The whole point of the theory is there is a correspondence between a 'no-arbitrage argument' and calculating an expectation under the 'risk-free' measure. The mathematical operation of expectation E is only a tool. This link between no-arbitrage and martingale theory is called the 'Fundamental Theorem of Asset Pricing'.
There is a nice (basic) explanation of this idea in the book by Baxter and Rennie "Financial Calculus" where they compare the 'expected value' approach of a bookmaker and the 'no-arbitrage' approach.
For a more advanced explanation, you can have a look at the book by Delbaen and Schachermayer (2006).
The no-arbitrage argument is another way of looking at it, but the two methods are equivalent. In particular, if the expected value of an option is higher than its price, you should buy the option - and if it's lower, you should sell it.
With just one transaction this would be statistical arbitrage rather than pure arbitrage, but if option prices regularly differed from the option's expected value, stat arb would be a fine strategy.
You are calculating an expected value in the sense of 'mathematical operation E' under some measure not in the sense of 'I expect the price to be...'
I don't want to pick a fight or anything, your product datanitro looks nice and it's cool you are writing articles on the topic.
Where did you learn this stuff from?
By expected value, I mean the price as you'd calculate it with a risk-neutral valuation based on some model of the underlying security.
For example, if you have a model that says an option is worth $4, and it's selling for $2, you should buy it if you're confident in your model. If you can do this repeatedly on a bunch of independent options you'll make money in the long run (assuming your model is correct and you're placing a large enough number of bets relative to the probability of making money on an individual option).
I learned this with a combination of practical experience, self-study, and coursework.