2. Quantifying model related risks. Content. • Formalizing the building blocks. • An
example: Black-Scholes nested into Heston. • Statistics for model specification ...
Quantifying model related risks
A nested-model framework
Quantifying model related risks
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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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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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Black-Scholes nested into Heston An example of nested models •
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The Black-Scholes model is parameterized as •
riskless rate r
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volatility s.
The Heston model is parameterized as •
riskless rate r ,
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initial volatility v(0),
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the parameters of the volatility, • • •
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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 Quantifying model related risks
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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. Quantifying model related risks
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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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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/leftcontinuous distributions • The same well-known tests can be applied Quantifying model related risks
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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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