ETFs, in this case, SPY, are second-order or third-order derivatives? What about ETFs that follow the VIX? Fourth-order?

Derivatives of what? These are not derivative instruments nor do they represent the mathematical derivative of anything. They track what they are designed to track. This does not make them any order derivative.

No, you're wrong. We're not talking Calculus class here smart guy. In the world of trading, a derivative is defined as any vehicle that derives its value from an underlying tradeable asset. SPY is a derivative of the S&P index. Options are a derivative of a stock or an index. The VIX is a derivative of the volatility on options. That makes it a derivative of a derivative of a derivative. DcWriter is correct and its an intelligent question, so quit trying to be some know it all responding with cute answers... cause you're not, and it makes you look like a huge rookie. Now, regarding the OP's question in another thread... "How many ways are there to short the VIX?". Several. But for you... use $SVXY. ... f'n ET... "Elite Traders" my ass. Edit.... lmfao.... https://en.wikipedia.org/wiki/Derivative_(finance)

His use of second or third order derivatives is mathematically relevant. I'm not a "smart guy", I just don't use terminology I don't understand. Gamma is a second order derivative, delta is a first order derivative. This is common terminology in both math and finance. A derivative instrument is never referred to in its order. Options are not only on stocks or indexes. An option is a contract that specifies the right, but not the obligation to buy or sell a specific unit of the underlying at (European) or at or before (American) it's expiration. You can trade options on anything as long as there is a market. We could even talk exotics like Bermudans if you'd like. In fact, insurance companies are basically option trading companies in the most broad definition. The VIX is an index that tracks the implied volatility of the S&P. You may have an argument that the VIX (index) is a derivative, and it's futures are certainly derivatives. By the way ETFs are not derivatives, smart guy. Let me repeat this to make it clear ETFs are not derivatives. Go back into Hull and tell me where his definition defines a derivative as "any element that derives it's value from an underlying tradeable asset". A derivative INSTRUMENT derives its value from the performance of an underlying asset not its value. Specifically, derivatives have contract level specifications on the performance of the tradeable asset they are written on. Are we going to call a mutual fund a derivative instrument now too? I'll concede leveraged ETFs may be able to considered a derivative in the loosest possible sense. But you are wrong in the general sense. What about hedge funds? Are they derivatives? How about trading spot gold? You'd be laughed at out of an interview for a banker job thinking like you do. Speaking of looking like a rookie - you don't even know what a derivative is. Pot calling the kettle black...hilarious. How about you cut back on the insults when you have no idea what you're talking about.

Lol. I don't know what a derivative is huh? There's a lot of things I don't know... for example when @GRULSTMRNN talks about stuff that totally impresses me... because I have no clue wtf he's talking about... https://elitetrader.com/et/threads/...-trading-operation.337140/page-2#post-4949114 ... but you know what hasn't changed in 200 plus years? Calculus. Fwiw... your favorite VZ made straight A's all the way thru 2nd order DEQ. BFD. But its safe to say, I do understand what a derivative is. The VIX is, options are, and so are index futures. ETF's... did I say that? Well, maybe not. Either way... you want to argue semantics. Below me. Move on.

The OP did call this a statistics question. The problem there, is that once you start using technical terminology, people will begin to interpret what you say technically. So him saying that SPY is a second order derivative implies that his assumption is incorrect. However, it is not clear where he went wrong. The question is not well posed.

SPY is a derivative of the S&P 500 index, in the financial definition of the word derivative. This S&P 500 index is a derivative of the stock prices of about 500 underlying companies. Each of these stock prices is multiplied by a weighing factor and then added, to reach the overall value of the S&P 500 index. So, using the word "derivative" not in a mathematical but a financial context, the SPY is a second order derivative of about 500 stock tickers. Is this the answer you're looking for?

SPY - itself is not a derivative. It's an actual fund holding the 500 issues. An SPX basket would also not be a derivative. Options and futures on the SPX are derivatives as are options on the SPY.

OP used xth-order derivatives and the title contained "statistics question" hence he was most likely referring to the technical derivatives terminology where even an etf that models spx is not considered a derivative. Vix, however, can be interpreted in this specific context as the derivative of spx as it is priced based on the implied volatility of specific spx index options. There is a high certainty that op refers to derivatives here as Greeks.