The volatility of the short-rate in the given equation is dependent on time. This is evident from the function itself, where the volatility term, denoted by $\sigma$, is a function of time, represented by $\sigma(t)$. This means that the volatility of the short-rate changes as time progresses, making it a time-dependent volatility. This is a key feature of many financial models, as it allows for the modeling of changing market conditions over time. The volatility of the short-rate is a crucial component in pricing options and other derivative securities, as it reflects the degree of uncertainty or risk associated with the price changes of these securities over a specific time period. Therefore, understanding how this volatility changes with time is essential for accurate pricing and risk management in financial markets.

Choice A is incorrect. The volatility of the short-term rate does not depend on the interest rate in this model. The equation given does not show any direct dependency of volatility ($\sigma$) on the interest rate ($\mathrm{d}r$). Instead, it shows that changes in the short-rate ($\mathrm{d}r$) are a function of a drift term ($\lambda(t)\,\mathrm{d}t$) and a diffusion term ($\sigma(t)\,\mathrm{d}w$), where $\sigma$ is dependent on time, not on the interest rate.

Choice B is incorrect. This statement is somewhat circular and misleading. In this context, $\sigma$ itself represents volatility; saying that "volatility depends on volatility" doesn't provide any meaningful insight or explanation about what factors influence changes in volatility over time.

Choice D is incorrect. While market prices can indeed influence perceived risk and thus impact implied volatilities in option pricing models, this specific model does not directly link short-rate volatility to changes in market prices of securities.