Stochastic Volatility with Levy Processes: Calibration and Pricing
Stochastic Volatility with Levy Processes: Calibration and Pricing
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Date
2005-11-24
Authors
Wu, Xianfang
Advisor
Madan, Dilip B.
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Abstract
In this thesis, stochastic volatility models with Levy processes are treated
in parameter calibration by the Carr-Madan fast Fourier transform (FFT) method and pricing
through the partial integro-differential equation (PIDE) approach.
First, different models where the underlying log stock price or volatility
driven by either a Brownian motion or a Levy process are examined
on Standard & Poor's (S&P) 500 data. The absolute percentage errors show that the calibration errors
are different between the models. Furthermore, a new method to estimate the standard errors, which can be
seen as a generalization of the traditional error estimation method, is proposed
and the results show that in all the parameters of a stochastic volatility model,
some parameters are well-identified while the others are not.
Next, the previous approach to parameter calibration is modified by making the volatility constrained under
the volatilty process of the model and by making the
other model parameters fixed. Parameters are calibrated over five consecutive
days on S&P 500 or foreign exchange (FX) options data. The results show that the absolute percentage errors do not get much larger
and are still in an acceptable threshold. Moreover, the parameter calibrating procedure is stabilized
due to the constraint made on the volatility process. In other words, it is more likely
that the same calibrated parameters are obtained from different initial guesses.
Last, for the PIDEs with two or three space dimensions,
which arise in stochastic volatility models or in stochastic skew models,
it is in general inefficient or infeasible to apply the same numerical technique to different parts
of the system. An operator splitting method is proposed to break down the complicated problem
into a diffusion part and a jump part. The two parts are treated with a finite difference and a finite element method, respectively. For the PIDEs in 1-D, 2-D and 3-D cases, the numerical approach by the operator splitting
is carried out in a reasonable time. The results show that the operator splitting method is numerically stable and has
the monotonicity perserving property with fairly good accuracy, when the boundary conditions
at volatility are estimated by Neumann conditions.