Applied Mathematics in Banking & Finance Mathematics in Banking & Finance ... “Risk is compensated vis-à-vis business. ... Pricing / Hedging vis-à-vis Credit Derivatives Desk CVA

Applied Mathematics in Banking & Finance

– Quantitative Models & Enterprise Analytics –

ดร.พูมใจ นาคสกุล

Poomjai Nacaskul, PhD, DIC, CFA

{FSVP, Quantitative Models & Enterprise Analytics, Siam Commercial Bank PLC;

มหาวิทยาลัยเทคโนโลยีมหานคร}

8 May 2013

Applied Math in Fin & Bank – Quantitative Models & Enterprise Analytics

Talk Agenda

I. Intro, a bit of 'philosophy' what exactly do we mean by (a) “applied” (b) “mathematics”, (c) “finance”, and (d) “banking”?

II. The 'Where' & 'How' (i) Financial Risk – Nonlinear Discriminant Analysis,

Probability Distribution, Stochastic Process, Extreme Value Theory, Copula Dependency

(ii) Financial Derivatives – Stochastic Differential Equation,

Equivalent Martingale Measure, Monte Carlo Simulation

(iii) Banking Business – Mathematical Programming, Evolutionary

Optimisation/Algorithms, Queueing Theory, Time-Series Forecast, Quantitative Analytics, Heuristics

(iv) Banking System – Network Centrality-Criticality Analysis,

Econometrics, Shock Propagation, Agent-Driven Simulation

III. The 'What' & 'Who' – psst … I am recruiting! QA: Research/Prototype/Development/Validation/Engineering

Intro

…

…

Applied Math in Fin & Bank – Quantitative Models & Enterprise Analytics

Mathematics • Definition Consequence Completeness

• Language for (re)presenting 'a/some' reality

Applied • 'beginning' of being 'useful' to 'people'

• this conundrum: can you apply the '+' operation to making a cup of coffee?

'Quant Analytics'

• (I/O Map.) Model

• Optimisation/Control

• Analytics, Diagnostics

• Visualisation/Informatics

…

Where & How

…

Applied Math in Fin & Bank – on Banking, Finance & Risks

Banking (as financial intermediary): how a monetary society fundamentally finances itself, i.e. raising money, principally by way of household (net savers’) deposits, in turn, contractually lending to those with more immediate financing needs

Finance: to raise money to do something, perhaps consume, perhaps make more money, i.e. promising to pay back and/or share the profits (and of course losses!)

Risk: Possibility + Probability + Preference Knightian Uncertainty + Notion of (Probabilistic) Likelihood

{{Set, Sigma Algebra }, Probability Measure } + Utility U, [ref: Von-Neumann & Morgenstern (1947)’s Utility Theory]

Applied Math in Fin & Bank – on Market, Credit & Operational Risks

Market Risk

• opportunity/possibility & probability of financially relevant gains/losses due to ‘movements’ of the financial-market and/or monetary-economic variables, namely interest/exchange rates, equity/commodity prices, etc. – “Risk is business.”

Credit Risk

• opportunity/possibility & probability of financially relevant losses (occasionally gains) due to ‘credit events’: obligor default, recovery, drawdown risks (respectively, PD, LGD, EAD); counterparty/settlement risks; rating-downgrade risks; credit derivatives risk, i.e. CDS (Credit Default Swaps), CDO (Collateralized Debt Obligations), etc. – “Risk is compensated vis-à-vis business.”

Operational Risk

• opportunity/possibility & probability of (partially) preventable occurrences of failures, errors, frauds, together with noncircumventable events in the form of random accidents, natural catastrophes, man-made disasters, whence resulting in material losses, disruptions, and/or various infractions, thereby severely and

adversely impacting financial condition, business conduct, and institutional integrity overall – “Risk just for being in business.”

Applied Math in Fin & Bank – Risk Management vs. Business Process

Identify (10%)

Measure (60%)

Mitigate (20%)

Report (10%)

Decide (10%)

Monitor (20%)

Market (10%)

Analyse (60%)

Applied Math in Fin & Bank – Upstream vs. Downstream Risk Analytics

Business

Model

Risk

Strategy

Credit

Decision

<< upstream analytics

downstream analytics >>

Risk

Measurement

Capital

Adequacy

Regulatory

Compliance

Nacaskul. Poomjai (2006), “Survey of Credit Risk Models in Relation to Capital Adequacy Framework for Financial Institutions”,

http://www.bot.or.th/English/FinancialInstitutions/New_Publications/QMFE/Folder1/Pages/Research-Publication-Presentation.aspx

Applied Math in Fin & Bank – Modelling Market-Credit-Operational Risks

• Old: Multivariate Normal Distribution N(,) (1952,29) Modern

Portfolio Theory Quadratic Programming, i.e. min w'w s.t. 'w {}; Value-at-Risk quartile, Expected Shortfall conditional expectation

• New: heavy-tailed, if still elliptical family, i.e. Multivariate Student; discrete jumps; asymmetric correlation, extreme co-movement…

Market Risk

• Old: Logistic Regression; Markov Chain; Asset Value Model

• New: Nonlinear Discriminant Analysis (see next); Markov Process; Default Intensity; Default 'Correlation' Copula, Distributional Mixture, Beta Distribution, i.e. for Bayesian PD prior, empirical LGD estimate…

Credit Risk

• Old is New: Cramer(1936)-Lundberg(1903) Ruin Theory (Actuarial Science), Compound Poisson Process, i.e. N-fold Convolution, where N is Poisson.

• New: Extreme Value Theory, Copula-EVT…

Operational Risk

Applied Math in Fin & Bank – Nonlinear Discriminant Analysis

Applied Math in Fin & Bank – Copula

Constructing a Gaussian Slug copula requires modification only w.r.t. the bivariate integrand:

)( )(

2

673/5

2

362

)( )(

.

11

1 1

1 1

2

12

2exp

),(

),(),(),,(

v u

v u

std

XY

XY

SlugGaussian

dtds

stts

dtdstsg

vuGvuC

(1)

From which the corresponding Gaussian Slug copula density is given by:

311

11.

673/2

)()(

),(),(2),,(

vfuf

vugvuc

YX

std

XY

antconstingnormalis

SlugGaussian

(2)

Figure 1: Standard Gaussian vs. ‘Gaussian Slug’ Copula Density – 3D Plots

Figure 2: Standard Gaussian vs. ‘Gaussian Slug’ Copula Density – Contour Plots

Nacaskul, P. & Sabborriboon, W.(2009) “Gaussian Slug – Simple

Nonlinearity Enhancement to the 1-Factor and Gaussian Copula

Models in Finance, with Parametric Estimation and Goodness-of-Fit

Tests on US and Thai Equity Data”, 22nd Australasian Finance and

Banking Conference, 16th-18th December, Sydney, Australia,

[http://papers.ssrn.com/abstract=1460576].

http://papers.ssrn.com/abstract=1460576


Applied Math in Fin & Bank – Financial Engineering (Derivatives Pricing)

3) A riskless portfolio should earn/grow as much as a money-market account.

Black-Scholes PDE (1976)

2) Assume continuous ‘Delta-neutral’ hedging possible.

Ito’s Lemma

1) From underlying asset, create a financial derivative.

Geometric Brownian Motion

2

222

2

1

S

CS

S

CSrCr

t

C

dtS

CSdS

S

Cdt

t

CdC

2

222

2

1

dWdtS

dS


Ito calculus – how do we do ‘Financial Engineering’? – Financial Derivatives:

>>> In the simplest form: given (observable) S0 and (contractual) CT = maxS0 – K, 0, C0 = ?

– Black & Scholes (1972): 1. continuous hedge ratio deltat = CtSt

2. riskless portfolio t = Ct – tSt should ‘grow’ like a money mkt account, Bt = B0e r t

3. St is assumed to follow a Geometric Brownian Motion (GBM)

4. Apply Ito’s Lemma expression for dCt

5. Altogether Black-Scholes PDE, a parabolic equation, as per Heat Diffusion

6. Apply Green’s function, boundary condition the famous ‘Black-Scholes formula’

– From which: 7. Note: entire edifice singly parameterised by the volatility parameter

8. Note: with Bt as numeraire, Ct – tStBt then becomes essentially a margingale

9. This connection is encapsulated quite elegantly by way of the Feynman-Kac

formula

10. Harrison & Kreps (1979); Harrison & Pliska (1981) Equivalent Martingale

Measure (EMM)

Applied Math in Fin & Bank – Method of Equivalent Martingale Measure

http://www.bot.or.th/English/FinancialInstitutions/New_Publications/QMFE/Folder1/Documents/BOT-QMFE-FinancialMathematicsFoundation-32.pdf


Dynamics

From GBM to Jump Diffusion to Lévy Process

From Multivariate Normal to Heavy-Tailed Distributions to Extreme Value Theory (EVT)

Calibration

From Spot Rate to Forward Rate to

Market Model

From scalar parameter to Volatility Surface

Hedging Static vs. Dynamic Hedging

Analytical Sensitivities vs. Monte Carlo Simulation

Applied Math in Fin & Bank – Evolutionary Optimisation/Algorithms

Nacaskul, Poomjai (1997), “Phenotype-Object Programming & Phenotype-Array Datatype: an Evolutionary Combinatorial-Parametric FX Trading Model”,

Proceedings of the 1997 International Conference on Neural Information Processing (ICONIP’97), Dunedin, New Zealand, [Singapore: Springer-Verlag].

Applied Math in Fin & Bank – Systemically Important Financial Inst.

Nacaskul, Poomjai (2012), “Systemic Importance Analysis (SIA)

– Application of Entropic Eigenvector Centrality (EEC)

Criterion for a Priori Ranking of Financial Institutions in Terms

of Regulatory-Supervisory Concern”, Bank for International

Settlements (BIS) Asian Research Financial Stability Network

Workshop, 29th March, Bank Negara Malaysia, Kuala Lumpur,

Malaysia, [http://papers.ssrn.com/abstract=1618693].

)(

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SIAAnalysisancemportISystemic

SVAAnalysisnerabilityVulSystemic

nerabilityvul

systemicequal

ityparticular

networkplus

ionconcentrat

volume

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tyconnectivi

anceimport

systemicequal

effect

networkplus

size

relative

captureswhich

objectscalar

captureswhich

objectscalar

captureswhich

objectscalar

objectvectorreigenvectoprincipal

objectmatrixsumrow

objectvector

captureswhichcaptureswhich

C

svsv

sv



…

…

What & Who

Quantitative Models & Enterprise Analytics (QMEA)

Mandate/ Modalities

I. Survey/Target – where/how (identify what is needed) to build (from zero), elevate (from working) and/or reboot (from stalled) quantitative models & enterprise analytics.

[priority: oppndatatechppl; buildrebootelevate]

II. Learn/Build – (i) survey the frontier build the capacity, (ii) map the problems match the complexity, (iii) acquire the technologies master the components.

[priority: techppldataoppn][sequencing: (i)(ii)(iii)]

III. Prototype/Perform –

Data Engineering track

Stochastic Modelling track

System Optimisation track

Enterprise Informatics track

Mathematical Finance track

OPPN

DATA

PPL

TECH

Standard Solution Problem

Mapping

Knowledge Application Knowledge

Acquisition

Technical Resolution Numerical

Experimentation

Resource Loading/Team Expertise

Resource Pooling/Domain

Expertise

Quant. Model. & Enterprise Analytics – Data Engineering

Engine Data Mining –

Nonlinear Regression Analysis

Data Mining – Unsupervised

Cluster Analysis

Machine Learning – Nonlinear Discriminant

Analysis

Machine Learning – Feature Detection, Pattern Recognition

Semi-/Unstructured Database Mapping

Application

Client-Business Rating / Scoring

Client-Profile Classification / Segmentation

Financial Time Series / Market Variable Prediction

‘Model-Ensemble’

Fraud Detection, Statistical Irregularity, Control Failure

Data Cleansing/Validation, ‘Big Data’ Analytics

Upshot

Higher Discriminatory Power

Better Product-Customer Marketing

Better Accuracy, Tighter Precision

Earlier Action, More Options

More Reliable, Wider Scope

Quant. Model. & Enterprise Analytics – Stochastic Modelling

Engine

Stochastic Process – Queueing Theory

Stochastic Process – Compound Poisson

Computer Simulation – Discrete Events,

Game-Theoretic Drivers

Computer Simulation – Discrete Shocks,

Network-Theoretic Drivers

Application

Client-Business Process – Branch Mgmt., HR, etc.

Operational Risk – Loss Distribution Approach

Business Continuity Plan., Vulnerability Analysis

Liquidity-Interbank Freeze, Systemic-Stress Events

Upshot

Better Slack Diagnostics

More Probabilistic Realism

Lower Fault Tolerance

More Reliable, Wider Scope

Quant. Model. & Enterprise Analytics – System Optimisation

Engine

Constrained Optimisation – Dynamic Programming

Constrained Optimisation – Nonlinear Programming

Application

Optimise Service Network Configuration/Parameter

Intra-Bank Capital Resource Portfolio

Upshot

Better Efficiency, Resource Allocation

Better Efficiency, Resource Allocation

Quant. Model. & Enterprise Analytics – Enterprise Informatics

Engine

Interactive Informatics – Enterprise Benchmarking

Interactive Informatics – Enterprise Evolution

Interactive Informatics – Enterprise Dashboard

Application

‘Bank as Service Provider’ Model

‘Bank as Profit Generator’ Model

‘Banking as Financial Intermediary’ Model

Upshot

Better Strategic, Decision Support

Better Tactical, Decision Support

Finer-tuned Managerial Control Lever

Quant. Model. & Enterprise Analytics – Mathematical Finance

Engine

Derivatives Pricing – Market Models

Derivatives Pricing – Credit Derivatives

Financial Mathematics – Copula Dependency

Financial Mathematics – Multi-Criteria Portfolio

Optimisation

Application

Pricing / Hedging vis-à-vis Interest Rate Desk

Pricing / Hedging vis-à-vis Credit Derivatives Desk

CVA Desk, Capturing Wrong-Way Risk

Strategic-Tactical Asset Mgmt. / Portfolio Allocation

Upshot

Efficient in Hedging, Competitive in Pricing

New Product Frontier

Efficient Counterparty Risk

Efficient Risk-Return Trade-off

Poomjai Nacaskul – Publication

2012 (w/ Janjaroen, K. & Suwanik, S.) “Economic Rationales for Central Banking: Historical Evolution,

Policy Space, Institutional Integrity, and Paradigm Challenges”, Bank of Thailand Annual

Symposium, 24th September, Bangkok, Thailand, [http://papers.ssrn.com/abstract=2156808]

[http://www.bot.or.th/Thai/EconomicConditions/Semina/symposium/2555/Paper_1_EconRational

esCentralBanking.pdf] (w/ Thai abstract) &

[mms://broadcast.bot.or.th/magstream/20120924_01.wmv] (video). 2012 “Systemic Importance Analysis (SIA) – Application of Entropic Eigenvector Centrality (EEC)

Criterion for a Priori Ranking of Financial Institutions in Terms of Regulatory-Supervisory

Concern”, Bank for International Settlements (BIS) Asian Research Financial Stability Network

Workshop, 29th March, Bank Negara Malaysia, Kuala Lumpur, Malaysia,


2011 “Relative Numeraire Risk and Sovereign Portfolio Management”, chapter 7 in Park, Donghyun

(ed., 2011), Sovereign Asset Management for a Post-Crisis World, pp. 71-84, London: Central

Banking Publications, [ISBN: 978-1-902182-71-1] [http://papers.ssrn.com/abstract=2156855]

[http://riskbooks.com/sovereign-asset-management].

2010 “Toward a Framework for Macroprudential Regulation and Supervision of Systemically Important

Financial Institutions (SIFI)”, SSRN Working Paper Series,


2010 “Financial Modelling with Copula Functions”, Lecture Notes,


2010 “The Global Financial (nee US Subprime Mortgage) Crisis –

12 Contemplations from 3 Perspectives”, SSRN Working Paper Series,




http://www.bot.or.th/Thai/EconomicConditions/Semina/symposium/2555/Paper_1_EconRationalesCentralBanking.pdf






mms://broadcast.bot.or.th/magstream/20120924_01.wmv










http://riskbooks.com/sovereign-asset-management













2009 (w/ Sabborriboon, W.) “Gaussian Slug – Simple Nonlinearity Enhancement to the 1-Factor and

Gaussian Copula Models in Finance, with Parametric Estimation and Goodness-of-Fit Tests on

US and Thai Equity Data”, 22nd Australasian Finance and Banking Conference, 16th-18th

December, Sydney, Australia, [http://papers.ssrn.com/abstract=1460576].

2009 “International Reserves Management and Currency Allocation: A New Optimisation Framework

based on a Measure of Relative Numeraire Risk (RNR)”, Joint BIS/ECB/World Bank Public Investors Conference, 16th-17th November, Washington, DC, USA, [http://papers.ssrn.com/abstract=1618692].

2006 “Adopting Basel II – Policy Responses in Case of Thailand”, chapter 12, pp. 80-97, in

Kim, H.-K. & Shin, H. S. eds., Adopting the New Basel Accord: Impact and Policy Responses of

Asia-Pacific Developing Countries, Proceedings of the Korea Development Institute (KDI) 2006

Conference, 6th-7th July, Seoul, Korea.

2006 “Survey of Credit Risk Models in Relation to Capital Adequacy Framework for Financial

Institutions”, [http://papers.ssrn.com/abstract=1625254].








1999 (w/ Dunis, et al.) “Optimising Intraday Trading Models with Genetic Algorithms”,

Neural Network World, v. 5, pp. 193-223.

1998 (w/ Dunis, et al.) “An Application of Genetic Algorithms to High Frequency Trading Models:

a Case Study”, chapter 12, pp. 247-278, in Dunis, C. & Zhou, B. eds., Nonlinear Modelling of

High Frequency Financial Time Series, [John Wiley & Sons, Chichester, UK].

1997 “Phenotype-Object Programming & Phenotype-Array Datatype: an Evolutionary Combinatorial-

Parametric FX Trading Model”, Proceedings of the 1997 International Conference on Neural

Information Processing (ICONIP’97), Dunedin, New Zealand, [Singapore: Springer-Verlag].

(version) Forecasting Financial Market (FFM) ’97, London, UK.

(version) Emerging Technologies Workshop ’97, University College London.

1996 “A Neuro-Evolutionary Framework for Fuzzy Soft-Constraint Optimisation: An FX/Futures

Trading Portfolio Application”, Proceedings of the 1996 International Conference on Neural

Information Processing (ICONIP’96), Hong Kong, [Singapore: Springer-Verlag].

(version) Forecasting Financial Market (FFM) ’96, London, UK.

(version) 1996 International Symposium on Forecasting (ISF), Istanbul, Turkey.

Some (germinating) ideas

• Nonlinear Discriminant Analysis

• Unsupervised Clustering Algorithm

• Bayesian approach to incorporating a priori factor loading

• Fuzzy-theoretic approach to incorporating domain expertise

• Game-theoretic approach to modelling client/counterparty behaviours

Individual Risk Models/Analytics

• ‘Plumbing’ model of economic capital usage

• ‘Connectivity’ model of risk concentration

• Copula >> cross-risk dependency modelling

• Wrong-way risk analysis/stress testing

• Compound Poisson Process >> Op Risk AMA

Portfolio Risk Models/Analytics

• Bank Network Service Optimisation as a Mathematica Programming Problem

• Transforming default prediction into an Optimal Control Problem

• Modelling staff turnover/recruitment as a Stochastic Process Queue

Banking-Enterprise Models/Analytics

1

2

3

4

5

6

7

8

9

3 Tier Cascade

Connectivity (outbound) BEC Vector || = 1.8013

EEC Vector || = 2.2695

,,, 24.37 30.49 ,,, 17.99 19.83 ,,, 13.06 12.64 , 6.91 5.49 , 5.93 4.45 , 6.50 5.00 4.18 2.64 7.52 5.97 13.53 13.49

Figure 1: Connectivity & Centrality for a ‘3-Tier Cascade’ Network

1

2

3

4

5

6

7

8

9

Inner & Outer Circles

Connectivity (outbound) BEC Vector || = 4.3468

EEC Vector || = 7.3397

,,, 13.89 14.17 ,,, 15.09 15.94 ,,, 15.09 15.94 ,,, 15.09 15.94 ,,, 15.09 15.94 , 6.43 5.52 , 6.43 5.52 , 6.43 5.52 ,, 6.43 5.52

Figure 2: Connectivity & Centrality for a ‘Inner & Outer Circles’ Network

Figure 1: Standard Gaussian vs. ‘Gaussian Slug’ Copula Density – 3D Plots

Figure 2: Standard Gaussian vs. ‘Gaussian Slug’ Copula Density – Contour Plots

(Cleanly) Linearly Separable 2-Population Data

-6

-4

-2

0

2

4

6

8

10

12

0 50 100 150 200 250 300 350 400 450 500

x_0 x_1

(Poorly) Linearly Separable 2-Population Data

-6

-4

-2

0

2

4

6

8

10

12

0 50 100 150 200 250 300 350 400 450 500

x_0 x_1 (Cleanly) Linearly Separable 2-Population Data (Poorly) Linearly Separable 2-Population Data

(Cleanly) Nonlinearly Separable 2-Population Data

-6

-4

-2

0

2

4

6

8

10

12

0 50 100 150 200 250 300 350 400 450 500

x_0 x_1

(Poorly) Nonlinearly Separable 2-Population Data

-6

-4

-2

0

2

4

6

8

10

12

0 50 100 150 200 250 300 350 400 450 500

x_0 x_1 (Cleanly) Nonlinearly Separable 2-Population Data (Poorly) Nonlinearly Separable 2-Population Data

Figure 1: Linearly vs. Nonlinearly Separable 2-Population Data

Ideal ‘Quant Analyst’

• Mathematics, Physics

• Operations Research, Industrial Engineering, System Science/Cybernetics

• Computer Simulation, Software Engineering

• Neural Network, Fuzzy Sets, Algorithmic Data Mining

• Game Theory, Agent-Based Modelling

Background

• Mathematica, R, MatLab

• Shareware Embedding

• GUI, Prototype

Skill

• Daring

• Learning

• Hardworking

• Experimenting

Ethos