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Model Risk for Pricing and Risk Models in Finance
1. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Model Risk and Model Control
Patrick H¨aner
H¨aner Consulting Berlin
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2. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
The
latest version of this document
additonal resources
examples
may be found on
https://github.com/haenerconsulting/modelrisk
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3. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Outline
1 Model Classes
2 Credit Risk Measures
3 Model Implementation
4 Back Testing
5 Model Control
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4. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Trade Models
Pricing Models
Overview
1 Model Classes
2 Credit Risk Measures
3 Model Implementation
4 Back Testing
5 Model Control
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5. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Trade Models
Pricing Models
What is a Model?
Definition (Model - Narrow)
A Model is a mathematical framework providing answers to a
specific set of Questions.
Refers only to mathematics
No reference to implementation
No relation to markets and trading activity of institution
Not a useful definition!
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6. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Trade Models
Pricing Models
What is a Model?
Definition (Model - Wider)
A Model provides answers to a specific set of Questions. It consists
of
Information input component
Processing component, applying mathematical
transformations
Reporting component, creating business information
Input
Market
Data
Static
Data
Processing
Mathematical
Model
Reporting
Business
Information
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7. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Trade Models
Pricing Models
Model - Wider Definition
Emphasis on usage
Covers data, software and mathematics
Context of institution, trading activity and market relevant
→ organizational impact for validation: roles and repsonsibilites
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8. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Trade Models
Pricing Models
What is the Question?
Types of Questions
How is the instrument defined
What is the value → pricing model
What will the prices in the future be → risk model
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9. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Trade Models
Pricing Models
What is the Question?
Dependencies
FutureNow
Pricing Model
Sensitivities
State Variables Transactions
VaR PFE
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10. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Trade Models
Pricing Models
Categorization
Types of Statements
Prescriptive Model independent, robust statements
Descriptive Explaining
Predictive Falsifiable
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11. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Trade Models
Pricing Models
Model Independent
Replication
Static Cashflow Replication
Floating leg of swap: replicate by long/short FRA.
Model Dependent Static Replication
Barrier options: static replication dependent of model.
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12. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Trade Models
Pricing Models
Model Independent
Relation
Across trade parameters
Across trade types
Trade Parameters
Price of knock-out option increases with barrier height.
Trade Types
Barrier option is cheaper than a Plain-Vanilla
Replicate Plain-Vanilla by in/out Barriers
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13. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Trade Models
Pricing Models
Trade Model
Requirements
Contractual details as in term sheet need computer-readable
representation:
trade representation
data exchange
auditing
Trade Repository
In US: Dodd-Frank regulation require DTCC data repository
(DDR) as a multi-asset class repository.
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14. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Trade Models
Pricing Models
Pricing Model
Usage
No liquid prices (Mark to Model)
Sensitivities
Valuation under future/hypothetical scenarios
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15. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Trade Models
Pricing Models
Pricing Model
Questions
Implied price w/o credit risk
Implied price w credit risk: CVA/DVA
Implied price range: incomplete markets
Bid/ask price: liquidity
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16. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Trade Models
Pricing Models
Pricing Model
Sensitivities
Which sensitivities should be reported?
Aggregation
How to aggregate vega sensitivities from two systems with
different models?
Lognormal model: σBS
Normal model: σnorm
Model Independence
Report sensitivities wrt. to market observables, i.e. instead
of sigmaBS, σnorm use option prices.
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17. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Types of Measures
Severity
Frequency
Pricing Credit Risk
Overview
1 Model Classes
2 Credit Risk Measures
3 Model Implementation
4 Back Testing
5 Model Control
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18. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Types of Measures
Severity
Frequency
Pricing Credit Risk
Likeliness vs Severity of Credit Events
Categories
Which dimensions to consider?
Severity How much will we lose?
Likeliness What’s the chance that we lose?
Granularity What does the measure refer to?
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19. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Types of Measures
Severity
Frequency
Pricing Credit Risk
Granularity of Measure
Based on Defaults
All Counterparties
Single Counterparty
Other Aggregations
Global/macro economic
Sector, country
Trade
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20. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Types of Measures
Severity
Frequency
Pricing Credit Risk
Exposure and Recovery
How to measure severity? Need to value trade:
Definition (Exposure at Default)
EAD(t) = max 0, p(t)|τ = t
τ : time at which CP defaults
Definition (Loss Given Default)
Loss at time t = LGD(t)EAD(t)
Definition (Recovery)
R(t) = 1 − LGD(t)
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21. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Types of Measures
Severity
Frequency
Pricing Credit Risk
Severity
Valuation Approaches
Accrual Banking book; rarely adjust; illiquid assets
Mark to market Trading book; frequently adjusted; traded assets
Mark to model Trading book; frequently adjusted; complex
structures
Example
CreditRiskMeasures.xlsx
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22. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Types of Measures
Severity
Frequency
Pricing Credit Risk
Severity
Accrual
Loan to Acme Ltd
value is face value
maximal loss is notional of loan
Mark to market
Buy bond of Acme Ltd; assume liquid market
value is mark to market of bond
value lower than in risk-free valuation
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23. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Types of Measures
Severity
Frequency
Pricing Credit Risk
Severity
Mark to model
Exotic interest rate swap with Acme Ltd. What is the value
risk free: assuming Acme may never default
risky: Acme may default
risky with own risk: Acme and we may default
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24. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Types of Measures
Severity
Frequency
Pricing Credit Risk
Forward Looking Measures
Assess exposure in future → model how state of the world evolves
Deterministic Evolution Scenario Analysis, Stress testing
Stochastic Evolution Model for risk factors
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25. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Types of Measures
Severity
Frequency
Pricing Credit Risk
Scenario Analysis/Stress Test
Meanings of Stress
change model parameters →
pick a single path → degenerate measure (Dirac measure)
Unified handling by Measure Transforms
Stochstic Process
(Langevin Equation)
Dual Model Representations
Probabiliy Measure
(Path Integrals)
: t ! x(t) Path
µ( ) ⇠ e S( )
S( ) ⌘
Z ⌧
0
L(x, ˙x) Action
L(x, ˙x) : Lagrangian
˙x = f(x) + ⇠
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26. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Types of Measures
Severity
Frequency
Pricing Credit Risk
Types of Stress Tests
Approaches
give economic scenario
given loss (inverse stress)
Inverse stresses
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27. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Types of Measures
Severity
Frequency
Pricing Credit Risk
Statistical Measures
Single Netting Set
Definition (Potential Future Exposure)
PFE(t) = max 0, p(t)|τ = t
τ : time at which CP defaults
Definition (Expected Exposure (EE))
EE(t) = E[PFE]
Definition (Expected Positive Exposure)
EPE(T) =
1
T
T
0
EE(t) dt
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28. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Types of Measures
Severity
Frequency
Pricing Credit Risk
Regulatory Measures
Definition (Effective Expected Exposure (EEE))
Maximum of EPE and past EEE: never decreasing.
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29. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Types of Measures
Severity
Frequency
Pricing Credit Risk
Statistical Measures
Multiple Netting Set
Definition (Losses across Netting Sets)
L(t) =
a
χτa≤tLGDa max 0, pa(τa)
a : Identifier of netting set
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30. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Types of Measures
Severity
Frequency
Pricing Credit Risk
Portfolio Measures
Meaningful risk measures for portfolios
Definition (Coherent Risk Measure)
Risk measure ρ: for portolio X:
Normalization ρ(∅) = 0 empty portfolio has no risk
Monotonicity X1 ≤ X2 → ρ(X1) ≥ ρ(X2)
Sub-additivity ρ(X1 + X2) ≤ ρ(X1) + ρ(X2) diversification/netting
Homogeneity ρ(αX) = αρ(X) α > 0
Translation invariance ρ(X + a) = ρ(X) − a adding cash a reduces
risk
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31. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Types of Measures
Severity
Frequency
Pricing Credit Risk
Portfolio Measures
Quantile
q% quantile: value, for which q% of outcomes are smaller/larger.
Quantiles are not coherent measures.
Expected Shortfall
Expected loss conditioned on the loss being larget than X. The
Expected Shortfall (Mean Excess Loss) is a coherent measure.
Example
PortfolioMeasure.xlsx
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32. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Types of Measures
Severity
Frequency
Pricing Credit Risk
Likeliness of Default
Example
LikelihoodExperiment.xlsx
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33. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Types of Measures
Severity
Frequency
Pricing Credit Risk
What does probability mean?
Average observers Implied ensemble Genuine ensemble
Probability and Measurement
Need to define
Ensemble
Measurement process
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34. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Types of Measures
Severity
Frequency
Pricing Credit Risk
Examples
Genuine Ensemble
Mathematics
Physics: Identically prepared experiment
Average observers
Consensus of observers:
Market prices
Betting quota
Implied Ensemble
Equivalence classes:
Names with same rating
Price returns in different time windows
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35. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Types of Measures
Severity
Frequency
Pricing Credit Risk
Measures for Probability of Default
Definition (Survival/Default Probability, Default Intensity)
Let τ be time of default
S(t) = p(τ > t)
S : survival probability
S(t) = e−λ(t)t
λ : term default intensity
D(t) = 1 − S(t)
D : default probability
Note: D is a CDF!
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36. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Types of Measures
Severity
Frequency
Pricing Credit Risk
Forward Intensity
Forward default intensity
Probability d(t) of defaulting between t and dt:
d(t) =
dD(t)
dt
(1)
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37. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Types of Measures
Severity
Frequency
Pricing Credit Risk
Estimating Probability of Default
Estimating λ
Credit Rating Typically using historical data
Market Prices Current credit spreads from bonds or CDS
Implied Default Intensity
Let s(t) be a credit spread
s(t) = (1 − R)λ(t)
R : recovery rate
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38. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Types of Measures
Severity
Frequency
Pricing Credit Risk
Unifiying Severity and Frequency Measures
High Severity/Low Frequency vs. Low Severity High Frequency
How to compare
Single large deal with good counterparty
Set of small deals with bad counterparties
Answer
Pricing including credit risk allows comparing!
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39. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Types of Measures
Severity
Frequency
Pricing Credit Risk
Approaches
Top-down vs Bottom-up
Top-down Pricing from first principles
Bottom-up Calculate price correction from building blocks:
Exposure (EE) and PE, LGD
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40. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Types of Measures
Severity
Frequency
Pricing Credit Risk
Bottom-Up approach
Assumptions
Risk-free prices known
Calculate EE
Estimate PE, LGD
Calculate correction to risk-free price
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41. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Types of Measures
Severity
Frequency
Pricing Credit Risk
Measuring the Corrections
Riskiness of counterparty reduces the price:
Definition (CVA)
Risky price p∗
A as seen from counterparty A with counterparty B:
p∗
= p − CVAB
p : risk-free price
CVAB : Credit Valuation Adjustment for counterparty B
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42. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Types of Measures
Severity
Frequency
Pricing Credit Risk
Measuring the Corrections
Does credit risk of counterparty A also affect price?
Definition (DVA)
Price pA as seen from counterparty A with counterparty B:
p∗ = p − CVAB + DVAA
p : risk-free price
DVAA : Debit Valuation Adjustment for counterparty A
DVA increases the price.
Accounting vs. Regulatory
DVA must be used for P&L but not for regulatory capital.
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43. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Types of Measures
Severity
Frequency
Pricing Credit Risk
Regulatory CVA
BCBS 189, paragraph 89:
Regulatory CVA
Similar to regulatory capital charge for default:
Assumes independence of exposure and default process.
CVA =
T
0
(1 − R)Df (t)EE(t)d(t) dt
where d is the default probability from equation (1),Df discount
factor
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44. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Types of Measures
Severity
Frequency
Pricing Credit Risk
CVA
Example
CVA.xlsx
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45. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Types of Measures
Severity
Frequency
Pricing Credit Risk
Regulatory CVA
Regulatory vs Trading CVA
Regulatory Historic measure for EE, implied for PD
Trading Both EE and PD in implied measure
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46. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Types of Measures
Severity
Frequency
Pricing Credit Risk
Pricing for Portfolio of Netting Sets
As for single netting sets: pricing combines severity and likeliness.
Requires knowing
prices of individual netting sets at default
probability of default P(χτ1≤t1 , χτ2≤t2 , . . . , χτN ≤tN
)
Additional useful quantity: in terms of total losses:
Definition (Loss distribution)
L(l, t) = P(L(t) ≥ l)
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47. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Types of Measures
Severity
Frequency
Pricing Credit Risk
Granularity of Measure in Regulatory Context
Metrics used for Regulatory Purposes
Focus on measures for individual counterparties. No proper
modelling of collective losses required.
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48. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Overview
1 Model Classes
2 Credit Risk Measures
3 Model Implementation
4 Back Testing
5 Model Control
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49. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Building Blocks
Model Building Process
Business Analysis Materiality, specification
Model choice Find adequate model
Software implementation Develop and roll out
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50. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Materiality
What to Model?
Which risk factors material for current portfolio?
How can we assess materiality without exposure model in place?
Approach
Simple estimation of exposure assuming
future portfolio prices normally distributed
estimation of first two moments
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51. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Gaussian Approximation
Need to estimate E[p(T)], E[p2(Ti )] at some future times T:
Performing Taylor expansion for price p around expected risk
factor:
p(x(T), T) ≈ p(x0(T), T) +
i
∂p(x0(T), T)
∂xi
∆xi (T)
+
1
2
ij
∂2p(x0(T), T)
∂xi ∂xj
∆xi (T)∆xj (T)
x0(T) ≡ E[x(T)]
∆xi (T) ≡ xi (T) − x0,i (T)
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52. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Gaussian Approximation
The expectation value M of the price is hence
M(T) ≈ p(x0(T), T) +
1
2
ij
γij (T)Ωij (T)
M(T) ≡ E[p(x(T), T)]
γij (T) ≡
∂2p(x0(T), T)
∂xi ∂xj
Ωij (T) ≡ E[∆xi (T)∆xj (T)]
For the variance V we obtain up to second order in ∆x:
V (T) ≈
ij
δi (T)δj (T)Ωij (T)
V (T) ≡ E[(p(x(T), T) − E[p(x(T), T)])2
]
δi (T) ≡
∂p(x0(T), T)
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53. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Gaussian Approximation
What can we learn?
Risk factor contributions
Matrix elements Ψij = δi (T)δj (T)Ωij (T) indicate contribution of
risk factors ij to total variance.
EE, PE
Knowning mean and variance of the Gaussian distribution, any
statistical quantity may be evalued.
Caveat
Depending on specifics of portfolio this approximation may be
more or less accurate: that is why we use Monte Carlo simulations
after all.
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54. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Practical Implementation
For t = 0: δ and γ from Market risk system. But: need
netting set level aggregation → deal level granularity
For t > 0 estimate future δ, γ by bumping
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55. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Trade Models
Requirements
Represent trades/products
Standardize for interoperability
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56. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Trade Models
Trade Parameters
Product represented by parameters
FpML
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57. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Trade Models
Trade Parameters
Pro/Con
⊕ standardized
logic in client
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58. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Trade Models
Cashflows
Product represented by casflows
Payoff macros
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59. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Trade Models
Cashflows
Pro/Con
⊕ simple
not expressive enough (just cash is exchanged)
single product (no interations)
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60. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Trade Models
Transaction Model
Approach
Multi agent simulation:
Time Map wall clock to simulation time
Market Events simulation time to events
Transactions events to transactions (e.g. cashflows)
Execution execute events
Pro/Con
⊕ general
⊕ all business logic in model → easy tooling
expensive
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61. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Pricing & Risk Models
Criteria for Model Choice
Categories
Independent of product Relate to Mathemathics or Physics
Dependent of product Specific to product type
Dependent of portfolio and market Context
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62. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Pricing & Risk Models
Independent of Product
Coordinate Sytems
From Physics we know: dynamics must not depend on choice of
coordinates → dimension analysis.
Interpolation
How to interpolate r, σ. Interpolate dimension-less quantities: rt
and σ2t.
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63. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Pricing & Risk Models
Product Dependent
State Variables vs. Parameters
Liquidity Hedge frequency, transaction costs, close-out period
Completeness Unhedgeable risk, uniqueness of price
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64. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Pricing & Risk Models
State Variables and Parameters
Indicators of Model Quality
Parameter Dimensionality Avoid overparamerization
Stability of Parameters Frequent recalibration: indicator of poor
model performance
GBM w termstructure vs Garch
TS GBM Garch
dimension ∞ 3
recalibration frequently for short end less frequent
time-homogeneous N Y
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65. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Pricing & Risk Models
Arbitrage
Risk Model for Volatility surface
Directly modelling surface w/o arbitrage not trivial. Alternatively
model option prices with HJM-like framework.
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66. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Pricing & Risk Models
Market
Liquidity & Completeness
Liquidity Hedge frequency, transaction costs, close-out period
Completeness Unhedgeable risk, uniqueness of price
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67. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Comparing Models
Assume state of the world evolves randomly:
Model as Process: Stochastic Differential Equation (Langevin
Equation)
dx
dt
= f (x) + g(x)ξ(t) Physics Notation
dx = f (x)dt + g(x)dW(t) Finance Notation
Wiener Process (SDE)
dx = dW (t)
W : Wiener Process
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68. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Comparing Models
Model as Measure P
P : Γ → µ(Γ) probability
Γ : t → x(t) some path
Wiener Process (SDE)
Γ ≡ {x1, . . . xN}
µ(Γ) ∼
i
G(xi , xi+1)
G : Gaussian
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69. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Pricing & Risk Models
Parametric Models
Error Analysis
Infer from parameter uncertainty price/risk uncertainty.
Parameter Uncertainty E.g. such that hedging instrument prices
still in bid-ask
Parameter Error Uncertainty of price/risk due to error in
parameters
GBM with vol uncertainty
(∆p)2 = ∂p
∂σ ∆σ
2
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70. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Benchmarking
Pricing/risk factor models Q, Q , empirical measure P
Comparing
Pricing Models Q vs Q
Risk Models P vs Q
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71. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Benchmarking
Distances
How far apart two models?
Need to define metric:
Expectation values E.g. differences of prices and EEs under
different measures
Distributions E.g. Kullback-Leibler entropy dP
P log dP
P dP.
Independent of quantity to average.
c 2015 H¨aner Consulting Model Risk 71 / 166
72. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Model Uncertainty
Benchmarking giving limited answer:
Calibration-Consistent Measures
Define metric d to quantify goodness of calibration:
pP
i : model price calibration instrument i
pi : market price calibration instrument i
dP
=
i
(pP
i − pi )2
C = {P|dP
≤ }
c 2015 H¨aner Consulting Model Risk 72 / 166
73. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Model Uncertainty
Non-uniqueness
For d = 0:
Multiple measures
For single parametric measure, multiple solutions for
calibration → ill behaved
Incomplete market
For d > 0:
For single parametric measure: parameter risk
c 2015 H¨aner Consulting Model Risk 73 / 166
74. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Beyond Benchmarking
Pricing model descriptive:
Replicates prices of hedging instruments
Determines no-arbitrage price of illiquit product
How to asses quality of model?
There are implied predictions:
State variables vs parameters Prediction: parameters are constant
Martingale Total price of deal and self-financing hedges should
be 0 at any point in time
c 2015 H¨aner Consulting Model Risk 74 / 166
75. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
State variables and parameters
State variables Temporal evolution or measure
Parameters Family of evolutions/measures
Analysis
Choice of state variables: qualitative assessment
Robustness of parameters: predicted are no changes
c 2015 H¨aner Consulting Model Risk 75 / 166
76. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Hedge Performance
If perfectly hedged: pathwise replication → P&L distribution
Unbiased
Sharply peaked (Dirac)
Hedge Simulations
Self Consistency Use state variables simulated with pricing model
Performance Historical state variables
c 2015 H¨aner Consulting Model Risk 76 / 166
77. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Exposure
Goal
Estimate Credit Risk measures → need to estimate exposure/price
distributions in future.
The exposure e(t) at time t of a netting set is given by
e(t) = max 0,
i
pi (x, t) − C(t) (2)
where
pi price of trade i
x risk factors
C(t) price of collateral
c 2015 H¨aner Consulting Model Risk 77 / 166
78. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Calculating Exposure, CVA/DVA and Losses
Risk Factors&Counterparty Default Times
t
x
t
x
RT1 RT2
Risk Factors
Trades
t
x
t
x
R1 R2
Collaterals
Portfolio Prices
t
p
t
p
P1 P2
Collateral Prices
t
c
t
c
C1 C2
PDF of
Exposures
Default Times of
Counterparties
PDF of
Exposures
at Default
Expected
Exposure
Potential
Future
Exposure
Bootom-up
CVA/DVA
Top-down
CVA/DVA
c 2015 H¨aner Consulting Model Risk 78 / 166
79. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Building Blocks
Components
Required for estimating risk measures for single and portfolios of
netting sets:
Pricing
Risk-factor
Collateral
Netting
Dependency
c 2015 H¨aner Consulting Model Risk 79 / 166
80. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Pricing Models
Requirements
Need to be fast!
Ideally same as front office
Perform well under stressed state variables
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81. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Pricing Models
Acceleration Techniques
Dumb lookup
Approximate price as function of few variables
define variables (e.g stock price)
define grid
recaluclate for each gridpoint price
interpolate
c 2015 H¨aner Consulting Model Risk 81 / 166
82. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Pricing Models
Acceleration Techniques
Smart lookup
Approximate price as function of few variables
define variables (e.g stock price)
prices on grid are side effect of pricing at spot; e.g. pricing on
tree or AMC
interpolate
c 2015 H¨aner Consulting Model Risk 82 / 166
83. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Risk Factor Models
Pricing vs Risk Models
Purpose
Pricing Model Fit liquid market instruments; arbitrage-free
Risk Model Predict
Challenges for Risk Model
Dependency Simultaneously simulate all asset classes
Calibrationl Global calibration
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84. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Risk Factor Models
Short vs Long term prediction
Long term prediction a challenge:
Reducing dimensionality
Economic macro factors
Co-integration
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85. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Risk Factor Models
Pricing model Dynamics
Arbitrage-free models used with risk calibration
GBM
HJM type of models
⊕ Well understood, tractable
Not intended for risk
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86. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Gaussian Dependency Modelling
Goal
Express random vector ξ with correlated ξi as
linear combination of
uncorrelated
random factors ηi :
ξ = Mη
E[ξi ξj ] − E[ξi ]E[ξj ] ≡ Ωij
E[ηi ηj ] − E[ηi ]E[ηj ] = λ2
i δij diagonal, pos. sem. def.
What to consider?
Ω?
correlation matrix?
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87. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Principal Component Analysis
Dimensional Analysis
Risk factors ξi not dimension-less!
interest rate :[T−1]
stock price :[Cash]
volatility: [T−1
2 ]
→ Ωij may have different dimensions,i.e. Ω in general not a
physically meaningful quantity!
c 2015 H¨aner Consulting Model Risk 87 / 166
88. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Principal Component Analysis
Solution
Consider instead of Ω following matrix Φ:
Φij ≡
∂f (ξ)
∂ξi
∂f (ξ)
∂ξj
Ωij
f : some function
For dimensionality [Φ]:
[Φij ] =
[f ]
[ξi ]
[f ]
[ξj ]
[ξi ][ξj ] = [f 2
] ∀i, j (3)
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89. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
GBM Risk Factor Model
Multivariate GBM
Xi (t + ∆t) = Xi e(µi −1
2
σi )∆t+σi
√
∆tξi (t)
µ : drift
σ volatility
ξi : Normal random
Cov(ln Xi (t + ∆t), ln Xj (t + ∆t)) = Ωij
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90. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Dependent Gaussian Random Variables
Given uncorrelated Gaussian random number vector ζ. Need
build η:
Cov(ηi , ηj ) = Ωij
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91. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Calibration
Definition
Calibration is the process to determine model parameters.
Approaches
Statistical Using historical data
Implied Market implied parameters
Economic Macro economical relation between rates, infaltion
Asumptions
Statistical Past is good predictor for future
Implied Information in spot market predicts future
Economic Some fundamental economic laws rule future
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92. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Statistical Calibration
For simple models: ad hoc parameter estimation
averaging
fitting
Example
SimpleEstimation.xls
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93. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Maximum Likelihood Estimation
Systematic way to calibrate
Approach
Parametric model with parameters α ↔ parametric measure µα:
µα(Γ) = e−Sα(Γ)
D[Γ]
Assume: historical path ΓH is the most likely one. Find α∗ such
that:
µα∗ (ΓH) = max
α
µα(ΓH)
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94. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Maximum Likelihood Estimation
Implementation
Assuming iid:
µα(Γ) = m(xi )
m(x) = e−s(x)
Γ = {x1, . . . , xn}
Maximizing m ↔ minimizing
i
s(xi ) : log-likelihood
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95. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Maximum Likelihood Estimation
Example
MLE.xls
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96. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Implied Parameters
Apply parameters used for pricing:
Drift and Volatility
Drift µ from T forward price (Covered Parity)
Volatility σ T years ATM implied volatility
Assumption
Risk neutral measure yield good predictor for real-world measure
Caveat
Carry trades
Supply/demand, risk premium
Perform analysis before using implied parameters!
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97. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Economic Calibration
Parities connect for instance
FX rates
Inflation rates
Real interest rates
Nominal interest rates
Purchansing power
Example
Parities.xlsx
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98. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Parities
Example (Relative Purchasing Power Parity)
pf (t1)(1 + if )X(t2) = pd (t2)(1 + id )
pd/f : domestic/foreign price
id/f : domestic/foreign 1 yr inflation rate
X : Exchange rate
Yields after averaging
E[X(t2)]
X(t1)
=
1 + Id
1 + If
where I is the expected inflation rate.
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99. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Parities
Example (International Fisher Effect (Uncovered Parity))
(1 + rd/f ) = (1 + ρd/f )(1 + id/f )
rd/f : domestic/foreign nominal 1 yr interest rate
ρd/f : real rdomestic/foreign 1 yr interest rate
Assumingρd = ρf gives
E[X(t2)]
X(t1)
=
1 + rd
1 + rf
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100. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Issues with standard GBM model
Issues
rigidity: calibration short vs long horizons → term structure of
parameters
dimensionality → factor models
underestimation of rare events and bursts (clustering) →
GARCH
not suitable where spread stationary process → cointegration
unable to capture some behabiour like regime-switches →
parametric models (Nelson-Siegel)
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101. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
GBM with Term Structure
Interpolation Principles
Interpolate dimension-less quantities
Forward Drift/Covariance
Dimensionality analysis → interpolate TΩ
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102. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Factor Models
Issues with general covariance matrix
N risk factors →∝ N2 parameters
over-parametrization
for empirical parameters: problems with positive definiteness
Idea
Split return r of riskfactors into contributions from
Indices fn shared by multiple risk factors
Idiosycratic factors unique to each risk factor
r = α +
n
βnfn +
and assume
indices uncorrelated to indosyncraticsc 2015 H¨aner Consulting Model Risk 102 / 166
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Model Implementation
Back Testing
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Analysis
Models
Software Development
Types of Factor Models
Classification
Macroeconomic Observables like changes in inflation, interest rate,
unemployment rate
Fundamental Portfolios associated to security attributes like
industry membership, book to market ratio, dividends
Statistical Factor analysis of covariance matrix
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104. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Macroeconomic Factor Model
Fast/Slow
Slow variables Macro-economic state of the economy: inflation,
unemployment rate, GDP
Fast Asset prices
Pros and Cons
⊕ Designed to predict long-term evolution
⊕ Able to reflect systemic macro risks
Empirical evidence not convincing
Theories controversial
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105. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Fundamental Factor Model
Sector/Region
1 Define for each sector/region pair an index
2 Associate stock to sector/region
3 Regress stock return vs index return → α, β
Example
FactorModel.xls
Pros and Cons
⊕ Designed to predict long-term evolution
⊕ Able to reflect systemic macro risks
Empirical evidence not convincing
Theories controversialc 2015 H¨aner Consulting Model Risk 105 / 166
106. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Choice of Factors
How to know whether factors appropriate?
Analyze variance explained by factors
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107. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Volatility Clustering
(a) Spot (b) Log-returns
Figure : GBPUSD spot
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108. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Autocorrelation
(a) Autocorrelation: log-returns
(b) Autocorrelation: squared
log-returnsc 2015 H¨aner Consulting Model Risk 108 / 166
109. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Garch Model
Let Xn be the log-return of some foreign exchange rate f at
time tn:
Xn = ln
fn
fn−1
(4)
we may then express the foreign exchange rate fN at some future
sampling point time tN by the initial value f0 at t0 and a series of
returns:
fN = f0e
N
i=1 Xi
(5)
The observation points ti are typically defined in terms of number
of business days ∆T between them. For short time horizon
predictions we choose ∆T = 1 for larger horizon, we may choose a
less granular time grid.
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110. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Garch Model
The dynamics of the returns is then assumed to follow a
Garch(1,1) process
Xn = µ + n t ∼ iid(0, σ2
n) (6)
σ2
n+1 = α + βσ2
n + γ 2
n (7)
The asymptotic value σ2
∞ = limn→∞ E[σ2
n] is then obtained by
equation (7) noting, that E[ 2] = σ2 and E[σ2
n+1] → E[σ2
n]:
σ∞ =
α
1 − β − γ
(8)
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111. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Garch Model: Limit
Weak limit:
Stochastic variance
Mean reverting variance
dXt = µXtdt +
√
vtXtdWt
dvt = α(vt)dt + β(vt)dZt
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112. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Copula
Dependence under Stress
In stressed markets correlations increase between
downward price movements → systematic risk
implied default probabilities → contagion
Definition (Copula)
Separate
Marginal distributions from
Dependency
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113. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Cointegration
Long-run Relationship
Variables moving together:
Macro-economic Consumption-Income
Prices-Wages
Domestic prices - fpreign prices
Exogeneous For instance managed currencies
How to model processes. which stay close to each other?
GBM with ρij 1 not? No!
Need dynamic, where difference is stationary
Definition
Stochastic processes x, y are cointegrated:
y(t) = a + bx(t) + ξ(t)c 2015 H¨aner Consulting Model Risk 113 / 166
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Models
Software Development
Implementation
1 find parameters a, b by regression
2 show residuals are stationary (e.g. Dickey-Fuller Test)
Example
Cointegration.xlsx
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115. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Risk Factor Models
Empirical Models
Nelson-Siegel model
r(T) = r∞ + a(T)r0 + b(T)rm
r∞ : rate for long maturities
r0 : rate for short maturities
rm : rate for intermediate maturities
a, b : decay functions
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116. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Risk Factor Models
Empirical Models
Nelson-Siegel model
Normal/inverted curves
But not arbitrage-free
How to introduce dynamics? E.g. PCA of (r∞, r0, rm)
Example
NelsonSiegel.xlsm
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117. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Wrong Way Risk
Types
Specific Legal connection between underlying and
counterparty
General Dependence between prob. of default of counterparty
and exposure
SFT Transactions
Lend cash to counterparty A accepting their stock as collateral.
Emerging Market CCY swap
We are long strong currency. Weakening of emerging market
currency, increased prob default → increase exposure
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118. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Modelling Wrong Way Risk
What is wrong with standard modelling?
p+ is not conditioned on default.
Need to add in price function default state χ of counterparty:
extending state of the world
Approaches
Given a model for default times either
Simulating counterparty’s default
Calculating price given default
Example
WrongWayRisk.xls
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119. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Collateral Modeling
Components
Margin Call Process Model margin calls with correct frequency
and close-out period
Collateral Price E.g. model bond price if collateral is bond
Simplification
Margin call process: just at spot → short-cut method
All collateral as cash → haircuts
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120. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Collateral Modeling
Short-Cut Method
Definition (Basel II Short-Cut Method)
EE and PE of collateralized trades given by EE and PE for
close-out period (5 days for SFT, 10d for OTC)
Benefits/Issues
⊕ Computationally cheap
⊕ No collateral exposure spikes at expity
Assumes exposures declining over time
Risk not accurately represented
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121. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Dependency Modelling
Among Risk Factors
Standard way to model dependence: Gaussian Copula.
Gaussian Copulas are Levy copulas. Replace Gaussian with other
Levy coupula and obtain Levy model.
Between Defaults
Simulate either
Default times τ E.g. by Marshall-Olkin Copulas
Default state at t:χτ≤t E.g. structural models
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122. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Dependency Modelling
Between a Default and Risk Factors
To caputure Wrong Way risk need to model dependence between
risk factor and default state
Example
WrongWayRisk.xls
Between a cross name Defaults and Risk Factors
Need modelling full state of the world (x(t), {χτ1≤t, . . . χτ1≤t}).
→ scenario consistency is system
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123. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Model Lifecycle
Organisation Execution
Problem
Definition
Analysis Implementation
Test
Deployment
MaintananceChanges
Figure : Model Development Lifcecyle
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124. Model Classes
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Back Testing
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Analysis
Models
Software Development
Specification
Approaches
Human readable Business and functional specs
Machine readable Specification ∼ test
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125. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Specification
Tools
ScalaTest Code
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126. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Specification
Tools
ScalaTest Output
Part of CI:
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127. Model Classes
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Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Implementation
Software
in-house
third-party
Require different validation strategies
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128. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Third Party
Strategies
Black-box, no code review
Reverse-engineering
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129. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Revision Control
Requirements
Audit Who changed what/when
Resurrect Roll-back to previous state
Collaborate Merge contributions from different authors
Approaches
Plain files Tag files/directories with version information
Local Local database contains version information (e.g
RCS)
Server Database on server (e.g. SVN)
Distributed Each developer has own databse with potentially
central db (e.g. Git)
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130. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Revison Control Tools
Approaches
MyDirectoryV1.0
MyDirectoryV1.1
MyDirectoryV1.2-bugfix1
(a) File based (b) Local VCS
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131. Model Classes
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Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Revison Control Tools
Approaches
(a) Centralized VCS (b) Distributed VCS
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Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Revision Control Tools
Git
Figure : Git Gui (SourceTree)
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133. Model Classes
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Model Implementation
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Model Control
Analysis
Models
Software Development
Documentation
Requirement
Contain enough information to reverse-engineer.
Tools
Automated API doc (Doxygen, ScalaDoc, . . .)
Internal wiki (e.g. Confluence)
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134. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Testing
Test Types
Unit Library level
Integration System level
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135. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Testing
Unit Test
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136. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Release
Requirements
Regression
Impact analysis
Sign-off
Auditing
Lock-down
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137. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Maintance
Bugs/Enhanements
Tracking system
Failing test cases
Metrics: severity, resolution time
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138. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Analysis
Models
Software Development
Integrated Development Process
Robust system should have
Components
Revsion Control system
Build System
Bug tracking system
Wikin
Components integrated to workflow with high degree of
automation
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139. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Regulatory Requirements
Measuring Model Performance
Overview
1 Model Classes
2 Credit Risk Measures
3 Model Implementation
4 Back Testing
5 Model Control
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140. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Regulatory Requirements
Measuring Model Performance
Motivation
Impact of Credit risk model
Trading activity limits set by PE
Capital charges regularity capital dependent of EEPE
P&L EE enters CVA/DVA
Model Risk
Back-testing should quantify model risk affecting these quantities.
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141. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Regulatory Requirements
Measuring Model Performance
Requirements
Back-testing Process
Should provide
Definition of measure for model risk
Monitoring of metrics
Mitigating actions for model deficiencies
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142. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Regulatory Requirements
Measuring Model Performance
BCBS Guidance
G1
G2
G3
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143. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Regulatory Requirements
Measuring Model Performance
BCBS Guidance
G4
G5
G6
c 2015 H¨aner Consulting Model Risk 143 / 166
144. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Regulatory Requirements
Measuring Model Performance
BCBS Guidance
G7
G8
G9
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145. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Regulatory Requirements
Measuring Model Performance
BCBS Guidance
G10
G11
G12
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146. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Regulatory Requirements
Measuring Model Performance
BCBS Guidance
G13
G14
G15
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147. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Regulatory Requirements
Measuring Model Performance
BCBS Guidance
G16
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148. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Regulatory Requirements
Measuring Model Performance
What is the Question?
Types of Investigation
Hypothesis testing (Answer in percentage or yes/no)
Estimation of model uncertainty (Answer in cash terms)
Analysis at different levels: figure 7
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149. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Regulatory Requirements
Measuring Model Performance
Domains
Economic Quantities
Regulatory
Capital
Limits CVA/DVA
Risk Measures
EEPE PE EE
Process Characterictics
Marginal
Distributions
Auto-Correlations N-Point Functions
Model-Dependent Quantities
Model Parameters Driver dynamic
Figure : Domains
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150. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Regulatory Requirements
Measuring Model Performance
Definition
A model is represented by a measure Q.
May be generated by a stochastic process.
Quantifying Difference of Models
Comparing expectation values
Comparing probability distributions
Note: PDFs and CDFs may be expressed as expectation values
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151. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Regulatory Requirements
Measuring Model Performance
Radon-Nikodym Derivative
Distance of model Q and end empirical measure P in terms of dP
dQ:
EP[f ] = EQ[
dP
dQ
f ] (9)
Compare P and Q
Direct dP
dQ ≈ id?
Expectation values Empirical expectation measures in terms of
model expectations
Relative Entropy Kullback-Leibler entropy → information
geometry (see [?])
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152. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Regulatory Requirements
Measuring Model Performance
Radon-Nikodym Derivative
Let ξ be a scalar stochastic variable (e.g. portfolio price π(t))
Definition
P empirical, Q model CDF
Ψ : [0, 1] → [0, 1] (10)
Ψ(α) = P(Q−1
(α)) (11)
Radon-Nikodym derivative ψ
EP[f ] = EQ[ψ(α)f ] (12)
ψ(α) =
dΨ(α)
dα
(13)
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153. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Regulatory Requirements
Measuring Model Performance
Example
(a)
0.0 0.2 0.4 0.6 0.8 1.0
α
0.0
0.2
0.4
0.6
0.8
1.0
Ψ(α)
[x]=100.00;σ=0.40
[x]=110.00;σ=0.40
[x]=90.00;σ=0.40
[x]=100.00;σ=0.44
[x]=100.00;σ=0.36
0.8
1.0
1.2
1.4
1.6
1.8
ψ(α)
[x]=100.00;σ=0.40
[x]=110.00;σ=0.40
[x]=90.00;σ=0.40
[x]=100.00;σ=0.44
[x]=100.00;σ=0.36
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154. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Regulatory Requirements
Measuring Model Performance
Cumulative Distribution Functions
Cumulative distribution function (CDF) for some state variable ξ
expressed as expectation:
Definition
P(ξ0) = EP[Θ(ξ − ξ0)] (14)
where Θ is the Heaviside function.
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155. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Regulatory Requirements
Measuring Model Performance
Estimating
Ensemble averages E estimated well by time averages if
ergodic
stationary
CDF
P(ξ0) ≈
1
N
N
i=1
Θ(ξ(ti ) − ξ0) (15)
Ψ
Ψ(α) ≈
1
N
N
i=1
Θ(ξ(ti ) − Q−1
(α)) (16)
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156. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Regulatory Requirements
Measuring Model Performance
Requirements for Estimation
Process neeeds to be
ergodic
stationary
iid price process
If empirical price process is iid, the ergodic.
iid process of underlying
Even if underlying process the price return process of the deal may
not be so, if deal not time homogeneus
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157. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Regulatory Requirements
Measuring Model Performance
Distances
Point Distance
di = |Ψ(qi ) − qi | (17)
Curve Distance
(Weighted) quadratic distance d between functions q → Ψ(q)
and q → q:
d(q, Ψ(q)) =
i
wi (Ψ(qi ) − qi )2
(18)
qi e.g (0.01, 0.05, 0.3, 0.5, 0.7, 0.95, 0.99)
c 2015 H¨aner Consulting Model Risk 157 / 166
158. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Regulatory Requirements
Measuring Model Performance
Hypothesis Testing
Null-Hypothesis
Null-Hypothesis, is that distances are 0.
Reject Null-Hypothesis p-values smaller than some threshold
Challenges estimating p-values
Temporal dependence: overlap of time-windows
Ensemble dependence: returns of netting sets not independent
Good p values get bigger
Bad Estimation tricky
Need some simplifications, like effective sample sizes
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159. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Regulatory Requirements
Measuring Model Performance
Problems using metrics for Ψ
Issues using metrics for Ψ
Opaque no cash denominated measure
Economics Product Dependent with same distance different
moments drive deviations in EE (see figure (9))
Limited usefulness Passes test if not enough data available
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160. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Regulatory Requirements
Measuring Model Performance
Problems using metrics for Ψ
20 40 60 80 100 120 140 160
strike K
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
EE
EE
[x]=100.00;σ=0.40
[x]=110.00;σ=0.40
[x]=90.00;σ=0.40
[x]=100.00;σ=0.44
[x]=100.00;σ=0.36
Figure : Comparing EEs for a forward using log-normal distributions with
different parameters
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161. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Regulatory Requirements
Measuring Model Performance
Comparison using Cash denominated Quantities
Economically Relevant Model Dependent Quantities
Regulatory Capital depends on EE(t) (through EEPE)
Limits impacted by CDF
P&L impacted by EE(t)
Measure
These three quantities are functions of EQ.
Their value under empirical measure P estimated through
equation (12) → difference in cash terms
c 2015 H¨aner Consulting Model Risk 161 / 166
162. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Overview
1 Model Classes
2 Credit Risk Measures
3 Model Implementation
4 Back Testing
5 Model Control
c 2015 H¨aner Consulting Model Risk 162 / 166
163. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Requirements
Control Processes should be
Complete
Accurate
Consistent
Timely
Appropriate and Relevant
Auditable
c 2015 H¨aner Consulting Model Risk 163 / 166
164. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Governance
Board of directors and senior management to establish a
strong model risk framework
Roles and responsibilities: clear reporting lines and incentives,
address conflicts of interest, sufficient authority to control staff
Firmwide model inventory should model use, products,
responsible parties and planned activities
Detailed documentation: understand how the model operates,
limitations and key assumptions
Developers, users, control and compliance units should
document their work including ongoing monitoring,
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165. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Audit
Review policies and compliance
Confirm and if appropriate challange validation work
Review records of model use, and confirm models are subject
to controls, and also account for limitations
Verify accuracy and completeness of the model inventory
Assess the process for establishing and monitoring limits and
usage
Asessments of operational systems and evaluate the reliability
of data used by models
Report findings to the board
c 2015 H¨aner Consulting Model Risk 165 / 166
166. Model Classes
Credit Risk Measures
Model Implementation
Back Testing
Model Control
Policies
Model Validation
Model Development
IT
Data Management
Backtesting
Stresstesting
c 2015 H¨aner Consulting Model Risk 166 / 166