The document discusses a nested-model framework for quantifying model-related risks, focusing on various aspects such as model specification and assumption risks. It provides an example of the Black-Scholes model nested within the Heston model and describes the statistical methods for testing model validity and assumptions. The framework has advantages like flexibility and economic interpretability, but also presents drawbacks such as increasing complexity in theoretical research.

1. Quantifying model related risks
A nested-model framework
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2. Content
• Formalizing the building blocks
• An example: Black-Scholes nested into Heston
• Statistics for model specification risk
• Statistics for assumption risks
• Non-parameterized models
• Advantages/draw-backs
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3. Formalizing the building blocks
• Let M(a_1,a_2,…,a_n) be a parameterized model
with finitely many parameters
• Let N(a_1,a_2,…,a_n,b_1,b_2,…,b_m) be an
extension of M(a_1,a_2,…,a_n) , i.e. let
N(a_1,…,a_n,0,…,0) = M(a_1,…,a_n)
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4. Black-Scholes nested into Heston
An example of nested models
• The Black-Scholes model is parameterized as
• riskless rate r
• volatility s.
• The Heston model is parameterized as
• riskless rate r ,
• initial volatility v(0),
• the parameters of the volatility,
• k (speed of mean reversion),
• t (the long term mean),
• x (the volatility of volatility)
• correlation coefficient r
• Setting v(0)=s², k=0, x=0 and leaving t, r arbitrary, the Heston
model coincides with the BS model
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5. Statistics for model specification risk
• Under the assumption that the model actually holds,
simulation of M(a) and N(a,0) should result in paths
from the same distribution.
• Using the Central Limit Theorem we transform the
simulated variables to be asymptotically normal
• Generating a very large number (millions) of paths
lead to sufficient samples for normal-distribution
based testing
• The differences between the paths can be quantified
via testing whether or not the expected value of
simulated variables are the same.
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6. Statistics for assumption risk
• Under the assumptions N(a,0) and the full model
M(a) equivalent
• The model N(a,0) is then nested into N(a,b)
• We can test the assumption b=0 (in the example
of B-S model the characteristics of volatility
equaling 0)
• Regular p-value and R-square statistics can be
applied to measure the “correctness” of the
assumption
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7. Non-parameterized models
• Model specification risk is ill defined
• Assumption risk on the other hand can be
measured
• Nesting the model can be done by relaxing the
assumptions
• Example: fitting smooth distribution to
insurance data versus fitting right/left-
continuous distributions
• The same well-known tests can be applied
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8. Advantages/draw-backs
• Advantages:
• The statistics are based on simple mathematics
• The framework is very flexible
• The quantified risks can be interpreted economically (e.g. in
relation to the underlying price, risk expressed in percentage)
• Draw-backs:
• Theoretical research is necessary to test each models one
by one
• The nesting models become intractably complicated very
quickly
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