Abstract:
Volatility is considered as a measure of risk of financial assets which is vital for prudent
financial decision of different stakeholders. GARCH models have been commonly
used to capture volatility dynamics of financial time series, particularly in modeling
volatility of stock returns and pricing of futures and options. Despite the generalization
of GARCH (p;q) model to assume different model orders, GARCH (1;1) model
continues to be widely used by practitioners when modeling volatility of financial assets
returns. A key assumption of the GARCH models utilized is that the processes are
stationary. This assumption allows for model identifiability. Financial asset returns,
however, often exhibit the volatility clustering property implying that assuming one
GARCH model is a poor fit. The IGARCH model may be perceived as a solution to
this problem as the assumption of stationarity is relaxed and thus the model is able
to model persistent changes in volatility. However, the IGARCH model is prone to a
shortcoming where the behavior of the process depends on the intercept. In this work,
change-point estimation is proposed as a solution to deal with this problem where observed
non-stationary series is assumed to be composed of a series of stationary series.
The main objective of this work is therefore to propose an estimator for the changepoint
which is considered as the point in time at which the series departs from one
stationary GARCH model with order (1;1) to another stationary GARCH model with
order (p;q). Given that plausible values for the model orders p and q can be arrived
at through inspection of sample autocorrelations and sample partial autocorrelations
of a squared returns series, a change-point estimator based on the Manhattan distance
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of sample autocorrelation of squared series is proposed. The estimator is given as
the first point in time at which the Manhattan distance is maximum. To facilitate
the detection of multiple change-points, binary segmentation technique is applied to
extend the single change-point detection algorithm.
The asymptotic consistency of the estimator is proven theoretically based on some
properties specific to sequence of stationary random variables with finite second and
fourth moments. The limit theory of the process generating the estimator is also established.
The general theory of the sample autocovariance and sample autocorrelation
functions of a stationary GARCH process forms the basis. Specifically the point processes
theory is utilized to obtain their weak convergence limit at different lags. This
is further extended to the change-point process. The limits are found to be generally
random as a result of the infinite variance.
Monte Carlo simulations is used to examine the performance of the estimator when
the sample size, size and position of change vary using the Adjusted Rand Indices.
It is established that ARI increases and tends to one as the size of change increases
irrespective of the sample size and of the source of change. Histograms are utilized to
assess the sampling distribution of the change-point estimator.
The research culminates with the application of the change-point estimator in pricing
American options. Comparison is made between the performance of the fitted GARCH
models and Black-Scholes model by examining plots of the option prices against moneyness.
The fitted piecewise GARCH model, following change-point detection, gives
higher prices compared to the Black-Scholes when the option is out-of-the-money indicating
that the volatility dymanics affect the prices of options. It is therefore important
for a investor trading in American options to consider change-points within the
volatility structure of a financial returns series when choosing an early exercise date.