How to draw geometric Brownian motions in tikz/pgf

Question

I am trying to replicate this excellent answer which explains how to draw Brownian motions in TiKZ:

\newcommand{\Lathrop}[6]{% points, advance, rand factor, options, end label, truncate from point
    \draw[#4] (0,0)
    \foreach \x in {1,...,#6} { 
        -- ++ (#2,rand*#3)
    }
    coordinate (tempcoord) {};
    \pgfmathsetmacro{\remainingwidth}{(#1-#6)*#2}; % changed to a custom number
    \def\remainingwidthcustom{1};
    \draw[#4] (tempcoord) -- ++ (\remainingwidthcustom,0) node[right] {#5};
}
\begin{tikzpicture}
    \def\ylength{3}
    \pgfmathsetseed{3} % control pseudo-random number
% Axis
\coordinate (y) at (0,3); \coordinate (x) at (6,0);
\draw[&lt;-&gt;] (y) node[above] {$S$} -- (0,0) --  (x) node[right] {$\mathbb{N}$};

% Brownian motion
\Lathrop{750}{0.02}{0.21}{blue!70!black}{strike price $S_N$}{250};


\end{tikzpicture}

My problem is that I cannot change it to the geometric Brownian motion for two reasons:

I cannot get exponential function via pgfmathparse
I do not fully understand the syntax inside the \foreach \x-loop.

Any help is appreciated.

For background: https://en.wikipedia.org/wiki/Geometric_Brownian_motion — Thruston, Jan 04 '23 at 09:47
IIRC, to generate an exponential random variable from [0,\infty) you compute -ln(1-x) where x is a uniform random varaible over [0,1). — John Kormylo, Jan 04 '23 at 16:02

score 2 · Accepted Answer · answered Jan 07 '23 at 07:58

I used the function defined by Mark Wibrow (see Gaussian random numbers) to obtain a Gaussian random number and then draw the geometric Gaussian motions with different variance values based on those Gaussian random numbers.

The code

\documentclass[11pt, margin=.5cm]{standalone}
\usepackage{tikz}
\usetikzlibrary{math}
%% Mark Wibrow's code
\newcount\gaussF
\edef\gaussR{0}
\edef\gaussA{0}
\makeatletter
\pgfmathdeclarefunction{gaussR}{0}{%
 \global\advance\gaussF by 1\relax
 \ifodd\gaussF
  \pgfmathrnd@%
  \ifdim\pgfmathresult pt=0.0pt\relax%
    \def\pgfmathresult{0.00001}%
  \fi
  \pgfmathln@{\pgfmathresult}%
  \pgfmathmultiply@{-2}{\pgfmathresult}%
  \pgfmathsqrt@{\pgfmathresult}%
  \global\let\gaussR=\pgfmathresult%radius
  \pgfmathrnd@%
  \pgfmathmultiply@{360}{\pgfmathresult}%
  \global\let\gaussA=\pgfmathresult%angle
  \pgfmathcos@{\pgfmathresult}%
  \pgfmathmultiply@{\pgfmathresult}{\gaussR}%
 \else
  \pgfmathsin@{\gaussA}%
  \pgfmathmultiply@{\gaussR}{\pgfmathresult}%
 \fi
}
\pgfmathdeclarefunction{invgauss}{2}{%
  \pgfmathln{#1}% <- might need parsing
  \pgfmathmultiply@{\pgfmathresult}{-2}%
  \pgfmathsqrt@{\pgfmathresult}%
  \let@radius=\pgfmathresult%
  \pgfmathmultiply{6.28318531}{#2}% <- might need parsing
  \pgfmathdeg@{\pgfmathresult}%
  \pgfmathcos@{\pgfmathresult}%
  \pgfmathmultiply@{\pgfmathresult}{@radius}%
}
\pgfmathdeclarefunction{randnormal}{0}{%
  \pgfmathrnd@
  \ifdim\pgfmathresult pt=0.0pt\relax%
    \def\pgfmathresult{0.00001}%
  \fi%
  \let@tmp=\pgfmathresult%
  \pgfmathrnd@%
  \ifdim\pgfmathresult pt=0.0pt\relax%
    \def\pgfmathresult{0.00001}%
  \fi
  \pgfmathinvgauss@{\pgfmathresult}{@tmp}%
}
\begin{document}
\tikzmath{%
  real \m, \s, \xBound, \yBound, \v0;
  \xBound = 3.5;
  \yBound = 7;
  \m = 1;
  \v0 = .2;
  function GBM(\t, \ini, \s) { % argument, initial value, standard deviation
    return {\iniexp((\m -\s\s/2)\t +\srandnormal/10))};
  };
}
\begin{tikzpicture}[xscale=2]
  \draw[gray, ->] (-.5, 0) -- (\xBound +.5, 0);
  \draw[gray, ->] (0, -1.2) -- (0, \yBound);
\foreach \s/\rgb in {.2/black, .5/violet, .7/blue,
    .9/green!70!black, 1.2/orange, 1.7/red}{%
    \draw[\rgb, thick] (0, \v0)
    \foreach \t in {.05, .1, ..., \xBound}{%
      -- (\t, {GBM(\t, \v0, \s)})
    };
  }
\end{tikzpicture}
\end{document}

How to draw geometric Brownian motions in tikz/pgf

1 Answers1