Admissible trading strategy

Last updated February 05, 2024

In finance, an admissible trading strategy or admissible strategy is any trading strategy with wealth almost surely bounded from below. In particular, an admissible trading strategy precludes unhedged short sales of any unbounded assets.^[1] A typical example of a trading strategy which is not admissible is the doubling strategy.^[2]

Mathematical definition

Discrete time

In a market with $d$ assets, a trading strategy $x\in \mathbb {R} ^{d}$ is admissible if $x^{T}{\bar {S}}=x^{T}{\frac {S}{1+r}}$ is almost surely bounded from below. In the definition let $S$ be the vector of prices, $r$ be the risk-free rate (and therefore ${\bar {S}}$ is the discounted price).^[1]

In a model with more than one time then the wealth process associated with an admissible trading strategy must be uniformly bounded from below.^[2]

Continuous time

Let $S=(S_{t})_{t\geq 0}$ be a d-dimensional semimartingale market and $H=(H_{t})_{t\geq 0}$ a predictable stochastic process/trading strategy. Then $H$ is called admissible integrand for the semimartingale $S$ or just admissible, if

the stochastic integral $H\cdot S$ is well defined.
there exists a constant $M\geq 0$ such that $(H\cdot S)_{t}\geq -M\,a.s.,\quad \forall t\geq 0$ .^[3]

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References

1 2 Föllmer, Hans; Schied, Alexander (2004). Stochastic finance: an introduction in discrete time (2 ed.). Walter de Gruyter. pp. 203–205. ISBN 9783110183467.
1 2 Frank Oertel; Mark Owen (2006). "On utility-based super-replication prices of contingent claims with unbounded payoffs". arXiv: math/0609403 .
↑ Delbaen, Schachermayer (2008). The Mathematics of Arbitrage (corrected 2nd ed.). Berlin Heidelberg: Springer-Verlag. p. 130. ISBN 978-3-540-21992-7.

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[FollmerSchied-1] 1 2 Föllmer, Hans; Schied, Alexander (2004). Stochastic finance: an introduction in discrete time (2 ed.). Walter de Gruyter. pp. 203–205. ISBN 9783110183467.

[OertelOwen-2] 1 2 Frank Oertel; Mark Owen (2006). "On utility-based super-replication prices of contingent claims with unbounded payoffs". arXiv: math/0609403 .

[3] Delbaen, Schachermayer (2008). The Mathematics of Arbitrage (corrected 2nd ed.). Berlin Heidelberg: Springer-Verlag. p. 130. ISBN 978-3-540-21992-7.

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