An empirical model of volatility of returns and option pricing

Abstract

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.

Martingale option pricing

“…Consider next the delta hedge, a portfolio long on Δ shares of stock and short a call w. If the portfolio is chosen to increase in value at the bank (money market, CD, …) interest rate then Δ=w', where prime denotes differential with respect to p, then we have the risk neutral portfolio where the Green function for the Black-Scholes pde [12,15] …”

“…In practice [12,15], one can impose a condition that fixes µ in the consensus price by the bank interest rate r.…”

“…If in the call price (9) we restrict to x=0, so that p≈p c and then approximate the drift R(x,t)=µ-D(x,t)/2≈constant in (3) and use the exponential density with H=1/2, then we obtain the formulae that we used much earlier [12,15] to price options empirically correctly. This means that we've approximated an arbitrary stock price p at present time t<T by the consensus price p c , a result would be expected to be of practical use only so long as stock prices don't vary too much.…”

“…dy = "E(y, t)dt / 2 + E(y, t)dB(t) (14) and yields the corresponding price sde (with x=lnS(t)/S c ) ! dS = S 2 e(S, t)dB(t) (15) with price diffusion coefficient e(S,t)=E(y,t)=D(x,t)=d(p,t). This shows that the risk neutral discounted price S=pe -rt is a Martingale.…”

“…We begin by reviewing the connection between the Black-Scholes pde and the Fokker-Planck pde describing the market Green function, a connection that we discovered and reported some years ago [12,15].…”