The VIX index, a mean-reverting proxy for risk aversion on the S&P500, is currently relatively low. This makes buying options, both puts or calls, relatively cheap. Usually volatility is negatively correlated to spot price of the underlying instrument (in this case the S&P500), but if the market remains flat and there's a significant spike in volatility, the position can still be P&L positive.

As for the convexity, assuming the puts he's buying are out of the money, his total position increases in $ value as it becomes in the money and vice versa. The opposite is true with an outright short.