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
