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\title{ Option pricing and hedging beyond Black-Scholes}
\author{Erik Aurell$^1$, Jean-Philippe Bouchaud$^{2,3}$,
Marc Potters$^3$ and Karol \.Zyczkowski$^{4}$}
\maketitle
\vspace {2.0cm}
\begin{center}
\begin{tabular}{ll}
$^1$ & Artificial Economy Project, \\
& Center for Parallel Computers, KTH,\\
& S-100 44 Stockholm, SWEDEN\\
& E-mail: eaurell@pdc.kth.se \\
$^2$ & Service de Physique de l'\'etat Condens\'e, \\
& Centre d'\'etudes de Saclay, Orme des Merisiers, \\
& 91191 Gif-sur-Yvette C\'edex, FRANCE \\
& E-mail: bouchau@amoco.saclay.cea.fr \\
$^3$ & Science \& Finance, 109-111 rue Victor Hugo,\\
& 92523 Levallois Cedex, FRANCE\\
$^4$ & Dept. of Physics, Jagiellonian University, \\
& ul. Reymonta 4, PL-30 057 Krak\'ow, POLAND\\
& E-mail: karol@castor.if.uj.edu.pl
\end{tabular}
\end{center}
\begin{abstract}
We introduce a prescription how to price options on
a stock or commodity, in a situation where
risk in options' trading
cannot be eliminated by hedging.
First, a hedging
strategy is found in such a way that risk is made as small as possible.
The remaining residual risk is an important quantity which we can
compute.
Second, the option price is determined by the condition that the
expected
return of an operator using the risk-minimizing hedging strategy is zero.
We compare our prescription
with
historical Bund call options on the LIFFE market, with very satisfactory agreement.
\end{abstract}
\centerline{ }
\vspace{1.0cm}
{\bf Keywords:} Option pricing, inherently risky options, mean-variance hedging, Bund call options
\vspace{1.0cm}
{\bf JEL classification:} G13, G14
\section{Introduction}
The famous Black and Scholes theory of option pricing has two remarkable
features:
The hedging strategy eliminates risk entirely, and
the option price does not depend at all on the average return of the underlying
asset.
Although everyone agrees that the geometrical Brownian motion
model of stock prices considered by Black and Scholes (1973) is only a
rough first
approximation to the real world, there is no consensus on how
to price an option in a more general,
hopefully more realistic, model.
It is important to realize that in
most models of stock price movements,
except a few special cases (of which the geometrical Brownian model is one),
risk in option trading
cannot be eliminated.
Even in the geometrical Brownian
model, risk
cannot be
eliminated if trading is
only allowed at discrete times,
as noted already by Black and Scholes themselves (1973, pp. 642-643).
Arbitrage arguments hence cannot be used to fix the option price in a general
price movement model.
The problem of pricing inherently risky options has been
addressed
quite frequently in the finance literature, and obviously also by
practitioners,
for whom it is a fact of life.
Broadly speaking, one can find two main approaches in the published
literature (see, e.g. Hull (1994)). The first consists of various more or less
ad hoc modifications of the basic Black-Scholes formula, where,
in particular, the volatility of the stock is adjusted to account
in some way for risk, and also for trading costs.
Three examples of such an approach are the well-known papers by
Leland (1985) and Figlewski (1989), and the recent contribution of Avellaneda (1996).
These prescriptions have the great
advantage of simplicity and computability, but they lack clear
theoretical support. Furthermore, the problem of the residual risk is rarely
discussed.
A second, more academic,
approach involves the introduction utility functions describing
agents on the market, and an option pricing procedure that follows
from maximization of the utility functions.
The whole basis of this construction is not entirely convincing.
Utility functions have an honoured place in the tradition of
theoretical economy, but it is not obvious that they are
relevant to the immediate practical problem, for a given
economic agent, of giving a price of an option on the market,
and this option price is furthermore somewhat arbitrary,
as arbitrary as the utility function considered. Nevertheless, advanced mathematical theory
has been developed along this path, see Davis (1993).
Its numerical implementation demand state-of-the-art techniques,
at present only available to a very select audience, c.f. Ishii \& Lions (1990).
The practical value of the results obtained in this approach so far is
questionable.
We want here to present a new approach to the pricing of risky options,
which works equally well when risk
cannot be eliminated,
and which is a natural generalisation of the classical Black-Scholes
recipe.
It can be described by two conditions:
$\bullet$ An optimal hedging strategy is found
by minimizing risk over all possible trading strategies.
$\bullet$ The price of the option is such that the
average profit of an agent using the optimal hedge is zero.
A mathematical formulation of these two conditions and what follows from
them is given in the next section.
The general idea of judging financial investment strategies by both their
expected return and their
risk gained theoretical support by the development
of the Capital Asset Pricing Model by Sharpe (1964,1965) and Litner (1965).
A recent overview can be found in the text-book by
Sharpe (1985).
To our knowledge, in the
context of the pricing of derivative securities,
risk-minimization arguments were first introduced
by Follmer and Sondermann (1986,1988).
Further development of this mathematical theory has been carried out by
CERMA (1988), Sch\"al (1994), Schweizer (1995) and Mercurio (1995,1996).
The prescription to price
options by minimizing risk was independently proposed by one of
us in Bouchaud \& Sornette (1994). It was thereafter also derived as the
equilibrium price on a market with different types of agents,
characterized by different
attitudes to risk in Aurell \&
{\.Z}yczkowski (1996).
A further discussion of the theoretical background of the prescription
is beyond the present paper. We will instead work out concrete
formulae and
consequences for practical option pricing, and compare with
empirical market data from the LIFFE Options Exchange.
Let us note that in our option price
prescription there is no fixed market price for residual risk.
However, from this observation it does not at all follow that our
prescription coincides with Black-Scholes.
When risk
cannot be eliminated, there will be
an optimal hedge that minimizes risk, but this hedge does not
need to be, and in general is not, a Black-Scholes portfolio.
The price fixed by the condition that the average profit of
the optimal hedge is zero, is thus in general different from
the Black-Scholes option price. For example, an important qualitative
difference with Black and Scholes is that
our option price prescription in general does depend on the average
return
of the asset. Finally,
the very existence of a non-zero risk
is related to the presence of (rather large) `bid/ask' spreads on option markets,
which reflect the need (for the writer of the option) to protect himself
at least partially against this risk. In other words, the value
of the residual risk can be used to rationalize the amplitude
of the bid/ask spread.
We will thus demonstrate that both the remarkable features of
the Black-Scholes option pricing, that risk can be eliminated,
and that the price does not depend on the average return, are
not inherent to rational option pricing as such, but are tied
to the specific log-Brownian price movement model.
\section{Global wealth balance}
\label{s:wealth-balance}
As the basic building blocks we take the average value
and the standard deviation of a {\it global wealth balance}.
Following Sharpe (1964) we refer to these two quantities as the
profit and the risk.
We will for concreteness write out the global wealth
balance of a writer of a European call option.
The balance for a buyer of a European call option is quite
similar, and can be found in Aurell \&
{\.Z}yczkowski (1996).
The generalization to other European
options is straight-forward,
and is covered in Bouchaud \& Sornette (1994) and Bouchaud et al (1996).
The generalization to American options is not completely obvious in theory, but it
can be shown that the rational price of an American call option is the
same as that of the corresponding European
call option\footnote{For the special case where the average rate of return on the stock
is the same as that of the bond, the relevant calculations
can be found in Bouchaud \& Sornette (1994).
In the general case the analysis is slightly more involved, but leads
to the same result Aurell \& Simdyankin (1996)},
while there is a difference
for put options in the presence of a non zero interest rate (Bouchaud et al (1996)). However, for
short term options, this difference is rather small.
Finally we will not here introduce transaction costs.
They could easily be included in the balance equation, but
the optimal hedge would have to be found either by
perturbation theory in the transaction costs or by a numerical
procedure, and not in explicit form.
For clarity of presentation we will therefore here stick to
the case of no market friction.
Let us now turn to the description of the balance equation.
At time $t=0$, the writer receives the
price of the option ${\cal C}[x_0,x_c,T]$, on a
certain asset the value of which is
$x(t=0)=x_0$. The strike price is $x_c$. Between $t=0$ and $t=T$,
the writer may trade
the asset at discrete times
$0,\tau,2\tau,...k\tau,N\tau=T$; his strategy is to hold
$\phi_k(x_k)$ assets if its price is $x(t)=x_k$ when the time is $t=k\tau$.
The change of wealth due to this trading is simply
given by:
\be
\Delta W_\subs{trading} = \sum_{k=0}^{N-1} \phi_k(x_k) [x_{k+1}-e^{r\tau}x_k]
%% K12 line below was: e^{T-t-\tau))\tau}
e^{r(T-t-\tau)}
\label{deltaWtrading}
\ee
The logic of (\ref{deltaWtrading}) is that the writer at time
$k\tau$ can decide to hold a portfolio of $\phi_k(x_k)$ shares of stock,
or to convert it into $x_k\phi_k(x_k)$ of cash or bonds.
At time $(k+1)\tau$, in the first case, the stock portfolio will be
worth $x_{k+1}\phi_k(x_k)$, while in the second case the bond
portfolio will be worth $e^{r\tau}x_k\phi_k(x_k)$, where
$r$ is risk-free rate.
The difference is a net profit or loss realized at time
$(k+1)\tau$. When carried forwards to time $N\tau$, that is $T$,
it gives a net profit or loss on account of
$\phi_k(x_k) [x_{k+1}-e^{r\tau}x_k]e^{r(N-(k+1))\tau}$.
Summing up all contributions from all trades gives (\ref{deltaWtrading}).
Finally, at time $T$, the writer looses the
difference $x-x_c$ if the option is exercized. Thus the complete wealth balance
reads:
\be
\Delta W = {\cal C}[x_0,x_c,T] e^{rT} - \max(x-x_c,0) + \sum_{k=0}^{N-1}
\phi_k(x_k) \delta x_ k e^{rT-r(k+1)\tau}
\label{deltaWtotal}
\ee
where we have introduced the notation: $\delta x_k \equiv [x_{k+1}-e^{r\tau}x_k] $.
Note that
$\delta x_k$ is posterior to the instant $k$ where $\phi_k$ is determined.
The average profit is
\be
P \equiv <\Delta W> = {\cal C}[x_0,x_c,T] e^{rT} - <\max(x-x_c,0)> + \sum_{k=0}^{N-1}
<\phi_k(x_k)> < \delta x_k > e^{rT-r(k+1)\tau}
\label{profit}
\ee
and the risk is
\be
R \equiv\sqrt{ <(\Delta W)^2> - (<\Delta W>)^2}
\label{risk}
\ee
We will not here write out explicitly the various terms in
$R$, which are similar to
those in (\ref{profit}). The interested
reader can find them, with all necessary details, in
Bouchaud \& Sornette (1994) and Aurell \&
{\.Z}yczkowski (1996).
The important properties which we want to use and stress are that $R$ is always
greater than or equal to zero and that the minimum is obtained
for a definite strategy $\phi^*$.
This will be our optimal hedging
strategy.
Inserting it in (\ref{profit}) we get the profit using the optimal
hedge, which by our second condition on option pricing should be
zero. That determines our option price prescription, which is thus
\be
{\cal C}[x_0,x_c,T] = e^{-rT} <\max(x-x_c,0)> - \sum_{k=0}^{N-1}
<\phi_k^*(x_k)> < \delta x_k >e^{-r(k+1)\tau}
\label{price}
\ee
\section{The case of zero excess average return}
\label{s:mu-zero}
In this section we consider
the case where the average return
of the stock over the bond, $\langle \delta x_k \rangle$, is
zero. This is not, strictly speaking, of practical relevance, as this situation
should not occur exactly in reality.
We show it nevertheless for two reasons: first the
pricing in this case is much simpler,
because one of the terms in (\ref{price})
vanishes, and second, it can be used as a starting point of
a perturbative calculation when the average return
of the stock over the bond is non-zero, but small. This latter case is of
immediate practical interest for all short to medium term options, and is treated in the next section.
The price of the option is given by (\ref{price}), but without the
last term, which is zero by assumption.
Let $P(x,T|x_0,0) dx$ be the probability that the asset value is $x$ at time
$T$, knowing that it was $x_0$ at time $0$.
That yields the option price
\be {\cal C}[x_0,x_c,T;m=0] = e^{-rT} \langle \max(x-x_c,0) \rangle \equiv
e^{-rT} \int_{x_c}^\infty dx (x-x_c) P(x,T|x_0,0)
\label{pricemequal0}
\ee
Equation (\ref{pricemequal0}) is of the same form as the Black-Scholes option prescription,
and agrees with Black-Scholes when the
price process described by the
probability $P(x,T|x_0,0) dx$ is a geometrical
Brownian motion.
The reader should however keep
in mind that the probability in (\ref{pricemequal0})
is that of the share variations themselves, so (\ref{pricemequal0}) is equivalent to
the Black-Scholes formula, for the geometrical Brownian
price process, only in the special case we consider in this section,
that is,
when $\langle \delta x_k \rangle$
is equal to zero.
In the case of a geometrical Brownian motion
with arbitrary average return the
extra term in (\ref{price}) will
take care of the difference
between (\ref{pricemequal0}) and Black-Scholes.
Even though residual risk is not priced at market
equilibrium in our prescription,
it is clearly an important thing to know.
There may also be some merit in displaying explicit formulae for residual
risk, which can be shown to be non-zero by inspection, and
we therefore now turn to the computation of this quantity, still assuming
zero excess average return.
In order to proceed, we shall argue that the price
increments $\delta x_k$ are independent random variables, which is a very good
approximation on highly liquid markets as soon as $\tau$ is larger than a few tens
of minutes (Arn\'eodo et al. (1996)). Assuming further that $\delta x_k$ are identically distributed is in
general not justified, since it is well documented now that `ARCH' effects (time
dependent volatility) must be
taken into account
(Engle (1982), Bollerslev (1986),
Gourieroux (1992), Potters et al (1996)). However, still for pedagogical
purposes, we shall discard this complication, and assume that $\langle \delta x_k \delta
x_\ell \rangle = D\tau \delta(k,\ell)$. In this case, the
risk being equal to $\sqrt{\langle \Delta W^2 \rangle}$, the relevant
formula is rather simple:
\ba\nonumber
\langle \Delta W^2 \rangle &=& \langle \Delta W^2 \rangle_0 + D\tau \sum_{k=0}^{N-1}
\int_{0}^\infty dx P(x,t=k\tau |x_0,0) \phi_k^2(x) e^{2r(T-t-\tau)} \\
&-& \sum_{k=0}^{N-1} 2\tau \int_{x_c}^{\infty} dx' (x'-x_c) P(x',T|x,k)
\int_{0}^\infty dx P(x,t=k\tau |x_0,0) \phi_k(x) \frac{x'-x}{T-t} e^{r(T-t-\tau)}
\label{risk0}
\ea
where $\langle \Delta W^2 \rangle_0$ is the unhedged ($\phi_k \equiv 0$) risk
associated to the option. Minimal risk is obtained by setting:\
\be
\frac{\partial \langle \Delta W^2 \rangle}{\partial \phi_k(x)} = 0
\ee
for all $k$ and $x$. This leads to the following explicit result for the optimal
hedging strategy:
\be
\phi_k^*(x)= \frac{1}{D} \int_{x_c}^{\infty} dx' (x'-x_c) P(x',T|x,t=k\tau ) \frac{x'-x}{T-t} e^{-r(T-t-\tau)}
\label{optimalstrategy0}
\ee
and when we insert (\ref{optimalstrategy0}) in (\ref{risk0})
we arrive at
\be
\langle \Delta W^2 \rangle^* = \langle \Delta W^2 \rangle_0 - D\tau \sum_{k=0}^{N-1}
\int_{0}^\infty dx P(x,t=k\tau |x_0,0) \phi_k^{*2}(x)
e^{-r(T-t-\tau)}
\label{residualrisk0}
\ee
In general, the left-hand side of
(\ref{residualrisk0}) is non-zero; in practice it is even quite high -- for
example, for typical one-month options on liquid markets, $\sqrt{\langle \Delta W^2
\rangle^*}$ represents as much as $25\%$ of the option price itself. As mentioned in the introduction, this accounts for the existence of the (sometimes large) bid-ask spreads observed on option markets.
However, in the special case where $P(x,t|x_0,0)$ is normal or log-normal, and in
the limit of continuous trading, that is, when $\tau \to 0$, one can show that the residual risk
$\langle \Delta W^2 \rangle^*$ vanishes
by a somewhat miraculous identity for Gaussian integrals (Bouchaud \& Sornette (1994)).
This is
one more check of the correctness and self-consistency of our approach.
\section{Small non-zero average return}
\label{s:mu-small}
Let us now consider the case where the average return of the stock with
respect to the bond $m \equiv \langle \delta x_k
\rangle$ is non zero, but small. Small here means that $m N \ll \sqrt{DN}$
($N=T/\tau$),
or,
stated in words, that the average return on the time scale of the option is small
compared to the typical variations, which is certainly the case for options up to a few
months{\footnote{Typically, $m=5
\%$ annual and $D=(15
\%)^2$ annual. The order of magnitude of the error made in neglecting the second
order term in $m$ is
$m^2 N/D
\simeq 0.1$ even for $N
\tau=1$ year.}}. The global wealth balance then includes the extra term which reads:
\be
\langle \Delta W_\subs{trading} \rangle = m \sum_{k=0}^{N-1} \int_0^\infty dx P(x,t=k\tau |x_0,0)
\phi_k^*(x) e^{-r(t+\tau)}
\label{deltw}
\ee
The advantage of considering the case of small average return is that one can
do a perturbation around the case of zero
average return, and still use the explicit optimal strategy of (\ref{optimalstrategy0})
as a first approximation to lowest order in $m$.
Compared to the case $m=0$, the option price is changed both because
$P(x,k|x_0,0)$ is {\it biased}, and
because $\langle \Delta W_\subs{trading} \rangle$
must be substracted off from Eq. (\ref{price}).
It is convenient to use the Fourier transform of the
probability distribution $\tilde{P}(y)=\int_{-\infty}^{\infty}
P(x,N|x_0,0) \exp(i x y)dx$ and to
expand it in a series introducing
the {\it cumulants} $c_n$. They are defined by
\be
\tilde{P}(y)=\exp\Bigr[\sum_{n=2}^{\infty}\frac{c_n (iy)^n}{n!}
\Bigl],
\label{cumul}
\ee
where $c_2=ND\tau$ is the variance, $c_4/c_2^2$ is
the kurtosis, etc... Applying the cumulant expansion to
the probability distribution in Eq.(\ref{optimalstrategy0})
we obtain the optimal strategy (Bouchaud et al (1996)):
\be
\phi_k^*(x) = \frac{1}{c_2}
\sum_{n=2}^\infty
\frac{(-1)^n c_n}{ (n-1)!} \frac{\partial^{n-1}}{\partial
x^{n-1}} {\cal C}[x,x_c,T-k\tau]
\label{strat2}.
\ee
Observe that in the Gaussian case ($c_n=0$ for $n>2$)
only the first term remains and gives the standard "$\Delta-$Hedging" as
predicted by the theory of Black and
Scholes.
Inserting the optimal strategy (\ref{strat2}) into Eq.(\ref{deltw}) and
integrating by parts one gets an expansion of the trading term
$\langle \Delta W_\subs{trading} \rangle$, which put into
Eq.(\ref{price}) gives the following
result for the option price
(Bouchaud et al (1996)):
\be
{\cal C}[x_0,x_c,T;m] = {\cal C}[x_0,x_c,T;m=0] - \frac{m}{c_2} \sum_{n=3}^\infty
\frac{c_n}{ (n-1)!} \frac{\partial^{n-3}}{\partial
x'^{n-3}}P_0(x',N|x_0,0)|_{x'=x_c}\label{m}
\ee
up to corrections of order $m^2$. In the Gaussian case, $c_n=0$ for all $n \geq 3$,
and one thus sees
explicitly
that ${\cal C}_m = {\cal C}_0$, at least to first
order in $m$. Actually, one can show that this is true to all orders in $m$ in the
Gaussian case, which is an alternative way to derive the result of Black and
Scholes
(Bouchaud~et al~(1996)) {\footnote{This result is
easy to obtain if one accepts Ito's calculus, which is only valid
for `quasi' Gaussian processes in the continuum time limit}}.
However, for even distributions with fat tails ($c_3=0$ and $c_4
>0$), it is easy to see from the above formula that a positive average return $m >0$
increases the price of out-of-the-money options
($x_c > x_0$), and decreases the
price of in-the-money options ($x_c < x_0$).
Hence, we see again
explicitly
that the independence of the option price on the
average return $m$, which is one of the
most important result of Black and Scholes,
does not survive for more general
models of stock fluctuations.
Note finally that Eq. (\ref{m}) can also be written as:
\be
{\cal C}[x_0,x_c,T;m] = e^{-rT}
\int_{x_c}^\infty dx (x-x_c) \hat P(x,T|x_0,0)
\label{BS_price_analogy}
\ee
with an effective distribution
$\hat P$ defined as:
\be
\hat P(x,T|x_0,0) = P_0(x,T|x_0,0) - e^{rT} \frac{m}{c_2}
\sum_{n=3}^\infty
\frac{c_n}{ (n-1)!} \frac{\partial^{n-1}}{\partial
x^{n-1}}P_0(x,N|x_0,0)
\label{effectivedistribution}
\ee
Note that the integral over $x$ of
$\hat P$ is one, but $\hat P$ is
not a priori positive everywhere.
Distribution $\hat P$
generalizes the `risk neutral probability' usually discussed
in the context of the Black-Scholes
theory, and also has the property that the excess average return (the integral of $x \tilde P$ over $x$) is zero, as
can easily be seen by inspection from (\ref{effectivedistribution}).
In fact, one can derive formula (\ref{BS_price_analogy}) without any restriction
on $m$, and the effective distribution still has the properties that it
has zero average excess return (Aurell \& Simdyankin (1996)).
\section{Comparison with empirical data}
\label{s:empirical}
For short term options, the simple formula
(\ref{pricemequal0}) is sufficient to obtain the price of options,
{\it provided} one has a good model for
$P(x,T|x_0,0)$. As discussed recently in Mantegna \& Stanley (1995),
Arn\'eodo et al (1996), the distribution $P_\tau$ of
elementary increments $\delta x$ is well modelled in terms of {\it truncated
L\'evy distributions}, i.e. a L\'evy stable distribution of index $\mu \simeq 1.5$ with exponentially truncated tails.
Interestingly, the value of $
\mu$ seems to be rather independent of the considered asset (exchange rates, stocks, etc..).
Since the increments are independent beyond a time scale $
\tau$ of the order of 30 minutes, $P(x,T|x_0,0)$ can be reconstructed by convoluting $P_\tau(\delta x)$ with itself
$N=T/\tau$
times {\footnote{This is again assuming that the $\delta x$ are identically distributed,
which is a good first approximation -- but see Potters et al (1996).}}.
We show in Fig.~\ref{f:JPB_data} some `experimental' prices for `Bund' call options of different maturities
(all less than a month) and strikes between January and June 1995.
The Bund is a future on long term German government bonds traded on the
LIFFE in London, from which the data was obtained.
The coordinate of each point is the theoretical price given by Eq. (\ref{pricemequal0})
on the $x$ axis, and the observed price on the $y$ axis.
The probabilities
$P(x,T|x_0,0)$ were
reconstructed
using historical data on $P_\tau(\delta x)$ in the same period. Since $30$ minutes
data were used, one has rather good precision on the
parameters of the `truncated' L\'evy distribution. The overall
agreement is satisfying: the linear regression
gives a slope of $0.9993 \pm 0.0009$
with a negligible intercept (0.02 base points),
to be compared with
a slope of one and a zero intercept, if theory and data were in perfect
agreement.
One can see that the difference between
observed prices and theoretical prices are actually
on
the order of the transaction costs.
The same analysis was done using Black-Scholes log-normal pricing with
zero interest rate (since futures are margin compensated) and
volatility measured historically over the same period. The resulting
graph (not shown) is visually similar to that of Fig.\ 1, but
with a somewhat larger spread around the line of slope one. Moreover,
a linear regression on the Black-Scholes data gives a slope of
$1.02 \pm 0.002$ and an intercept of 6 base points.
The fits on the slope and the intercept are, respectively,
one and two orders of magnitude worse than in our procedure,
and reflect the fact
that the Black-Scholes theory systematically
misprices mainly out-of-the-money options, while working
relatively better for at-the-money
and in-the-money options.
A more detailed analysis,
on other time periods, however
reveals that
part of the theoretical {\em vs}\/ observed
differences seen in Fig.~\ref{f:JPB_data} are systematic (Potters et al (1996)),
and related to the fact that
these time-series are non-stationary:
volatility is actually itself time dependent,
a feature
also recognized to be important by traders.
We have also compared the theoretical residual risk with the results
of a Monte-Carlo simulation, where optimally hedged portfolios were
generated using real option prices. The residual risk was found to be
somewhat
(about 50\%)
larger than the one predicted theoretically,
Eq.~(\ref{residualrisk0}), which again reflects the fact that an extra
source of risk comes from the non-stationarity of the time series, the
so-called `volatility risk'.
\begin{figure}[hbt]
\iffigs
% \centerline{\psfig{file=full_path_to_poscript_figure.ps,width=10cm,clip=}}
\psfig{file=Aurell.eps,width=14cm,clip=}
\else
\drawing 100 10 {Our LIFFE data}
\fi
\caption{ `Experimental' prices for Bund call options of different maturities (all less than a month)
and strikes between January and June 1995.
The data was obtained from the LIFFE.
The coordinate of each point
is the theoretical price and the observed price.
$P(x,t|x',t')$ was
reconstructed
using historical data in the same period.
The differences with Black and Scholes are actually not very large ($ \sim 2\%$)
for this very liquid market; however they are {\it systematic} (see text),
and reflect the existence of a `volatility smile' (i.e. the need to change the
volatility used in the BS formula with the strike price),
which is well captured by a truncated L\'evy process description.}
\label{f:JPB_data}
\end{figure}
\section{Discussion}
\label{s:discussion}
In the last twenty-five years financial derivatives have grown from
a somewhat marginal activity to occupy center-stage position
in economical theory and practice. At the same time, mathematical
finance has grown to be one of the main branches of applied
mathematics, and in even more simple terms, a sizeable fraction of
graduates in mathematics and related subjects now go
on to work on option pricing.
The single largest credit for
these remarkable developments are due to Fisher Black and Myron Scholes,
who in their classic paper of 1973 gave a theory of how to price options
in a simple model. Without this prescription, option pricing would have
remained more of an art than a science, and trading in options would have
arguably have been less liquid and important, as traders would have had
a less firm idea on how to hedge the bet on the future, which options in
the end come down to be. It is certain that without the Black-Scholes' formula,
the mutual interest of mathematicians in finance and financial operators
in advanced mathematical techniques would not have been what they
are today.
Nevertheless, like all clear-cut and useful results, the Black-Scholes
theory holds only under certain definite conditions. If these are
not fulfilled, it is an open
question whether to price an option using the Black-Scholes formula
is the best possible strategy.
The economically sensible meaning of best and better
in this context must be, that a strategy is better
if those operators who use it in the long
run do better than the rest.
Empirical and theoretical modifications to the Black-Scholes
theory amount today to a small industry. We surveyed some parts
of that literature in the introduction, and will not repeat that
discussion here, except to remark that the less academic side
of this activity goes
on in a commercial environment, and would not prosper if there were
not profit to made fram capitalizing on the short-comings of
the Black-Scholes theory.
Successful financial operators must effectively use pricing
procedures at least as good as Black-Scholes, to offset the
additional costs of time and money.
Those pricing procedures are of course not made available
directly to the public, but, to judge from the published
literature, they seem at least partly based on
arguments which are ad hoc, or in any
case difficult to penetrate.
It is therefore quite concieveable that pricing procedures
on the market effectively come down to approximations,
stated in terms which can be very complicated, and perhaps not fully
rationalized, to some other prescription, which
is not necessarily known by the operators themselves.
A similar situation probably pertained twenty-five years ago
before the Black-Scholes theory had been published and
become widely known.
We have in this paper presented an approach to option pricing which
is almost as simple as the Black-Scholes theory. Indeed, from the
conceptual point of view it is even simpler, and can be described
entirely in elementary terms: first a special
hedging strategy is chosen such that
risk (measured by the variance) is minimized. Then the price is determined
by the condition that the mean profit of an operator using the special
hedge is zero.
In theoretical terms the advantage of our prescription is that it only
demands that we minimize risk, not that we eliminate it entirely.
In Black-Scholes theory, the crucial argument is the absence of
arbitrage on effective markets, which means that there can only be one
risk-free rate of return. Hence, to use these arguments, it must be possible
to hedge away all risk
entirely.
Since risk must always be non-negative, our prescription agrees with Black-Scholes
in all cases where the latter can be applied, but it can also be applied
in a much wider setting.
That a strategy is
determined by risk-minimization is intuitively appealing,
and certainly very reasonable for risk-sensitive operators.
A further theoretical argument can however also be made from equilibrium considerations
on a market with operators with
different attitudes to risk (Aurell \& \.Zyczkowski, 1996).
In computational terms our prescription can be used to derive analytical
formulae, of similar complexity as those in the Black-Scholes theory.
Those
formulae
are valid when the excess rate of return of the stock over the bond
is small compared to stock volatility. The precise meaning of small is
clarified above in section~\ref{s:mu-small}, suffice it to say that the formulae hold with good
accuracy for options with time to expiration up to a few months.
Longer term options are in any case more difficult since the assumption of
stationarity of the time series becomes more and more questionable as the time
to maturity increases.
The prescription can also be used directly as a specification of a minimization
problem, which could be used for options with long time to expiration.
This minimization problem can be handled with standard numerical tools.
In empirical terms, we have compared our prescription to real financial data
from the LIFFE exchange, a very liquid market where bid-ask spreads are
expected to be small, and found quite good agreement.
Let us separate the discussion of the
general potential practical usefulness
of our prescription into the general
idea to minimimize risk, and the concrete
formulae derived above in sections~\ref{s:mu-zero} and~\ref{s:mu-small}.
We believe that the general framework we have presented, to determine
prices of derivative securities by
minimizing risk, has very wide applicability.
It can also be used in situations where variance is not an adequate
measure of risk, because the tail of the distribution is so massive that
the variance is formally infinite (Bouchaud et al, 1996,1997).
A relevant measure of risk is then the probability of large losses,
and the prescription says that the probability of these losses
(or the {\it Value-at-Risk}) should be minimized.
We have in this paper looked at the simplest implementation of the
general idea of option pricing by risk minimization.
We assumed that price increments are independent, identically
distributed
random variables and that there are no transaction
costs, and thus derived the formulae presented in sections~\ref{s:mu-zero} and~\ref{s:mu-small}.
The direct practical usefulness of these formulae then hinges on that the
assumptions
are sufficiently well
fulfilled,
and that the resulting price
prescription is systematically different from the one that can be derived
from Black-Scholes, which reflects the need of operators to use a volatility `smile'.
However, an even closer agreement with market prices is obtained when one explicitely
takes into account what is known as `volatility risk' (Potters et al. (1996)).
\section{Acknowledgements}
We thank J.-P.~Aguilar, R.~Cont,
and D.~Sornette for important
discussions.
This work was supported by the Swedish Natural
Science Research Council through grant
S-FO 1778-302 (E.A.), by a grant from the
Swedish Board of Research and Development NUTEK (E.A.), and by
the Polish State Committee for Research (K.\.Z.).
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