No-Arbitrage

def. Arbitrage Portfolio. Portfolio with value $\Pi (t)$ at time $t$ is an arbitrage portfolio if:

$$\Pi (t)=0⟹\underset{\text{ price doesn’t go down }}{\underbrace{\mathbb{P}(\Pi (T)\ge 0)=1}}\text{ and }\underset{\text{ price guarateed to rise }}{\underbrace{\mathbb{P}(\Pi (T)>0)>0}}$$

Alternatively we can define it as:

$$\underset{\text{ if started negative... }}{\underbrace{\Pi (t)<0}}⟹\underset{\text{ ...will be non-negative }}{\underbrace{\mathbb{P}(\Pi (T)\ge 0)=1}}$$

def. Law of One Price (LOP). Two portfolios with the same future value must have the same value to begin with. if time $t<T$ then:

$${\Pi }_A(T)={\Pi }_B(T)⟹{\Pi }_A(t)={\Pi }_B(t)$$

thm. (Arbitrage is equivalent to LOP) If there is no arbitrage portfolio, then the law of one price holds.

Proof. Contrapositive: If law of one price doesn’t hold, there is an arbitrage portfoilo. Let portfolios $A,B$ such that

$${\Pi }_A(T)={\Pi }_B(T)\text{ and }{\Pi }_A(t)<{\Pi }_B(t)$$

Then construct the following new portfolio that:

• Long position on $A$
• Short position on $B$ Then

1. At time $t$, value is $\underset{\text{ Asset }}{\underbrace{{\Pi }_A(t)}}-\underset{\text{ Cash Debt }}{\underbrace{{\Pi }_A(t)}}-\underset{\text{ Asset Debt }}{\underbrace{{\Pi }_B(t)}}+\underset{\text{ Cash }}{\underbrace{{\Pi }_B(t)}}=\underset{\text{ Assets }}{\underbrace{{\Pi }_A(t)-{\Pi }_B(t)}}=0$
2. At time $T$, value is $\underset{\text{ Asset }}{\underbrace{{\Pi }_A(T)}}-\underset{\text{ Cash Debt }}{\underbrace{{\Pi }_A(t)e^{r(T-t)}}}-\underset{\text{ Asset Debt }}{\underbrace{{\Pi }_B(T)}}+\underset{\text{ Cash }}{\underbrace{{\Pi }_B(t)e^{r(T-t)}}}=\underset{\text{ =0 }}{\underbrace{{\Pi }_A(T)-{\Pi }_B(T)}}+\underset{\text{ >0 }}{\underbrace{({\Pi }_B(t)-{\Pi }_A(t))e^{r(T-t)}}}>0$
   1. The cash investments are risk free, thus this is determined.
3. Therefore this portfolio is an arbitrage portfolio.■

One can similarly prove the other way. Thus LOB is equivalent to No Arbitrage condition.