  In a seminal paper in 1973, Black and Scholes argued how expected
distributions of stock prices can be used to price options. Their model assumed
a directed random motion for the returns and consequently a lognormal
distribution of asset prices after a finite time. We point out two problems
with their formulation. First, we show that the option valuation is not
uniquely determined; in particular, stratergies based on the delta-hedge and
CAMP (Capital Asset Pricing Model) are shown to provide different valuations of
an option. Second, asset returns are known not to be Gaussian distributed.
Empirically, distributions of returns are seen to be much better approximated
by an exponential distribution. This exponential distribution of asset prices
can be used to develop a new pricing model for options that is shown to provide
valuations that agree very well with those used by traders. We show how the
Fokker-Planck formulation of fluctuations (i.e., the dynamics of the
distribution) can be modified to provide an exponential distribution for
returns. We also show how a singular volatility can be used to go smoothly from
exponential to Gaussian returns and thereby illustrate why exponential returns
cannot be reached perturbatively starting from Gaussian ones, and explain how
the theory of 'stochastic volatility' can be obtained from our model by making
a bad approximation. Finally, we show how to calculate put and call prices for
a stretched exponential density.
