Arbitrage Definitions in One- and Multi-Period Binomial Models
Summary
The discussion examines an apparent mismatch between two definitions of arbitrage in Björk’s treatment of binomial models. One definition for a single-period model appears stricter because it excludes any outcome where the terminal portfolio value is zero, while the multi-period formulation can allow zero value in some states. The exchange highlights that the distinction may depend on whether such states have positive probability.
One answer lists equivalent formulations requiring zero initial cost, nonnegative terminal payoff, and a strictly positive payoff with positive probability (or nonzero payoff under the stated setup). Another argues that a zero-payoff state is permitted when it has probability zero. The exchange raises a useful point about almost-sure conditions, but does not provide the original textbook definitions or settle the interpretation fully; readers should check the precise assumptions and wording in the source.
Key ideas
- Arbitrage definitions commonly require zero initial value and a nonnegative terminal payoff.
- A strict formulation may rule out zero terminal payoff in every state.
- An alternative formulation permits zero payoff in some states when positivity holds with positive probability.
- Whether a zero-payoff state matters depends on its probability under the model.
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# Inconsistent Definition of Arbitrage in Bjork?
# Inconsistent Definition of Arbitrage in Bjork?
In Tomas Bjork's Arbitrage Theory in Continuous Time (or here), $\exists$ what seems to be 2 inconsistent definitions of arbitrage:
The first definition is for the single period Binomial model
The second definition is for the multi period Binomial model
The second suggests that there is a possibility of the portfolio value ending up zero while the first does not...
...Why?
Edit: Oh, I forgot to mention: My prof uses the latter definition to replace the first definition for the one-period. E said something about different conditions or something. (I'll ask about it during next consultation hours.)
## Answer by DoubleTrouble (score 0, accepted)
https://quant.stackexchange.com/a/12914
My initial answer was incorrect, I was thinking to quickly (or slowly!?)
I agree with you that these two definitions are not consistent. The first definition is much more strict since it does not allow for any outcome $\omega \in \Omega = \{\omega_1, \omega_2\}$ such that $V_1^h(\omega)=0$. We only have 2 outcomes since we are considering the single period Binomial model.
As a side note, here are three equivalent definitions of an arbitrage portfolio $h$ (same notation as in Björk).
- $V_h^0 = 0$, $V_h^1 \geq 0$, and $\ E[V_h^1]>0.$
- $V_h^0 = 0$, $P(V_h^1 \geq 0)=1$, and $\ P(V_h^1 > 0)>0.$
- $V_h^0 = 0$, $V_h^1 \geq 0$, and $\ V_h^1 \neq 0$
## Answer by KaapstadKwant (score 2)
https://quant.stackexchange.com/a/12915
I agree with the question and not with the answer. Definition (2.2) means that $\omega$ for which $V^h_1(ω )=0$ is such that $P(\omega) = 0$, ie an event with measure (probability) 0.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.