%Paper: ewp-fin/9607009
%From: jchalupa@ix.netcom.com
%Date: Wed, 24 Jul 96 14:29:25 CDT


\documentstyle[12pt]{article}

\title{Option Valuation and the Price of Risk}
%\thanks{First Draft: July 18, 1996}}
\author{J. Chalupa \\ 
Box 82 \\
Princeton, MA 01541 USA \\
jchalupa@ix.netcom.com}
 \date{}
\maketitle
\begin{abstract}
\noindent
A valuation model is presented for options on stocks for 
which Black-Scholes arbitrage does not entirely eliminate risk. 
The price dynamics of a portfolio of options and the 
underlying security is quantified by requiring that the 
excess reward-to-risk ratio of the portfolio be identical 
to that of the underlying stock:

$$\left. \frac{\hbox{excess return}}{\hbox{risk}}
\right\vert_{\hbox{portfolio}}  = 
~\left. \frac{\hbox{excess return}}{\hbox{risk}}
\right\vert_{\hbox{stock}} . $$

\noindent  The nonlinear 
evolution equation for the portfolio value is homogeneous 
of degree one. A representative distribution is obtained 
from recent stock-history time series; numerical solutions 
for European calls are usually close to the Black-Scholes 
values, but naked and covered calls have different valuations. 
For infinitesimal time steps and a lognormal stock-price 
distribution, the evolution equation reduces to 
the Black-Scholes form. An analytically tractable non-lognormal 
distribution is analyzed near option expiration, and a formula 
expressing the deviation from the lognormal case is obtained 
for an out-of-the-money call. The present 
model is discussed in the context of previous work, and the 
effect of nonlinearity on the valuation of a portfolio 
of derivative securities is considered. 

\end{abstract}

\begin{document}


\newpage
\section{Introduction}

The Black-Scholes analysis of option 
pricing\cite{blackscholes} assumes
that the logarithm of the stock price changes with 
time  according to a Wiener process. The ensuing 
distribution is lognormal. In this case a unique 
option price can be determined by use of arbitrage 
arguments: any price other than the Black-Scholes 
value permits a riskless prounction of the stock price which leads to such 
an instantaneous return and satisfies the boundary 
conditions at expiration. It turns out to be 
independent of the first moment of the stock's 
lognormal distribution. 

When the stock-price dynamics is not a Wiener 
process\cite{mantegna,mantegnastanley}, the 
higher moments of the distribution 
are not negligible. By taking an offsetting 
position in the stock, an option holder typically 
can reduce risk but may not be able to eliminate 
it altogether.
 
Thus, the issue arises of how to assign an 
option valuation for stocks to which the 
Black-Scholes arbitrage argument and risk-neutral 
valuation are not applicable. It is of interest for 
mathematical 
economics\cite{hp,follmersonder,schweizer} and for recent 
calculational 
valuation methods
\cite{bouchaudjphys,mikheev,bouchaudrisk,bouchaudbook,aurell}.
For a Wiener process, the 
option--or any portfolio composed of the stock and 
its derivatives--have the same excess-reward-to-risk 
ratio as the stock itself\cite{merton,hull}. 
A natural valuation procedure is to 
require that this hold for the general case:

\begin{equation}
\frac{\hbox{excess portfolio return}}
{\hbox{portfolio risk}} = 
\frac{\hbox{excess return of underlying security}}
{\hbox{risk of underlying security}} . 
\end{equation}

\noindent In other words, the price of risk for 
an option portfolio is identical to the price 
of risk of the underlying security. The work 
reported here examines some consequences of 
this assumption. The formulation is designed 
to be readily impleentable in computations and 
calculations, and the presentation is more 
intuitive than deductive. 

The paper is organized into five sections  
of which this Introduction is the first. 
Section 2 develops the reward-to-risk valuation
model, applies it to recent time-series data for 
Merck and Oracle stock, and finds for thiese cases 
that the corresponding values of 
European values are close to the Black-Scholes 
results. Section 3 treats
stock-price distributions of the 
Chapman-Kolmogorov class with nonnegligible 
higher cumulants and presents an analytically 
tractable example. The behavior of the 
option value is examined in the short-time 
regime and deviations from the Gaussian 
approximation are exhibited. Section 4 
treats the relation of this work to other
analyses of nongaussian stock distributions
\cite{bouchaudjphys,mikheev,bouchaudrisk,bouchaudbook,aurell}. 
It also considers how the 
nonlinearity of the equation of motion affects 
the valuation of a portfolio of derivatives. 
Section 5 ends the paper with a summary and 
conclusions. 


\section{Valuation Equation}

This section presents the equation of motion 
which determines the valuation of a 
portfolio $\Pi$ of consisting of shares of a 
stock or stock index of price S and options 
on that stock. As discussed 
in the Introduction, the basic assumption is that 
the excess-return-to-uncertainty ratio of the 
portfolio equals that of the stock. Consider the 
portfolio at a time t. In terms of the logarithm 
$z = \ln S$ of the stock price, Equation 1 
indicates that the portfolio's 
value a time increment $\Delta t$ later is 
determined by the relation 

\begin{equation}
\frac{\left\langle \Delta \Pi\right\rangle - 
r \, \Delta t \, \Pi(z,t)}
{{\cal R}[\Pi]} = 
\frac{\left\langle e^{\Delta z} \right\rangle   - 
(1 + r~ \Delta t)}
{{\cal R}[e^{\Delta z}]}
\label{equationofmotion}\end{equation}\begin{equation}
\Delta \Pi \equiv \Pi(z+\Delta z, t+\Delta t) - \Pi(z,t)
\end{equation}

\noindent where the expectation value 
$\left\langle...\right\rangle$ 
is taken over the distribution for stock price changes. 
The risk metric ${\cal R}$ will be discussed further below. 
The time steps $\Delta t$ are taken as finite but, for 
simplicity,  
small enough that a linearized approximation is 
valid for interest accrual at the riskless rate $r$. 

The evolution of the stock-price distribution 
and option value will be traced from the initial time 
$t_0 = 0$ and initial price $z_0$.  
The probability distribution z at time 
t is taken as $ P(z,t; z_0,t_0)$. 
For times $t_1$ between $t_0$ and 
$t$, the Chapman-Kolmogorov condition\cite{merton}

\begin{equation}
P(z,t; z_0,t_0) = 
\int dz_1 \, P(z,t ; z_1,t_1) \,  P(z_1,t_1; z_0,t_0) 
\end{equation} 

\noindent is assumed. The distribution is specified to 
have the form 

\begin{equation}
P(z_2,t_2;z_1,t_1) = 
P(z_2-z_1,t_2-t_1; 0,0) \qquad  (t_2 \geq t_1). 
\end{equation}

\noindent The shorthand $P(z,t)$ will be used for $P(z,t;0,0)$. 
Under these conditions, Equation \ref{equationofmotion} 
predicts that a $z$-independent portfolio, i.e. one consisting 
of cash, will compound at the riskless interest rate, and the 
equation admits the stock price $S=e^z$ as a solution. 

The immediate task is to transform the evolution equation into 
a form suitable for performing computations.  Consider a portfolio 
$\Pi$ consisting of cash, the stock S, and European options 
depending on S. For simplicity let all the derivatives mature 
at the same time $T$. The expectation value of the change in 
the portfolio's value in time $\Delta t$ is 

\begin{equation}
\left\langle \Delta \Pi \right\rangle = 
\left\langle \Pi(z+\Delta z, t+\Delta t )\right\rangle_
{P(\Delta z,\Delta t)} - \Pi(z,t).  
\end{equation}

\noindent The standard deviation 

\begin{equation}
{\cal R}[\Pi] = 
\left\langle \left\vert \Pi(z+\Delta z, t+ \Delta t) - 
\left\langle \Pi (z+\Delta z, t+\Delta t)
\right\rangle\right\vert^2\right\rangle^{1/2} 
\end{equation}

\noindent sugests itself as a choice of risk metric. 
(Actually, the sign of ${\cal R}$ is the sign of the derivative 
$\partial \Pi /\partial z$, which will be taken as positive, as 
will $\Pi$, unless specified otherwise.) However, more 
general forms like 

\begin{equation}
{\cal R}[\Pi] = \left\langle \left\vert 
\Pi(z+\Delta z, t+\Delta t) - 
\left\langle \Pi(z+\Delta z, t+\Delta t)
\right\rangle \right\vert^p\right\rangle^{1/p} 
\label{riskmetricform}
\end{equation}

\noindent can also be used. The form of Equation 
\ref{riskmetricform} 
ensures that the portfolio's value scales linearly if the 
quantity of each asset is changed by a common multiplicative factor, 
i.e. that the evolution equation, although nonlinear, is 
homogeneous of degree one. 
The $p=1$ alternative has appeal because 
a risk can be assigned to a $\delta$-source 
(Arrow-Debreu security) 
whereas $p=2$ yields the integral of the square of a 
$\delta$-function 
in this case; moreover, the integral of $e^{p\,z}P(z,t)$ 
must converge 
at large z for the risk metric to be finite, and the $p=1$ case  
accommodates distributions with stronger tails than does the 
$p=2$ case. For both the $p=1$ and $p=2$ cases, a portfolio's 
risk metric is independent of its amount of cash; the cash part 
compounds at the riskless rate while the rest of the portfolio 
is governed by the risk metric. 

Equation \ref{equationofmotion} allows the portfolio's value 
(as a function of stock price) at a 
given time to be determined from its value at the 
{\it subsequent} time step: 

\begin{equation}
\Pi(z,t) = 
\frac{\left\langle \Pi(z+\Delta z, t+\Delta t )\right\rangle_
{P(\Delta z,\Delta t)}-\omega{\cal R}
[\Pi(z + \Delta z, t + \Delta t)]}
{1 + r~\Delta t}
\label{formalsolution}
\end{equation} 

\noindent where the constant $\omega$ is the Sharpe ratio for the 
stock:

\begin{equation}
\omega = \frac{\left\langle e^{\Delta z} \right\rangle   - 
(1 + r~ \Delta t)}
{{\cal R}[e^{\Delta z}]}.
\end{equation} 

\noindent Equation \ref{formalsolution} can be iterated in 
backward time steps. Thus, one can begin at the expiration time $T$ 
when the dependence of $\Pi$ on the stock price is known, determine 
the portfolio value at the prior time step, and repeat the process 
until the required time to expiration is reached. If the 
evolution equation were linear, the Chapman-Kolmogorov 
condition might be used to self-convolve the distribution 
and price the option on the basis of the price distribution 
for the time to expiration. Such procedures are not possible 
for Equation 
\ref{formalsolution} because of its nonlinearity (unless the 
time step is deliberately increased--even to be as large as the 
time to expiration--because a valuation only over longer time 
intervals is of interest). Thus, the numerical solution 
procedure can be time-consuming. 

%the computations reported in this 
%paper were performed on a 21164/300 MHz Alpha workstation.

There are countervailing effects in Equation \ref{formalsolution}. 
For example, if the distribution is leptokurtotic, 
i.e. has excess weight in the 
tails, the first term raises the price of a well-out-of-the-money 
call from its Black-Scholes value. However, the risk of the call 
in (\ref{formalsolution}) presumably also increases, which tends 
to decrease the option's value because of the risk discount. 
Moreover, it is not clear {\it prima facie} whether the risk of the 
call increases or decreases relative to the risk of the stock. In 
short, a complicated situation ensues and the value of an option 
may increase or decrease when weight is added to the 
underlying security's distribution tails. 

%\begin{table}\begin{center}\begin{tabular}{l}
%$C_1 = 2.$ \\
%$C_2 = 2.$ \\
%$C_3 = 2.$ \\
%$C_4 = 2.$ \\
%\end{tabular}\end{center}
%\caption{The cumulants for the daily MRK time series}
%\end{table}

\begin{table}[p]
\begin{center}
\begin{tabular}{|c|c|c|c|}\hline
Strike  &  
  $\frac{\hbox{Naked Call}}{\hbox{Black-Scholes Call}}$ & 
$\frac{\hbox{Covered Call}}{\hbox{Black-Scholes Call}}$ & 
Black-Scholes \\ \hline\hline
30 & .999 & .999 & 32.44 \\ \hline
35 & .998 & .999 & 27.67 \\ \hline
40 & .998 & .998 & 22.94 \\ \hline
45 & .996 & .998 & 18.36 \\ \hline
50 & .995 & .997 & 14.08 \\ \hline
55 & .995 & .996 & 10.32 \\ \hline
60 & .993 & .995 & 7.22 \\ \hline
65 & .987 & .997 & 4.84 \\ \hline
70 & .984 & .997 & 3.11 \\ \hline
75 & .981 & .997 & 1.93 \\ \hline
\end{tabular}
\end{center}
\caption{The January 1997 MRK call relative to the Black-Scholes 
value. The stock price is $62.125$ and \$1.02 in dividends will 
be paid before expiration.}
\end{table}

In order to demonstrate the use of Equation \ref{formalsolution}, 
it was applied to the time series of Merck closing prices from
1 September 1988 to January 2-April 10. Under the assumption that 
the series is generated by 
a markovian process, the empirical cumulative distribution 
associated with the series was generated in histogram 
form. 
Results of the Kolmogorov-Smirnov and Kuiper tests
\cite{numericalrecipes} indicated 
that, at least for demonstration purposes, the time series may 
plausibly be taken as quasistationary.  
Table \nopagebreak 
1 shows the 
ratio of the $p=2$ Equation \ref{formalsolution} values 
to the Black-Scholes price of naked and covered European calls. 
The covered call value was determined by applying the expiration 
boundary condition to the portfolio (long stock and short call), 
back-stepping to compute the portfolio value at previous times, 
and subtracting this value from the stock price. The numerical 
results were checked by decreasing the histogram 
bin size; a stability analysis\cite
{numericalrecipes} of the valuation equation was not undertaken.
In this particular situation the effects of nonlinearity are weak, 
the Central Limit Theorem gives rise to Gaussian behavior, and 
the deviation from the Black-Scholes result is small. 

Some deviations from the Black-Scholes values were found for 
Oracle (ORCL) stock. The time series from 
1 September 1988 to 30 June 1996 was examined, and the valuation 
computations were carried out notwithstanding that evidence for 
stationarity was not strong. Table 3 shows the results for calls 
with 15 days to expiration given a stock price of 39.5. Despite 
the weak stationarity, the discrepancies with Black-Scholes make 
the results of interest. The differences between the values of 
naked and covered calls are due to the nonlinearity of the 
evolution equation: the risk of a covered call couples 
to the risk of the stock in the portfolio. 

\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}\hline 
\multicolumn{1}{|c|}{$\hbox{Strike}\atop\hbox{Price}$} &   
\multicolumn{2}{|c|}{
$\frac{\hbox{Naked Call}}{\hbox{Black-Scholes Call}}$ } &
\multicolumn{2}{|c|}{ 
$\frac{\hbox{Covered Call}}{\hbox{Black-Scholes Call}}$ } &
\multicolumn{1}{|c|}
           {$\hbox{Black-Scholes}\atop\hbox{Call}$} \\ \hline
$~$ & p=1 & p=2 & p=1 & p=2 & $~$  \\ \hline\hline
35 & .990 & .990 & .997 & 1.010 & 5.1116 \\ \hline
37.5 & .976 & .971 & .988 & 1.003 & 3.3201 \\ \hline
40 & .960 & .935 & .970 & .983 & 1.9787 \\ \hline
42.5 & .942 & .867 & .969 & .968 & 1.0814 \\ \hline
45 & .956 & .776 & .988 & .956 &  0.5437 \\ \hline
47.5 & 1.015 & .630 & 1.153 & 1.060 & 0.2530 \\ \hline
50 & 1.197 & .436 & 1.252 & 1.044 & 0.1097 \\ \hline
\end{tabular}
\end{center}
\caption{ORCL call relative to the Black-Scholes 
value. The stock price is $39.5$ and there are fifteen (15) 
days toexpiration.}
\end{table}


\section{Model Stock-Price Distributions} 

Using the time series to value the option, as was done in 
the previous section, is attractive because the price is 
determined directly from the data rather than from a fabricated 
model for the stock-price dynamics. On the other hand, analyzing 
pretailored models offers insights not available from raw data. 
This section considers such a model and applies it to option 
behavior near expiration. 

For distributions satisfying Equations 2 and 3, the Fourier 
transform 

\begin{equation}
P(k,t)  = \left\langle e^{-ikz} \right\rangle = 
\int dz \, e^{-ikz}P(z,t)
\end{equation}

\noindent obeys the relation 

\begin{equation}
P(k,t) = P(k,1)^t. 
\label{fourierchapman}
\end{equation}

\noindent If $P(z,t)$ is known for a given value of $t$, say $t=1$, 
Equation \ref{fourierchapman} allows it to be determined for any 
time. The lognormal distribution for geometric Brownian 
motion satisfies this equation. Before adopting $P(z,t < 1)$,
it is necessary to check that it is a 
positive-semidefinite function of $z$.

Evaluating the option behavior near expiration requires the 
evolution equation for small time steps. In the limit
$\Delta t \rightarrow  0$, the $p=2$ Sharpe ratio is 

\begin{eqnarray}
{\omega} &=& \lim_{\Delta t \rightarrow  0}
\frac{P(k=i,\Delta t)-(1+r)\Delta t}
{(P(k=2i,\Delta t) - P(k=i,\Delta t)^2)^{1/2}} \\
{~} &=& \Delta t^{1/2} \frac{\ln P(i,1)-r}
{\ln \left[P(2i,1))/P(i,1)^2\right]. } 
\end{eqnarray} 

\noindent in terms of the distribution's characteristic function. 

The stock-price distributions under consideration have stronger 
tails than a gaussian, but the weight in the tails is small 
enough that the risk ${\cal R[S]}$ is finite. A natural 
possibility is the exponential distribution 

\begin{equation}
P(z,1) = \frac{\sqrt{2}}{2 \sigma}
e^{-\left\vert z \right\vert \sqrt{2}/\sigma}, 
\end{equation}

\noindent The mean $\left\langle e^z \right\rangle$
and mean square $\left\langle e^{2 \, z} \right\rangle$ 
of the stock price are finite for sufficiently small 
values of the standard deviation $\sigma$. 
$P(z,1)$ has the Fourier transform

\begin{equation}
P(k,1) = \frac{1}{1+\frac{1}{2}k^2 \sigma^2}. 
\end{equation}

\noindent For arbitrary time t, Equation 
\ref{fourierchapman} gives\cite{gradshteyn}

\begin{eqnarray}
{P(z,t)} &=& \int_{-\infty}^\infty \frac{dk}{2\pi} 
e^{ikz}P(k,1)^t \\
{~} &=& \frac{\sqrt{2/\pi}}{\sigma\Gamma(t)} 
\left( \frac{\left\vert z \right\vert}
  {\sigma \sqrt{2}}\right)^{t-\frac{1}{2}} 
K_{t-\frac{1}{2}} 
(\left\vert z \right\vert \sqrt{2}/\sigma),
\label{evenbessel}
\end{eqnarray}

\noindent where
$K_{t-\frac{1}{2}}(\left\vert z \right\vert \sqrt{2}/\sigma)$ 
is a Bessel function.  This distribution is an even 
function of z and therefore unskewed. For small times 
$t \rightarrow 0$, $P(z,t)$ approaches the $\delta$-function 
$P(z,t=0) = \delta(z)$ in a different manner than the Gaussian 
associated with geometrical Brownian motion. Near the origin, 
the distribution diverges as 
$P(z,t) \sim t/\left\vert z \right\vert^{1-2t}$ ; its asymptotic 
behavior far from the origin is dominated by the exponential 
decay of the Bessel function, and the amplitude is proportional 
to $t \rightarrow 0$. (Thus, the expectation value 
$\overline{\left\vert x \right\vert} \sim t$ for $t \sim 0$, but 
the Central Limit Theorem indicates that 
$\overline{\left\vert x \right\vert} \sim t^{1/2}$ for 
$t \rightarrow \infty$. If an exponent $\widetilde{H}(t)$ 
is defined by 

\begin{equation}
\widetilde{H}(t) = \frac
{\partial \ln \overline{\left\vert x \right\vert} }
{\partial \ln t},
\end{equation}

\noindent $\widetilde{H}(t)$ will depend on the time scale. 
The expectation value $\overline{\left\vert x \right\vert} \sim t$ 
is similar to the quantity which Peters\cite{peters}
analyzed to assign a Hurst exponent to economic time series, 
and exhibits a similar crossover downward to the Gaussian value.)

It is useful to generalize Equation \ref{evenbessel} 
to a symmetric distribution 
which reduces to the exponential form at an arbitrary 
time $\tau$. This can be done by adopting the relation

\begin{eqnarray}
{P(z,t)} &=& \int_{-\infty}^\infty \frac{dk}{2\pi} 
e^{ikz} \frac{e^{-i k v t}}
{(1+\frac{1}{2}k^2 \sigma^2\tau)^{t/\tau}} \\
{~} &=& \frac{\sqrt{2/\pi}}{\sigma\Gamma(t/\tau)} 
\left( \frac{\left\vert z - v t \right\vert}
{\sigma\sqrt{2\tau}}
\right)^{\frac{t}{\tau}-\frac{1}{2}} 
K_{\frac{t}{\tau}-\frac{1}{2}} 
\left(\frac{\left\vert z - v t \right\vert}{\sigma} 
\sqrt{\frac{2}{\tau}}\right).
\label{secondevenbessel}
\end{eqnarray}

\noindent A drift velocity $v$ also has been introduced in 
Equation \ref{secondevenbessel}. The parameters $v, \sigma, 
\tau$ can be assigned by matching the first, second, and 
fourth cumulants of the time series for stock-price changes. 
For the Merck and Oracle series analyzed in 
Section 2, the values of $\tau$ are respectively 
$\tau_{MRK} \simeq .4~\hbox{day}$ and 
$\tau_{ORCL} \simeq 5~\hbox{day}$. Having these 
characteristic time scales on the order of days is consistent 
with the numerical results that the Merck call values are very 
close to Black-Scholes prices while Oracle shows some deviation. 

The overall effect of a nongaussian shape of the distribution 
function has long been understood\cite{merton,jarrowrudd}.
For the distribution of Equation \ref{evenbessel}, on general 
grounds one expects at- or out-of-the-money calls near maturity 
to be 
proportional to the time $T-t$ to expiration. The farther a 
call is 
out of the money, the smaller the proportionality constant 
will be. 
A Black-Scholes call, on the other hand, has the limiting 
behavior $\sim\sqrt{T-t}$ near the strike price and $\sim 
e^{-\frac{ln^2(S/X)}{2\sigma^2(T-t)}}$ far out of the money. 
Near expiration, the near-the-money Black-Scholes call is more 
valuable than that from Equation \ref{evenbessel}, and far from 
the money the Black-Scholes call is worth less.

In fact, limiting forms of option valuations associated with 
$P(z,t)$ can be derived. For example, suppose that the risk 
metric is approximated as the risk metric of the gaussian 
with the same first two moments as $P$:

\begin{equation}
\frac{R_{gauss}[\Pi]}{R_{gauss}[S]} = 
\frac{1}{S}\frac{\partial\Pi}{\partial z}. 
\end{equation}

\noindent In the continuous-time limit this leads to a 
risk-neutral form of $P$, denoted 
$P_{rn}$, which is identical to the original distribution 
except that the replacement 

\begin{equation}
v \rightarrow r - \frac{1}{\tau} \ln(1-\frac{1}{2}\sigma^2\tau) 
\end{equation} 

\noindent is made in the first moment. For an out-of-the-money 
call ($S << X = e^z$), the expression 

\begin{equation}
e^{r(T-t)}C(z,T-t) = 
\left\langle (e^{z+\Delta z} - X) 
\Theta(e^{z+\Delta z} - X)\right\rangle_{P_{rn}(\Delta z,T-t)}, 
\end{equation}
 
\noindent where $\Theta(x) \equiv \frac{1}{2}
(1+ \frac{\left\vert x \right\vert}{x})$, 
can be evaluated by using the 
asymptotic form 

\begin{equation}
P(\Delta z,T-t) \simeq 
\frac{(T-t)/\tau}{\left\vert z \right\vert } 
e^{-\left\vert \Delta z \right\vert \sqrt{2/\tau}/\sigma}
\qquad (T-t\rightarrow 0^+,
   \left\vert\Delta z\right\vert \rightarrow \infty)
\end{equation}

\noindent of the Bessel function. This yields 

\begin{equation}
C(S,T-t) \simeq 
\frac{\sigma^2(T-t)}{2(1-\sigma\sqrt{\frac{\tau}{2})} \ln(X/S)} 
\left(\frac{S}{X}\right)^{\sqrt{2/\tau}/\sigma} 
\left(1 + {\cal O}\left(\frac{1}{\ln(X/S)}\right)\right).
\end{equation}

\noindent The effect of the exponential tail in $P$ makes the 
shows up in the exponential decrease with $z$ of the 
out-of-the-money 
call. This decrease is larger than the Gaussian decrease 
associated 
with the Black-Scholes call. Table 3 shows results for ORCL; the 
parameters of the model distribution were computed by matching 
the first, second, and fourth cumulants from the time series. 
This procedure is straightforward, but a cogent 
alternative\cite{rubinstein,dermankanichriss} is to 
determine the implied stock distribution from market 
prices of options.

The Gaussian analysis motivates examination of the corresponding 
$p=1$ case. On the basis of the 
Gaussian-risk approximation, a solution of the form 

\begin{equation}
C(z,T-t) = C(z,T-t=0)+ (T-t)f(z)
\label{ansatz}
\end{equation}

\noindent is assumed in the limit $T-t \rightarrow 0$. 
Substituting this into the evolution equation and 
matching leading terms in $\Delta t$ and $T-t$ gives 

\begin{eqnarray} 
{C(z,T-t)} &\simeq& (1-2\omega) 
\frac{\sigma^2 (T-t)}{2(1-\sigma\sqrt{\frac{\tau}{2}})\ln(X/S)} 
\left(\frac{S}{X}\right)^{\sqrt{2/\tau}/\sigma} \\
{\omega} &=& \frac{v-r-\frac{1}{\tau} 
\ln(1-\frac{1}{2}\sigma^2\tau) }
{2 \ln(1+\sigma\sqrt{\frac{\tau}{2}})}\tau
\end{eqnarray}

\noindent for $S-X \ll 0$. This is essentially the Gaussian form 
reduced by an additional price-of-risk discount for the higher 
moments of the distribution.

If the stock growth rate $v$ is 
high enough, the calculated value of a naked call becomes negative. 
A potential 
buyer of a naked call who prices with the $p=1$ risk metric 
concludes 
that any positive call value gives an insufficient 
return-to-risk 
ratio. Such a buyer will refuse to pay even an infinitesimal 
price for the call. This decision is a consequence of the choice 
of risk metric. In particular, the uncertainty of the return at 
time $t+\Delta t$ is defined with respect to the expectation 
value of 
the return at $t+\Delta t$. Section 4 will indicate how this 
convention is necessary for the methodology to reduce to the 
arbitrage results for binomial discrete-step stock motions. 
Conveniently, use of the expectation value at $(t+\Delta t)$ 
greatly 
simplifies the numerical solution of the evolution equation by 
enabling backward-stepping in time. However, careful 
interpretation is needed of the unexpected consequences, 
particularly in asymptotic 
limiting cases, which can arise from the nonlinearity of 
the option valuation method. Legitimate reward-to-risk 
criteria may not invariably 
lead an investor to place a positive value on a call. 

For the $p=2$ risk metric, the situation can be even more extreme. 
For an out-of-the-money naked European call near expiration at 
strike price X, one finds

\begin{equation}
\left\langle \Delta C(S e^{\Delta z},\Delta t) \right\rangle
  \simeq 
\frac{\Delta t}{2 \ln(X/S)} 
\left(\frac{S}{X}\right)^{\sqrt{2/\tau}/\sigma} 
\frac{\sigma^2}{1-\sigma\sqrt{\tau/2}}
\end{equation}

\noindent and 

\begin{equation}
{\cal R}\left[ \left(C(S e^{\Delta z},\Delta t)\right)^2 
\right]   \simeq
\frac{(\Delta t) X^2}{\ln(X/S)} 
\left(\frac{S}{X}\right)^{\sqrt{2/\tau}/\sigma} 
\frac{\sigma^3\sqrt{\frac{\tau}{2}}}
{(1-\sigma\sqrt{2\tau})(1-\sigma\sqrt{\frac{\tau}{2}})}. 
\end{equation}

\noindent Use of these two expressions and 
Equation \ref{ansatz} gives 

\begin{equation} \textstyle{
C(S,X,T-t) \simeq 
(T-t)X\left[\frac{\sigma^2 (S/X)^{\sqrt{2/\tau}/\sigma}}
{2(1-\sigma\sqrt{\frac{\tau}{2}})\ln(\frac{X}{S})} - 
\omega
\sqrt{\frac{(S/X)^
{\sqrt{2/\tau}/\sigma} \sigma^3\sqrt{\frac{\tau}{2}}}
{(1-\sigma\sqrt{2\tau})(1-\sigma\sqrt{\frac{\tau}{2}})
\ln(\frac{X}{S})}}
\right]} .
\end{equation}

\noindent The (negative) risk term above is 
asymptotically of the form 
${\cal R}[C]
\approx (S/X)^{\sqrt{2/\tau}/(2\sigma)}$, 
whereas the first term 
is of the form  
$\left\langle \Delta C \right\rangle 
\approx (S/X)^{\sqrt{2/\tau}/\sigma}$. 
For sufficiently large $X$ (and positive $\omega$), there 
ensues 

\begin{equation}
\frac{{\cal R}[C]}
{\Delta C}
\gg 1 , 
\end{equation}

\noindent and the expression for $C(S,X,T-t)$ is 
negative in this limit. No matter how small the excess return, 
the uncertainty in C drives the far-out-of-the-money valuation 
negative. 

The distribution studied above is an even function of z whereas 
empirical stock returns are skewed. By considering a distribution 
of the form 

\begin{equation}
P_{skew}(z,\tau) = 
\left\{ 
\begin{array}{cr}
\displaystyle{
\frac{\kappa_p \kappa_n }{\kappa_p + \kappa_n}
  e^{-\kappa_p (z-v\tau)} }
 \quad (z \geq v\tau ) \\
\scriptscriptstyle{{~} {~}} \\
\displaystyle{
\frac{\kappa_p \kappa_n }{\kappa_p + \kappa_n}
e^{\kappa_n (z-v\tau)}  }
 \quad (z \leq v\tau) 
\end{array}
\right. 
\end{equation}

\noindent the model can be generalized to incorporate skewness. 
A drift term $v$ and a timescale factor $\tau$ 
have also been included. The Fourier transform 
is\cite{gradshteyn} 

\begin{equation}
P_{skew}(z,t) = \int_{-\infty}^\infty \frac{dk}{2\pi}
\frac{e^{ik(z-v \, t)}(\kappa_p \kappa_n)^{t/\tau}}
{\left(\frac{1}{4}(\kappa_p + \kappa_n)^2 + 
 (k - \frac{i}{2}(\kappa_p - \kappa_n))^2 \right)^{t/\tau}} 
\end{equation}
\begin{equation}  
\textstyle{=
\sqrt{\frac{\kappa_p + \kappa_n}
{\pi \left\vert z - v \, t \right\vert }} 
\left(\frac{\kappa_p \kappa_n \left\vert z - v \, t \right\vert}
{\kappa_p + \kappa_n}\right)^{t/\tau} 
\frac{ e^{-(\kappa_p - \kappa_n)(z - v \, t)/2}  } 
  {\Gamma(t/\tau)} 
K_{\frac{t}{\tau}-\frac{1}{2}} 
(\frac{1}{2}(\kappa_p + \kappa_n) 
    \left\vert z- vt \right\vert)}.
\label{horriblebessel}
\end{equation}

\noindent The four quantities $(v,\kappa_p,\kappa_n, 
\tau)$ can be assigned to characterize 
a data set, e.g. so that the 
first four moments of $P_{skew}$ match the
first four moments of an empirical distribution. 
Whether the matching procedure yields a solution 
and the quality of the fit must be determined on 
a case-by-case basis.  

Finally, it should be noted that the infinite value of 
$P(z=0,t < 1/2)$ is due to the fact that the Fourier transform 
$(1+\frac{1}{2}k^2\sigma^2)^{-t}$ is not integrable for 
$t < \frac{1}{2}$. An alternative like $P_{alt}(k,t=1) = 
\hbox{sech}(k/\kappa)$ gives\cite{gradshteyn} 

\begin{eqnarray}
{P_{alt}(z,t)} &=& \int_{-\infty}^\infty \frac{dk}{2\pi} 
e^{ikz}\hbox{sech}^t(k/\kappa)  \\
{~} &=& \frac{2^{t-2}\kappa}{\pi \Gamma(t)}
\left\vert \Gamma\left(\frac{1}{2}(t + i \kappa x)
\right)\right\vert^2
\label{alternativedistribution}, 
\end{eqnarray}

\noindent which is manifestly positive for all finite x and t, 
and is therefore an acceptable model distribution. For small t, 
$P_{alt}(z=0,t \rightarrow 0^+) \approx 1/t$. Thus, in the 
$t \rightarrow 0$ limit, $P_{alt}(z,t)$ is larger for 
$z \approx 0$ 
and $z \rightarrow \infty$ than a Gaussian of the same mean and 
variance. Because Equation \ref{alternativedistribution} is a 
special case of an integral representation of a hypergeometric 
function, generalizations of $P_{alt}(z,t)$ can be essayed to 
incorporate skewness. 

To illustrate the use of the valuation equation in the 
continuum 
limit, this section has presented calculations utilizing 
well-characterized special functions. The exponential form 
of the distribution (5) at $t=\tau$ was chosen for its 
analytical tractability. However, application of the 
Chapman-Kolmogorov condition to many distributions, e.g. 
the exponentially modulated L\'evy form considered by Montegna 
and Stanley, will be a computational task. Because a distribution 
which is positive-semidefinite for $t=\tau$ may have negative 
values for $t < \tau$, candidate distributions must be analyzed 
on a case-by-case basis. 


\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|}\hline
Strike  &  
  $\frac{\hbox{Risk-Neutral Call}}{\hbox{Black-Scholes Call}}$ & 
% $\frac{\hbox{Covered Call}}{\hbox{Black-Scholes Call}}$ & 
Black-Scholes \\ \hline\hline
30 & 1.002 & 9.6321 \\ \hline
32.5 & 1.002  & 7.2633  \\ \hline
35 & .997  & 5.1116 \\ \hline
37.5 & .981  & 3.3201 \\ \hline
40 & .962 & 1.9787 \\ \hline
42.5 & .966 & 1.0814 \\ \hline
45 & 1.022 & .5437 \\ \hline
47.5 & 1.155 & .2530 \\ \hline
50 & 1.399 & .1097 \\ \hline
\end{tabular}
\end{center}
\caption{Risk-neutral ORCL call relative to the Black-Scholes 
value. The call value was computed with a time-evolved exponential 
distribution. The stock price is $39.5$ and there are fifteen 
(15) days to expiration.}
\end{table}

%  \includeonly{data00a}

\section{Discussion} 

This section covers three topics. First, it will be indicated 
that the present formalism reduces to the Black-Scholes case for 
geometrical Brownian motion. Second, the relationship of the 
present work to the papers of Bouchaud and Sornette and 
subsequent workers will be discussed. Finally, some issues 
arising from adoption of a nonlinear 
valuation equation will be briefly explored. 

The reduction to Black-Scholes will be demonstrated in a concise 
but indirect way. Consider a portfolio $\Pi$ such that 
$\Pi,\frac{\partial\Pi}{\partial z} \geq 0$. The stock price 
changes at discrete time intervals $\Delta t$. 
The distribution of changes z is taken to be binary: 

\begin{equation}
P(z) = q \delta(z-a) + (1-q)\delta(z-b) \qquad (b > a),
\end{equation}

\noindent where $\delta(z)$ denotes a Dirac delta-function. 
For this distribution the risk metric 

\begin{eqnarray}
{{\cal R}[\Pi]} &=& \left\langle \left\vert 
\Pi(z+\Delta z, t+\Delta t) - 
\left\langle \Pi(z+\Delta z, t+\Delta t)
\right\rangle \right\vert^p \right\rangle^{1/p}  \\
{~} &=& \left((q^p + (1-q)^p \right)^{1/p} 
\left(\Pi(z+b,t+\Delta t) - \Pi(z+a,t+\Delta t)\right)
\end{eqnarray} 

\noindent is linear in $\Pi$. Thus a perfect hedge for a 
call can be constructed in the standard way by taking a 
compensating short position in the stock, and the standard 
binomial option valuation\cite{coxrossrubinstein} 
is recovered. Since geometrical 
Brownian motion and Poisson\cite{coxross} hopping processes 
can be 
extracted from the binomial distribution by appropriately 
taking the continuum limit $\Delta t \rightarrow 0$, the 
riskless-hedge option pricing results for these cases are 
recovered as well. Note that the risk metric 

\begin{equation}
\tilde{{\cal R}}[\Pi] = 
\left\langle\left\vert\Delta\Pi
        \right\vert\right\rangle^{1/p}=
\left\langle \left\vert 
\Pi(z+\Delta z, t+\Delta t) - 
\left\langle \Pi(z,t)
\right\rangle \right\vert^p 
     \right\rangle_{P(\Delta z,\Delta t)}^{1/p}
\end{equation}

\noindent is {\it not} linear in $\Pi$ for the binomial 
distribution and therefore does not lead to consistency 
with the arbitrage valuations. 

The methodology in this paper was complete when prior studies 
by Bouchaud and Sornette\cite{bouchaudjphys,mikheev}
and subsequent workers\cite{aurell} came to my attention 
in the financial literature\cite{bouchaudrisk}. While full 
details of their approach will be described 
in a forthcoming monograph\cite{bouchaudbook}, 
my understanding of it is as follows. Bouchaud and 
Sornette's time-integrated formalism takes the viewpoint 
of an option seller who minimizes risk by taking an offsetting 
position in the underlying stock. The trading strategy, i.e. 
the time dependence of the number of shares owned, is 
determined by minimizing the uncertainty in the seller's 
wealth at option expiration. The option offering price is 
determined by the risk premium which the seller requires. 
Aurell and \.Zyckowski have built on this formalism and 
developed a trading strategy to optimize the gain of an investor 
willing to accept a given amount of risk. (They use the term 
"value" to characterize portfolios whose uncertainty vanishes; 
in this paper, an option's value to an investor is 
determined from the expiration boundary condition on the 
portfolio in which the option resides.)
On the other hand,  
the starting point of this paper is a risk-to-reward balance 
for the upcoming time step ("equation of motion"), not an 
integrated trading history. 
Moreover, the price of risk is the cornerstone of the present 
approach and is contained in all hedging strategies based on it; 
in Ref. \cite{bouchaudrisk} an independent parameter is 
introduced to characterize the option seller's risk aversion. 

This paper focuses on valuation of options embedded in a 
given portfolio, but the methodology has also been used to 
derive formal results about hedging. These can facilitate 
comparison with other work. A portfolio consisting of N 
shares of stock S and short one call C has the $p=2$ risk 
metric 

\begin{equation}
{{\cal R}(\Pi)} = \left\langle\left(
\Pi(z+\Delta z,t+\Delta t)- 
\left\langle \Pi(z+\Delta z,t+\Delta t) \right\rangle 
\right)^2 \right\rangle^{1/2}.
\end{equation}

\noindent The portfolio with minimal absolute risk is found 
from the variational condition 
$\partial{\cal R}[\Pi]/\partial N = 0$, which has 
the solution 

\begin{equation}
N_{abs} = \frac{\left\langle \delta C \delta S\right\rangle}
{\left\langle \delta S^2 \right\rangle}
\end{equation}\begin{equation}
\delta \Pi \equiv \Pi(z+\Delta z,t+\Delta t) - 
\left\langle \Pi(z+\Delta z,t+\Delta t) \right\rangle, 
\end{equation}

\noindent and, in terms of the risk 
$\left\langle \delta S^2 \right\rangle $ of the stock, 
the residual risk of the portfolio is 

\begin{equation}
\frac{{\cal R}_{abs}[\Pi]}
{\left\langle \delta S^2 \right\rangle^{1/2}} = 
\sqrt{\frac{\left\langle \delta C^2\right\rangle}
{\left\langle \delta S^2\right\rangle }- 
\left(\frac{\left\langle \delta C \delta S \right\rangle}
{\left\langle\delta S^2\right\rangle}
\right)^2 } 
\end{equation}

\noindent If this risk is neglected, the evolution equation for 
$C(z,t)$ becomes linear. For this approximation, it is convenient 
to treat the call as a linear combination of options whose values 
are bounded in $z$ so that the Fourier transform exists. If 
$f(z,t)$ is such a constituent option and $f(k,t)$ is its Fourier 
transform, the linearized evolution equation for $f(k,t)$ is 

\begin{equation}
\frac{\partial f}{\partial t} + 
\left(\ln P(-k,1) 
 + \frac{r-\ln P(i,1)}{\ln\frac{P(2i,1)}{P(i,1)^2}}
\ln\frac{P(-k+i,1)}{P(-k,1)P(i,1)} \right)f = rf.
\end{equation}

\noindent The problem has been reduced to a first-order 
differential equation whose solution is straightforward. 
Section 3 has noted that for $p=2$ the risk metric may 
be so large that positive option values are not always 
compatible with the risk-to-reward tradeoff.

Minimizing the {\it fractional} 
uncertainty by $\frac{\partial {\cal R}[\Pi]/\Pi}{\partial N} = 0$ 
yields the solution 

\begin{equation}
N_{frac} = \frac{C \left\langle \delta C \delta S\right\rangle - 
S \left\langle \delta S^2\right\rangle}
{C \left\langle \delta S^2\right\rangle - 
S \left\langle \delta C \delta S \right\rangle}
\end{equation}

\noindent $N_{frac}(t+\Delta t)$, which is computed from the 
option value at time $t + \Delta t$, is used to solve for $\Pi$ 
at time $t$; 
$N_{frac}(t)$ can be calculated from $\Pi(t)$  
and the solution process iterated backward in time.
In the Black-Scholes analysis, $N = \partial C/\partial S$; 
$N_{abs}$ and $N_{frac}$ are homogeneous of order 1 in C and 
of order -1 in S. After some rearrangement, the residual risk 
is found to be

\begin{equation}
\left. \frac{\left\langle\delta \Pi^2\right\rangle/\Pi^2}
{\left\langle\delta S^2\right\rangle/S^2}\right\vert_{frac} = 
S^2\frac{
\frac{ \frac{\left\langle \delta C^2\right\rangle}
{\left\langle \delta S^2\right\rangle }- 
\left(\frac{\left\langle \delta C \delta S \right\rangle}
{\left\langle\delta S^2\right\rangle}
\right)^2} 
{\left(S \frac{\left\langle \delta C \delta S\right\rangle}
{\left\langle \delta S^2 \right\rangle} - C\right)^2}
 }
{1+ S^2\frac{ \frac{\left\langle \delta C^2\right\rangle}
{\left\langle \delta S^2\right\rangle }- 
\left(\frac{\left\langle \delta C \delta S \right\rangle}
{\left\langle\delta S^2\right\rangle}
\right)^2} 
{\left(S \frac{\left\langle \delta C \delta S\right\rangle}
{\left\langle \delta S^2 \right\rangle} - C\right)^2} }
\label{horrendousrisk}
\end{equation}

\noindent In view of the correspondence of the functional 
form above to the minimized absolute risk, the approximation

\begin{equation}
\left. \frac{\left\langle\delta \Pi^2\right\rangle/\Pi^2}
{\left\langle\delta S^2\right\rangle/S^2}\right\vert_{frac} 
\simeq
\frac{\left. \frac{\left\langle\delta \Pi^2\right\rangle/\Pi^2}
{\left\langle\delta S^2\right\rangle/S^2}\right\vert_{abs} }
{1+\left. \frac{\left\langle\delta \Pi^2\right\rangle/\Pi^2}
{\left\langle\delta S^2\right\rangle/S^2}\right\vert_{abs} }
\label{approximaterisk}
\end{equation}

\noindent suggests itself. 
This equation is {\it not} exact. The reason is 
that the call values in Equation \ref{horrendousrisk} are 
the solutions to the $\Pi$ valuation equation with $N=N_{frac}$ 
whereas the valuation of calls with minimal absolute risk is 
determined with $N = N_{frac}$. Still, because the call value 
appears in the numerator and denominator of $N_{frac}$ but 
only in the numerator of $N_{abs}$, the minimal-risk valuation 
equation is structurally simpler for absolute risk than for 
fractional risk. Equation \ref{approximaterisk} suggests that 
$N_{frac} \simeq N_{abs}$ may be a good approximation when 
the portfolio uncertainty is small compared to the uncertainty 
of the stock. The simpler valuation equation can be used under 
these conditions, although the difference may not be critical 
if the portfolio dynamics is being determined numerically. 

However, the distinction illustrates the effect of nonlinearity, 
which is the final issue to be discussed in this section. The 
equation of motion is a first-degree homogeneous nonlinear 
equation. The solution depends on the boundary conditions 
at option expiration, 
and the use of those boundary conditions for option valuation must 
be done with deliberation. For example, investors with different 
amounts of 
the underlying stock will not value a single call identically even 
if they use the same risk metric; ensuing differences in option 
valuations are typically small but important in principle. Because 
the equation of motion is 
nonlinear, the portfolio's risk cannot be expressed as the sum of 
the risks of the components. (In fact, the equation can be 
linearized 
for a portfolio consisting of stock and an infinitesimal number 
of options.) 

There is a situation in which the nonlinearity of the 
risk metric is clearly important. The 
risk term has the sign of $\partial\Pi/\partial z$ \cite{hull}, 
and it has been assumed in this paper that the sign of 
$\partial\Pi/\partial z$ does not 
change. However, portfolios are commonly constructed for which this 
assumption is invalid. For example, a straddle is a long call and 
short put with the same strike price. Because the risk changes sign 
and does not vanish, it must be discontinuous as a function of 
stock price. This discontinuity finds its way into the equation of 
motion and, for finite time steps, into the portfolio value. 

For the model Bessel distribution studied in this paper, 
the discontinuity generated by a quadratic minimum in $\Pi$ 
is of order $\Delta t^{1/p}$ and vanishes in the continuum limit
$\Delta t \rightarrow 0$. Moreover, the discontinuity is "smeared" 
when the equation of motion is solved by backward propagation 
(although a new discontinuity arises at each time step). After 
the continuum limit is taken, some feature--kink or twist or 
cusp--may well remain as the remnant of the discontinuity. 

Discontinuities persist if Equation (2) is formulated for 
discrete time steps. Coarse-grained valuations for 
portfolios with extrema can be obtained heuristically, but a 
fully satisfactory treatment will probably be grounded in an
understanding of the continuum limit
\cite{runggaldier}. 

\section{Summary and Conclusions} 

This paper has examined the problem of option valuation when 
the underlying stock does not execute geometrical Brownian 
motion. The central assumption  has been that option portfolios 
should be valued so that their excess return per risk equals 
that of the stock; the methodology can be used to construct 
optimally hedged portfolios but is not restricted to such 
portfolios. The ensuing equation of motion is 
nonlinear and first-degree homogeneous; a solution has been 
displayed which appears well suited 
for numerical implmentation. The method has been applied to 
an empirical distribution obtained from time series for Merck 
and Oracle stock. 
An analytically tractable model of the stock motion has been 
utilized to determine the asymptotic form of option prices near 
expiration. Equivalence to the Black-Scholes, Poisson and 
binary-tree results has been noted. The relationship to the
related work of Bouchaud, Sornette and others has been discussed. 

Stock-price distributions deviate from the 
lognormal form, and option prices are in only partial agreement 
with the Black-Scholes value. It is believed that these two effects 
may be linked, and the present work supports this position. 
The model distribution analyzed in this paper was generated 
by beginning 
with an exponential distribution and, in effect, applying 
Central-Limit-Theorem arguments in reverse; as expected, deviations 
from the Gaussian form were most pronounced for shortest times. 
Sample results were presented for discrete- and continuous-time 
formulations of the valuation methodology. 
Despite the utility of discrete-time calculations, 
understanding the short-time continuum limit 
is highly desirable. 

The paper has explored how nonlinearity in option 
valuation models increases their level of complexity. The 
nonlinearity is important in principle even when its effects 
are small or negligible. Because of its intuitive basis, the 
present methodology is plausible, and its consequences appear 
coherent. The intuitive foundation provides guidance when 
literal solu\-tions have anomalous features, and it will 
provide guidance for modifications or generalizations. 
\pagebreak
%~\\
%~\\

\section*{Acknowledgments} 

I am grateful to J. McGarity for helpful discussions. I thank my 
former colleagues on the MSX Experiment Data Certification 
Team--T.L. Murdock and others too numerous to list--for 
understanding my interest in the work reported here. 

\begin{thebibliography}{99}

\bibitem{blackscholes}
F. Black and M. Scholes, J. Political Economy 
$\underline{\hbox{3}}$, 637 (1973).

\bibitem{mantegna}
R.N. Mantegna, Physica A $\underline{\hbox{179}}$, 232 (1991)

\bibitem{mantegnastanley} 
R.N. Mantegna and J.E. Stanley, Nature $\underline{\hbox{376}}$, 
46, 1995; and references therein.

\bibitem{hp}
J.M. Harrison and S.R. Pliska, Stochastic Processes and their 
Applications $\underline{\hbox{11}}$, 215 (1981). 

\bibitem{follmersonder}
H. F\"ollmer and D. Sondermann, in {\it Contributions to 
Mathematical Economics}, W. Hildebrand and A. Mas-Colell eds., 
North-Holland, Amsterdam, 1986. 

\bibitem{schweizer}
M. Schweizer, Stochastic Processes and their Applications 
$\underline{\hbox{37}}$, 339 (1991).

\bibitem{bouchaudjphys}
J.P. Bouchaud and D. Sornette, J. Phys. I France, 
$\underline{\hbox{4}}$, 863 (1994).

\bibitem{mikheev}
L.V. Mikheev, J. Phys. I France, $\underline{\hbox{5}}$, 
217 (1995); J.P. Bouchaud and D. Sornette, ibid., 
$\underline{\hbox{5}}$, 219 (1995). 

\bibitem{bouchaudrisk}
J.P. Bouchaud, G. Iori, and D. Sornette, RISK 
$\underline{\hbox{9}}$,61 (1996).

\bibitem{bouchaudbook}
J.P. Bouchaud, M. Potters and D. Sornette, 
{\it Finance in a Risky World}, forthcoming.

\bibitem{aurell}
E. Aurell and K. \.Zyczkowski, Economics 
Working Paper Archive (http://netec.wustl.edu/)
Working Paper ewp-fin/9601001 (1996), 
submitted to J. Political Economy. 

\bibitem{merton}
R.C. Merton,{\it Continuous-Time Finance},
Blackwell, Cambridge, 1994. Among the 
papers included are R.C. Merton, Bell 
J. Econonomics and Management Science 
$\underline{\hbox{4}}$, 141 (1973); J. Financial 
Economics $\underline{\hbox{3}}$, 125 (1975); 

\bibitem{hull}
J.C. Hull, {\it Options, Futures, and Other 
Derivative Securities}, Prentice Hall, Englewood Cliffs, 1993.

\bibitem{numericalrecipes}
W.H. Press, S.A. Teukolsky, W.T. Vetterling and 
B.P. Flannery, {\it Numerical Recipes in C}, 
Cambridge University Press, New York, 1992. 

\bibitem{gradshteyn}
I.S. Gradshteyn and I.M. Ryzhik, 
{\it Table of Integrals, Series, and
Products} Fourth Edition, Academic Press, New York,
1965.

\bibitem{peters}
E.E. Peters, {\it Chaos and Order in the Captial Markets}, 
Wiley, New York, 1991; {\it Fractal Market Analysis}, 
Wiley, New York,1994. 

\bibitem{jarrowrudd}
R. Jarrow and A. Rudd, J. Financial Economics 
$\underline{\hbox{10}}$, 347 (1982).

\bibitem{rubinstein}
M. Rubinstein, J. Finance $\underline{\hbox{49}}$, 771 (1994).

\bibitem{dermankanichriss}
E. Derman, I. Kani and N. Chriss, J. Derivatives 
$\underline{\hbox{3}}$, 7 (1996).

\bibitem{coxrossrubinstein}
J.C. Cox, S.A. Ross and M. Rubinstein, J. Financial 
Economics, $\underline{\hbox{7}}$, 229 (1979).

\bibitem{coxross}
J.C. Cox and S.A. Ross, Working Paper No. 2-75, Rodney L. 
White Center for Financial Research, University of 
Pennsylvania, 1975. 

\bibitem{runggaldier}
For example, see W.J. Runggaldier and M. Schweizer, 
in {\it Seminar on Stochastic Analysis, Random Fields and 
Application}, E. Bolthausen, M. Dozzi and F. Russo eds., 
Birkh\"ser, Basel, 1995. 

\end{thebibliography}

\end{document}





































































