Does the market price of risk, be it of stochastic volatility, interest rate or equity return, have to be positive? What is the rationale if it does?
No, it can be negative. The price of risk is what you agree to receive on average in exchange for positive returns when the risk measure is high, and determined by the covariance of the risk measure with your marginal utility of consumption. That said, stochastic volatility risk is negatively priced: you happily agree to a negative return on average in exchange for positive compensation in times of higher variance, because it is these times when your marginal utility is high. Correlation risk is negatively priced. Equity market risk is positively priced: when the stock market is bullish, you tend not to need positive returns as much as you need them when the market is bearish, so you require a positive compensation on average to be exposed to this risk.