Synthetic option replication with transactions costs: Arbitrage bounds on the volatility smile and hedging performance.

摘要

This dissertation consists of two essays. The first essay examines arbitrage bounds around the volatility smile and the second essay examines the impact of market imperfections on the performance of synthetic option replication.;The first essay shows that the volatility smile is not necessarily inconsistent with the Black-Scholes analysis. Specifically, when transaction costs are present, the absence of arbitrage opportunities does not dictate that there exists a unique price for an option. Rather, there exists a range of prices within which the option's price may fall, and still be consistent with the Black-Scholes arbitrage pricing argument. This essay uses the binomial model and a linear program (LP) to determine the smallest possible range of prices for S&P 500 index options which are consistent with no-arbitrage in the presence of transaction costs, and shows that the volatility smile can exist within these bounds. The LP method, based on a binomial model, employs dynamic trading in the underlying and riskless assets as well as fixed positions in other options which trade on the same underlying security. One-way transaction cost levels, inclusive of the bid-ask spread, would have to be below 0.1% for the volatility smile to present an arbitrage opportunity which has a reasonable probability of being profitable.;The second essay examines the impact of market imperfections and volatility mis-estimation on the performance of synthetic option replication. A linear programming replication technique is used, which determines the optimal mix of static and dynamic replication and can account for transaction costs on all securities in the replicating portfolio. Monte Carlo simulations are used in conjunction with the linear programming technique to assess the model's performance, measured by the deviation of the actual payoff of the replicating portfolio less the target payoff, in a variety of dimensions. The central result is that volatility mis-estimation can cause large deviations in the expected distribution of hedging errors, but by including traded options in the replicating portfolio, the Vega of the target security can be matched to the Vega of the replicating portfolio, and the effects of volatility mis-estimation can be mitigated.