An Adaptive Method for Valuing Derivatives on Assets with Stochastic Volatility

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An Adaptive Method for Valuing Derivatives on
Assets with Stochastic Volatility

• Sergei Fedotov
• Stephanos Panayides
• School of Mathematics
• The University of Manchester
• UK

WEHIA 2005, University of Essex, 2005
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Contents
WEHIA 2005, University of Essex, 2005
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Stochastic Volatility Models
WEHIA 2005, University of Essex, 2005
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Uncertain Volatility Models
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Adaptive Processes and Adaptive Control
Bellman, Kalaba (1960) On adaptive control
processesElliott, Aggoun, Moore (1995) optimal
control of stochastic systems under incomplete
informationKaratzas, Zhao (2001) Bayesian
adaptive portfolio optimizationRunggaldier and
his colleagues (1999,2004) Bayesian adaptive
control in binomial model and discontinuous
market models
WEHIA 2005, University of Essex, 2005
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Discrete Stochastic Model
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Stochastic Volatility
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The Adaptive Method
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The Adaptive Method
The idea is to use an adaptive procedure by which
the uncertainty regarding un can be reduced by
Bayesian updating
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The Adaptive Method
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The Adaptive Method
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Risk Minimization Procedure
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Risk Minimization Procedure
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The Adaptive Method
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Adaptive Decision Processes
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The Adaptive Method
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Numerical Results
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Conclusions

• We applied an adaptive control procedure which
  allows us to revise the stochastic
  characteristics of latent volatility during
  decision making.
• By using Bayesian analysis, we derived the
  recurrence equation for the variance of
  innovation term. This equation describes a
  reduction of uncertainty about volatility which
  is crucial for option pricing.
• We implemented the idea of adaptive procedure by
  using the risk-minimization analysis and
  stochastic dynamic programming.
• We showed that the adaptation leads to a decrease
  in the option price compared to the standard
  models without learning.

WEHIA 2005, University of Essex, 2005