Think of a binomial tree. The greater the assumed volatility, the wider the future price swings, so the more likely that the bond will be called; the smaller the assumed volatility, the narrower the future price swings, so the less likely that the bond will be called.
When the bond is called, the bondholder receives a lower cash flow than he would if the bond weren’t called. Therefore, the greater the assumed volatility, the lower the future cash flows; the smaller the assumed volatility, the higher the future cash flows (though never greater than for a noncallable bond).
The lower the cash flows, the lower the discount rate required to get a specified PV: today’s market price; the higher the cash flows, the higher the discount rate. The lower the discount rate, the lower the spread (OAS) added to the treasury rates; the higher the discount rate, the higher the spread (OAS) added to the treasury rates.
So:
- Greater volatility, wider price swings, higher chance of being called. lower cash flows, lower discount rate, lower OAS.
- Smaller volatility, narrower price swings, lower chance of being called. higher cash flows, higher discount rate, higher OAS.
(I encourage you to go through the same sequence for putable bonds.)
Switching gears …
Z-spread is the spread – additional yield above that of a risk-free bond – paid to cover all risks,
including option risk. OAS is the spread – additional yield above that of a risk-free bond – paid to cover all risks
except option risk. The difference, therefore, is the spread to cover exactly the option risk: it’s the price of the option, measured in basis points of yield.