ESTIMACION DE MODELOS DE VOLATILIDAD ESTOCASTICA

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ESTIMACION DE MODELOS DE VOLATILIDAD ESTOCASTICA
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Stochastic Volatility Models in Finance

• Black-Scholes model (73)
• Cox model (Cox 85)
• Time deformed model (Mandelbrot 67, Ghysels 95)
• Micro market model (OHara 95, Bouchard 98,
  Farmer 98)

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Inference problems in Volatility Models

• I Estimation of diffusion models with
  complete discrete observations

Given a model for an asset price P
the inference problem to solve is
given a time series of asset price
Estimate
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Inference problems in Volatility Models (cont)

• II- Identification of diffusion models with
  partial discrete observations

Given a model for an asset price P
the inference problem to solve is
given a time series of asset price
Estimate
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Inference problems in Volatility Models (cont)

• III- Identification of diffusion models from
  partial and noisy discrete observations

Given a continuous model for an asset price P
and a discrete observation equation
the inference problem to solve is
given a time series of asset price
Estimate
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State Space formulation of the models
Continuous model equation
Discrete observation equation
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Inference methods for diffusion processeswith
complete discrete observations
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Inference methods for diffusion processeswith
noisy partial discrete observations
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Maximum Likelihood method
where and the discrete
innovation and its variance

• Practical computation of
• Kalman filter (Schweppe 65)
• EM algorithm (Singer 93)

are consistent and
asymptotically normal (Jensen Pedersen 99)
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Prediction-based estimating functions for SDEs
with noisy observations (Nolsoe et al., 2000)
The EF is defined by where the
optimal weights are obtained by
considering that

has the same statistical and computational
properties as the prediction-based estimators for
models with noisy free observations

• Applications in Finance
• None up to now (only simulations with the CIR
  model)

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Innovation Method
Nonlinear model
Linear observations
Discrete innovation
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Innovation Method

• Properties of the estimators
• is consistent and asymptotically normal (Ljung
  Caine 79)
• coincides with ML estimator for linear models
  with additive noise

• Practical computation of and
• Recursive filters
• - Local Linearization Filters (Ozaki 94, Shoji
  98, Jimenez Ozaki 02)
• - Second order filters (Nielsen et al. 98, 00)
• Simulated filters
• - Particle filter (del Moral, 01)

• Properties of the approximate estimators
• Recursive filters produce biased estimators, but
  the bias is negligible in most applications with
  small and moderate Dt. They are computationally
  efficient.
• Simulated filters produce unbiased estimators
  for any Dt, but they are computationally expensive

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Innovation Method
Comparative study (Nielsen et. al, 00)
For simulations of the CIR model, the Innovation
method provides similar or better results of
those obtained by the Prediction-based estimating
function with much lower computational cost.

• Applications in Finance
• estimation of CIR model (Brigo Hanzon 98,
  Geyer Pichler 99, Nielsen et al. 00)
• estimation of Black-Scholes-Coutadon model for
  the US stock market (Nielsen et. al, 00)
• estimation of a micro-market model and a
  time-deformed model for the currencies rates
  market (Ozaki Iino 01, Ozaki Jimenez 02)
• analyze the volatility structure of LIBOR
  markets (To Chiarella 03)

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Comparacion de propiedades de distintos metodos
de estimacion de parametros de difusiones
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Aplicaciones de distintos metodos en finanza
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Validation of a financial models from actual data

• Micro market model

Given Yen-Dollar Exchange Rate
estimate
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Numerical results
ftt
logltt
Data
Innovation
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Innovations as a source of information for
improving the model
Histogram of the Innovation processes
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Referencias
-- J. C. Jimenez, R. J. Biscay, T. Ozaki.
(2006). Inference methods for discretely observed
continuous-time stochastic volatility models A
commented overview. Asian-Pacific Finantial
Marketing 12, 109-141.