Dominant Trading Strategies
Definition (Dominant Trading Strategy)
Introduction to Mathematical Finance:
Part I: Discrete-Time Models
The trading strategy Ĥ is said to be dominant if there exists another
trading strategy H̄ such that
and V̂1 (ω) > V̄1 (ω) ∀ω ∈ Ω
V̂0 = V̄0
AIMS and Stellenbosch University
April-May 2012
Financially speaking, both strategies Ĥ and H̄ start with the same initial
investment amount but the dominant strategy Ĥ leads to a higher gain
under all possible states of the world.
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Exercise (Dominant Trading Strategies)
and
V1 (ω) > 0
∀ω ∈ Ω
V0 < 0
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and
V1 (ω) ≥ 0
∀ω ∈ Ω
A dominant trading strategy is one that can transform strictly
negative wealth at t = 0 into nonnegative wealth at t = 1.
Proof. [=⇒] Let H = (a, b) s.t. V0 (H) = 0 and V1 (H) > 0 (Deduce:
Using previous Question). We note that b = −aS0 . Let H̄ = (ā, b̄)
defined by ā = a and b̄ = b − δ where δ > 0 is given by
δ(1+r ) = min(aS u +b(1+r ), aS d +b(1+r )) = min(V1u (H), V1d (H)) > 0 ,
i.e.
[⇐=] H dominates the trading strategy αH, for any α ∈ [0, 1). In
particular, 0, i.e. this trading strategy is dominant since it dominates
the strategy which starts with zero value and does no investment
at all.
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A dominant trading strategy Ĥ exists if and only if there exists a
trading strategy H s.t
A dominant trading strategy is one that costs nothing to set up
and will definitely make money (with probability one). Obviously a
dominant trading strategy is an arbitrage.
Proof.
[=⇒]Suppose Ĥ dominates H̄, we define a new trading strategy
H̃ = Ĥ − H̄. Let Ṽ0 and Ṽ1 denote the portfolio value of H̃ at t = 0
and t = 1. By construction, we have Ṽ0 = 0 and Ṽ1 (ω) > 0 for all
ω ∈ Ω.
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Exercise
A dominant trading strategy Ĥ exists if and only if there exists a
trading strategy H s.t
V0 = 0
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δ = min(a
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Su
Sd
− S0 , a
− S0 ) > 0 .
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Link between Dominant Trading Strategies and Arbitrage
Clearly, we have
V̄0 = −δ < 0
and V̄1 (ω) = V1 (ω) − δ(1 + r ) ≥ 0,
∀ω ∈ Ω .
Exercise
Conclude that the market is arbitrage free, if there is no portfolio s.t.
V0 ≤ 0,
[⇐=] If ∃ a H = (a, b) s.t. V0 < 0 and V1 (ω) ≥ 0 for all ω ∈ Ω. Then
Ĥ := (â, b̂) = (a, b − V0 ) = (a, b) + (0, −V0 ) satisfies
V̂0 = 0
and V̂1 (ω) = V1 (ω) − V0 (1 + r ) > 0
and P(V1 > V0 ) > 0.
1. If V0 = 0, we have an arbitrage by definition,
2. If V0 < 0, we use the previous result on dominant strategy.
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∀ω ∈ Ω
Proof.
∀ω ∈ Ω
since V1 ≥ 0 and V0 < 0 (i.e. −V0 (1 + r ) > 0).
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V1 (ω) ≥ 0
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Exercise (Tutorial Questions)
Consider the following 1-Period Binomial Market (under the
arbitrage-free condition):
B
M=
S
t=0
1
S0
B
D=
S
up
1+r
Su
down
p
1+r
B 1+r
=
Sd
S
Su
1−p
1+r
Sd
Solution
I
(i) Consider the Call Option C := (S1 − K )+ for a strike price K . Using
the law of one price, show that the fair-price π0 (C ) satisfies
(S0 −
At time t = 1, the following cash flows inequalities
0 ≤ P = (K − S1 )+ = max(K − S1 , 0) ≤ K imply (via Law of One
Price)
K
0 ≤ π0 (P) ≤
1+r
K +
) ≤ π0 (C ) ≤ S0 .
1+r
(ii) Consider the Put Option P := (K − S1 )+ for a strike price K . Using
the law of one price, show that the fair-price π0 (P) satisfies
0 ≤ π0 (P) ≤
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K
.
1+r
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