\documentclass[12pt,epsf]{article}

 \usepackage{mathpazo}


\usepackage{ccaption} \captiondelim{}\renewcommand{\thefigure}{} \renewcommand{\figurename}{}

\renewcommand\floatpagefraction{.9}
\renewcommand\topfraction{.9}
\renewcommand\bottomfraction{.9}
\renewcommand\textfraction{.1}   
\setcounter{totalnumber}{50}
\setcounter{topnumber}{50}
\setcounter{bottomnumber}{50}

 
\usepackage{hyperref} \hypersetup{breaklinks=true, pagecolor=white, colorlinks=true, linkcolor=black, hyperfootnotes= false, urlcolor=blue } \urlstyle{rm}
 

\usepackage{graphicx}

\usepackage{amsmath}

 \usepackage{amssymb}


\reversemarginpar
  \topmargin  -.3in
  \oddsidemargin -.1in
  \textheight  8.5in
  \textwidth 7in
  \baselineskip 16pt


 
 

         \begin{document}

 

%42 pages in Nov 2004 submission. 
 
%46 pages in October 2005  currently.

 
%Cut the literature discussion.   

% I can cut out Figure 1. 

%I can cut out some references-- the applications and motivation. 



 

\titlepage 


\begin{raggedright}


 \parindent 24pt

 \parskip 10pt


\begin{center}

\begin{LARGE}

 \textbf{When Does Extra Risk Strictly Increase  an Option's Value? }
\\

\end{LARGE}

\bigskip

 \parindent 24pt

\parskip 10pt


17 November 2005 (reformatted October 14, 2006) \\

\bigskip

Eric Rasmusen \\

\bigskip

The Review of Financial Studies,   20(5): 1647-1667 (September 2007)\\

\bigskip

\textit{Abstract}

\end{center}


 \begin{quotation}

It is well known that  risk increases the value of options. This paper makes that
precise in a new way. The conventional theorem  says that the   value of an option
does not fall if the   underlying asset becomes riskier in the conventional
sense of the mean-preserving spread. This paper uses two new definitions of
``riskier'' to show that the value of an option strictly increases  (a) if the
underlying asset becomes     ``pointwise riskier,''  and  (b)  only if the
underlying asset becomes  ``extremum riskier.''

\end{quotation}

 A number of my results in this paper are false. See the counterexamples and discussion in 
James Huang  Deyuan Zhang `` Some New Results on When Extra Risk Strictly Increases an Option's Value,''  {\it The Journal of Future Markets}, Volume 33, Issue1
January  2013, 44-54, which nicely corrects me. 


 {\small \hspace*{20pt} Dan R. and Catherine M. Dalton Professor, Department  of
Business Economics and Public Policy,   Kelley School of Business, Indiana
University,  BU 456, 1309 E. 10th Street, Bloomington, Indiana, 47405-1701.
Office: (812) 855-9219. Fax: 812-855-3354. Erasmuse@indiana.edu. \url{http://rasmusen.org}, \url{http://rasmusen.org/papers/options_rasmusen.pdf}.}

  {\small I thank Maria Arbatskaya,  Jack Meyer,  Peter Rangazas,  Steve
Russell,  Yacheng Sun, an anonymous referee,  and participants in talks at the 2003 BLISS Conference
and Georgia State and DePaul Universities  for their comments.
  }



\newpage


 



\noindent
 \textbf{I. Introduction}

  A call option is the right to buy the asset at a strike price, $P$.    It has
been  well known at least since Merton (1973) that the value of a call option
increases with the riskiness of the underlying asset.  If  extra risk  increases
the probability that the  market price exceeds $P$, then  the value of the
option increases. A standard finance text   says
\begin{quotation}
\begin{small}
``The holder of a call option will prefer more variance in the price of the
stock to less. The greater the variance, the greater the probability that the
stock price will exceed the exercise price, and this is of value to the call
holder.'' (Copeland \& Weston,  3rd edition, p. 243)
 \end{small}
\end{quotation}

 But this is not quite correct,   despite  being  the sort of thing that even
experts say in conversation and in textbooks. As I am sure   Copeland and
 Weston knew as they wrote this passage, it quite possible for the   risk
and variance of the underlying asset to increase while the value of the option
remains does not increase. The value will not fall, but it might remain
unchanged.    Suppose the strike price is \$50 and the current price of the
asset  is \$40.  If the probability of the price being between \$10 and \$15 or
\$45 and \$49 increases, while the probability it is between \$38 and \$42
falls, the asset has become riskier, but there is no effect on the option value,
because the probabilities of asset values above the strike price of \$50 have
not changed.      This, too, is well-known, but  it leaves open the question of
what kind of risk does actually increase the value of options. It is false,
strictly speaking,  to say that additional risk increases the value. On the
other hand it is true  but uninteresting  to say that additional risk does not
reduce the value.  A great many variables do not reduce the value of an option,
usually because they never affect the value either way.   For introductory
textbooks no great harm is done in stating a   risk-value proposition loosely,
but it is worth thinking about how we can come up with a proposition for this
basic intuition  that is  both interesting  and true.   One way out is to
surrender generality in the kinds of asset distributions that we describe. Bliss
(2001), noting the  problem of coming up with a rigorous proposition, points out
that a sufficient condition for option value to increase with risk  is that the
underlying asset value  has a   two-parameter distribution  such as the normal
or lognormal. The relationships between option value and risk, however,
clearly holds for much more general distributions. 

The options literature has travelled down the route of studying particular
stochastic processes for asset returns--- diffusion or jump processes--- rather
than looking at general distributions for end-states as     Merton (1973) did.
This began with  the log-normal diffusion processes of   Black \& 
Scholes (1973) and  continued with such generalizations as   Cox \&  
Ross (1976),  Merton (1976), and Heston (1993).  More recent entries in the literature that look at option properties as well as pricing include 
 Bergman,  Grundy \&  Wiener  (1996) and    Kijima (2002).

Other papers look at other considerations absent in the simplest model of one
underlying asset, risk-neutral investors, and zero transaction costs.  
Jagannathan   (1984), for example,  looks at values when investors are not risk
neutral  and value wealth more in particular states of the world. In such a
situation, a riskier  asset might not have a higher option value because the
option might yield its highest returns in a state of the world when investors
are wealthier anyway and hence value the return less.  While  the ``extreme
value theory'' of, e.g., Chavez-Demoulin   \&  Embrechts (2004) has turned to
looking at the effects of  unusual events on financial valuation, it is oriented
towards estimation of the value of particular assets.

In this article I    return to the original problem of how risk affects option
value,  for very  general distributions of the underlying asset but without looking at how values evolve slowly over time.   First,  we will see that if the
underlying asset becomes riskier, then  we can at least say that for {\it some}
strike prices  a call or put  option will become more valuable--- a very simple result,
but worth noting.    Second,  I will show that    only if
the underlying asset becomes riskier in the special way I call ``extremum
riskier''  will  every   option   rise in value regardless of the strike
price--- a necessary condition for a rise in value. Third, I will show that if the
underlying asset becomes riskier in the special way I call ``pointwise riskier''
then  every    option will rise in value regardless of the strike price--- a
sufficient condition for a rise in value.


   The article's main contribution is to tidy  up of one of the fundamental
ideas in finance theory. This will    be useful for those
analysts who do not wish to assume normality of asset returns, particularly in
real option  theory,  where option value enters only as part of  a larger model
of business  decisionmaking (see, e.g. Dixit \& Pindyck [1994] or section 25.6
of Gollier [2001]). The definitions here  may also be useful in other areas of economics.
 Arrow \& Fischer  
(1974) applied  the idea of changing option value to cost-benefit analysis in environmental projects
with irreversibility. Search theory is another application; see  Weitzman
(1979) for a classic model in which the value of searches increases with
uncertainty, or Varian (1999) for a more recent article. In such models  it
may be  useful to  identify assumptions on changes in distributions  so that
propositions can be found that say when a change in uncertainty strictly
increases the payoff from the option-creating action  rather than just not
reducing the payoff.

\bigskip

\noindent
\textbf{II. The Model}

      Let there be an asset  which has terminal value $x_i$ with probability $f(x_i)$,  where the  values of $x_i$ with  positive probability are $x_1< x_2< ...<x_m$.       A call option entitles its owner to buy the asset at price $p$ at the terminal time if he wishes.   Denote  by $V_{call}(f, p)$ the current  value to a risk-neutral owner of a  call option  on that asset with strike price $ p$ such that $x_1 < p< x_m. $  This rules out strike prices of $x_1$ or below and $x_m$ and above, because they would lead to riskless options which would    be exercised  always or never. It does allow a strike price that does not happen to equal any of the $x_i$.  Similarly, denote   by $V_{put}(f, p)$  the value of  a put option  that entitles its owner to sell the asset at price $p$ at the terminal time if he wishes. Our focus will be  on seeing how   option  values   change if the underlying asset  changes to  follow a different distribution $g(x) $ which has the same mean as $f(x)$,  so
\begin{equation} \label{e1}
 E x = \sum_{i=1}^m f(x_i) x_i =   \sum_{i=1}^m g(x_i) x_i + \sum_{i=  m+1  }
^{n} g(x_i) x_i,
 \end{equation}
 where $x_{m+1}<x_{m+2}<  ...< x_n$ are  points in the support of $g$ but not
$f$. This  allows, for example,    $x_{m+1}  <x_1$:    $g$
can have positive probability on $x$ values  below or above   the
support of $f(x)$, or  values between the  $x$'s in  $f(x)$'s support. We will 
denote the cumulative distributions by $F(x)$ and $G(x)$.

Denote  the discount factor, the  present value of a dollar received at the terminal date, by  $\beta$.
The value of a   call option   is
\begin{equation} \label{e2}
 \begin{array}{lll}
 V_{call}(f, p)& = & {\displaystyle  \beta \sum_{i=1}^m  f(x_i) Max \{0,  (x_i-  p)\}}
\\
 & & \\
  & = & {\displaystyle \beta   \sum_{i=j}^m     f(x_i) (x_i-p) \; where \; j:  x_{j-1}
<p< x_{j }}\\
  \end{array}
  \end{equation}
The value of a put option   is
\begin{equation} \label{e2aa}
 \begin{array}{lll}
 V_{put}(f, p)& = & {\displaystyle  \beta \sum_{i=1}^m  f(x_i) Max \{0,  (   p-x_i)\}}
\\
 & & \\
  & = & {\displaystyle  \beta   \sum_{i=1}^{j-1}     f(x_i) ( p-x_i) \; where \; j:  x_{j-1}
<p< x_{j }}\\
  \end{array}
  \end{equation}

   Typically, as the  option's maturity increases (the difference    between the current date and terminal date),   the  dispersion of the  possible values  of the underlying asset also increase.   In that case, another way to put the question of this paper (time discounting aside) is whether option value  strictly increases with maturity.   If the option is  a real option  where the value of the project becomes known after a certain date, then the riskiness will not increase over time--- indeed, it will become zero once the uncertainty is resolved.  Other kinds of assets, however, have price distributions which do become riskier in one or more of the senses to be defined below.  If  maturity  increases risk, then the propositions below will apply directly to how an option's value changes with its maturity. 
 
Since the model employs  only two  dates, the current date and the terminal   date, it applies to   ``European''  options, which cannot be exercised early,  rather than   ``American'' options, which can.   European options are more appropriate here   because early exercise occurs only when ``option value'' (in distinction from the ``value of an option security'')  is unimportant because some other benefit of the option determines its value.  Our task is to look at whether option value increases strictly or weakly with riskiness, but  early exercise occurs only when small changes in riskiness are irrelevant.  For example, two common reasons for early exercise are dividend payments (for calls) and re-investment value (for puts). If a stock is about to pay a dividend,  it is possible that a call option should be exercised early to receive that dividend,  despite the loss of  the option value of waiting. If a put option is ``in the money''  with  a high exercise price and the underlying   stock's price close to zero (so it cannot fall much further), the put should be exercised immediately so the profit can be reinvested and earn a positive return. In both situations, the value of the option is equal to the immediate cash gain, and since the   distribution of the future price of the  underlying asset is not determining the option's  value, small changes in riskiness will have no effect.  Thus,  the question of  whether the value of options increases strictly with risk  rather than possibly remaining unchanged is only interesting if we rule out early exercise.   

 Also, we will not be considering  
exotic  options  that convey purchase or sale rights over  ranges of prices
that do not  slice the real line in two (e.g., the right to buy if the price is
either  in  the interval [3, 5.6] or in [7, 26]). Neither the intuition nor the
  propositions extend to that kind of option, since an exotic option
such as   my  parenthetic example can increase  in value when probability
shifts from the extremes to the middle, a  reduction in  risk.


 
\bigskip

\noindent
\textbf{  Defining Risk }

The standard definition of risk is  based on the idea of the ``mean-preserving spread.''\footnote{Definition 1a is specialized to  discrete probability distributions  and   uses
  the  3-point mean-preserving spread  of  Petrakis \&
Rasmusen (1994) rather than the    4-point mean-preserving spread 
of Rothschild \& Stiglitz (1970), which has negative probability at two middle
points rather than one. The two  definitions of spread lead to   equivalent
definitions of risk.    The  3-point definition is
simpler, as well as allowing an easy fix
of an error in    Rothschild \& Stiglitz's main proof).}

 

\noindent
{\bf  Definition 1a: }{\it A {\bf mean-preserving spread}  consists of three
numbers   $s(y_1)$, $s(y_2)$, and $s(y_3)$   for  $y_1<y_2<y_3 $ such that }
\begin{equation} \label{e2a}
 s(y_1)y_1 +  s(y_2) y_2 + s(y_3)y_3 =0, \; {\rm (the \; mean \; is \;
preserved)}
\end{equation}
\begin{equation} \label{e2c}
 s(y_1)  +  s(y_2) + s(y_3)    =0, \; {\rm (the \;  new \; probabilities \;  sum
\; to  \; zero)}
\end{equation}
 { and }
\begin{equation} \label{e2b}
 s(y_1)>0,\;   s(y_2)<0,\;    s(y_3)>0 \;   \;
{\rm (the \;  probability \; is \;spread)}
\end{equation}

 

\noindent
{\bf  Definition 1b: }{\it Distribution $g(x)$ is {\bf riskier}  than $f(x)$
iff $g(x)$ can be reached from $f(x)$ by a  sequence of mean-preserving spreads.
}

  This definition of risk has long been conventional, since it is equivalent to
saying that the asset becomes less attractive to a risk- averse investor (one
with a concave utility function) or that $f$ is like $g$ with noise added,
although Definition 1b  is only a partial ordering, and many pairs of
distributions cannot be ranked  by it.  In the option context,  Bliss (2001)
shows the importance of using Definition 1b   instead of defining risk as simply
higher variance, which   is not an equivalent definition.   Variance can
increase without making an asset less attractive to a risk-averse investor,
and option values do not change in a uniform direction with changes in
variance.\footnote{An example to show that increased variance can increase
utility for a risk-averse person is the following. Let the utility be $ U = x$
for $x \leq 10$ $U = 10 + \frac{x}{2}$ for $x  \geq 10$, which is weakly concave.
Suppose wealth is initially distributed as $f$: (.8:7, .2:12) , which has mean 8,
variance $   4 (= .8*1^2+.2*4^2)$, and utility $ 7.8 (=.8*7 + .2* 11)$. If the
distribution is changed to $g$: (.2:0, .8:10),  the  mean  is still 8, the
variance increases to $16 (=.2*8^2+.8*2^2) $, and utility rises to  $8 (=.2 *0 +
.8* 10)$. Kurtosis, which increases when moving weight to the tails of the
distribution,  is equally unreliable for ranking  the riskiness of
distributions; it  starts as   $    52 (= .8*1^4+.2*4^4)$   in this example and
rises  to $832 (=.2*8^4+.8*2^4).$  Option value, too,  can fall with variance:    in this example,   $V_{Call}(f,11) = .2
(12-11) =.2$ but $V_{Call}(g,11) = 0$. }
 
\bigskip

 

 A fundamental
proposition in the theory of    options  is   Proposition
1:   option value is weakly increasing in risk--- or, rephrased, optional value does not decline with risk. 

\noindent
{\bf  Proposition 1 (Merton [1970] Theorem 8, p. 149): }{\it    If $g$ is riskier
than $f$, then $    V_{call}(g,p) \geq  V_{call}(f, p) $ and $   V_{put}(g,p) \geq  V_{put}(f, p) $  for any $p$.}

\noindent
{\bf  Proof: } We will demonstrate the result if $f$ and $g$ differ by a single mean-preserving spread, which by induction implies it is true if they differ by a series of them.  From (\ref{e2}),  we must prove   the following two inequalities: 
\begin{equation} \label{e2d1}
\begin{array}{l}
  V_{call}(g, p)-V_{call}(f, p) = \beta \{
  s(y_1)Max (y_1-p,0) +  s(y_2) Max (y_2-p,0) + s(y_3)
Max (y_3-p,0)\}  \geq 0   \\
\end{array}
\end{equation}
  and 
\begin{equation} \label{new1}
\begin{array}{l}
  V_{put}(g, p)-V_{put}(f, p) = \beta \{
  s(y_1)Max (p-y_1,0) +  s(y_2) Max (p-y_2,0) + s(y_3)
Max ( p-y_3,0) \}  \geq 0.   \\
\end{array}
\end{equation}

 From   equation (\ref{e2a}),    the  spread is mean preserving, so
$s(y_1)y_1 +  s(y_2) y_2 + s(y_3)y_3 =0$, and  by equation (\ref{e2c})     the
spread's probabilities add to zero,  so  $[s(y_1)  +  s(y_2)   + s(y_3)]=0$.
Adding or subtracting two zeroes results in zero, so we obtain an expression to be used later: 
\begin{equation} \label{e101}
\begin{array}{ll}
  {\displaystyle \sum_{i=1}^3  s(y_i) (y_i-p)   = \sum_{i=1}^3  s(y_i) (p-y_i)= 0,  }  \\
\end{array}
 \end{equation}
 

\noindent
   (i) If  $p \leq  y_1$,   inequality (\ref{e2d1})    becomes 
\begin{equation} \label{e100}
\begin{array}{l}
  V_{call}(g, p)-V_{call}(f, p) =  \beta \{ s(y_1) (y_1-p) +  s(y_2)  (y_2-p ) + s(y_3)(y_3-p )\}  \geq 0,    \\
\end{array}
\end{equation} 
 which is true by equation (\ref{e101}),  and inequality (\ref{new1}) becomes the obviously true expression, 
\begin{equation} \label{new2}
\begin{array}{l}
    V_{put}(g, p)-V_{put}(f, p) = \beta \{
  s(y_1)(0) +  s(y_2) (0) + s(y_3)(0)\}
  \geq 0.   \\
\end{array}
\end{equation}  

    

\noindent
  (ii)   If  $p \geq  y_3$, the reasoning is analogous to case (i). Inequality (\ref{e2d1}) becomes  the obviously true $ V_{call}(g, p)-V_{call}(f, p) = \beta  \sum_{i=1}^3 s(y_i)(0) 
  \geq 0 $  and inequality (\ref{new1}) becomes    $V_{put}(g, p)-V_{put}(f, p) =   \beta  \sum_{i=1}^3 s(y_i)( p - y_i) \geq 0$, which is true by equation (\ref{e101}).

  

\noindent
(iii)  If $p \in (y_1, y_3)$ the options have positive value.  Then, since $Max (y_1-p,0)=
0$ and $Max (y_3-p,0) =  y_3-p$, we can rewrite expression (\ref{e2d1})
as
\begin{equation} \label{e2f}
 V_{call}(g, p)-V_{call}(f, p) = \beta \{ 0  +  s(y_2) Max (y_2-p,0) + s(y_3) (y_3-p ) \}  \geq 0.
\end{equation}
 The last term of (\ref{e2f}) is positive, and the middle term is either zero (if $y_2\leq p$) or negative (if $y_2 >p$, since $s(y_2)<0$).  Equation (\ref{e101}) tells us that  $s(y_1)  (y_1-p) +s(y_2) (y_2-p) +
s(y_3) (y_3-p ) = 0$, so since  $s(y_1)>0$,  when $y_1-p<0$, as in case (iii), it  must be that  $ s(y_2) (y_2-p) +
s(y_3) (y_3-p ) >  0$,  and (\ref{e2f}) is true even if $y_2>p$. 

Analogously, we  can rewrite inequality (\ref{new1}) as
\begin{equation} \label{new3}
\begin{array}{l}
  V_{put}(g, p)-V_{put}(f, p) = \beta \{
  s(y_1)  (p-y_1 ) +  s(y_2) Max (p-y_2,0) + s(y_3)
 ( 0) \}  \geq 0.   \\
\end{array}
\end{equation}
  The first term of (\ref{new3}) is positive,  and the middle term is either zero (if $y_2 \geq p$) or negative (if $y_2 <p$). Equation (\ref{e101}) tells us that  $s(y_1)  (p-y_1 ) +s(y_2) (p-y_2 ) +
s(y_3) (p-y_3 ) = 0$, so since  $s(y_3)>0$,  when $p-y_3<0$, as in case (iii), it  must be that  $ s(y_2) (p-y_2 ) +
s(y_3) (p-y_3  ) >  0$,  and (\ref{new3}) is true even if $y_2<p$. Thus the call and put options either increase in value after the spread or are unchanged. 
 Q. E. D.

  
  Compare  Proposition 1 with Proposition 1a, which differs only in
the
strength
of the inequality.

\noindent
{\bf  Proposition 1a (false): }{\it   If $g$ is riskier than $f$, then $    V_{call}(g,p) > V_{call}(f, p) $ and $   V_{put}(g,p) <  V_{put}(f, p) $  for any strike price $p$.}

\noindent
{\bf Disproof. }     Consider a call option with an exercise price of 4.5  and
the asset price distributions  shown in  Figure 1.   $V_{call} (f, 4.5)  =
V_{call}(g,4.5)$, even though $g$ is riskier than $f$.  The increase in risk has
no effect   because only changes in the probabilities of terminal values greater
than 4.5   matter to the value of the call, and there are no such changes in
the example. (Similarly, $V_{put} (f, 4.5)  =
V_{put}(g,4.5)$.) 

 \begin{figure}
\centering
%\includegraphics[width=80mm]{options1.jpg}
\caption{Figure 1: Risk Does Not Increase Option Value}  
\end{figure}

 Propositions 1  and    1a  differ only in the weakness of the inequality.  That
is enough, however, for ``Proposition 1a: Option value increases with risk'' to
be false.  Instead, we are left with ``Proposition 1: Option value does not fall
with risk,''  which although true, is very weak. That kind of statement can be
made of any variable outside the model: ``Option value does not fall with
wealth,'' or ``Option value does not fall with unemployment,'' or ``Option value
does not fall with the temperature in Bloomington.'' The statement ``Option
value does not fall with risk,''   however, though it does translate the
mathematical notation of Proposition 1, is unnecessarily weak. We can instead
say that ``Option value does not fall with risk, and for at least one value of
the strike price it increases.'' Propostion 1b expresses this in  mathematical
notation.

\noindent
   {\bf  Proposition 1b: }{\it   If $g$ is riskier than $f$, then there exists
some exercise price $p'$ such that the associated call and put  options are more valuable
under $g$ than under $f$,    but no exercise price $p''$  such that they are
  more valuable under $f$:\\   
$\;\;\;\;\; \exists\; p': V_{call}(f, p')  <   V_{call}(g, p') \; and \; V_{put}(f, p')  <   V_{put}(g, p')
 $; \\
    but   \\ 
  $ \;\;\;\;\;\not\exists \;p'': V_{call}(f, p'' )  >   V_{call}(g,p'' ) \; or \; V_{put}(f, p') >  V_{put}(g, p') $.  }

\noindent
{\bf  Proof: }  
 Proposition 1's proof showed that if  $p \in (y_1, y_3)$,  where $y_1$ and $y_3$ are from one of the spreads that makes $g$ riskier than $f$,  then
the
values of the call  strictly increases. Thus, simply pick $p'$  from inside
$(y_1, y_3)$.
 That there exists no value $p''$  for which option value declines is
a
direct corollary of Proposition 1. QED.

\bigskip

\noindent
\textbf{III.  New Definitions of Risk}


 Another approach is to find a definition of risk under which
something like the false Proposition 1b becomes true, and the value of the option
does strictly  increase with ``risk'' regardless of the strike price.

\noindent
    {\bf  Definition 2: }{\it  Distribution $g(x)$ is   {\bf pointwise
riskier} than $f(x)$  iff $f$ and $g$  have the same mean and there exist points
$\underline{x} $ and $\overline{x} $ in $(x_1, x_m)$ such that \\
    $\;\;$ \\
(a)  if $x < \underline{x}$,  then $g(x)  \geq f(x)$  and if $f(x)>0$ then $g(x)
> f(x)$;\\
  $\;\;$     \\
(b) if  $x   \in [ \underline{x}, \overline{x}] $, then   $g(x)  \leq f(x)$
and if $f(x)>0$ then $g(x)  < f(x)$;\\
  $\;\;$    \\
 (c) if $x  >  \overline{x}$, then $g(x) \geq f(x)$  and if $f(x)>0$ then $g(x)
> f(x)$.}

 Definition 2 says   that $g(x)$ is  pointwise riskier than $f(x)$ if it takes
probability away from {\it each point} in  the middle of the distribution and
adds probability to {\it each point} at the two ends, while preserving the mean.
Distribution $g(x)$ in Figure 2 is an example.   Definition 2  also allows
$g(x)$ to add probability  to points outside the  interval $[x_1, x_m]$--- that
is, beyond the two extremes of the    support of $f(x)$. Pointwise riskiness
captures something of the same intuition as the idea of the mean-preserving
spread-- that probability is to be moved from the middle to the ends of the
distribution.  If $g$ is pointwise riskier than $f$, it is also riskier in the
conventional sense.  Note that the change  from $f$ to $g$ need not be symmetric
around the mean of the distribution.   Every point on the ``sides'' must gain probability, but not
necessarily the same amount of
probability, nor must more extreme points gain more probability than
less extreme ones. Note, too, that the definition applies directly to continuous distributions,  where it has the additional implication that the densities of $f$ and $g$ cross twice, at $\underline{x}$  and  $\overline{x}$, as  density $f$ does with $g$ and $h$ in  Figure 4 below.  


 \begin{figure}
\centering
%\includegraphics[width=80mm]{options2.jpg}
\caption{Figure 2:    Pointwise and Extremum Riskiness}  
\end{figure}

 

 Pointwise riskiness will be sufficient but not necessary   for option value to
increase with risk for all strike prices, as we will see   later 
once we have derived other results useful in proving
sufficiency.\footnote{\label{strict}  Since  pointwise riskiness and strict second 
order stochastic dominance both    can    be defined in terms of  functions that
cross a limited number of times, the reader may wonder if  they are the same.   Distribution $F$  strictly second-order
stochastically dominates $G$ if $  \int_0^t G(x)dx > \int_0^t  F(x)dx      $
for all   values of $t$ such that $G(t)>0$ and $G(t)<1$. It could be, however,
that $G$ is pointwise riskier than $F$ without $F$ strictly second-order
dominating $G$. Suppose, for example, that $F$ is uniform, with $F(1) = 0.25,
F(2) = 0.5, F(3)=0.75, F(4) = 1$ and $G$ moves weight from the middle to the
tails  and is pointwise riskier so  $G(1) = 0.30, G(2) = 0.5, G(3)=.7, G(4) =
1$.  If we define $D_F(t) \equiv \int_0^t F(x)dx$
(and similarly for $G)$  then $D_F(1) = 0.25, D_F(2) = 0.75, D_F(3)=1.5$
and $D_G(1) = .30, D_G(2) = .8, D_G(3)=1.5$. Since  $D_F(3)= D_G(3)$,
$F$ does not strictly  dominate $G$. $F$ does weakly dominate $G$, as
we would expect since $G$ is riskier in the conventional sense. }
Distribution $h(x)$ in Figure 2 is an example in which $h$ is not
pointwise riskier than $f$, but $ V_{call}(h,p) >V_{call}(f, p)   $ 
for all $p$ nonetheless.  Pointwise riskiness   is
stronger than  the  standard risk of  Definition 1b; if $g$ is pointwise riskier than $f$ it is riskier too, but $g$ could be riskier without being pointwise riskier.\footnote{ Other strengthened definitions of riskiness have been proposed
in the context of comparative statics--- for example,  how an agent's investment portfolio responds   to an increase in
risk.
Often the response to an increase in standard risk is ambiguous,and clear predictions can
only be made for increases in stronger, nonstandard,  forms of riskiness.
Gollier (1995) has shown what definition is  both necessary and sufficient for
making predictions about the behavior of any risk-averse agent when riskiness
increases. As one might expect from such a strong result, the definition of
riskiness he uses is complicated enough to be difficult to use in applications.
Another approach,  transforming the random variable, was pioneered in
Sandmo (1971).  In this approach, the   distribution
function $F(x)$ is transformed to a new distribution by replacing $x$  with
$t(x)$ for a suitable function $t$ that keeps the mean constant  but increases
risk.  Sandmo (1971)
uses a transformation in which  $t(x)-x$  is linear.  Meyer \&
Ormiston (1989) generalize   to the ``simple  transformation,'' in which
$t(x)-x$ can be any monotonic function.  This approach provides a way to think about ``stretching out'' distributions, but  it will   increase the support of the distribution, something not necessarily true of pointwise riskiness.  }  


 

\bigskip

Our other new definition of  risk is one which will be   {\it necessary}  for extra
risk to increase option value: extremum risk.

\noindent
  {\bf Definition 3: }{\it  Distribution $g(x)$ is   {\bf  extremum riskier } than $f(x)$  iff $G(x_1+\epsilon)> F(x_1+\epsilon)$ and $ G(x_m-\epsilon)<   F(x_m-\epsilon)$   for arbitrarily small $\epsilon>0$.   }

 This is stated in terms of the cumulative distributions of $f$ and $g$, but in discrete distributions it has  easy-to-understand implication that (a) either $f(x_1) < g(x_1)$ or $g(x)> 0$ for some $x < x_1$, and (b) either  $f(x_m) < g(x_m)$ or $g(x)> 0$ for some $x > x_m$.  Definition 3 is slightly stronger than this, however, because it says that if  $g$ extends to more extreme values of $x$ it must also increase the total probability of values of $x$ beyond  $x_1$ and $x_m$. 


 In Figure 2 (above),   distribution $h(x)$    is extremum riskier than  $f(x)$ or $g(x)$.  In Figure 3 (below),  $g(x)$ is  both riskier and  extremum riskier than $f(x)$, but not riskier, since the probability of both  the mean   and  the extreme values   have increased.

 \begin{figure}
\centering
%\includegraphics[width=80mm]{options3.jpg}
\label{options1.jpg }
\caption{Figure 3:  Extremum Riskiness   versus Conventional Risk}  
\end{figure}

 


   The definition of  extremum
riskiness  applies to    continuous  as well as discrete distributions.  Cumulative distributions must be used  because if 
$f$ is a continuous density then  each of the extrema has  zero probability,
even if positive density, and to change the value of  an option it is necessary
to change probabilities over an interval  of  $f$'s support, not just over one
point.  The density $g$ must put more  probability on the intervals $[-\infty,
x_1+\epsilon] $ and $[x_m-  \epsilon,  \infty]$.  If    the   density $f$'s support is unbounded (which cannot happen with a discrete distribution) then  the  numbers  $x_1$ and $x_m$ are no longer the bounds of $f$'s support.  We can continue to define them, however, as   the bounds of the interval in which the strike price lies, so    $x_1 <p<x_m$, for arbitrarily large and small  bounds.    The definitions and propositions  apply within any such range of strike prices, thus allowing.

 Figure 4 shows how the definitions apply to an example of a continuous distribution. All four  densities $g(x)$, $h(x)$, $l(x)$, and $m(x)$ are riskier than $f(x)$, which is shown in  gray in all five diagrams.    Densities $g(x)$  and $h(x)$  are both pointwise riskier and extremum riskier than $f(x)$. Density $l(x)$ is extremum riskier, but not pointwise riskier, since it leaves the density of some points unchanged. Density $m(x)$ is  riskier than $f(x)$, since it   spreads probability from the center to  the two peaks on each side,  but  it is neither pointwise nor extremum riskier, since it leaves the extreme densities  unchanged.  

 \begin{figure}
\centering
%\includegraphics[width=80mm]{options4.jpg}
\label{options1.jpg }
\caption{Figure 4:  Riskiness in  Continuous Distributions}  
\end{figure}


 

 

\bigskip


\noindent
\textbf{IV. The Effects of Extremum and Pointwise Riskiness on Option Value}



    {\bf  Proposition 2:  }{\it  Consider two distributions $f$ and  $g$.  A
necessary condition for it to be true that $V_{call}(g, p) > V_{call}  (f,p)$
for  any strike price $p$    is that  $g$ be extremum-riskier than  $f$. That $g$ be extremum-riskier than  $f$ is also a necessary condition for  $V_{put}(g, p) > V_{put}(f,p)$
for  any strike price $p$.   }



\noindent
{\bf  Proof: }   
   Let us begin with calls.  Since we must show that for any call, $V_{call}(g, p) > V_{call}  (f,p)$, that must be true for $p= x_1+\epsilon$ and $p=x_m -\epsilon$, for arbitrarily small $\epsilon>0$. We need to show  that  $V_{call}(g, p)- V_{call}(f, p)>0$, so from equation (\ref{e1})'s notation for $g$'s support  and     equation (\ref{e2}) for call value,   we must show that
\begin{equation} \label{e200}
\begin{array}{l}
     {\displaystyle     
   \beta \left( \sum_{i=1}^m g(x_i)  Max \{0,  (x_i-  p)\} + \sum_{i=  m+1}^{n} g(x_i) Max \{0,  (x_i-  p)\} \right)  - \beta \sum_{i=1}^m  f(x_i) Max \{0,  (x_i-  p)\}   >0  } \\
\end{array}
 \end{equation}
 where $x_{m+1}<x_{m+2}<  ...< x_n$ are  points in the support of $g$ but not
$f$.    If $p=x_m-\epsilon,$   expression (\ref{e200})  becomes
   \begin{equation} \label{e200a}
\begin{array}{l}
   {\displaystyle     
   \beta  \left( g(x_m)  (x_m-  [x_m-\epsilon]) + \sum_{i=j}^{n} g(x_i) (x_m-  [x_m-\epsilon]) \right) - \beta  f(x_m)    (x_m-   [x_m-\epsilon]) >0 } \;\; {\rm or} \\
  \\
   {\displaystyle   \left(    \left[g(x_m) + \sum_{i=j}^{n} g(x_i)\right] - f(x_m)   \right)  (\epsilon)>0},  \\  
\end{array}
 \end{equation}
 where  $j$ is chosen so  $x_{j} <x_m-\epsilon <  x_{j+1}$:   points $x_i  \geq x_j$  are the  points in $g$'s support   but not in $f$'s support    that make the option worth exercising (a possibly empty set).  Since $f(x_m) = 1 - F(x_m-\epsilon )$ and  $\left[g(x_m) + \sum_{i=  j}^{n} g(x_i) \right] = 1-  G(x_m-\epsilon)$, the     last inequality in    (\ref{e200a}) is true if and only if    $ F(x_m-\epsilon)>G(x_m-\epsilon)$, one of the two conditions in Definition 3. 

If   $p=x_1+ \epsilon$, the call will be exercised except when $x_i \leq x_1$. Thus, we can rewrite the value of the call as   the    value of the underlying asset minus the discounted exercise price  minus   the discounted expected  value of exercising the option when $x_i  \leq x_1$ (which is negative). 
Then the difference in the call values,   $V_{call}(g, x_1+\epsilon) -  V_{call}(f, x_1+\epsilon)$, is 
\begin{equation} \label{e200b}
\begin{array}{l}
    {\displaystyle  
     \beta  \left(   (E x  - p) -  \left(  g(x_1)  (x_1- [x_1+\epsilon])  +  \sum_{i=  m+1  }^{k} g(x_i) (x_1- [x_1+\epsilon]) \right)  \right)}\\
 \\
  {\displaystyle - \beta \left(   (E x  - p) -  f(x_1) (x_1- [x_1+\epsilon])    \right)  } \;\; {\rm or}  \\
   \\
     {\displaystyle  \left(  \left[g(x_1) +  \sum_{i=m+1}^{k} g(x_i)  \right]  - f(x_1) \right)  (\epsilon)  } 
\end{array}
 \end{equation} 
 where $k$   is chosen so $x_k < x_1+\epsilon < x_{k+1}$;  points $x_i  \leq x_k$  are the set of points in $g$'s support but  not in $f$'s  that   make the option not  worth exercising (a possibly empty set).   Since $f(x_1) =   F(x_1+\epsilon)$ and  $\left[g(x_1) + \sum_{i=m+1}^{k} g(x_i) \right] = G(x_1+\epsilon)$, expression    (\ref{e200b}) is positive  if and only if    $ F(x_1+\epsilon)<G(x_1+\epsilon)$, the second condition  in Definition 3. 

  The proof for the put option is similar. If $p= x_1+\epsilon$, the put will only be exercised if $x \leq x_1$,  so, using the same definition of $k$ as in inequality (\ref{e200b}) to indicate small values of $x$,   $V_{put}(g, p)- V_{put}(f, p)>0$ if 
\begin{equation} \label{e200c}
\begin{array}{l}
   {\displaystyle     
   \beta  \left( g(x_1)  ([x_1+\epsilon]  -x_1) + \sum_{i=m+1}^{k} g(x_i) ([x_1+\epsilon]  -x_1) \right) - \beta  f(x_m)    ([x_1+\epsilon]  -x_1) >0 } \;\; {\rm or} \\
  \\
   {\displaystyle   \left(    \left[g(x_m) + \sum_{i=m+1}^{k} g(x_i)\right] - f(x_m)   \right)  (\epsilon)>0},  \\  
\end{array}
 \end{equation}
  Inequality (\ref{e200c}) is true if and only if  $ F(x_1+\epsilon)<G(x_1+\epsilon)$, the second condition  in Definition 3. 
 
   If $p= x_m-\epsilon$, on the other hand, the put will always be exercised unless $x \geq x_m$.  Thus,  using the same definition of $j$ as   in inequality (\ref{e200a}) to indicate large values of $x$, $V_{put}(g, p)- V_{put}(f, p)>0$ if 
 \begin{equation} \label{e200d}
  \begin{array}{l}
    {\displaystyle  
     \beta  \left(   (p- E x  ) -  \left(  g(x_m)  ([x_m- \epsilon] - x_m  )  +  \sum_{i=  j  }^{n} g(x_i) ([x_m- \epsilon] - x_m  ) \right)  \right)}\\
  {\displaystyle - \beta \left(   (p- E x   ) -  f(x_m) ([x_m- \epsilon] - x_m  )    \right) >0  } \;\; {\rm or}  \\
   \\
     {\displaystyle  \left(  \left[g(x_m) +  \sum_{i=j}^{n} g(x_i)  \right]  - f(x_m) \right)  (\epsilon) >0 } 
\end{array}
 \end{equation}

      Inequality (\ref{e200d}) is true if and only if  $  1-  G(x_m-\epsilon) > 1 - F(x_m-\epsilon )$, which is equivalent to      $ F(x_m-\epsilon)>G(x_m-\epsilon)$, one of the two conditions in Definition 3.  Thus, both of the conditions in Definition 3 are also  necessary for all puts to increase in value. 
 QED.
    
 

\bigskip

    Even if Proposition 2 were stated only in terms of increasing the value of calls, not of both calls and puts, both conditions  for extremum riskiness in  Definition 3  would be  necessary. The density 
  $g$  must add probability to $f$ 
at  {\it both}  extremes, not just  at the maximum.   This was part of the proof of Proposition 2, but  the 
numerical  example  illustrated in Figure 5 is helpful in understanding why. 
In  Figure 5,      $g$ is made  riskier than $f$
by shifting probability away from $x=  2$, the mean, to $x = 1  \;\frac{1}{3}$ and $x =
 4\;\frac{2}{3}$.   As a result,  $g$   has more  probability  than $f$ on the maximum,  $x=4\;\frac{2}{3}$, but no more  probability on the  minimum,    $x=0.$  
A call with a strike price above  $1  \;\frac{1}{3}$  is more valuable under $g$ than under $f$. 
  But think about a     call with a
strike price of 1.  It will have equal value under $f$  and $g$, because the
mean of the distribution conditional on $x$ being  greater than 1 has not
changed. The  probability of the  state  of
the world ($x=0$) in which the call is  {\it not} exercised  is the same with either distribution. 


 \begin{figure}
\centering
%\includegraphics[width=80mm]{options5.jpg}
\caption{ Figure 5:   Why Extremum Risk Needs  to Spread Probability at Both Ends}  
\end{figure}


 

  The general problem is that unless both extrema are increased in  $g$, it is
possible to find a strike price such that the  total amount of  probability on
prices above the strike price is unchanged. If the minimum does  not
increase, as in Figure 5, then choose the strike price to be  very
low, just  above the minimum. The call is then   a bet that the price will
exceed the  minimum, and the probability of winning that bet is the same
for  $f$ and  $g$.   If, on the other hand,  the maximum does not
increase,  choose the  strike  price to be very high, just below the maximum.

\bigskip

    Why is extremum risk just a necessary,  not sufficient? Look  back at
Figure 3. In Figure 3,  $g(x)$ has more probability at the  extremes than $f(x)$
does-- the probability of each extreme is .25  instead of .20--- but it is not
riskier in the conventional sense,  because it cannot be reached from $f(x)$ by
a sequence of mean-preserving spreads.  If the strike price is 4.5, then the
call's value  is higher under distribution $g(x)$, because  the outcome $x= 5$
occurs with probability .25 instead of  $f(x)$'s .20. $V_{call}(f,  4.5) =
.20(5-4.5)= .10 < V_{call}(g, 4.5) =  .25 (5-4.5) = .125$.  If the strike price
is 3.5, however, the call's value is higher under  distribution $f(x)$, because
under $g(x)$   the outcomes  $x=4$ and  $x= 5$ together occur  with probability
.25 instead of .40 and  $V_{call}(f, 3.5)  =  .20(4-3.5) + .20 (5- 3.5) = .40 >
V_{call}(g,  3.5) =   .00(4- 3.5) + .25 (5-3.5) =  .375$.

\bigskip

 Extremum riskiness already implies that $g$ is  not less risky than $f$, since
more weight is in the far tail of the  distribution in $g$, but it might be that
$f$ and $g$ are not ordered  by  risk.  Although  neither conventional nor
extremum riskiness  is  by  itself sufficient to make calls more valuable, in
combination  they do  yield a sufficient condition, as stated in Proposition
3.\footnote{This combination of conventional risk and extremum  riskiness is the
same idea as the   ``strong increase in risk'' of Meyer \& Ormiston (1985),
although  their formal statement allows a strong increase to be no change at
all--- they use weak inequalities in their definition.     Thus,  loosely
speaking,    a ``strong increase in risk''  strictly increases  the value of an
option.}

\noindent

    {\bf  Proposition 3:  }{\it  Consider two distributions $f$ and  $g$.  A
sufficient condition for it to be true that $V_{call}(g,p)> V_{call}  (f, p)   $
for  any strike price $p$  is that  \\
 (a) $g$   is  extremum-riskier than  $f$;
and \\
 (b) $g$ is riskier than $f$.\\
 This is also a  sufficient condition for it to be true that $V_{put}(g,p)> V_{put}  (f, p).   $}

\noindent
{\bf  Proof: }  
 From Proposition  1  we know that if condition (b) is true, then  $V_{call}(g,
p) \geq V_{call}(f, p)$  and $V_{put}(g,
p) \geq V_{put}(f, p)$  , that is,    Proposition 3's  inequalities are true at
least weakly. Thus, all that we need  to   show is  that condition (a)  makes   the
inequalities   strict.  The proof of Proposition 1 showed that if a mean-preserving spread  that made $g$ riskier than $f$  changed probability on three
points  $y_1< y_2< y_3$, then  if the option's strike price were $p \leq  y_1$
or     $p \geq  y_3$, the option's value would be the same under $f$  as under
$g$.

 Since $g$ may be derived from $f$ by a series of mean-preserving  spreads, let
$y_1^*$ be the lowest $x$ value that is changed  and  $y_3^*$ the highest. Consider either a call or a put.  If
the option's strike price were $p \leq  y_1^*$  or     $p \geq  y_3^*$, an 
option's value  would be the same under $f$  as under $g$.  That is the
possibility we are trying to rule out. But  condition (a) says that $g$ is
extremum riskier. That implies that the  probability of $x_i$ less than or equal
to $x_1$ increases, so  $y_1^*  \leq x_1$, and that the probability of $x_i$
greater than or equal to  $x_m$ increases, and   $y_3^* \geq x_m$. Thus, it is
impossible (since  we rule out the riskless options with $p= x_1$ or $p = x_m$)
that $p  \leq  y_1^*$ or     $p \geq  y_3^*$.  As a result,  the option values
cannot be equal for any $p$ and it must be that  both   $V_{call} (g,p) >  V_{call}
(f,p)$  and $V_{put} (g,p) >  V_{put}
(f,p)$.  Q.E.D.
\bigskip

You might ask  why I did not write  Proposition 3   to say that  conditions (a)
and (b) are jointly  necessary and sufficient, rather  than just    sufficient.
If options on $g$ are to be always more  valuable than  options on $f$, isn't it
necessary that $g$ be both riskier and  extremum-  riskier than $f$?  No, as we will see by demonstrating the falsity of Proposition 3a. 

 \noindent
    {\bf  Proposition 3a: (False)  }{\it  Consider two distributions $f$ and  $g$.   The
following two conditions are    necessary and  sufficient   for it to
be true that $V_{call}(f,  p) < V_{call}(g,p)$  or  $V_{put}(f,  p) < V_{put}(g,p)$   for  any strike price $p$: \\
 (a) $g$   is extremum-riskier than  $f$; and \\
 (b) $g$ is riskier than $f$.  }

\noindent
 {\it Disproof:}  Proposition 3 tells us that Conditions (a) and (b) are jointly   sufficient for  options on $f$ to be less valuable.  Proposition 2 tells us that Condition (a) by itself is necessary for  options on $f$ to be less valuable.    Thus, what we
need to show to  prove Proposition 3a is false  is that there exist distributions $f$ and
$g$  such that Condition (b) is violated but nonetheless $V_{call} (f, p) <
V_{call}(g,p)$ for any $p$--- i.e., that $g$'s options are always
more valuable  but $g$ is not riskier than $f$.
The counterexample in Figure 6 will do this. Distribution $g$ is extremum riskier than
  $f$, but it is not riskier, because it has more probability at the
mean, $x=5$.  The  distributions $f$ and $g$  both have mean $Ex=5$ and cannot be
ordered by risk, yet we will see that all options on $g$ are more valuable. 

 

 The value of  the options on an asset with  distribution $f$ and   strike price $p
\in (2,8)$ are, from equation  (\ref{e2}), 
  \begin{equation} \label{e20}
\begin{array}{lll }
V_{call}(f, p) & = &  Max \{0, .25 (2- p)   \}  +   Max \{0, .25 (4- p)  \}  +
Max \{0, .25 (6- p)   \} \\
  & & + Max \{0, .25 (8- p)   \}    \\ 
 && \\
 V_{put}(f, p) & = &  Max \{0, .25 (p-2)   \}  +   Max \{0, .25 (p-4)  \}  +
Max \{0, .25 (p-6)   \} \\
  & & + Max \{0, .25 (p-8)   \}    \\ 
 \end{array}
  \end{equation} 
 and      if the distribution is $g$ they are 
\begin{equation} \label{e21} 
 \begin{array}{lll} 
  V_{call}(g, p)& = & Max \{0,
.30 (1- p)   \}  +   Max \{0, .40 (5- p)  \}    + Max \{0, .30 (9- p)   \}    \\
 && \\
 V_{put}(f, p) & =& Max \{0,
.30 (p-1)   \}  +   Max \{0, .40 (p-5)  \}    + Max \{0, .30 (p-9)   \}    \\
\end{array}
 \end{equation}

  The possible values of $p$ for our comparison go from $p=2$ to $p=8$, where the  endpoints are
not possible (as the option would then be always or never exercised).  We will
split this up into four intervals and examine each in turn.

\noindent
   $p \in (2,4]. $ In this case,    $V_{call}(f, p)  =         .25 (4- p) + .25 (6- p)  +  .25 (8- p) =   4.5 - .75 p <  V_{call}(g, p)  = .40 (5- p) +   .30 (9- p)    =  4.7 - .70p$. The put values are   $V_{put}(f, p)=         .25 (p-2)  =   .25p - .5 <  V_{put}(g, p)  = .3 (p-1)   =  .3p - .3$.  Thus, $g$  has the more valuable options.

 \noindent
    $p \in (4,5]. $
In this case,   $V_{call}(f, p)  =    .25 (6- p) +   .25 (8- p)  =  3.5 -
.50p,     $  while  $V_{call}(g, p)  =    .40 (5- p) + .30  (9- p)  =
4.7 - .70 p $.   It is true that $ 3.5 - .50p <4.7 - .70 p$ if $.20p  <  1.2$,
which is true if $p < 6$, and in particular  for   $ p \in [4, 5]$.  The put values are   $V_{put}(f, p)=     .25 (p-2)  +.25 (p-4) =   .5p - 1.5$, while     $V_{put}(g, p)  = .3 (p-1)   =  .3p - .3$. The put on $g$ is more valuable if $p<6$, just as with the call, so 
   $g$
has the  more valuable options.

 \noindent
   $p \in [5,6]. $
 In this case,  $V_{call}(f, p)  =       .25 (6- p) = .3.5 - .5p,   $  while   $V_{call}(g, p)  =     .30 (9- p)    = 2.7 - .30p.   $  It is true  that
$ 3.5 -  .5p < 2.7 - .30p$ if $.8 <  .2p$, which is true if $p >4$, and in
particular  if   $ p \in [5, 6]$. The put values are   $V_{put}(f, p)=     .25 (p-2)  +.25 (p-4) =   .5p - 1.5$, while     $V_{put}(g, p)  = .3 (p-1) + .4 (p-5)  =   .7p - 2.3$, which is  the more valuable if $p>4$, just as with the call. 
 Thus,  $g$ has the more valuable options.

 \noindent
    $p \in [6,8). $
  In this case,   $V_{call}(f, p)  =   .25 (8-p) = 2 - .25p$, while 
$V_{call}(g, p)  =   .30 (9- p)   = 2.7 - .30 p.$ It is true that $2  - .25p
<2.7 - .3 p$ if    $p < 14$, and in particular     if  $ p \in  [6, 8]$.
The put values are   $V_{put}(f, p)=     .25 (p-2)  +.25 (p-4) + .25 (p-6) =   .75p - 3$, while     $V_{put}(g, p)  = .3 (p-1) + .4 (p-5)  =   .7p - 2.3$, which is  the  more valuable if $p<14$, just as with the call. 
 Thus,  $g$ has the more valuable options.


 

  Combining all four cases, we see that
for any $p \in (2,8)$,   $g$  has more valuable options. Q. E. D. \\

\medskip

 To understand  Proposition 3a's falseness, start with the simpler idea  that an  option with
price $p$ can be more valuable under distribution $g$ even  if $g$ is not
riskier than $f$. That is true because for some  particular $p$, the call's
value is  $\sum_{i=j}^m  f(x_i)    (x_i-  p)$ for $j: x_{j-1} <p < x_j$, which
depends on all of the $f$  distribution for every  $x_i>p$ but not on every
$x_i$ individually.  Thus,  it is possible  that  $g(x_k) < f(x_k)$ for some
particular  value of $x_k >p$ in a way that makes it impossible to rank $f$ and
$g$ by risk,  but for that to be outweighed by $g$'s greater weight on  most
high values of $x_i$.  We can generalize  this to the idea  that an
option can be more valuable for {\it any} price $p$ even  though risk does not
rise.  We can find a $g$ that  puts so much  weight on its
extrema compared to $f$ that  $g$'s expected  values over $x_i >p$ will be greater
even if it puts   more weight  on  the mean of $x$  too.


 \begin{figure}
\centering
%\includegraphics[width=80mm]{options6.jpg}
\caption{Figure 6:    Conventional  and Extremum  Riskiness Are Not Jointly  Necessary  
  to Increase  Option Value }  
\end{figure}

  

  Now let us leave extremum riskiness and look back to the second  new
definition of ``riskier'': ``pointwise riskiness''.   In 
applications, it is convenient  to specify a simple  sufficient  condition for
options on one distribution to have higher value than those on another.  Proposition 4 says that pointwise
riskiness is   such a condition. 

\noindent
{\bf  Proposition 4: }{\it  If $g$ is pointwise riskier than $f$, then   for any
$p$,    $V_{call}(g,p) >V_{call}(f, p)   $ and $ V_{put}(g,p)> V_{put}(f, p).$ }

\noindent
{\bf  Proof: } If $g$ is pointwise riskier than $f$, then it is also  riskier
and extremum riskier. It is riskier because we can move from $f$ to  $g$ by a
series of mean-preserving spreads that take probability away from the  middle
interval $[ \underline{x}, \overline{x}]$   and move it to the  extremes.  It
is extremum riskier because $x_1< \underline{x}$ and $ x_m>  \overline{x}$, so
$g$   puts more probability on $x_1$ and $x_m$ than $f$ does.  It  follows from
Proposition 3   that  calls  and puts on $g$ will be more valuable  than
calls  and puts on $f$.  Q.E.D. 

\bigskip

 We have  already found  one sufficient condition for  options on $g$
to be  more valuable  than  options on $f$, the combination of  riskiness
and  extremum riskiness  in Proposition 3.    Proposition 3, in
fact, provides a tighter   sufficient condition than Proposition 4. If $g$  is pointwise riskier than $f$
it is always  both riskier and extremum  riskier---  but $g$  can  be  riskier
and  extremum riskier   without being pointwise  riskier.  
Pointwise riskiness is  nonetheless a useful concept, because it is  simpler and  more
intuitive than the combined conditions.

 

 
\bigskip

\noindent
\textbf{V. Concluding Remarks}

If distribution $g$ is riskier than distribution $f$, then any call option  on
an asset whose value has distribution $g$ will  be at least as  valuable as the
equivalent option on an asset with distribution $f$. But the option on $g$ might
not be {\it more} valuable, because the values might be equal. This paper has
developed  a necessary condition for all call options on an asset whose value
has distribution $g$ to be  strictly more valuable than the equivalent option on
an asset with distribution $f$, and two sufficient conditions for it,  differing
in  strength and convenience.   The necessary condition is that $g$ be
``extremum riskier'': it must put more probability on the extreme values of the
asset.   One sufficient condition is that $g$ be not only extremum riskier, but
also riskier under the conventional definition of risk-- that $g$ can be reached
from $f$ by a series of mean-preserving spreads. A second sufficient condition,
more restrictive but   simpler, is that $g$ be ``pointwise riskier'':   asset
values in  the middle of  $g$ have higher probability than under $f$, and  asset
values outside the middle have lower probability.

\newpage



 \noindent
 \textbf{References}

\begin{description}

\item[]
Arrow, K. \& A. Fischer (1974) ``Environmental Preservation,  Uncertainty, and
Irreversibility,'' {\it The Quarterly Journal of  Economics}, 88(2): 312-319 (May 1974).

\item[]
 Bergman, Yaacov Z.,  Bruce D. Grundy \& Zvi Wiener (1996) ``General Properties
of Option Prices,'' {\it The Journal of Finance},      51(5): 1573-1610
(December  1996).

\item[]
Black, Fischer \& Myron S. Scholes (1973) ``The Pricing of Options and Corporate
Liabilities,'' {\it The  Journal of Political Economy},  81(3):  637-654 (May 1973).



\item[]
 Bliss, Robert R. (2000) ``The Pitfalls in Inferring Risk from Financial Market
Data,'' Federal Reserve Bank of Chicago WP 2000-24 (December 21, 2000).
 \url{http://ideas.repec.org/p/fip/fedhwp/wp-00-24.html}.



\item[]
 Chavez-Demoulin, V. \&  Embrechts, P.  (2004) ``Smooth Extremal Models in
Finance and Insurance,'' {\it  The Journal of Risk and Insurance} 71(2): 
183-199.

\item[]
Copeland, Thomas \& J. Fred Weston (1988) {\it Financial Theory and  Corporate
Policy}, Reading: Addison-Wesley.

\item[]
Cox, John \&  Stephen Ross (1976) ``The Valuation of Options for  Alternative
Stochastic Processes,'' {\it The  Journal of Financial Economics}, 3(1-2):  145-166
(January-March 1976).

\item[]
Dixit, Avinash \& Robert Pindyck (1994)   {\it Investment Under Uncertainty},
Princeton: Princeton University Press, 1994.

\item[]
Gollier, C. (1995)  ``The Comparative Statics of Changes in Risk  Revisited,''
{\it The Journal of Economic Theory}, 66(2):  522-535 (August 1995).

\item[]
  Heston, S.  (1993) ``A Closed Form Solution for Options with Stochastic Volatility
with Applications to Bond and Currency Options,'' {\it  The Review of Financial
Studies},  6(2): 327-343.

\item[]
 Jagannathan, Ravi (1984) ``Call  Options  and the Risk of Underlying
Securities,''  {\it  The Journal  of  Financial Economics}, 13(3): 425-434  (September
1984).

\item[]
 Kijima, Masaaki (2002) ``Monotonicity and Convexity of Option Prices
Revisited,'' {\it Mathematical Finance,}   12(4):  411-425 (October 2002).

\item[]
  Merton, Robert C. (1973) ``Theory of Rational Option Pricing,'' {\it  The
Bell Journal of Economics and Management Science}, 4(1): 141-183  (Spring
1973).

\item[]
 Merton, Robert C.   (1976)  ``Option Pricing When Underlying Stock  Returns Are
Discontinuous,''  {\it The Journal of Financial Economics},  3(1-2):  125-144
(January-March 1976).

\item[]
 Meyer,  Jack   \& M. B. Ormiston (1985) ``Strong Increases in Risk and Their
Comparative Statics,'' {\it  International Economic  Review}, 26(2): 425-437 (June 1985).


\item[]
Meyer, Jack \&  M. B. Ormiston  (1989) ``Deterministic Transformations of Random
Variables and the Comparative Statics of Risk,''   {\it The  Journal of Risk and
Uncertainty}, 2(2): 179-188 (June 1989).

\item[]
New School for Social Research (2004) ``Riskiness,'' 
 \url{http://cepa.newschool.edu/het/essays/uncert/increase.htm}.

\item[]
Petrakis, Emmanuel   \& Eric Rasmusen (1992)  ``Defining the Mean-Preserving
Spread: 3-pt versus 4-pt,''  {\it Decision Making under Risk and Uncertainty:
New Models and Empirical Findings}, edited by John Geweke (Amsterdam: Kluwer,
1992).

\item[]
 Rothschild, Michael \& Joseph Stiglitz (1970)  ``Increasing Risk I,''   {\it The 
Journal of Economic Theory},    2: 225-243 (September 1970).


\item[]
Sandmo, Agnar  (1971) ``On the Theory of the Competitive Firm under Price
Uncertainty,'' {\it The American Economic Review},   61(1):   65-67 (March
1971).

\item[]
Varian, Hal R.  (1999) ``Economics and Search,'' Plenary address at ACM-SIGIR
1999, held in Berkeley, California. {\it  SIGIR Forum}, (Fall  1999). 
 \url{http://www.sims.berkeley.edu/~hal/Papers/sigir/sigir.html}.

\item[]
 Weitzman, Martin (1979) ``Optimal Search for the Best Alternative,''    {\it
Econometrica}, 47(3):654 (May 1979).


\end{description}

  \end{raggedright}


\end{document}


