Calibration errors in risk management (A06#)

• Subject Area: Mathematics
• Term: from 2011 to 2013
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 68236791

With the dissemination of quantitative methods in risk management andintroduction of complex derivative products, statistical methods have come to playan increasingly important role in financial derivatives making, especially in thecontext of calibration of derivative instruments. While the use of such methods hasundeniably led to better managing of market risk, it has in turn given rise to a newtype of risks linked to the unknown error bounds for the quantities delivered bythese methods. When the pricing model is specified the aim of calibration is toestimate parameters of the model using the prices of liquidly traded options suchas call and put options on major indices, exchange rates and major stocks. For suchan option the price is determined by supply and demand on the market. Because ofthe 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 treatedcarefully. For example, in the case of jump diffusion Merton model the bid-askspread of order 1% and the number of vanilla call options as large as 50 can lead toa relative error in the parameters estimate of an order up to 20% if the calibration isnot accompanied with a proper regularization. Moreover, the use of differentcalibration procedures (for example, based on different error measures) can lead todifferent calibration results and give rise to the so-called calibration uncertainty orcalibration risk. The unknown error bounds can not only lead to the mispricing ofderivative products but also make this mispricing sometimes unnoticeable for along time. While this type of risk is acknowledged by most operators who make useof quantitative methods, most of the discussion on this subject has stayed at aqualitative level. The aim of this project is to quantify (construct error bounds,investigate worst-case scenarios) statistical and numerical errors arising duringcalibration and propose new computationally efficient algorithms for calibrationwhich can reduce these errors.

• DFG Programme: Collaborative Research Centres
• Subproject of: SFB 823: Statistical Modelling of Nonlinear Dynamic Processes
• Applicant Institution: Technische Universität Dortmund
• Co-Applicant Institution: Universität Duisburg-Essen
• Project Head: Professor Dr. Denis Belomestny