UNDERSTANDING BID-ASK SPREADS OF DERIVATIVES UNDER UNCERTAIN VOLATILITY AND TRANSACTION COSTS

The classical option valuation models assume that the option payoff can be replicated by continuously adjusting a portfolio consisting of the underlying asset and a risk-free bond. This strategy implies a constant volatility for the underlying asset and perfect markets. However, the existence of nonzero transaction costs, the consequence of trading only at discrete points in time and the random nature of volatility prevent any portfolio from being perfectly hedged continuously and hence suppress any hope of completely eliminating all risks associated with derivatives. Building upon the uncertain parameters framework we present a model for pricing and hedging derivatives when the volatility is simply assumed to lie between two bounds and in the presence of transaction costs. It is shown that non-arbitrageable prices for the derivatives can be derived by a non-linear PDE related to the convexity of the derivatives. We use Monte Carlo simulations to investigate the way our hedging strategy behaves within a realistic environment. We show that it is possible to derive robust prices for derivatives although we are in the presence of imperfect and incomplete markets. We explain how gamma diversification can be used to improve the efficiency of derivative markets and to recover bid-ask spreads of derivatives compatible with the tight spreads observed on the market. This has important implications for price determination in options markets as well as for testing of valuation models. Market data are used to corroborate our findings and to explain how market prices can be used to infer the implicit degree of diversification on the market.