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You can use a matrix type seperability condition as well. This is similar but the equation has more flexibiliity. The rates are then markovian in some combinations of the Brownian motion. See More Mathematical Finance for details.

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The following paper, Interpolation Schemes in the Displaced-Diffusion LIBOR Market Model and the Efficient Pricing and Greeks for Callable Range Accruals, addresses this issue: We introduce a new arbitrage-free interpolation scheme for the displaced-diffusion LIBOR market model. Using this new extension, and the Piterbarg interpolation scheme, we study ...

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The subject is interesting and not so easy if you want to interpolate in an arbitrage-free way, to my knowledge a good paper on the subject is this one

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Re 2: It's a mathematical trick. Insisting on the separability of volatility function makes LMM useless. Its power lies in its powerful calibration abilities. If you constrain the vol function to separable form, you throw that ability out of the window. You might just as well use LGM then, and it will be more intuitive and faster.

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You might want to set $a= \epsilon - d$ and write $\epsilon>0$ as a constraint. I guess $\textbf{lsqnonlin}$ is the suitable fonction for what you intend to do. I personnally like to use and play around with $\textbf{fmincon}$, which allows more flexibility and performs well, if you are willing to provide Jacobian and/or Hessian in algorithms options

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You could bootstrap a curve based on the forward rates you get, plus your standard interpolation scheme. That's certainly what you'd do if the rates were presented to you as a set of market quotes for FRAs. It does ignore the evolution of the future forward rates though, so I'd expect it to work best if the intetpolated rate is close to one of your ...

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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 ...

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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: $dZ_t\sim N(0,dt),$ $dW_t,\tilde{dW_t}\... 2 For a swap, we have a sequence of re-setting and payment dates. The # of forward rates corresponding to the # of payment dates. For example, let us assume that we have$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$n$"... 2 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 ... 1 If I have read the question correctly then I will assume that$a$,$b$,$c$,$d$,$T_i$, and$k_i$are constants. If this is the case then the only term which we need to show is bounded is $$\big(a + b(T_i - t)\big)\exp\big(-c(T_i-t)\big).$$ If we assume that we are only considering the temporal domain$0 \leq t \leq T_i$such that$T_i - t \geq 0 $then ... 1 The forward Libor rate at time$t$is the forward rate over a certain accrual period$[T, T+\Delta]$, where$\Delta, in years, can be 3 months or 6 months, and is defined by \begin{align*} L(t, T, T+\Delta) = \frac{1}{\Delta}\left(\frac{P(t, T)}{P(t, T+\Delta)}-1 \right), \end{align*} whereP(t, u)$is the price at time$t\$ of a zero coupon bond with unit ...

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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 ...

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