<< Trader: (digging in heels, determined not to be wrong) Right now the positions I have offset each other in non-linear ways not captured by covariance used in VaR. The risk that I am carrying is much lower than you think. >>
Remember Tommy Lee Jones in [i]The Fugitive[/i]? Harrison Ford says, "I didn't kill my wife." Tommy Lee Jones, slightly surprised, replies "I don't care." His job isn't to pass judgment on murderers, it's to bring in the people he's told to bring in.
When I'm on the desk, my job isn't to rewrite risk management models on the fly. VaR is not one model, it's a host of models aggregated together with a bunch of assumptions. There are lots of inaccurate models and bad assumptions, but it still works. I trust it because I have looked carefully at its performance in many situations over a long period of time; I trust it far more than I trust trader's assertions, especially when embellished with mathematical terms. Asserting there is one new bad assumption I don't know about isn't going to make me throw away years of backtests.
Remember, I don't believe in VaR limits. I'm not telling the trader he has to cut back positions. I'm asking him to explain his profit potential in a way that justifies $5 million VaR. He's welcome to use his "non-linear ways" to analyze the center of the distribution, to claim a high expected return and low standard deviation. I might or might not believe him. But I'm not going to accept it as a way to estimate the 1% tail without a lot of evidence, which he doesn't have and probably doesn't exist. In any case, that's a question for a different time and place.
It's true that the conversation sometimes breaks down along the lines you suggest. The trader claims it's mathematically impossible for him to lose more than $1 million. So the argument gets kicked upstairs. I say I think the trader is out of control, making trades without realistic consideration of risk. It's not a question of him being right or wrong about the loss potential, it's his insistence that the firm take his word without going through the process of presenting a model and having it validated, then analyzed for risk.
If you're betting your own money, it's enough to be right. If you're part of a modern trading organization, chaotic profits without risk control require so much economic capital and such a high discount rate that they're rarely positive EV from a shareholder value perspective.
Anyway, the risk manager doesn't always win these arguments. But that's okay, it's not his job to win all the time. His job is to manage the risk as best he can. If he gets overruled, that's okay too. He wasn't appointed CEO or head trader, those people can decide to throw away the models and trust someone's instinct, like Luke Skywalker turning off his computer before bombing the Death Star in the first [i]Star Wars[/i]. It always works in the movies, it may sometimes work in real life.
Going back to the original question, however, the point is that VaR is a specific to argue about. Without it, or with a model about unobservables, you don't even get to the stage where the trader asserts non-linear effects not captured in the covariance matrix.