  Identifying unambiguously the presence of a bubble in an asset price remains
an unsolved problem in standard econometric and financial economic approaches.
A large part of the problem is that the fundamental value of an asset is, in
general, not directly observable and it is poorly constrained to calculate.
Further, it is not possible to distinguish between an exponentially growing
fundamental price and an exponentially growing bubble price. We present a
series of new models based on the Johansen-Ledoit-Sornette (JLS) model, which
is a flexible tool to detect bubbles and predict changes of regime in financial
markets. Our new models identify the fundamental value of an asset price and
crash nonlinearity from a bubble calibration. In addition to forecasting the
time of the end of a bubble, the new models can also estimate the fundamental
value and the crash nonlinearity. Besides, the crash nonlinearity obtained in
the new models presents a new approach to possibly identify the dynamics of a
crash after a bubble. We test the models using data from three historical
bubbles ending in crashes from different markets. They are: the Hong Kong Hang
Seng index 1997 crash, the S&P 500 index 1987 crash and the Shanghai Composite
index 2009 crash. All results suggest that the new models perform very well in
describing bubbles, forecasting their ending times and estimating fundamental
value and the crash nonlinearity. The performance of the new models is tested
under both the Gaussian and non-Gaussian residual assumption. Under the
Gaussian residual assumption, nested hypotheses with the Wilks statistics are
used and the p-values suggest that models with more parameters are necessary.
Under non-Gaussian residual assumption, we use a bootstrap method to get type I
and II errors of the hypotheses. All tests confirm that the generalized JLS
models provide useful improvements over the standard JLS model.
