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3

For Q1, the function $a(t)$ is the instantaneous correlation. The form given by (2) is basically the Cholesky decomposition. Of course, you may directly show, uisng Levy's characterization, that $$\widetilde{W}(t) = \int_0^t\bigg[\frac{1}{\sqrt{1-||a(t)||^2}} dZ(t) -\frac{a(t)^T}{\sqrt{1-||a(t)||^2}} dW^B(t) \bigg]$$ is a standard scalar Brownian motion ...

3

Q1: $$(1)\rightarrow(2)$$ (1): $a(t)$ is the instantaneous correlation of $\rho(Z_t,W_t)$ because: $$\rho(dZ_t,dW_t)=\dfrac{Cov(dZ_t,dW_t)}{\sigma_{dZ_t}\sigma_{dW_t}}=\dfrac{E(dZ_t\cdot dW_t)}{\sqrt{dt} \sqrt{dt}}=\dfrac{\langle dZ_t, dW_t\rangle}{t}=a(t)$$ $\Rightarrow$ (2) holds as following, in the 1-dim case: $dZ_t\sim N(0,dt),$ ...

2

For a swap, we have a sequence of re-setting and payment dates. The # of forward rates corresponding to the # of payment dates. For example, let us assume that we have $n$ payment dates $t_1, \ldots, t_n$, where $0< t_1 < \cdots < t_n$. Then there are $n$ forward rates. During the simulation, for time steps prior to $t_1$, there exist $n$ ...

1

There are two things that might be confusing you. The time step in Time dimensions and time steps along the forward curve. The first is given a time t from today until a certain day in the future, this dt usually is the next reset date. The the other is tau representing a tenor for the forward curve maturing in tau days ahead. Dtau could vary ...

1

Thanks to my research leader, I found what I missed. $V_{0,1}$ is vol of swaption that matures at $T_0$ which is not 0 (as I thought), rather it is maturity of the first libor. So $V_{0,1}$ is the closest available point on market. And now this is all clear with table on page 323 in section 7.4. $V_{0,2}$ is realy vol of swaption that matures at $T_0$=1y ...

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