Relationship between asset volatility and debt and equity value

So how I understand it, higher asset volatility implies a higher call option price. The Merton Model holds that the value of equity is a call option. This therefore implies that the equity value must increase as well. Assuming this is correct (which it very well might not be), how is the value of debt affected? Does it stay the same? By Merton’s model, debt is a ‘short put’, so theoretically, it should also increase?

Also, if my interpretation of the relationship between asset volatility and equity value is correct, does this mean that equity holders will partake in riskier projects to maximise their value? Or is there something i’m missing.

Sorry if I sound like I have no idea what I’m talking about – as that might actually be the case…