A stock option trader taught me yesterday that the correlation between the spot price of asset X and the variance of asset X is approximately -1. Can anyone give me a explanation why this is true?
I think the trader was referring to the leverage effect: stock price changes (not stock price) are negatively correlated with volatility. Stock price and stock price change (i.e. returns) are not the same thing.
If we consider variance and option pricing, then there are pricing models (example: Heston model) that represent the stock price evolution and the evolution of the variance of the stock price. The correlation between the two quantities is also considered and often it will be negative. This may be what the stock option trader was referring to, but assuming always a -1 correlation is a big approximation.
The statement is not correct as it stands for several reasons. First and foremost, it is only remotely meaningful if asset X is an option (or a similar type of derivative contract). Second, I think he meant 1, not -1. Third, correlation is probably not the best word to use.
The grounds for your trader friends statement is the work on option pricing theory done in the 1970's by Fischer Black, Myron Scholes, and Robert Merton (see here) (although possibly Ed Thorpe beat them to it by a decade).
The model they developed demonstrated that if stock prices follow a continuous-time semi-martingale (essentially a mathematical construct that implies you can't predict the direction in which stock prices will move), then the only relevant factor in determining the current price of an option is the volatility of the underlying security (that the option is written on). Strictly speaking, it is the volatility of the underlying security between the present time and the expiry time of the option.
So, if the model is correct, then an increase in volatility of the underlying will always simultaneously increase the present value of the option, while a decrease in volatility of the underlying will lead to a decrease in the value of the option. This implies the two are positively correlated, which is why I asserted in the first paragraph that your trader friend probably meant 1, not -1.
Now, correlation is a measure of linear response. But if you click through the link I provided above and have a look at the Black-Scholes-Merton formula, you'll notice that the relationship between the present value of the option and volatility is non-linear. So correlation is not really the right word to use.
Two final thoughts: First, empirically, the Black-Scholes-Merton formula only holds approximately, so the word "approximately" in your traders sentence is important. Second, remember in all of this that true volatility is unobservable, even ex-post.
Hope this helps. I've deliberately avoided going into the details of the Black-Scholes-Merton model and equation as I don't think they're really necessary to answer the question.