With the dissemination of quantitative methods in risk management and introduction of complex derivative products, statistical and numerical methods have come to play an increasingly important role in financial derivatives making, especially in the context of calibration, pricing and hedging of derivative instruments. While the use of such methods has undeniably led to better managing of market risk, it has in turn given rise to a new type of risks linked to the unknown error bounds for the quantities delivered by these methods.
When the pricing model is specifed the aim of calibration is to estimate parameters of the model using the prices of liquidly traded options such as call and put options on major indices, exchange rates and major stocks. For such an option the price is determined by supply and demand on the market. Because of the bid-ask spread and small number of daily available options on the given stock (or interest rate) the calibration is an ill-posed problem and has to be treated carefully. For example, in the case of jump diffusion Merton model the bid-ask spread of order 0:1 and the number of vanilla call options as large as 50 can lead to a relative error in the parameters estimate of a order up to 20% if the calibration is not accompanied with a proper regularization (see Belomestny and Reiß (2006)). Moreover, the use of different calibration procedures (for example, based on different error measures) can lead to different calibration results and
give rise to the calibration uncertainty or calibration risk. Detlefsen and Härdle (2006a) provide evidence for such a risk in a time series of DAX implied volatility surfaces.
Upon calibrating the model one usually has to compute the prices of exotic or illiquid options which are issued over-the-counter and for which a market price is often unavailable. American-style derivatives are the most popular exotic derivatives. They can be exercised at any point in time before maturity. As a result, part of the valuation problem consists of identifying the optimal exercise policy, i.e., the exercise time that maximizes the value for the holder of the security. Misspecifed exercise strategies (for example, due to small number of basis functions used for approximation) can lead to a loss in the value up to 200% (see, for example, Bender, Kolodko and Schoenmakers (2005)). The situation with Greeks or option sensitivities is even more dramatic.
The unknown error bounds can not only lead to the mispricing of derivative products but also make this mispricing sometimes unnoticeable for a long time. While this type of risks is acknowledged by most operators who make use of quantitative methods, most of the discussion on this subject has stayed at a qualitative level. The aim of this project is to quantify (construct error bounds, investigate worst-case scenarios) statistical and numerical errors arising during calibration (pricing) and propose new computationally effcient algorithms for calibration (pricing) which can reduce these errors.