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8

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.


7

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 ...


4

this is a well-known problem. One solution is to make volatility zero when rates exceed a certain high level. It's less problematic than it looks because any cash-flows generated will be divided by a rolling money market account which has huge value and so the deflated cash-flows are very small.


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: $dZ_t\sim N(0,dt),$ $dW_t,\tilde{dW_t}\...


3

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

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


2

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*} where $P(t, u)$ is the price at time $t$ of a zero coupon bond with unit ...


2

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 ...


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 ...


2

We assume that, under the $T_j$-forward probability measure $P_{T_j}$, \begin{align*} \frac{dP(t, T_j)}{P(t, T_j)} = \mu_P(t, T_j) dt + \sigma_P(t, T_j) dW_t^{T_j}, \end{align*} where $\mu_P(t, T_j)$ and $\sigma_P(t, T_j)$ are the respective drift and volatility functions. Let $Q$ be the risk-neutral probability measure. Then \begin{align*} \frac{dQ}{dP_{...


2

Just to be precisely clear, your mathematical formulation will not necessarily capture the nuances of the physical dates that libor is valued between, due to holiday calendars and modification rules. Take GBP for example. The LIBOR in that currency is subject to a Modified Following rule as well as a Month End Consistency rule. For example: Generally 6M ...


1

For example a Caplet with Expiry of 3year with tenor = 0.5 has to be priced (following the analytical formula) with the LIBOR rate L(0,2.5,3). Am I getting it right ? Thats right. The caplet hast a tenor of half a year and expires in 3 more years, therefore it starts at T =2.5 and ends at T = 3. (Which in this case is the forward rate)


1

From a practitioner standpoint, we know the prices of non accreting swaptions. The price of the accreting swaption in any model calibrated to these non accreting swaptions, is heavily dependent on the intra curve correlation assumptions in the model. We check that these correlations are consistent with other correlation dependent markets such as curve ...


1

The explosion of the forward rates in the log-normal LMM simulated in the spot measure seems to be related to the explosion of the Eurodollar futures prices in this model which was studied in this paper http://www.tandfonline.com/doi/abs/10.1080/1350486X.2017.1297727 The Eurodollar futures prices are given by the expectation of the Libor in the spot ...


1

The rates will explode in the current low rates environment my friend where empirically they are at a too low level to use a log-normal model if you want to preserve your log-normality please use a shifted log normal distribution instead to a convenient rate cut off of around 2%. This happens mainly on EUR market. Hope this help


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 ...


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