0

Let $(\Omega, \mathcal{F}, \mathbb{P})$ a probability space. Consider a random variable $R \in \mathbb{L}^1$ such that $\mathbb{E}[R] > 0$ and $\mathbb{P}( R < 0) >0$. Define a 1-period market with riskless asset $S^0_0 = 1$, $S^0_1 = 1$ and risky asset $S^1_0 = 1$, $S^1_1 = 1 + R$. In this model an arbitrage opportunity is a vector $(\alpha, \beta) \in \mathbb{R}^2$ such that:

  • $\alpha + \beta \leq 0$;
  • $\alpha + \beta + \beta R \geq 0$ $\mathbb{P}$-a.s.;
  • $\mathbb{P}(\alpha + \beta + \beta R > 0) > 0$.

It seems to me that taking as $\alpha = -1$ and $\beta = 1$ we get a (possible) arbitrage opportunity (to rule it out we would need more hypothesis on $R$. Hence, we can not conclude a priori that this market is arbitrage-free.

Is there anything wrong in this argument? If so how can I prove that this market is arbitrage free?

leobgg
  • 193
  • Hint: by the assumption $\mathbb P(R<0)>0$ you cannot guarantee the truth of the second bullet point when you take $\alpha=-1$ and $\beta=1,.$ – Kurt G. Jun 21 '23 at 17:38
  • I agree but this does not prove that the model is arbitrage free. Indeed we would need more assumptions on $R$ to have that. Without more assumptions on $R$ it seems to me that nothing rules out the fact that $\mathbb{P}(R > 0) >0$. – leobgg Jun 21 '23 at 18:02

1 Answers1

0

If we add the standard assumptions that $$ 1+R\ge 0\quad \mathbb P\text{-a.s. ( asset prices are nonnegative )} $$ $$ \mathbb E[1+R]=1\quad \text{( $S^1$ is a martingale )} $$ Then for any $\alpha,\beta$ with $\alpha+\beta\le 0$ and $\alpha+\beta+\beta R\ge 0$ $\mathbb P$-a.s. we have $$ \mathbb E[\alpha+\beta+\beta R]=\alpha+\beta\,\mathbb E[1+R]=\alpha+\beta\le 0\,, $$ and $$ \mathbb E[\alpha+\beta+\beta R]\ge 0\,. $$ Therefore, $$ \mathbb E[\alpha+\beta+\beta R]= 0\,. $$ Due to $\alpha+\beta +\beta R\ge 0$ it is not possible that $$ \mathbb P\{\alpha+\beta+\beta R>0\}>0\,. $$ Therefore, not arbitrage opportunity exists.

Kurt G.
  • 14,198
  • I am confused by your answer. Even if we add that additional assumption I don’t understand why you assume that $\mathbb{E}[1+R] = 1$. This is if and only if $\mathbb{E}[R] = 0$ but I am assuming that R has positive expectation. – leobgg Jun 21 '23 at 18:28
  • The martingale assumption is here $\mathbb E[1+R]=1$ because $S^1_0=1$ and the "numeraire" $S^0$ is always one. I did indeed overlook your assumption $\mathbb E[R]>0,.$ If you read the minimum about no arbitrage theory you will find that no-arbitrage still holds if another probability measure $\mathbb Q$ exists that is equivalent to $\mathbb P$ (has the same null sets) under which $S^1$ is a martingale. Can you repeat the entire proof with $\mathbb Q,?$ – Kurt G. Jun 21 '23 at 18:34
  • But to ensure that exists an equivalent martingale measure you would need no arbitrage (it’s the first fundamental theorem of asset pricing) hence we are back at the beginning. – leobgg Jun 21 '23 at 18:37
  • 1
    True. You asked for extra conditions that guarantee no arbitrage and we have collected the standard ones. Where does that question come from if it asks to show no arbitrage differently ? – Kurt G. Jun 21 '23 at 18:39
  • It’s a bit hard to explain where does my question comes from. But basically it boils down to “with just the assumptions listed can I prove that the market is arbitrage free?”. From what I understand by your reply you think that the answer is “No, you can’t”. Am I right? – leobgg Jun 21 '23 at 18:41
  • 1
    Before giving up completely let's see if we can prove something :) – Kurt G. Jun 21 '23 at 18:43