The volatility of the short rate in one year is 125 basis points while the volatility of the short rate in two years is 128 basis points per year. In the context of time-dependent volatility functions, the volatility of short rates is expressed as a function of time. The function $\sigma(t)$ represents the volatility of the short rate at time $t$. Therefore, $\sigma(1) = 1.25\%$ represents the volatility of the short rate in one year, and $\sigma(2) = 1.28\%$ represents the volatility of the short rate in two years. The values $1.25%$and `$1.28\%$ are expressed in basis points, where 1 basis point is equal to $0.01`\%$. Thus, the volatility of the short rate in one year is 125 basis points, and the volatility of the short rate in two years is 128 basis points. This interpretation is consistent with the concept of time-dependent volatility, where volatility can change over time.

Choice A is incorrect. The values $\sigma(1) = 1.25\%$ and $\sigma(2) = 1.28\%$ do not represent the volatility of the short rate in 6 months and one year respectively. Instead, they represent the volatility of the short rate in one year and two years respectively.

Choice B is incorrect. This choice incorrectly suggests that the volatility of the short rate increases from one year to two years, which contradicts with given values where $\sigma(1)$ is less than $\sigma(2)$.

Choice D is incorrect. These values do not signify time-dependence of short rates but rather they represent volatilities at different points in time.

Things to Remember

• Time-dependent volatility functions are used to model how volatility changes over time, allowing for more flexibility in pricing options.
• Volatility is a key parameter in option pricing models, as it represents the degree of variation of a trading price series over time.
• Basis points are commonly used in finance to describe interest rates, bond yields, and volatility changes. One basis point is equal to 0.01%.
• Volatility functions can be deterministic or stochastic, with deterministic functions assuming a specific form of volatility evolution over time.
• Stochastic volatility models incorporate randomness in the volatility process, allowing for more realistic and dynamic representations of market conditions.