I see a lot of academic papers talking about accuracy in pricing American Options (and finding analytic solutions). Why is there so much interest in this topic? Isn't the option price set by the market? From what I can understand, accuracy in pricing gives accuracy in implied volatility, which gives you a sense of which options are expensive relative to others (eg. A listed puts for the same underlying, same maturity but varying strike may have different volatilities, even though they should be the same). I don't see why as an option trader accuracy is important, since the market already decided the fair price.
The market creates price quotes only for some standard products. But maybe you have some products in your portfolio that are non standard (e.g. because the market has changed since you bought it).
Now you want to know what your assets are worth. What to write to your balance sheet? Or maybe you want to sell it. how much should you be payed? The market doesn't tell you directly. You have to to calculate it. And typically you want to know it exactly.