# 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?

The Mathematics of Arbitrage, 2008, Delbaen

Stochastic Finance, 2015, Follmer

• One thing to keep in mind is that, especially in deregulated physical power markets with localized nodal power prices and transmission constraints, the price of electricity your asset can realize will greatly decrease as your market area becomes oversupplied. So one could argue that your payoff function is also monotonically decreasing as generation capacity increases based on complex transmission networks and demand sources. Feb 21 at 4:48
• Great point Mason. In my case I'm working with a 'price-taker' assumption so this dynamic is considered out-of-scope for my modelling but I'm defintely going to incorporate that obesrvation in my manuscript ;) cheers! Feb 26 at 14:03

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.

• Cheers Enrico !! Yes I've been using a CVAR objective function which works quite well as long as it is risk averse enough. When performing portfolio optimization in finance which kind of utility functions would you say are the most common? It seems like the quadratic is very popular in general, but then for pure hedging optimization, people have been using the CVAR more recently? (Sorry if it is innappropriate to add on additional questions) Feb 26 at 13:54
• In the academic literature on portfolio optimization, variance (i.e. quadratic utility) is very common, but so is power utility. But I'd go through papers with problems similar to yours, and look what kind of functions they have used. Feb 28 at 18:05
• Thanks Enrico! :) Mar 7 at 16:55

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.

• Awesome explaination! Thanks Archrbot, that makes it very intuitive for me. Feb 26 at 13:59