This section of the book develops the theory of simulation in the
LMRW. Although the theory for this is developed similarly to that for the Gaussian
HJM model, the results here are somewhat more complicated than those. In
particular, the drift term in the LMRW has an additional feature that makes it
different from that in the Gaussian model. Moreover, the methods for reducing
dimensionality and constructing the drift term for use in simulation are different
from those for the Gaussian model. Readers are recommended to review the
corresponding results in Chapter 6 to more deeply understand the properties of
the LMRW.
Most of the arguments in this chapter are based on Yasuoka (2013a); Section
9.2 is newly written to describe maximum likelihood estimation for the market
price of risk.
Some numerical examples will be shown in Chapter 10, using the same historical
data as in the example for the Gaussian HJM model.
Keywords: Complete market, Dimensionality reduction, Eigenvalue, Full-factor
model, Interpretation of the market price of risk, Least squares problem,
LIBOR market model, LMRW, Log-scale observable trend, Log-scale rolled
trend, Market price of risk, Monte Carlo simulation, Maximum likelihood estimation,
MPR score, Negative price tendency, PCA, Principal component
score, Properties of simulation, Real-world measure, Real-world simulation,
State space setup, Volatility component.
