In this presentation I give a nested model framework to quantify model risk. While it is more of academic interest, it can be efficiently used to validate model assumptions.
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Quantifying model related risks A nested-model framework Quantifying model related risks 1
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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 Quantifying model related risks 2
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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) Quantifying model related risks 3
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Black-Scholes nested into HestonAn 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 Quantifying model related risks 4
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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 5
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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 Quantifying model related risks 6
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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/left- continuous distributions• The same well-known tests can be applied Quantifying model related risks 7
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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 Quantifying model related risks 8
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