ERCIM News No.38 - July 1999

## Model Risk Analysis for Discount Bond Options

by Denis Talay

Resarchers of the Omega research group at INRIA Sophia Antipolis and of the University of Lausanne have started in 1998 a study on model risk for discount bond options. This research is funded by the Swiss Risklab institute. The aim of the project is to see how models risk affects the risk management of interest rate derivatives and how to manage this risk.

RiskLab is a Swiss inter-university research institute, concentrating on precompetitive, applied research in the general area of (integrated) risk management for finance and insurance. The institute, founded in 1994, is presently co-sponsored by the ETHZ, the Crédit Suisse Group, the Swiss Reinsurance Company and UBS AG. Several research projects are being funded by Risklab. Among them, the project on model risk analysis for discount bond options proposed by researchers at the University of Lausanne (Rajna Gibson and François-Serge Lhabitant) and the Omega Research group at INRIA Sophia Antipolis (Mireille Bossy, Nathalie Pistre, Denis Talay, Zheng Ziyu).

Model risk is an important question for financial institutions. Indeed, trading, hedging and managing strategies for their books of options are derived from stochastic models proposed in the literature to describe the underlying assets evolutions. Of course these models are imperfect and, even if it were not, their parameters could not be estimated perfectly since, eg, market prices cannot be observed in continuous time. For discount bond options, additional mispecifications occur: for example, it seems difficult to discriminate models and to calibrate them from historical data of the term structure. Thus a trader cannot make use of perfectly replicating strategies to hedge such options. The purpose of the study is to provide an analytical framework in which we formalize the model risk incurred by a financial institution which acts either as a market maker — posting bid and ask prices and replicating the instrument bought or sold — or as a trader who takes the market price as given and replicates the transaction until a terminal date (which does not necessarily extend until the maturity of his long or short position).

The first part of the study is to define the agent’s profit and loss due to model risk, given that he uses an incorrect model for his replicating strategy, and to analytically (or numerically) analyse its distribution at any time. This allows us to quantify model risk for path independent as well as for path dependent derivatives. The main contributions of the study is to decompose the Profit and Loss (P&L) into three distinct terms: the first representing a pricing freedom degree arising at the strategy’s inception (date 0), the second term representing the pricing error evaluated as of the current date $t$ and the final term defining the cumulative replicating error which is shown to be essentially determined by the agent’s erroneous ‘gamma’ multiplied by the squared deviation between the two forward rate volatilities curve segments’specifications. We furthermore derive the analytical properties of the P&L function for some simple forward rate volatilities specifications and finally conduct Monte Carlo simulations to illustrate and characterize the model error properties with respect to the moneyness, the time to maturity and the objective function chosen by the institution to evaluate the risk related to the wrong replicating model. A specific error analysis has been made for the numerical approximation of the quantiles of the P&L.

Aside from providing a fairly general yet conceptual framework for assessing model risk for interest rate sensitive claims, this approach has two interesting properties: first, it can be applied to a fairly large class of term structure models (all those nested in the Heath, Jarrow, Morton general specification). Secondly, it shows that model risk does indeed encompass three well defined steps, that is, the identification of the factors, their specification and the estimation of the model’s parameters. The elegance of the HJM term structure characterization is that those three steps can all be recast in terms of the specification and the estimation of the proper forward volatility curve function.

The second part of the study concerns the model risk management. We construct a strategy which minimizes the trader’s losses universally with respect to all the possible stochastic dynamics of the term structure within a large class of models. This leads to complex stochastic game problems, hard to study theoretically and to solve numerically: this is in current progress.

Risklab: http://www.risklab.ch/

Omega Research team: http://www.inria.fr/Equipes/OMEGA-fra.html

Please contact:Denis Talay - INRIA

Tel: +33 4 92 38 78 98

E-mail: Denis.Talay@sophia.inria.fr

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