Among many risk management methods, option replication corresponds to a self-financing method to replicate the payoffs of an option instrument with some other instruments. We can hedge option by taking delta equivalent in the underlying, then by the same token so can its payoff profile be synthesized via the underlying asset. But the trouble with options is that 'resolved' asset flows keep changing because their delta changes.
The option replication is based on the Black-Scholes model and so theoretically the hedging costs of dynamic replication should equal the premium of the option. But many empirical works show that many assumptions, which the Black-Scholes model has, are only an idealization, and they must be properly modified to reflect better the reality. So we need to relax the assumptions of perfect markets conditions and introduce proportional transaction costs and volatility change to assess the impact of hedge efficiency for market reality.
The aim of this paper is to provide insights into the nature of the hedging problems associated with dynamic option replication where markets imperfections like transaction costs and volatility change may exist.
First, we have to consider transaction costs for dynamic replication. Since the original paper by Leland (1985), transaction costs in option replication have been extensively investigated in the literature. He has proposed a very simple modification to the Black-Scholes model for vanilla calls and puts, which can be extended to portfolios of options, that introduces discrete revision of the portfolio and transaction costs by using modified volatility.
Second, the volatility is not constant. A general and promising model is the one, which allows the stock price volatility to be stochastic. Hull and White introduced stochastic volatility model that the volatility as well as stock price follows the geometric Brownian motion and have proposed modified PDE of option portfolio.
We examine via Monte Carlo Simulation, the hedging results and variability that occur from traditional dynamic hedging in perfect and imperfect markets to evaluate the usefulness of dynamic replication strategy. In general the stochastic volatility makes large the standard deviation of difference between hedging cost and option premium and hedging costs raised when transaction costs are considered.
A market participant needs to choice the proper adjustment gap under consideration whip-saw costs for the effective dynamic replication because hedging costs is proportional to the frequency of readjustment. Moreover they have to consider liquidity, market shock and gamma risk for the effective dynamic replication. The lack of buyers tends to coincide with large market moves and liquidity drying up so that the limitations of dynamic replication have more to do with curvature risks rather than spot risks.
