\documentclass[letterpaper,10pt,conference]{cssconf} % % cdc04v8c17.tex: cdc04 paper, version 8 using code 17. % \IEEEoverridecommandlockouts % use with \thanks \overrideIEEEmargins % \usepackage{graphicx} \usepackage{amsmath,amssymb,amscd,euscript} %\usepackage{truetype} % for windows only else \usepackage{times} \usepackage{qtimes} % for unix/maxosx only, else use \usepackage{truetype} \usepackage{cdc04} %\usepackage[normal]{subfigure} \usepackage{url} \newtheorem{thm}{Theorem}[section] \newtheorem{cor}{Corollary}[section] %% \title{\LARGE \bf Comparison of Market Parameters for Jump-Diffusion Distributions Using Multinomial Maximum Likelihood Estimation} %% \author{Floyd B. Hanson and Zongwu Zhu% <-this % stops a space \thanks{The work is supported by the National Science Foundation under Grant DMS-02-07081. The content of this material is that of the authors and does not necessarily reflect the views of the National Science Foundation. Published in Proc. 42nd IEEE Conference on Decision and Control, 2004, pp.~3919-3924. }% \thanks{Department of Mathematics, Statistics, and Computer Science, M/C 249, University of Illinois, Chicago, IL 60607-7045 {\texttt{\small hanson@math.uic.edu} and \texttt{\small zzhu@math.uic.edu}}% }% endthanks }% endauthor %% \begin{document} %% \maketitle \thispagestyle{plain}%WEB\thispagestyle{empty} %\pagestyle{plain} %% when page numbers are desired \pagestyle{plain}%WEB\pagestyle{empty} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % ZZ/FH 8/29: remove central from moments \begin{abstract} Previously, we have shown that the proper method for estimating parameters from discrete, binned stock log returns is the multinomial maximum likelihood estimation, and its performance is superior to the method of least squares. Useful formulas have been derived for the jump-diffusion distributions. Numerically, the parameter estimation can be a large scale nonlinear optimization, but we have used techniques to reduce the computation demands of multi-dimensional direct search. In this paper, three jump-diffusion models using different jump-amplitude distributions are compared. These are the normal, uniform and double-exponential. The parameters of all three models are fit to the Standard and Poor's 500 log-return market data, constrained by the data first and second moments. While the results for the skew and kurtosis moments are mixed, the uniform jump distribution has superior qualitative performance since it produces genuine fat tails that are typical of market data, whereas the others have exponentially thin tails. However, the log-normal model has a big advantage in computational time of parameter estimation compared with the others, while the double-exponential is most costly due to having one more model parameter. % \end{abstract} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction} % Despite the great success of Black-Scholes % Zhu 8/29 options: Black-Scholes model or Black-Scholes option price formulae% model \cite{BS73} % Zhu 8/29: , % end % FH 8/29 Good. in option pricing, this pure log-normal diffusion model fails to reflect the three empirical phenomena: (1) the large random fluctuations such as crashes and rallies; (2) the non-normal features, that is, negative skewness and leptokurtic (peakedness) behavior in the stock log-return distribution; (3) the implied volatility smile, that is, the implied volatility is not a constant as in the Black-Scholes model. Therefore, many different models are proposed to modify the Black-Scholes model so as to represent the above three empirical phenomena. Merton \cite{Merton76} introduced the jump-diffusion model in financial modeling, using a Poisson process for the jump timing and a log-normal process for the jump-amplitudes to describe the market crashes and rallies. Some models are proposed to incorporate the \textit{volatility smile}, for example, Andersen, Benzoni and Lund \cite{ABL02} have made elaborate estimations to fit jump-diffusion models with log-normal jump-amplitudes, stochastic volatility and other features. Some models are proposed to incorporate the asymmetric features of the stock log-return distributions. Recently, Kou \cite{Kou02} proposed a jump-diffusion model with a log-double-exponential process for the % Zhu 8/29 jump-amplitude--> % FH 8/29 OK jump-amplitudes. % end Since crashes and rallies are rare events, so the Poisson process is reasonable for the timing of jumps. However, there is a problem in choosing the log-normal or log-double-exponential process for the % Zhu 8/29 jump-amplitude--> % FH 8/29 OK jump-amplitudes % end since the exponentially small tails of the log-normal and log-double-exponential distributions are contrary to the flat and thick tails of the long time financial market log-return data. Around the near-zero peak of the log-double-exponential and the log-normal, the jumps are small, so are not qualitatively different from the continuous diffusion fluctuations. %When the jumps are large, then the density tails are exponentially small, %but the large jumps of the data are more persistent. Moreover, an infinite jump domain is unrealistic, since the jumps should be bounded in a real world financial markets and an infinite domain leads to unrealistic restrictions in portfolio optimization \cite{CDC02FM}. So, Hanson and Westman \cite{MTNS02FMT} proposed one jump-diffusion model with log-uniform jump-amplitude. Most recently, Hanson, Westman and Zhu \cite{CONMATH04} showed that for IID simulations % Zhu 8/29: that % end % FH 8/29 OK the binned distribution is multinomial. They estimated the market parameters for this log-uniform model by subsequent multinomial maximum likelihood method to fit financial market distributions such as the Standard and Poor's 500 stock index. %In this paper, the value of the skewness %of the log-uniform model is greatly improved using more accurate %computations here. The main purpose of this paper is to compare the performance of three jump-diffusion models whose jump-amplitudes are the log-normally, log-uniformly and log-double-exponentially distributed. % FH 8/22: %The measures of performance are the skewness, kurtosis and computational costs. % % Zhu 8/29: \section{Some Jump-Diffusion Models Theoretcial Results} --> \section{Some Theoretical Results} % end / No need to emphasize it since the reader can understand it by default. % FH 8/29 OK % \subsection{Stock Return Process, $\mathbf{S(t)}$} % The following stochastic differential equation (SDE) is used to model the dynamics of the asset price, \begin{small}$S(t)$\end{small}: % \begin{small} \eq \label{dS(t)} dS(t) = S(t)\left(\mu_d dt + \sigma_d dW(t) + J(Q)dP(t)\right),% \nq% \end{small}% % \noindent{where} $\mu_d$ is the drift coefficient, $\sigma_d$ is the diffusive volatility, \begin{small}$W(t)$\end{small} is the stochastic diffusion process, \begin{small}$J(Q)$\end{small} is the Poisson jump-amplitude, \begin{small}$Q$\end{small} is its underlying Poisson amplitude mark process, % \begin{small}$J(Q)dP(t)$\end{small} is just a symbol for the counting \begin{small}$\sum_{i=1}^{dP(t)}J(Q_{i})$\end{small}, % Zhu 8/29: Q_{i} --> J(Q_{i} % end % FH 8/29 Good correction! %% \begin{small}$P(t)$\end{small} is the standard Poisson jump process with joint mean and variance \begin{small}$\E[P(t)] = \lambda{t} = \Var[P(t)]$\end{small}. % \subsection{Stock Log-Return Process, $\mathbf{ln(S(t))}$} % The stock log-return $\ln(S(t))$ can be transformed to a simpler jump-diffusion stochastic differential equation (SDE) upon % Zhu 8/29: use--> % FH 8/29 OK the use % end of the stochastic chain rule \cite{Book04}, % \begin{small} \eq \label{d[ln(S(t))]} d[\ln(S(t))] = \mu_{ld}dt + \sigma_d dW(t) + QdP(t),% \nq% \end{small}% % \noindent{where} \begin{small}$\mu_{ld} \equiv \mu_d - 0.5\sigma_d^2$\end{small} is the log-diffusive (ld) drift and for simplicity the log-jump-amplitude is taken as the mark \begin{small}$Q = \ln(J(Q)+1)$\end{small}. % \subsection{Log-Normal Jump Distribution} % Let the density of the jump-amplitude mark $Q$ be normal % \begin{small} \eq \label{phi_normalQ(q)} \phi_Q(q) = \phi_n(q;\mu_j, \sigma_j^2),% \nq% \end{small}% % \noindent{where} \begin{small}$\phi_n(q;\mu_j, \sigma_j^2)$\end{small} is the normal density with mean $\mu_j$ and variance $\sigma_j^2$. The log-normal jump-amplitude jump-diffusion model was used in \cite{Merton76}, \cite{ABL02}, \cite{ACC02FM} and others. For the density for this jump-diffusion model with log-normal jump-amplitude, Hanson and Westman \cite{ACC02FM} proved, with corrections given here in terms of the distribution truncated to the second order approximation in terms of the Poisson distribution, % the following corollary: \\ \textbf{Corollary:} The second order approximation to $[x_{1},x_{2}]$ bin probability distribution for the linear jump-diffusion log-return increment $\D[\ln(S(t))]$ with log-normal jump-amplitude is given by % \begin{small} \eq \label{ThNormal_Phijd(x1,x2)} \Phi_{\njd}(x_{1},x_{2})\!\!&\!\!\simeq\!\!&\!\!\frac{\sum_{k=0}^{2}p_k(\lambda\Dt) \Phi_n(x_{1},x_{2};\mu\!+\!k\mu_j, %\nonumber\\\nonumber\!&\!&\! \sigma^2\!+\!k\sigma_j^2)}{\sum_{j=0}^{2}p_j(\lambda\Dt)},% \nonumber\\*[-1ex]\!&\!&\!~% \nq% \end{small}% % \noindent{for} \begin{small}$-\infty < x < +\infty$\end{small}, where \begin{small}$\Phi_n(x_1, x_2; \mu, \sigma^2)$\end{small} is the normal distribution on the \begin{small}$[x_1, x_2]$, $\mu \equiv \mu_{ld}\Dt$, $\sigma \equiv \sqrt{\sigma^2_d\Dt}$, $p_k(\Lambda) = e^{-\Lambda}\Lambda^k/k!$\end{small} is the Poisson distribution with parameter \begin{small}$\Lambda = \lambda\Dt$\end{small} and $k$ jumps and \begin{small}$\Dt$\end{small} is the corresponding trading time increment in years. This corollary is based upon the law of total probability \cite[Chapter 0]{Book04} resulting in the sum over all $k$ Poisson jumps, the convolution theorem \cite{Book04} yielding the density of the log-jump-diffusion conditioned on there being $k$ IID jumps, and the fact that the convolution of two normals is also normal \cite{Book04}. %The corollary %is posed as the log-return increment rather than for the infinitesimal, %because the time between trading data is small but not infinitesimal. For the purpose of comparison, we use two terms of the expansion to provide more accurate estimations since we are dealing with small but moderately small time steps. The additional contribution of the third order approximation is only 1.5\% whereas the 2nd order approximation contributed 23\% to the 1st. Note that an appropriate 2nd order renormalization is used to preserve the distribution property. %The scale of these time steps can be magnified by a jump rate that includes many %small jumps that are indistinguishable from the fluctuations of the %diffusion process. % \subsubsection{Basic Moments $M_i^{(\njd)}$ of Log-Return Increments $\mathbf{\D[ln(S(t))]}$ for Log-Normal Jumps for $i=1\!\!:\!\!4$} % %\begin{itemize} % %\setlength{\itemsep}{-1ex} % %\item 1st moment: \begin{small} \equnn M_1^{(\njd)}\!\!&\!\!\equiv\!\!&\!\!\E[\D[\ln(S(t))]]\!=\!(\mu_{ld}\!+\!\lambda\mu_j)\Dt. %\nqunn % %\item 2nd moment: %\equnn \\ M_2^{(\njd)}\!\!&\!\!\equiv\!\!&\!\!\Var[\D[\ln(S(t))]] \!=\!(\sigma_d^2\!+\!\lambda(\sigma_j^2\!+\!\mu_j^2))\Dt. %\nqunn % %\item 3rd moment: %\equnn \\ M_3^{(\njd)}\!\!&\!\!\equiv\!\!&\!\!\E\left[(\D[\ln(S(t))]\!-\!M_1^{(jd)})^3\right] \!=\!(3\sigma_j^2\!+\!\mu_j^2)\mu_j\lambda\Dt. %\nqunn % %\item 4th moment: %%FH6 reformat %\equnn \\ \hspace*{-1em}M_4^{(\njd)}\!\!&\!\!\equiv\!\!&\!\!\E\left[(\D[\ln(S(t))]\!-\!M_1^{(jd)})^4\right] \\\!&\!=\!&\!(\mu_j^4\!+\!3\sigma_j^4\!+\!6\mu_j^2\sigma_j^2)\lambda\Dt \!\!+\!3(\sigma_d^2\!+\!\lambda(\sigma_j^2\!+\!\mu_j^2))^2(\Dt)^2. \nqunn \end{small} ~\\*[-3em] % %\end{itemize} % % \subsection{Log-Uniform Jump Distribution} % Let the density of the jump-amplitude mark $Q$ be uniform % \begin{small} \eq \label{phi_uniformQ(q)} \phi_Q(q) = \frac{H(b-q) - H(a-q)}{b - a},% \nq% \end{small}% % \noindent{where} $a < 0 < b$ and \begin{small}$H(x)$\end{small} is the Heaviside unit step function. The mark \begin{small}$Q$\end{small} has moments, % Zhu 8/29:\E_Q[Q]-->\E[Q] and etc. \begin{small}$\mu_j \equiv \E[Q] = 0.5(b + a)$, $\sigma_j^2 \equiv \Var[Q] = (b - a)^2/12$\end{small}. % end Zhu: keep the symbols consistent. % FH 8/29 OK The original jump-amplitude \begin{small}$J$\end{small} has mean \begin{small}$\E[J(Q)] = (\exp(b)-\exp(a))/(b - a) - 1$\end{small}. For the distribution of the jump-diffusion model with log-uniform jump-amplitude, the following corollary follows from the density in~\cite[Chapter 5]{Book04}, \\ \textbf{Corollary:} The second order approximation to $[x_{1},x_{2}]$ bin probability distribution for the linear jump-diffusion % Zhu 8/29: , % end % FH 8/29 OK log-return increment $\D[\ln(S(t))]$ with log-uniform jump-amplitude is given by % \begin{small} \eq \label{ThUniform_Phijd(x1,x2)} \Phi_{\ujd}(x_{1},x_{2})\!&\!\simeq \frac{\sum_{k=0}^{2}p_k(\lambda\Dt)\Phi_{\ujd}^{(k)}(x_{1},x_{2})}{\sum_{j=0}^{2} p_j(\lambda\Dt)}, \nq \end{small}% % \noindent{for} \begin{small}$-\infty < x < +\infty$\end{small}, where \begin{small}$\Phi_{\ujd}^{(0)}(x_1, x_2) \equiv \Phi_n(x_1, x_2; \mu, \sigma^2)$\end{small}, % \begin{small} % Zhu 8/29: *, \cdot are deleted. Keep the formulae more consistent. % % FH 8/29 OK \equnn \Phi_{\ujd}^{(1)}(x_1, x_2) \!&\!\!=\!&\!\frac{1}{b\!-\!a}\biggl( (x_2\!-\!x_1)\Phi(x_2\!-\!b,x_2\!-\!a;\mu,\sigma^{2}) \\\!&\!&\!-(x_1\!-\!b\!-\!\mu)\Phi_n(x_1\!-\!b,x_2\!-\!b;\mu,\sigma^2) \\\!&\!&\!+(x_1\!-\!a\!-\!\mu)\Phi_n(x_1\!-\!a,x_2\!-\!a;\mu,\sigma^2) \\\!&\!&\!\left.\!+\frac{\sigma}{\sqrt{2\pi}}\! \left(e^{-\frac{x_{1b}^2}{2}}\!-\!e^{-\frac{x_{2b}^2}{2}} \!-\!e^{-\frac{x_{1a}^2}{2}}\!+\!e^{-\frac{x_{2a}^2}{2}}\right)\right) ,% \nqunn% % end Zhu \end{small}% % \noindent{where} \begin{small}$x_{1a} = (x_1\!-\!a\!-\!\mu)/\sigma$, $x_{2a} = (x_2\!-\!a\!-\!\mu)/\sigma$, $x_{1b} = (x_1\!-\!b\!-\!\mu)/\sigma$, $x_{2b} = (x_2\!-\!b\!-\!\mu)/\sigma$\end{small}, % \begin{small} \equnn \Phi_{\ujd}^{(2)} (x_1,x_2)\!\!&\!\!=\!\!&\!\!\frac{0.5\sigma^2}{(b\!-\!a)^2}\biggl( (x_{2A}^2\!-\!x_{1A}^2)\Phi_n(x_2\!-\!C,x_2\!-\!A;\mu,\sigma^2) \\\!&\!&\!+(x_{1B} ^2\!-\!x_{2B} ^2)\Phi_n(x_2\!-\!B,x_2\!-\!C;\mu,\sigma^2) \\\!&\!&\!+(x_{1A}^2\!+\!1) \Phi_n(x_1\!-\!A,x_2\!-\!A;\mu,\sigma^2) \\\!&\!&\!+(x_{1B} ^2\!+\!1) \Phi_n(x_1\!-\!B,x_2\!-\!B;\mu,\sigma^2) % Zhu 8/21% bad v \\\!&\!&\!-2(x_{1C} ^2\!+\!1) \Phi_n(x_1\!-\!C,x_2\!-\!C;\mu,\sigma^2) \\\!&\!&\!\left.\!+\frac{1}{\sqrt{2\pi}}(2X_{1C,2C}\!-\!X_{1A,2A}\!-\!X_{1B,2B}) \right),% \nqunn% \end{small}% % \noindent{where} \begin{small} $A = 2a$, $C = a\!+\!b$, $B =2b$, $x_{1A} = (x_1\!-\!A\!-\!\mu)/ \sigma$, $x_{2A} = (x_2\!-\!A\!-\!\mu)/ \sigma$, $x_{1C} = (x_1\!-\!C\!-\!\mu)/ \sigma$, $x_{2C} = (x_2\!-\!C\!-\!\mu)/ \sigma$, $x_{1B} = (x_1\!-\!B\!-\!\mu)/ \sigma$, $x_{2B} = (x_2\!-\!B\!-\!\mu)/ \sigma$, $X_{1A,2A} = x_{1A}e^{-x_{1A}^2/2}-x_{2A}e^{-x_{2A}^2/2}$, $X_{1B,2B} = x_{1B}e^{-x_{1B} ^2/2}-x_{2B}e^{-x_{2B} ^2/2}$, $X_{1C,2C} = x_{1C}e^{-x_{1C} ^2/2}-x_{2C}e^{-x_{2C} ^2/2}$. \end{small} {For} the distribution (\ref{ThUniform_Phijd(x1,x2)}), the first and second order term formulas, \begin{small}$\Phi_{\ujd}^{(i)}(x_1, x_2)$\end{small} for $i\!=\!1\!\!:\!\!2$, have been reduced by integration by parts to single normal distribution integrals to minimize the computational costs, making it comparable % Zhu 8/21 that % for the normal jump case in (\ref{ThNormal_Phijd(x1,x2)}). Also, the terms are arranged to minimize the effects of catastrophic cancellation. \subsubsection{Fourth Central Moment of Log-Return Increments $\mathbf{\D[ln(S(t))]}$ for Log-Uniform Jumps} % %\begin{itemize} % %\setlength{\itemsep}{-1ex} % %\item 1st moment: %\equnn %M_1^{(\ujd)} %= (\mu_{ld} + \lambda\mu_j)\Dt. %\nqunn % %\item 2nd moment: %\equnn %M_2^{(\ujd)} %\!&\!=\!&\! (\sigma_d^2 + %\lambda (\sigma_j^2 + \mu_j^2))\Dt. %\nqunn % %\item 3rd moment: %\equnn %M_3^{(\ujd)} %\!&\!=\!&\! %(3\sigma_j^2+\mu_j^2)\mu_j\lambda\Dt. %\nqunn % %\item 4th moment: \begin{small} \equnn \hspace*{-1em}M_4^{(\ujd)} \!\!&\!\!=\!\!&\!\!\left(\mu_j^4\!+\!1.8\sigma_j^4\!+\!6\mu_j^2\sigma_j^2\right)\lambda\Dt \\\!&\!&\!\!+\!3\left(\sigma_d^2\!+\!\lambda(\sigma_j^2\!+\!\mu_j^2)\right)^2(\Dt)^2.% \nqunn% \end{small}% % %\end{itemize} % \noindent{Note} that the formulas for the first three moments are the same for both log-normal and log-uniform jumps. % \subsection{Log-Double-Exponential Jump Distribution} % Let the density of the jump-amplitude mark $Q$ be double-exponential % \begin{small} \eq \label{phi_dblexp_Q(q)} \phi_Q(q) = \frac{p_1}{\mu_1}\exp\left(\frac{\dps q}{\dps \mu_1}\right)I_{\{q<0\}} \!+\!\frac{p_2}{\mu_2}\exp\left(\frac{\dps -q}{\dps \mu_2}\right)I_{\{q\geq 0\}},% \nq% \end{small}% % \noindent{where} $\mu_1>0$ and $\mu_2>0$ are one-sided means, and $0 3$,% %\nqunn% \end{small}% % where $\beta_4^{(n)} = 3$ is the normal distribution value and $M_4^{(\sp)}$ is the 4th central log-return moment of the data. % \end{itemize} % % \subsection{Multinomial Maximum Likelihood Estimation} % In a previous paper \cite{CONMATH04}, the multinomial maximum likelihood estimation of model parameters is justified for binned financial data, but applies to very general binned data. The main idea for this method is the following: % \begin{itemize} % \item Step 1: % Zhu 8/29: Sample Data is sorted into $nb$ bins --->% FH 8/29 OK Sort the sample data into $nb$ bins % end Zhu % and get the sample frequency $f_b^{(sp)}$, for $b=1\!\!:\!\!nb$. % \item Step 2: Get the theoretical jump-diffusion frequency with parameter vector $\xv$: \begin{small} $$ f_b^{(\jd)}(\xv) \equiv ns \int_{B_b} \phi^{(\jd)}(\eta ;\xv) d\eta\:, % Zhu 8/29: \phi^{(\jd)}(\eta ;\xv) no definition, need some changes here??!! % $$% \end{small}% % \noindent{where} $\phi^{(\jd)}(\eta ;\xv)$ is some jump-diffusion density in $\eta$ and $B_b$ is the $b$th bin. % FH 8/29: \noindent{where} and $B_b$ is the $b$th bin. % \item Step 3: Minimize the objective function: % \begin{small} \eq \label{EstObjective} y(\xv) \equiv -\sum_{b=1}^{nb}\left[f_b^{(\sp)}\ln\left(f_b^{(\jd)}(\xv) \right)\right]\:,% \nq% \end{small}% % \noindent{where} the negative of the likelihood is minimized, %FH 8/29 Killed another rephrase corresponding to the minimizing % Zhu 8/29: \texttt{fminsearch} function ---->% FH 8/29 OK, but the rest is a bad long sentence. MATLAB %FH 8/29 function \texttt{fminsearch} % end Zhu % % FH 8/29: to get the optimal parameters $\xv^*$ for the three compared models, respectively. This MATLAB function is an implementation of the Nelder-Mead down-hill % Zhu 8/29: simplex direct search method in MATLAB. %The Nelder-Mead method \cite{NelderMead} is used --> simplex direct search method~\cite{NelderMead}. % in MATLAB % end Zhu% %FH 8/29 % to get the optimal parameters %$\xv^*$ for the three compared models, respectively. The Nelder-Mead is usually faster than other optimization methods when it works. Some comparisons with our multidimensional golden section search method for the financial parameter estimation problem are given in \cite{CONMATH04}. % \end{itemize} % \subsection{Jump-Diffusion Moment Estimation Constraints} % For the jump-diffusion model with log-normal and log-uniform jump-amplitude, there are five (5) free jump-diffusion parameters: % $\{\mu_{ld}, \sigma_d^2, \mu_j, \sigma_j^2, \lambda\}$. % For the model with log-double-exponential jump-amplitude, there are six (6) free jump-diffusion parameters: % $\{\mu_{ld}, \sigma_d^2, \mu_1, \mu_2, p_1, \lambda\}$. % So, to reduce this set to a reasonable number, the multinomial maximum likelihood estimation is subjected to the mean and variance constraints: % \begin{small} \eq \label{MeanCons} M_1^{(\sp)} = M_1^{(\jd)} \AND \label{VarCons} M_2^{(\sp)} = M_2^{(\jd)}\:.% \nq% \end{small}% % \noindent{Two} diffusion parameters, $\mu_{ld}$ and $\sigma_d$, are eliminated by % \begin{small} \eq \label{MeanElim} \hspace*{-3em}\mu_{ld}\!&\!=\!&\!\left(M_1^{(sp)} - \mu_j\lambda\Dt\right)/\Dt, \\ \label{VarElim} \hspace*{-3em}\sigma_d^2\!&\!=\!&\!\max\left[\left(M_2^{(sp)} - \left(\sigma_j^2 + \mu_j^2\right)\lambda\Dt\right)/\Dt,\varepsilon\right],% \nq% \end{small}% % \noindent{subject} to positivity constraints with sufficiently small $\varepsilon > 0$. Although $\sigma_d^2$ normally should be positive, but not necessarily for the first argument of the $\max$ in (\ref{VarElim}). % {F}or the log-normal and log-uniform jump-diffusion model, only three free parameters are left: % \begin{small}$\xv=\{\mu_j, \sigma_j^2, \lambda\}$\end{small}. % {F}or the log-double-exponential jump-diffusion model, four free parameters are left: % \begin{small}$\xv=\{\mu_1, \mu_2, p_1, \lambda\}$\end{small}%, % with significantly more computational estimation costs and it is subject to an exponential form of catastrophic cancellation unless the one-sided exponentials are appropriately collected. \section{Numerical Results, Figures and Discussion} % We use the MATLAB 7.0 \cite{MATLAB} % Zhu 8/29: But the reference given is about version 6 (2000)??% end Zhu % % FH 8/29: I knew that but the version 7 manual was not distributed yet! to program our codes. The multinomial maximum likelihood estimation given here is used to estimate the jump-diffusion parameters. The numerical optimization was performed using the \texttt{fminsearch} function \cite{MATLAB}. % Zhu 8/29: , an implementation of the down-hill %simplex direct search method of Nelder and Mead \cite{NelderMead} %Zhu: In Section B, we already gave an explanation. % end FH 8/29 Good For the normal distribution integrals \begin{small}$\Phi_n(x_1,x_2;\mu,\sigma^2)$\end{small} or \begin{small}$\Phi_n(x_1;\mu,\sigma^2)$\end{small}, the use of %Yan the the fast and reliable MATLAB complementary error function \texttt{erfc} was critical, since more standard integration functions for unscaled arguments failed small variance $\sigma^2$ tests due to poor detection of the main probability mass. For the log-normal and log-uniform model, the same starting point $\xv_0$ is used. For the log-double-exponential model, the different starting point $\xv_0$ is used: initial $\mu_1$ and $\mu_2$ are from an initial estimation of the $\mu_j$ and using $p_1 \simeq 0.6 > 0.5$ and the $\lambda\Dt$ value are the same as the log-normal or log-uniform. The empirical data used in the estimation are the S\&P500 daily closing log-returns from the decade 1992-2001. This data is displayed here in Figure~\ref{FigHist92-01} with 100 equally spaced bins. The ragged appearance of the histogram resembles the random simulation of a density using a moderate, but inadequate sample size. The crashes are represented by the extreme negative tails and the rallies by the extreme positive tails, but these extreme events are difficult to see since they are rare events with small frequency counts, i.e., the extreme tails are sparsely populated, with the extreme negative events more widely separated than the extreme positive events. Note that extremeness is measured by the values of the log-returns on the horizontal axes. % \begin{figure}[!hbtp] \centering {\includegraphics[width=3.25in]{cdc04normal17sphistbw.eps}} % {\includegraphics[width=3.25in]{cdc04normal16f32sphistbw.eps}} \caption{ Histogram of S\&P500 log-return frequencies for the decade 1992-2001, using 100 centered evenly spaced bins. } \label{FigHist92-01} \end{figure} % However, if the histogram frequencies are multiplied by the centered value of the evenly spaced bin log-returns, then the extreme jumps are clearly visible. This moment-histogram is called a \textit{hysteriagram} \cite{MTNS02FMT} since it magnifies the count of the larger jumps and corresponds to the extreme reaction of investors. The hysteriagram for the S\&P500 is given in Figure~\ref{FigXHist92-01} using the same data as in the previous figure, using the same 100 evenly spaced bins. % \begin{figure}[!hbtp] \centering {\includegraphics[width=3.25in]{cdc04normal17spxhistbw.eps}} % {\includegraphics[width=3.25in]{cdc04normal16f32spxhistbw.eps}} \caption{ Hysteriagram of S\&P500 log-return frequencies multiplied by the centered bin log-return values for the decade 1992-2001, using 100 evenly spaced bins. } \label{FigXHist92-01} \end{figure} % Figure~\ref{FigNormalXHist} shows that the log-normal jump-amplitude fitted model hysteriagram exhibits too thin tails that decay too fast with the jump magnitude. From (\ref{ThNormal_Phijd(x1,x2)}) the bin distribution will be a Poisson sum of normal distributions, so will have the thin Gaussian exponential tails. % The corresponding histogram for the log-normal, % not shown here, does not show enough visual detail to sufficiently % distinguish it from the other jump-amplitude models. % \begin{figure}[!hbtp] \centering {\includegraphics[width=3.25in]{cdc04normal17jdxhistbw.eps}} % {\includegraphics[width=3.25in]{cdc04normal16f32jdxhistbw.eps}} \caption{ Hysteriagram of the predicted log-returns frequencies multiplied by the centered bin log-return values for the log-normal jump-amplitude jump-diffusion model, % Zhu 8/29: using 100 evenly spaced bins using estimated parameters.---> using 100 evenly spaced bins and the estimated parameters. % end Zhu % FH 8/29 Good } \label{FigNormalXHist} \end{figure} % Figure~\ref{FigUniformXHist} shows that the log-uniform jump-amplitude fitted model hysteriagram exhibits thicker tails that decay more slowly with the jump magnitude, especially in the shoulders of the hysteriagram. The convolution of the diffusion with multiple jump uniform distributions in (\ref{ThUniform_Phijd(x1,x2)}) help counter the normal distribution tendency to having exponential thin tails, but not for very large values of the log-returns. The positive tails are thicker since the extreme positive jumps are more closely spaced than the negative jumps. The uniform model mainly compensates for the non-normal data with significant lumps in the shoulders of the jump-diffusion distribution. % \begin{figure}[!hbtp] \centering {\includegraphics[width=3.25in]{cdc04uniform17b100m0jdxhistbw.eps}} % {\includegraphics[width=3.25in]{cdc04uniform16f32jdxhistbw.eps}} \caption{ Hysteriagram of the predicted log-returns frequencies multiplied by the centered bin log-return values for the log-uniform jump-amplitude jump-diffusion model, % Zhu 8/29: using 100 evenly spaced bins using estimated parameters.---> using 100 evenly spaced bins and the estimated parameters. % end Zhu % FH 8/29 Good*2 } \label{FigUniformXHist} \end{figure} % Figure~\ref{FigDblExpXHist} shows that the log-double-exponential jump-amplitude fitted model hysteriagram exhibits too thin tails that decay too fast with the jump magnitude that is very similar to the log-normal jump-amplitude model. The convolution of normal and exponential distributions in (\ref{ThDblExp_Phijd(x1,x2)}), like the normal jump-amplitude model, can only lead to exponential thin tails. % \begin{figure}[!hbtp] \centering {\includegraphics[width=3.25in]{cdc04dblexp17b100m0jdxhistbw.eps}} % {\includegraphics[width=3.25in]{cdc04dblexp16f32jdxhistbw.eps}} \caption{ Hysteriagram of the predicted log-returns frequencies multiplied by the centered bin log-return values for the log-double-exponential jump-amplitude jump-diffusion model, % Zhu 8/29: using 100 evenly spaced bins using estimated parameters.---> using 100 evenly spaced bins and the estimated parameters. % end Zhu % FH 8/29 Good*3 } \label{FigDblExpXHist} \end{figure} % % FH 8/24: % The log-uniform model is a qualitatively, but not necessarily quantitatively, % better model for the S\&P500 data, since uniform jump model has thicker tails. From Table~\ref{TAB1Moments1}, we can have a quantitative estimate of the derived distribution parameters $\{\mu_d$, $\sigma_d$, $\mu_j$, $\sigma_j$, $\lambda$\}. Since the trading days per year are about 250 days, it is not likely that the jumps rate is more than 100 per year because the finance market should be kept stable, so, $\lambda \simeq 64$ for the log-uniform is more reasonable. The near-zero peaks of the normal and double-exponential lead to jump rates that are double or more than the uniform jump rate. Note that the jump rate includes all size jumps, including those hidden under the log-normal diffusion. In the table the overall jump mean $\mu_j$ is given for the purpose of comparison, but for the double-exponential, the negative jump mean is $-\mu_1 =$ -6.88e-3 and the positive jump mean is $\mu_2 =$ +6.35e-3. For the double-exponential, the probability of negative jumps is $p_1=0.504$ and that for positive jumps is $p_2=0.496$. The overall jump-diffusion parameters $\{\mu_d, \sigma_d, \mu_j, \sigma_j, \lambda\}$ have somewhat different distributions among the three jump models, with the diffusive means and volatilities being the closest among the parameters. The uniform distribution gives more weight to the negative jumps with the largest $-\mu_j =$ 12.18e-4 and largest $\sigma_j \simeq$ 1.52e-2. % \begin{table}[!hbtp] \centering \caption{ Comparison summary of derived distribution parameters for the log-normal, log-uniform and log-double-exponential jump-diffusion models, respectively. } \label{TAB1Moments1} \begin{tabular}{|c||c|c|c|c|c|}\hline Model & $\mu_d$ & $\sigma_d$ & $\mu_j$ & $\sigma_j$ & $\lambda$\\\hline\hline Normal & 0.191 & 0.088 & -7.09e-4 & 1.19e-2 & 121. \\\hline Uniform & 0.184 & 0.100 & -12.18e-4 & 1.52e-2 & 64.0 \\\hline Dbl-Exp & 0.170 & 0.085 & -3.21e-4 & 0.94e-2 & 202. \\\hline \end{tabular} \end{table} % Table~\ref{TAB2Moments2} shows the differences of the variance-normalized higher moments of skewness $\beta_3$ and kurtosis $\beta_4$ between the estimate value and the observed values. % Zhu 8/29: The skewness difference %is lowest for the log-uniform jump model, while the kurtosis difference %is lowest for the log-double-exponential jump model. ---> FH 8/29 OK The absolute skewness difference is the lowest for the log-uniform jump model, while the absolute kurtosis difference is the lowest for the log-double-exponential jump model. On the other hand, the skewness difference is the worst for the log-normal jump model and the kurtosis difference is the worst for the log-uniform jump model, % FH 8/24 but the log-normal is only 5\% lower than the log-uniform. % The skewness and kurtosis differences % are lowest for the log-double-exponential jump model. % The skewness difference is almost the same for % the log-uniform as for the log-double-exponential jump model. % On the other hand, the skewness difference is the worst for the log-normal jump model % and the kurtosis difference is the worst for the log-uniform jump model. Of course, the numerical calculations of the third and fourth moments are of doubtful computational reliability for data. The final multinomial maximum likelihood values using the negative of minimum values are essentially the same for all models, since the same stopping criterion was used. % \begin{table}[hbtp] \centering \caption{ The skewness and kurtosis coefficients for the three models are compared to S\&P500 values, respectively, and Multinomial Maximum Likelihood (MML $\simeq -\min[y(\mathbf{x})])$. } % FH 8/24:% \label{TAB2Moments2} \begin{tabular}{|c||c|c|c|c|c|}\hline Model & $\beta_3$ & $\%$diff. & $\beta_4$ & $\%$diff. & MML \\\hline\hline Normal & -0.147 & -49.5 & 5.98 & -23.4 & 1107. \\\hline Uniform & -0.219 & -24.7 & 5.57 & -28.7 & 1106. \\\hline Dbl-Exp & -0.183 & -37.6 & 6.80 & -12.8 & 1108. \\\hline\hline S\&P500 & -0.291 & 0.0 & 7.80 & 0.0 & --- \\\hline \end{tabular} \end{table} % Table~\ref{TAB3Perf} shows that % Zhu 8/21 % %the log-normal and log-uniform %models take somewhat more iterations and function evaluations, the log-normal model takes somewhat more iterations and function evaluations than the log-uniform model does, %end 8/21 % but the log-normal model parameter estimate takes about 60\% of the time to execute. One reason is that the log-normal requires only similar normal distribution calculations for each % Zhu 8/29:jump--> jump count % size %FH 8/29 size usually means amplitude %end Zhu % $k$ in (\ref{ThNormal_Phijd(x1,x2)}), while the others require more complex combinations of normal distributions, powers and exponentials. The extra parameter needed for the double-exponential means the iteration count, the function evaluation count and the timings are much greater. However, the computational effort for both the uniform and double-exponential models have been greatly reduced by using integration by parts and more to get single integrals. % An added advantage of such a reduction also can improve accuracy and speed of computation. If the parameter estimation is done off-line for an application instead of on-line % Zhu 8/29: than---> FH 8/29 Good then % the computational saving would not be too much of an advantage. % \begin{table}[hbtp] \centering \caption{ Comparison summary of computational performance measures: } % FH 8/24: \label{TAB3Perf} \begin{tabular}{|c||c|c|c|c|}\hline Jump & Number & Number & Function & Timings \\ Model & Parms. & Iters. & Evals. & (sec) \\\hline\hline Normal & 3 & 131 & 238 & 4.7 \\\hline Uniform & 3 & 71 & 128 & 7.8 \\\hline Dbl-Exp & 4 & 205 & 343 & 21. \\\hline \end{tabular} % \\*[0.5em] \begin{small} Combined Legend for Table~\ref{TAB1Moments1}, Table~\ref{TAB2Moments2} and Table~\ref{TAB3Perf}: % \begin{itemize} % %\item Normal: Log-normal jump-amplitude. % %\item Uniform: Log-uniform jump-amplitude. % %\item Dbl-Exp: Log-double-exponential jump-amplitude. % GY 8/31 Change: \item Maximum Number of Iterations: 400.%$^*$. % GY 8/31 Changes: \item Using same fminsearch tolerances: tolx $=$ 5e-7 \\ and toly $=$ 5e-7. % \item Using a dual G5@2GHz CPU computer processor with MATLAB 7.0. % \end{itemize} \end{small} \end{table} % % \section{Summary and Conclusions} % From the above theoretical and data analysis, we can get the following conclusions: % \begin{itemize} % \item The log-uniform model is the qualitatively best overall among the three models, in terms of genuinely representing the fat tail property, better approximation to the empirical skewness and more reasonable jump rates of real-world market distributions. % \item The log-normal model runs faster than the other two models due to simpler normal bin integrals. On the other hand, the integration by parts technique has been used to reduce the computational effort for the log-uniform and log-double-exponential models. However, the deficiencies of the log-normal model demonstrates that the distribution that is better analytically is not necessarily a better model for financial markets. % , i.e., finding a better model may be counter to the desire % to obtained closed form solutions. % \item The results for the log-normal and log-double-exponential jump amplitude models are qualitatively similar, having exponentially small tails and near-zero peaks in the jump distribution making small jumps more likely. % \item However, all three models give reasonable quantitative, although somewhat mixed, results. % \item For the future research and considerations. \begin{enumerate} \item To improve the log-uniform and other jump models, % Zhu 8/29 the stochastic volatility should be considered.---> % FH 8/29 does not read properly: consider the stochastic volatility the stochastic volatility will be considered with % and other factors. % end Zhu % % since in the real world the implied volatility curve is not a % constant, but has a curve with a `smile'. % \item To consider the option price and optimal portfolio applications, % Zhu 8/29 getting ---> % Fh 8/29 ? get the % end Zhu % approximate solutions to these problems % FH 8/29? will be obtained. \end{enumerate} % \end{itemize} % \bibliographystyle{amsalpha} \begin{thebibliography}{00} % \bibitem{ABL02} T. 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