The Extended Black-Scholes Model with-LAGS-and “Hedging Errors”

Abstract

The Black-Scholes model is derived under the assumption that heding is done instantaneously. In practice, there is a “small†time that elapses between buying or selling the option and hedging using the underlying asset. Under the following assumptions used in the standard Black-Scholes analysis, the value of the option will depend only on the price of the underlying asset S, time t and on other Variables assumed constants. These assumptions or “ideal conditions†as expressed by Black-Scholes are the following.

• The option us European,

• The short term interest rate is known,

• The underlying asset follows a random walk with a variance rate proportional to the stock price. It pays no dividends or other distributions.

• There is no transaction costs and short selling is allowed, i.e. an investment can sell a security that he does not own.

• Trading takes place continuously and the standard form of the capital market model holds at each instant.

The last assumption can be modified because in practice, trading does not take place in-stantaneouly and simultaneously in the option and the underlying asset when implementing the hedging strategy. We will modify this assumption to account for the “lagâ€. The lag corresponds to the elapsed time between buying or selling the option and buying or selling - delta units of the underlying assets. The main attractions of the Black-Scholes model are that their formula is a function of “observable†variables and that the model can be extended to the pricing of any type of option. All the assumptions are conserved except the last one.