Abstract
A new method for calibrating the Black-Scholes asset price dynamics model is proposed. The data used to test the calibration problem included observations of asset prices over a finite set of (known) equispaced discrete time values. Statistical tests were used to estimate the statistical significance of the two parameters of the Black-Scholes model: the volatility and the drift. The effects of these estimates on the option pricing problem were investigated. In particular, the pricing of an option with uncertain volatility in the Black-Scholes framework was revisited, and a statistical significance was associated with the price intervals determined using the Black-Scholes-Barenblatt equations. Numerical experiments involving synthetic and real data were presented. The real data considered were the daily closing values of the S&P500 index and the associated European call and put option prices in the year 2005. The method proposed here for calibrating the Black-Scholes dynamics model could be extended to other science and engineering models that may be expressed in terms of stochastic dynamical systems.
Highlights
The Black-Scholes formulae 1 used to price European call and put options are based on an asset price dynamics model
The volatility and drift parameter values are necessary to apply the model to asset and option price forecasting; in practice, the values of these parameters must be determined prior to using the Black-Scholes dynamics model and the option pricing formulae derived from it
The problem of estimating the asset price dynamics model parameters must be considered based on the available data. This problem is a calibration problem and is an inverse problem for a stochastic dynamical system defined by a stochastic differential equation
Summary
The Black-Scholes formulae 1 used to price European call and put options are based on an asset price dynamics model. The solution to the Black-Scholes asset price dynamics model is a stochastic process called geometric Brownian motion, which depends on two parameters: the drift and the volatility. From this fact, it follows that the asset price at any given time may be modeled as a random variable with a log-normal distribution and, the log return of the asset price is normally distributed. The mean and variance of these Gaussian random variables can be expressed using elementary formulae as a function of the drift and volatility parameters of the Black-Scholes asset price dynamics model and of the time interval between observations. The synthetic data considered were generated by numerically integrating the stochastic differential equation that defines the asset price dynamics in the Black-Scholes model, for several choices of the parameter values.
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