sparse grid method in the libor market model. option valuation and the

1. Tenor structure of N interest rates
2. Instantaneous volatility function σ i (t), i ∈ [1..N]
3. Correlation structure of LIBOR rates.
Tenor structure of interest rates is available to us immediately but volatility and correlation
require some extra work.
3.4.1 Calibration to caplets
LIBOR Market Model assumes that forward LIBOR rates follow a lognormal distribution,
i.e L i (T i ) = L i (t) exp(Z i+1 ), where Z i+1 ∼ N (− 1 2 κ2 , κ 2 ) under Q T i+1
,
and:
κ 2 =
∫ Ti
t
||σ(s)|| 2 ds
One can see that, in order to get a value for variance of a LIBOR process, we need to
integrate an instantaneous volatility function, that is unfortunately is not at our disposal
by default. What we do have is Black implied volatilites that are available to public.
Calibration to caplets implies that we should take quoted caplet volatility values as a
template/skeleton and fit the curve that matches the view of the market. Discussion on
approximating the instantaneous volatility function can be found in [Rebonato, 1998].
In this thesis, a simplified approach of a flat instantaneous volatility function will
be adopted. To be more specific, we shall assume LIBOR rate volatility to be constant
for a given tenor. Calibration to caplets in this case is particularly easy. By setting:
or
σ 2 BLACK(T i − t) = κ 2
√
1
σ BLACK =
T t − t
∫ Ti
t
||σ(s)|| 2 ds
we can accept Black volatilities as an average of an instantaneous volatility function
throughout the given tenor and use them directly.
3.4.2 Correlation structure of forward rates.
Having calibrated to caplet volatilities, the setup of the model is still not complete. In
our discussion of pricing under terminal measure we have neglected any correlation
that might exist between the LIBOR processes. As for this moment, any given process
for a LIBOR rate is driven solely by its Brownian Motion with drift, imposed by
change of measure. If we are modelling such tenor structure of N LIBOR rates, the
resulting processes can be shown in the following matrix-vector form with matrix A as
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