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2

The Feynman-Kac theorem can be used in both directions. That is, If we know that $r_t$ follows the Ito process as described by the following stochastic differential equation \begin{align} d{{r}_{t}}=\mu ({{r}_{t}},t)dt+\sigma ...

2

Bond Prices Assume that the short rate $r_t$ follows the Ito process as described by the following stochastic differential equation \begin{align} d{{r}_{t}}=\mu ({{r}_{t}},t)dt+\sigma ({{r}_{t}},t)d{{W}_{t}^{P}} \end{align} we assume the bond price to be dependent on $r_t$ only, independent of default risk, liquidity and other factors. If we write the bond ...

3

There is actually a lot of art involved. The most simplistic framework is as follows: The first step is to obtain a list of FOMC meeting dates. These are available currently for 2015 and 2016 here. If you're interested for rate expectations beyond 2016, you'd need to "guess" the meeting dates in the future based on past patterns. The next step is to ...

3

I am not sure how that probability was computed. However, the standard approach is to use fed futures to proxy for the "unexpected change" of FED rate. The most prominent reference is Bernanke and Kuttner (2005). What they do, is to estimate the unexpected FED target rate change by doing: $\Delta i^u = \frac{D}{D-d}(f_{m,d}^0-f_{m,d-1}^0)$, where ...

0

For a two-factor option pricing model with underlying variables $S$ and $r$ defined as above, if we assume there is no correlation between the two Wiener processes $W_1$ and $W_2$, one finds the generalized Black-Scholes PDE \begin{align} V_t+\frac{1}{2}\sigma^2V_{SS}+r\,S\,V_S-r\,V+\frac{1}{2}\Sigma\,^2\,V_{rr}+\kappa(\theta-r)V_r=0 \end{align} This ...

5

Mean reversion speed $\kappa$ is better interpreted with the concept of half-life, which can be calculated from $\text{HL} = \ln(2) / \kappa$. For example, if the mean reversion coefficient is $\kappa = 1.5$, then the half-life of the process is $\ln(2) / 1.5 = 0.46209812$ years, or about 6 months. Let's assume that the current interest rate is 1% and the ...

0

In the Mean-Reverting Models like C.I.R \begin{align} &dr_t=\kappa(\theta-r_t)dt+\sigma\sqrt {r_t} dW_t \end{align} speed of mean reversion ($\kappa$) is not negative.If the condition $2\kappa\theta> \sigma^2$ holds, then the drift is sufficiently large for the process to be guaranteed positive and not reach zero. This condition is known as the Feller ...

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