\documentstyle[12pt]{article}

\title{Multifactor Generalization of\\
"Discount-Bond Derivatives on\\
a Recombining Binomial Tree"\\}

%\thanks{First Draft: May 29, 1997}}
\author{J. Chalupa \\ 
Box 82 \\
Princeton, MA 01541 USA \\
jvic@tiac.net}
 \date{}
\maketitle
\begin{abstract}
\noindent


The security dynamics described by the Black-Scholes equation 
with price-dependent variance can be approximated as a damped 
discrete-time hopping process on a recombining binomial tree. 
In a previous working paper, such a nonuniform tree was 
explicitly constructed in terms of the continuous-time variance. 
The present note outlines how the previous procedure could be 
extended to multifactor Black-Scholes with price- and 
time-dependent coefficients. The basic idea is to derive new 
coordinates which give a Black-Scholes equation with all the 
$\sigma$'s equal to unity. In 
the discrete-time tree corresponding to this equation, nodes are 
uniformly spaced and the hopping probabilities are not constant. 
When the new coordinates are mapped back onto prices, the ensuing 
tree is nonuniform. A derivative can be valued with the new 
coordinates or the original prices. 
 
\end{abstract}
\newpage

\begin{document}

%\section*{~}

A recent analysis\cite{jcbukas} 
of the B\"uhler-K\"asler discount-bond model examined the 
security dynamics in the discrete-time formulation. For a 
price-dependent variance, a recombining binomial 
tree with the correct continuous-time limiting behavior 
was explicitly constructed.

Left open in that paper was the issue of whether the 
tree construction procedure could be extended to 
time-dependent variance and to multifactor continuous-time 
processes. This note discusses how to go about this. 
A recent discussion\cite{findif} of the 
finite-difference method vis-\`a-vis the implied-tree 
approach\cite{dermankanizou} stresses the interest 
of the issue. The present version of this paper 
is intended to be read in conjunction 
with Ref. \cite{jcbukas}. Equations for the tree 
structure will be derived, but only a general discussion 
of the expected solutions will be presented here. 

In terms of the price variables $z_i\; (i=1,...,N)$, the 
valuation equation of interest for a derivative security 
$f$ is 

\begin{equation}
\frac{\partial f}{\partial t} + 
\frac{1}{2} \sum_{i,j} \rho_{ij}\sigma_i \sigma_j
\frac{\partial^2 f}{\partial z_i \partial z_j} + 
\sum_i v_i \frac{\partial f}{\partial z_i} - r f = 0 , 
\label{zequation}
\end{equation}

\noindent where $\rho_{ii}=1$ holds; the quantities 
$\sigma_i, \rho_{ij}, v_i$ and the interest rate 
$r$ are functions of the price variables $z_i$ and time $t$. 
The Black-Scholes no-arbitrage argument leads to equations 
of this form. 

Suppose that the price variables $z_i$ can be transformed 
to new variables $\xi_i$. In the resulting valuation 
equation, 

\begin{equation}
\frac{\partial f}{\partial t} + 
\frac{1}{2} \sum_{i,j} \widetilde{\rho}_{ij}
\frac{\partial^2 f}{\partial \xi_i \partial \xi_j} + 
\sum_i \widetilde{v}_i \frac{\partial f}{\partial \xi_i} 
- r f = 0 , 
\label{xiequation}
\end{equation}

\noindent $\widetilde{\rho}_{ii}=1$ holds 
as before, and the 
coefficients are functions of the $\xi_i$'s and $t$. 
Discussion of the transformation is deferred. The 
expression  $\frac{\partial f}{\partial t}$ has 
different meanings 
in equations (\ref{zequation}) and (\ref{xiequation}) 
because the partial derivative is taken with different 
quantities held constant. 
A discrete-time multinomial-tree analog of 
(\ref{xiequation}) is sought. For a time increment 
$\tau$, consider the expression

\begin{equation} 
f(\mbox{\boldmath $\xi$},t)=
\frac{1}
   {1+\tau\,r(\mbox{\boldmath $\xi$},t)}
\sum_{\mbox{\boldmath $\epsilon$}} 
p(\mbox{\boldmath $\epsilon$}) 
f(\{\xi_i+\epsilon_i a\},t+\tau) \qquad 
\epsilon_i = \pm 1 ,
\label{hopequation} 
\end{equation}

\noindent which describes a time-reversed 
(from $t+\tau$ to $t$)  hopping process on a 
multinomial recombining tree with lattice spacing 
$a$. The notation 
$\mbox{\boldmath $\xi$}=\left\{\xi_i\right\}$ is 
used. The probabilities 
$p(\mbox{\boldmath $\epsilon$})$ are the 
likelihoods of given price changes. 

As usual, equation (\ref{hopequation}) is Taylor-expanded 
to second order in {\boldmath $\xi$} and first order in 
$\tau$ and matched to (\ref{xiequation}). The 
assignment $a=\sqrt{\tau}$ leads to a set of equations 
for the $p(\mbox{\boldmath $\xi$})$'s in terms of the 
coefficients of (\ref{xiequation}). This set is 
underdetermined when the number $N$ of price 
variables exceeds two.

The set of constraints for 
the $2^N$ $p$'s must be closed. A possible starting point is  
the observation that the multinomial tree can be viewed as 
a particular type of finite-difference scheme: the nodes 
and time increments are chosen so that, if all works out, the 
weights of all the $f(\mbox{\boldmath $\xi$},t+\tau)$'s are 
nonnegative and a probabilistic interpretation exists. 
The $p(\mbox{\boldmath $\xi$})$'s might be constructed 
directly from a finite-differencing scheme. In fact, 
consider a finite-difference solution of 
(\ref{xiequation}) in which the partial derivatives are 
computed from the $f$-values at the corners of a hypercube 
of side $2\sqrt{\tau}$. This corresponds to the 
probability

\begin{equation}
p(\mbox{\boldmath $\epsilon$})
=2^{-N}\left(\sum_ {m\le n}^N  
\epsilon_m \epsilon_n \widetilde{\rho}_{mn}+ 
\sum_{m=1}^N \epsilon_m \widetilde{v}_m \sqrt{\tau} 
\right)
\end{equation}

\noindent  proposed by Boyle, Evnine and Gibbs\cite
{boyleevninegibbs}. They caution 
that the non-negativity of the 
$p(\mbox{\boldmath $\epsilon$})$'s must be checked in 
each given case. Ideally, the domain of 
{\boldmath $\widetilde{\rho}$} in which the correlation 
matrix has 
no negative eigenvalues would coincide with the domain 
of {\boldmath $\widetilde{\rho}$} 
in which all the $p(\mbox{\boldmath $\epsilon$})$'s 
are nonnegative\footnote{If the 
$\xi_i$'s are strongly correlated, the $p$'s differ strongly 
in magnitude; the short-time continuum dynamics 
might help indicate the tree paths with significant 
statistical weight.}.

It remains to present a scheme for determining 
{\boldmath $\xi(z)$} or {\boldmath $z(\xi)$}. 
Let the notation 
$\left.\partial f/\partial z_i\right\vert_
{\mbox{{\boldmath{$z'$}},$t$}}$ denote the partial 
derivative of $f$ with respect to $z_i$ with $t$ and 
$z_{j\ne i}$ fixed. 
When the chain rule  

\begin{eqnarray}
\left.\frac{\partial f}{\partial z_i}\right\vert_
{\mbox{{\boldmath{$z'$}},$t$}} &=& \sum_m 
\left.\frac{\partial \xi_m}{\partial z_i}\right\vert_
{\mbox{{\boldmath{$z'$}},$t$}} 
\left.\frac{\partial f}{\partial \xi_m}\right\vert_
{\mbox{{\boldmath{$\xi'$}},$t$}}\\
\left. \frac{\partial f}{\partial t}\right\vert_
{\mbox{\boldmath $z$}} &=& 
\left.\frac{\partial f}{\partial t}\right\vert_
{\mbox{\boldmath{$\xi$}}}  +  \sum_m 
\left.\frac{\partial \xi_m}{\partial t}\right\vert_
{\mbox{\boldmath{$z$}}}
\left.\frac{\partial f}{\partial \xi_m}\right\vert_
{\mbox{{\boldmath{$\xi'$}},$t$}}
\end{eqnarray}

\noindent is substituted into equation (\ref{zequation}), 
the second-order term becomes 

\begin{equation}
 \sum_{i,j} \rho_{ij}\sigma_i \sigma_j
\frac{\partial^2 f}{\partial z_i \partial z_j}=
\sum_{m,n} \left\{\sum_{i,j} 
\rho_{ij}\sigma_i\sigma_j 
\left.\frac{\partial \xi_m}{\partial z_i}\right\vert_
{\mbox{{\boldmath{$z'$}},$t$}}
\left.\frac{\partial \xi_n}{\partial z_j}\right\vert_
{\mbox{{\boldmath{$z'$},}$t$}} \right\}
\frac{\partial^2 f}{\partial\xi_m\,\partial\xi_n}
+...
\end{equation}

\noindent Comparing to equation (\ref{xiequation}) gives 

\begin{equation}
\sum_{i,j} 
\rho_{ij}\sigma_i\sigma_j 
\left.\frac{\partial \xi_m}{\partial z_i}\right\vert_
{\mbox{{\boldmath{$z'$}},$t$}}
\left.\frac{\partial \xi_m}{\partial z_j}\right\vert_
{\mbox{{\boldmath{$z'$}},$t$}} = 1 
\qquad\qquad m=1,...,N .
\label{treeequation}
\end{equation}

\noindent The boundary condition can be taken, 
for example, as 
$\xi_i\left(\{z_{n\ne i}\},z_i=0\right)=0$, or it 
can be fine-tuned to the expiration boundary 
conditions on the derivatives of interest. 
The $\xi$-lattice has the points 

\begin{equation}
\xi_i = n_i \sqrt{\tau} 
\qquad i=1,...,N;\quad  n_i = 0,\pm 1,\pm 2,...
\end{equation}

\noindent The corresponding points of the $z$-lattice 
are the intersections of the N $z$-hypersurfaces 
$\xi_i(\mbox{\boldmath $z$}) = n_i \sqrt{\tau}$. 
Alternatively, Jacobian matrices could be 
used to express 
(\ref{treeequation}) in terms of partial derivatives 
at constant $\mbox{\boldmath{($\xi'$}},t)$, and the 
ensuing expressions solved for \boldmath{$z(\xi)$}. 
Also, (\ref{treeequation}) could be solved for the 
$\sigma$'s in terms of a posited z-tree 
structure and the $\rho$'s or $p$'s. 

For the tree description 
to be used as an economic model and not only as an 
approximant to the continuous-time limit, no-arbitrage 
conditions for the z-jumps must hold on the 
portion of the tree used to value derivatives. 
The existence of a path to the continuous-time 
limit enhances confidence in heuristic utilizations of 
nonuniform multinomial trees. 

\begin{thebibliography}{9}

\bibitem{jcbukas}
J. Chalupa, Economics 
Working Paper Archive
ewp-fin/9702003 (1997)
{\tt <}http://econwpa.wustl.edu{\tt >}; 
and references therein.

\bibitem{findif} 
R. Lagnado and S. Osher, RISK 
$\underline{\hbox{10}}$, No. 4, p. 79 (1997).

\bibitem{dermankanizou}
E. Derman, I. Kani and J.Z. Zou, Financial Analysts Journal 
July-August 1996, p. 25.

\bibitem{boyleevninegibbs}
P.P. Boyle, J. Evnine and S. Gibbs, Rev. Financial Studies 
$\underline{\hbox{2}}$, 241 (1989).

\end{thebibliography}

\end{document}