An Empirical Model for Volatility of Returns and Option Pricing

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 ,strategies based on the delta-hedge and CAPM (the 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. 1. The CAPM portfolio selection strategy The Capital Asset Pricing Model (CAPM) is very general: it assumes no particular distribution of returns and is consistent with any distribution with finite first and second moments. Therefore, in this section, we generally assume the empirical distribution of returns but also will apply the model to Gaussian returns (lognormal prices) in part 2 below. The CAPM is not, as is often claimed, an equilibrium model because the distribution of returns is not an equilibrium distribution. We will exhibit the time-dependence of some of the parameters in the model in the familiar lognormal price approximation. Economists and finance theorists (including Sharpe [1| and Black [2]; see also Bodie and Merton [3])) have adopted and propagated the strange notion that random motion of returns defines ‘equilibrium’, which disagrees with the requirement that in equilibrium no averages of any moment of the distribution can change with time. Random motion in the market is due to trading and the excess demand of unfilled limit orders prevents equilibrium at all or almost all times. Apparently, what many economists mean by ‘equilibrium’ is more akin to assuming the EMH (efficient market hypothesis), which has nothing to do with vanishing excess demand in the market. The only dynamically consistent definition of equilibrium is vanishing excess demand: if p denote the price of an asset then excess demand is defined by dp/dt= (p,t) including the case where the right-hand side is drift plus noise, as in stochastic dynamical models of the market. These issues have been discussed in detail in a previous paper [4]. Bodie and Merton [3] claim that vanishing excess demand is necessary for the CAPM, but one sees in part 2 below that no such assumption comes into play during the derivation and would even cause all returns to vanish in the model! The CAPM [5] can be stated in the following way: Let Ro denote the risk-free interest rate, (1) is the fluctuating return on asset k where pk(t) is the price of the kth asset at time t. The total return x on the portfolio of n assets relative to the risk free rate is given by (2) where fk is the fraction of the total budget that is bet on asset k. The CAPM minimizes the mean square fluctuation (3) subject to the constraints of fixed expected return R, (4) and fixed normalization xk ln( pk (t t) / pk( t)) x Ro fi( i 0 n xi Ro ) 2 fi fj i, j (xi Ro )(xj Ro) fi f j i, j ij R Ro (x Ro ) fi i , j (xi Ro ) fi(Ri Ro) i, j