Positive Expected Value, Unbounded Investment, and Arbitrage

# Is this arbitrage? Infinite payoff / infinite loss (energy generation investment problem)


# Is this arbitrage? Infinite payoff / infinite loss (energy generation investment problem)












I'm a student using stochastic optimization in energy systems and I have a particular phenomena in an optimization problem that I think must occur in finance aswell, so I have been trying to find allegorical cases and terminology in finance literature. Please bear with my amateurism!

Simplified problem: I make a decision at $t=0$ to install capacity $x^G$ units of an electricity generator for an discounted annualised CAPEX cost of (say) $C^G=70€$ per unit installed. The electricity will be sold on the market at price $\pi_t$, at time $t=1$.

The payoff at $t=1$ is thus:

$V_1= (\pi_1 - C^G) \cdot x^G$

However the market price is uncertain with two possible scenarios $w_1$, $w_2$:

$\pi_1(w_1)=300€$, giving a positive payoff $V_1(w_1)=230€ \cdot x^G$ with probability $P(w_1)=0.25$

$\pi_1(w_2)=10€$, giving a loss of $V_1(w_2)=-60€ \cdot x^G$ with probability $P(w_2)=0.75$

As such, the expected value of my payoff is positive:

$E[V_1]= 6.25 \cdot x^G$

If I am risk neutral, my objective function will be:

$\underset{x^G}{\max} E[V_1]$

Which will lead the optimizer to simply increase the capacity installed $x^G$ to its heart's content because increasing $x^G$ always increases the expected value.

In this case, the expected payoff will go to infinity (unless limited by the upper bound of $x^G$, or if you put in place an additional budget constraint). But in this case I will either have an infinite profit or an infinite loss.

MY QUESTION - I suspect that this behaviour might fit the definition of an arbitrage in finance theory, however I'm hesitant because it seems like it does not fulfil the properties of (i) having a net-zero initial cost, (ii) having strictly positive payoff (it only has strictly positive expected value).

In terms of the existence of a martingale measure proof, I'm struggling to understand the theory here, but it seems like you could construct one by changing the probabilities of the two scenarios, and so such a measure could exist...? Meaning that this is not technically an arbitrage?

Thanks in advance for your help!

What I have been reading:

The Mathematics of Arbitrage, 2008, Delbaen

Stochastic Finance, 2015, Follmer

## Answer by Enrico Schumann (score 1)

https://quant.stackexchange.com/a/78419

Arbitrage means that you can a profit (in at least some states of the world), without the risk of losing. IIUC, in your state 2, you'd make a loss, and the bigger your investment x, the bigger the loss. So it's not an arbitrage.

A positive expected value is not sufficient for arbitrage.

You'd rather handle such a case by a budget constraint or, more indirectly, by a utility function that penalizes risk.

## Answer by Achrbot (score 0)

https://quant.stackexchange.com/a/78423

Consider your scenario as a market with 1 traded asset $V$, with state-dependent payoff $\pi_1 - C^G$. We can construct an equivalent martingale measure $\mathbb{Q}$ by $\mathbb{Q}(w_1) = 60/290$, so the market does not contain arbitrage.

As you mention, arbitrage requires the existence of a portfolio of zero initial cost, with guaranteed non-negative (and positive expected) returns. Intuitively, this means that all agents with increasing utility functions (not just risk-neutral) would buy the portfolio.