ABSTRACT
The LIBOR Market Model has become one of the most popular models for
pricing interest rate products. It is commonly believed that Monte-Carlo
simulation is the only viable method available for the LIBOR Market
Model. In this article, however, we propose a lattice approach to price
interest rate products within the LIBOR Market Model by introducing a
shifted forward measure and several novel fast drift approximation
methods. This model should achieve the best performance without losing
much accuracy. Moreover, the calibration is almost automatic and it is
simple and easy to implement. Adding this model to the valuation toolkit
is actually quite useful; especially for risk management or in the case
there is a need for a quick turnaround.
Key Words : LIBOR Market Model, lattice model, tree model,
shifted forward measure, drift approximation, risk management,
calibration, callable exotics, callable bond, callable capped floater
swap, callable inverse floater swap, callable range accrual swap.
an interest rate model based on evolving LIBOR market forward rates
under a risk-neutral forward probability measure. In contrast to models
that evolve the instantaneous short rates (e.g., Hull-White,
Black-Karasinski models) or instantaneous forward rates (e.g.,
Heath-Jarrow-Morton (HJM) model), which are not directly observable in
the market, the objects modeled using the LMM are market observable
quantities. The explicit modeling of market forward rates allows for a
natural formula for interest rate option volatility that is consistent
with the market practice of using the formula of Black for caps. It is
generally considered to have more desirable theoretical calibration
properties than short rate or instantaneous forward rate models.
In general, it is believed that simulation is the only viable numerical
method available for the LMM (see Piterbarg [2003]). The simulation
is computationally expensive, slowly converging, and notoriously
difficult to use for calculating sensitivities and hedges. Another
notable weakness is its inability to determine how far the solution is
from optimality in any given problem.
In this paper, we propose a lattice approach within the LMM. The model
has similar accuracy to the current pricing models in the market, but is
much faster. Some other merits of the model are that calibration is
almost automatic and the approach is less complex and easier to
implement than other current approaches.
We introduce a shifted forward measure that uses a variable substitution
to shift the center of a forward rate distribution to zero. This ensures
that the distribution is symmetric and can be represented by a
relatively small number of discrete points. The shift transformation is
the key to achieve high accuracy in relatively few discrete finite
nodes. In addition, we present several fast and novel drift
approximation approaches. Other concepts used in the model are
probability distribution structure exploitation, numerical integration
and the long jump technique (we only position nodes at times when
decisions need to be made).
This model is actually quite useful for risk management because normally
full-revaluations of an entire portfolio under hundreds of thousands of
different future scenarios are required for a short time window (see
FinPricing (2011)). Without an efficient algorithm, one cannot properly
capture and manage the risk exposed by the portfolio.
The rest of this paper is organized as follows: The LMM is discussed in
Section I. In Section II, the lattice model is elaborated. The
calibration is presented in Section III. The numerical implementation is
detailed in Section IV, which will enhance the reader’s understanding of
the model and its practical implementation. The conclusions are provided
in Section V.
- LIBOR MARKET MODEL
Let (,,,) be a filtered probability space satisfying the usual
conditions, where denotes a sample space, denotes a -algebra, denotes a
probability measure, and denotes a filtration. Consider an increasing
maturity structure from which expiry-maturity pairs of dates (,) for a
family of spanning forward rates are taken. For any time , we define a
right-continuous mapping function by . The simply compounded forward
rate reset at t for forward period (,) is defined by
(1)
where denotes the time t price of a zero-coupon bond maturing at
time T and is the accrual factor or day count fraction for period
(,).
Inverting this relationship (1), we can express a zero coupon bond price
in terms of forward rates as:
(2)