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\title{Stochastic volatility}

\author{Oleg Makhnin\\ New Mexico Tech\\ Dept. of Mathematics }
\date{February 25, 2011}

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\begin{document}

\frame{\titlepage}

\section{Intro}
\frame{
\frametitle{}
{\Large
  \hilite<2> Volatility and applications \vsp \\
 %
  \hilite<3> Evidence of non-constant volatility \vsp \\
  %
  \hilite<4> Model and its estimation \vsp \\ 
   % 
  \hilite<5> Extensions, possible work \vsp \\
 %
}

}

\section{Volatility and applications}
\frame{
\frametitle{Model for stock price}

 Price of the stock at time $t$ is $X_t$ governed by {\em \purple geometric Brownian motion}
 %
   $$ d(log X_t) = \mu dt + \sigma dB_t$$ 
 Here $\mu dt$ is the deterministic component and  $\sigma dB_t$ is the stochastic component. \vsp

$B_t$ is Brownian motion: \\
  \hsp $dB_t$ is Normal with mean 0 and st.dev. $\sqrt{dt}$. \vsp

% \omnote{possibly use some of my slides from {\em Stochastic integration} talk}


 Volatility $\sigma$ determines the amplitude of the process fluctuations. \vsp


 Other applications: commodities, foreign currency exchange rate etc.
  

%\end{itemize} 
}


\frame{
\frametitle{Applications}

 \begin{itemize}[<+->] 

 \item \hnsp   Volatility forecasting: important for portfolio management, risk assessment etc

 \item \hnsp
  {\navy Black-Scholes}: A formula to obtain an Option price \vsp 

 {\em Call Option} is a contract to buy a security at time T (= maturity time) for the price S (=strike price). \vsp

Value of the call option: 
  $$ C(X_t, t, T, S) = \Phi(d_1) X_t - \Phi(d_2) S e^{-r(T-t)} $$
where
  $$ d_{1,2} = \frac{\log (X_t/S) + (r \pm \sigma^2/2)(T-t)}{\sigma \sqrt{T-t}}$$ 
 $r$ is the risk-free rate (at which you can borrow money) and $\Phi$ is the Normal CDF. 
  

\end{itemize} 
}  

\section{Non-constant volatility}

\frame{
\frametitle{Evidence of non-constant volatility}

  S\&P500 index  \vnsp \vnsp \\
 \includegraphics[height = 3in, width = 4in]{S-P1950}
}


\frame{
\frametitle{}

  S\&P500 index daily returns $Y_t = (X_t - X_{t-1})/X_t$, in \%  \vnsp \vnsp \\
 \includegraphics[height = 3.2in, width = 4.3in]{S-Pret1950}
}


\frame{
\frametitle{Evidence of non-constant volatility}

\hnsp  \includegraphics[height = 3in, width = 5in]{DJw.jpg}


}



\section{Model and its estimation}

\frame{
\frametitle{State-space model}
% $$ \begin{array}
 $\bullet \hsp  
        Y_t = \beta exp(h_t/2) \eps_t$  \vsp 

 $\bullet \hsp 
h_t = r h_{t-1} + \sigma_\eta \eta_t, \hsp t = 2,...,T$  \vsp

Hidden state $h_t$ is {\em log volatility} and it follows an autoregressive model (r = correlation between today's state and yesterday's state). \vsp

 $\eps_t$ and  $\eta_t$ are uncorrelated, standard Normal shocks \vsp


Efficient estimation methods since late 90's: Markov Chain Monte Carlo.
Alternates between simulating from distribution of $h_t$ and other unknown parameters ($\beta, r, \sigma_\eta$). Represents distribution of  $log(Y_t^2)$ (which is not Normal) as a mixture of Normals.
 
} 



\frame{
\frametitle{Estimation results}

  MCMC output for parameters: r (autocorrelation)

 \hnsp \includegraphics[height = 2in, width = 5in]{mc-r}


}


\frame{
\frametitle{Estimation results}

  MCMC output for parameters: $\beta$ (mean absolute return)

 \hnsp \includegraphics[height = 2in, width = 5in]{mc-beta}


}



\frame{
\frametitle{}  %Estimation results}


$ $ \vnsp \vnsp \vnsp \vnsp \\
 \hnsp \includegraphics[height = 3.4in, width = 4.5in]{sv-Volume2a}

}





\section{Extensions, possible work} 

\frame{
\frametitle{Extensions}

Rich, exponentially growing literature, including: \vsp

 $\bullet$ Continuous/intraday trading \vsp

 $\bullet$ Non-symmetry of returns     \vsp

 $\bullet$ Non-normal increments     \vsp

 $\bullet$ Models with jumps     \vsp

  $\bullet$  Multivariate: looking at several stocks at once   \vsp

  $\bullet$ Dimension reduction: hidden factor models
       (a few factors to explain SV behavior of several stocks)

}


\frame{
\frametitle{Multiscale model:  Molina, Han \& Fouque } 
  \includegraphics[height = 3in, width = 4.2in]{Multiscale}
}



\frame{
\frametitle{Implied volatility, jumps: Lopes \& Polson } 
  \includegraphics[height = 2in, width = 4.5in]{XLF} 

2007 data. XLF: financial stocks index. VIX: volatility implied from VIX index
}





\frame{
\frametitle{}

 \hnsp \includegraphics[height = 3.3in, width = 5in]{FactorLAW} 

% Factor SV: Lopes, Aguilar \& West 


}





\frame{
\frametitle{Possible research:}
  \begin{itemize}[<+->] 

   \item  Hidden factor models:

  \begin{itemize}[<+->] 
    \item  how do volatilities from different sources correlate? \vsp 
    \item  e.g. entire market $\to$ industries $\to$ individual stocks \vsp
    \item  factor models: several common causes that drive the behavior of many stocks (under-explored?) \vsp
 \end{itemize}
     \item Predictability (maybe use multiscale models)
     \item Development of particle-filtering methods (active area of research)
     \item Use in trading

  \end{itemize}

}


\frame{
\frametitle{Bibliography}


   $\bullet$ {\tt Kim, Shephard \& Chib, 1998. {\em Stochastic Volatility: Likelihood Inference and Comparison with ARCH Models}, Review of Economic Studies, vol. 65(3),  361-93} \vsp 

   $\bullet$  {\tt  {\em Stochastic Volatility: Selected Readings}, edited by N.Shephard, Oxford, 2005} \vsp 

 

 $\bullet$ {\tt Lopes \& Polson {\em Bayesian Inference for S.V. modeling,} {\tiny in Bocker, K. (Ed.) {\em Rethinking Risk Measurement and Reporting: Uncertainty, Bayesian Analysis and Expert Judgement}, 2010, 515-551} } \vsp 

  $\bullet$ {\tt Lopes \& Polson {\em Extracting S\&P500 and NASDAQ volatility:
The Credit Crisis of 2007-2008} {\tiny in {\em The Handbook of Applied Bayesian Analysis}, Oxford 2009}}



}







\frame{
\begin{center} \vspace{.25in} \LARGE{QUESTIONS?} \end{center}
}

\frame[containsverbatim]{
\begin{center} \vspace{.25in} \LARGE{\red THANK YOU!} \vsp \\

  see \verb"www.nmt.edu/~olegm/talks/SV" 
\end{center}
}

\end{document}



  http://ideas.repec.org/e/psh10.html     Neil Shephard at IDEAS

 McMC Estimation of Multiscale Stochastic Volatility Models
German Molina, Chuan-Hsiang Han and Jean-Pierre Fouque

  1/f noise? 


