Model Dependency of the Digital Option Replication Replication under an Incomplete Model

A digital option is a special type of financial derivative with a non-linear discontinuous payoff function. In spite of this, the payoff is simple enough to allow (relatively) easy valuation of these contracts. It is the reason why digital options can be (and regularly are) applied to decompose and hedge statically the positions in many options with more complicated and usually discontinuous payoff functions; see for example (Andersen – Andreasen – Elizier, 2002), (Carr – Chou, 1997), (Carr – Ellis – Gupta, 1998) or (Derman – Ergener – Kani, 1995). It is clear, that in order to ensure efficient risk management of complicated exotic options, the procedures for pricing and hedging digital options must be no less efficient. However, the valuation of digital options is easy only in the Black and Scholes (1973) setting. By contrast, relaxing some Black and Scholes restrictions can cause an incompleteness of the model, either by stochastic volatility, presence of jumps or non-normally distributed returns (non-normality of returns is commonly modeled by suitable Lévy models with an infinite intensity of jumps). Another problem arises if a trader is not sure about the underlying model. Hence, he or she can only guess and, therefore, probably applies the incorrect one. Obviously, it can also happen that the only model available to apply is the Black and Scholes model. Although the trader can know the true evolution, it can be comprised of such complicated features that the application can be impossible. This situation can again lead to incorrect results.