Advances in Risk Management

FRANÇOIS-SERGE LHABITAN T 211

thoseoutputs. Inalltherecentderivativeslosses, managementcanbefaulted
for a lack of understanding the problem. Murphy’s Law holds; what can go
wrong will go wrong. You can only tell when a model is wrong. It will always
be more difficult to tell when a model is right.

NOTES

1. The Black – Scholes option-pricing formula, for example, can also be expressed as the
   solution to the heat-diffusion equation.


2. Note that the consequences of model risk are also visible in non-financial areas. For
   instance, a simple programming error – trying to store a 64-bit number into a 16-
   bit space – exploded the European Agency rocket Ariane 5 shortly after take off,
   destroying $7 billion of investment and 10 years of work.


3. The implied volatility is the volatility figure that one would need to plug in the Black
   and Scholes formula to obtain a theoretical price equal to the market price.


4. For instance, Derman and Kani (1994) construct implied binomial trees from an
   observed volatility smile and use it for pricing and hedging both standard and exotic
   options. Dupire(1994)providesanalgorithmtorecoverauniquerisk-neutraldiffusion
   process consistent with observed (or fitted) option prices.


5. WorkingintheBlackandScholesframeworkleadstoimportantanalyticsimplification
   without any loss of generality. The equivalent derivation in the case of a more general
   model can be found in Bossyet al.(1998).


6. See for instance Lhabitant, Martini and Reghai (2001) for options on a zero-coupon
   bond.


7. Marking to market is the process of regularly evaluating a portfolio on the basis of its
   prevailing market price or liquidation value.


8. See for instance Gibson, Lhabitant and Talay (2001) who survey more than 60 different
   models of interest rates.


9. In a strip issue, a bond is stripped into its regular coupon annuity payment (interest
   only) and principal repayment (principal only).

REFERENCES

Bachelier, L. (1900) “Theorie de la Speculation” (Thesis),Annales Scientifiques de l’École
Normale Superieure, III-17: 21–86. [Cootner (ed.) (1964)Random Character of Stock Market
Prices, Massachusetts Institute of Technology, pp. 17–78 or Haberman, S. and Sibett,
T.A. (eds) (1995)History of Actuarial Science, VII, pp. 15–78, London].
Black, F. and Scholes, M. (1973) “The Pricing of Options and Corporate Liabilities”,Journal
of Political Economy, 81: 637–59.
Cox, J.C. and Ross, S.A. (1976) “The Valuation of Options for Alternative Stochastic
Processes”,Journal of Financial Economics, 3: 145–66.
Derman, E. (April, 1996) “Model Risk”, Goldman Sachs Quantitative Strategies Research
Notes.
Derman, E. and Kani, I. (January, 1994) “The Volatility Smile and Its Implied Tree”.
Goldman Sachs Quantitative Strategies Research Notes.
Gallus, C. (1996) “Exploding Hedging Errors for Digital Options”, Working Paper,
Deutsche Morgan Grenfell, October.