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

with ɛ i being independent standard normal random variables. In practice, we need to
model the dynamics of a tenor structure of N forward rates. Pricing under terminal
measure in LMM introduces a drift-correction term into an individual LIBOR process.
So, discretization of (3.17) yields:
L i (T n+1 )
L i (T n )
= exp
{(
−σ i (T n )
N∑
k=i+1
α k ρ ik σ k (T n )L k (T n )
− σ i(T n ) 2 )∆t+σ i (T n ) √ }
∆tɛ i
1 + α k L k (T n ) 2
(4.3)
Having simulated the relevant LIBOR paths for the underlying rates with (4.3), we
can compute sample payoff value for a derivative at maturity. In other words, we get a
sample value of an independent random variable V (T ) that represents an option payoff.
Next step, by recipe, is to repeat the sampling procedure n number of times and take the
average over the all sampled payoff values. In the limit, by the Law of Large Numbers,
this average will converge to the expectation value.
Recall that, in order for the risk-neutral valuation principle to hold, we must operate
in terms of numeraire-rebased quantities:
[ ]
V (t) V (T )
N(t) = EQ N(T ) ∣ F t
where N is the numeraire asset and Q is the unique equivalent martingale measure
imposed by the choice of N. Thus, we have to make sure that every sample payoff
V (T ), is denominated in terms of the terminal bond D, representing our choice of
numeraire in LMM. Then, by averaging over all D-rebased sample payoff values, we
shall obtain the value of the Q-expectation and, equivalently, V (t)/D(t, T ).
Standard Error If we are able to obtain the approximate value for expectation, it is
easy to get a number for standard deviation ω. As in [Hull, 1999], the expression for a
standard error of Monte Carlo is:
ω
√ n
A convenient fact from [Hull, 1999] also states, that a 95% interval of trust for the
derivative lies within approximately 2 standard errors:
µ − 1.96ω √ n
< c t < µ + 1.96ω √ n
Sometimes the standard error of MC simulation can be reduced by means of variance
reduction techniques, such as antithetic variables or control variates. Yet, this subject is
outside the scope of present work. A useful reference to such techniques is [Rebonato,
1998].
Correlated Brownian Motions If we recall the discussion from the previous chapter
on calibration to correlation between the rates, an individual LIBOR process can be
driven by a composite Brownian motion ˆɛ i such, that:
ˆɛ i =
i∑
α ik W k
k=1
31