Lets see what we can do by holding a combination of stocks and bonds,
namely
φ
of stock
s
1
ψ
of bond
B
0
The value of this portfolio today is
V
(
0
) =
φ
s
1
+
ψ
B
0
and the value of this portfolio at time 1 is
φ
s
3
+
ψ
B
0
e
r
δ
t
if the stock goes up
φ
s
2
+
ψ
B
0
e
r
δ
t
if the stock goes down
Now remember that we want to price a derivative that pays
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