applied mathematics in banking & finance - quantitative ... · ˇthe (some) mathematics of (in)...

Applied Mathematics in Banking &

Finance - Quantitative Models

& Financial Engineering

ดร.พูมใจ นาคสกุล (DIC, CFA)ทมีแบบจําลองเชิงปรมิาณและวิศวกรรมการเงิน

สายนโยบายสถาบันการเงิน ธนาคารแหง่ประเทศไทยการประชุมวชิาการคณิตศาสตร์บริสุทธิ์และประยุกต์

ประจําปี ๒๕๕๔ (๑๙ พฤษภาคม)

ˇจุดประสงค์วันนี้ Applied Mathematics in Banking & Finance

Yes, it sure(ly) does!

From pure to applied, and back!?

Cross fertilisation of ideas or just brain drains

(perhaps twilight career option/rejuvenation for some)?

Quantitative Models & Financial Engineering

อะไร ทาํไม ทีไ่หน อย่างไร (แล้วไง)

ตัวอย่างเห็น ๆ (4-5 คงจะพอ)

อนาคต (อันไมใ่กล้ไมไ่กล)

“Show ’n Tell …”




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ˇQuantitative Models & Financial Engineering - อะไร?

แบบจําลองเชิงปริมาณ (Quantitative Models – QM)

Model – satisfactorily realistic, manipulatable, simplifiedrepresentation of a real, costly-to-manipulate systems

Quantitative – mathematical, analytical, probabilistic, statistical, empirical, computational, simulation-based

(2 1/2) Problems – (i) System Representation/Modelling, (ii) Optimisation & (Optimal) Control (iii)

Central Banking Cybernetics* (vis-à-vis the monetary-financial economy) – ผูว้่าการ ~ skipper, ดอกเบี้ย ~ lever, มาตรการ ~ steerage

*coined 1948 by Norbert Wiener (1894-1964) from Gk. kybernetes "steersman" (metaphor. "guide, governor") + -ics; perhaps based on 1830s Fr. cybernétique "the art of governing" [www.etymonline.com]

ˇQuantitative Models & Financial Engineering - อะไร?

วิศวกรรมการเงิน (Financial Engineering – FE)

Engineering – designing, testing, developing, optimising, productionising, pricing a product

Financial – in the sense of financial instruments/assets, returns/risks, and markets/institutions

(2) Problems – (i) Financial Options/Derivatives, i.e. pricing/valuation, calibration/testing, hedging/replication, (ii) (Financial) Risk Management, i.e. measurement/mitigation

FE QM – models required to price/calibrate/hedge financially engineered products are quantitative in nature

FE QM – the tasks involved in and the processes of engineering financial products go beyond those of quantitative modelling

ˇThe (Some) Mathematics of (in) Banking & Finance Arithmetic equations – what exactly is ‘Banking’? ‘Finance’? ‘Capital’? Household (HH) balance: Earning – Spending = Saving fi ‘Capital Investment’

Business (BUS) financing: Asset = Equity + Liability ‹ ‘Investment Capital’

Commercial Bank (Bank)’s ‘Intermediary Channel’: BUS liability ‹ Bank Loan ‹ Bank Balance Sheet fi Bank Deposit ‹ HH Saving

Investment Bank (IB)’s ‘Non-intermediary Channel’: Stocks (securitised BUS Equity) + Bonds (securitised BUS Liability) + IB fees ‹ HH Saving

Non-determinism – what happens in the (Secondary) ‘Fin. Market’? Stock Mkt.? random variable: SDt = S0e r Dt , r ~ N(m,s2) fi SDt ~ LN(mDt,s2Dt) stochastic process: {St , t > 0} , dSt/St = mdt + sdWt , dWt ~ N(0,dt) Markov Chain/process: states = credit rating grades, w/ transition probabilities/intensities.

Ornstein–Uhlenbeck process: stochastic interest rate exhibits mean reversion.

Lévy process: thnk Lévy–Itō decomposition fi ‘trend vs. continuous vs. small jumps vs. big’

Forecasting & Optimisation – what do ‘Fund Managers’ do all day? Time-Series Econometrics: predict trends, account for noises, disentangle series, etc.

Quadratic Programming (QP): min (quadratic) risk s.t. (linear) target return constraint …

ˇThe (Some) Mathematics of (in) Banking & Finance Probabilistic analysis – what exactly do we mean by ‘Risk’? Risk นิยามโดย: triplet {possibility, probability, preference}, i.e. probability space {W,√,R} together w/ ‘˝’

Why manage risk? In a word, utility, as per von Neumann & Morgenstern (1944).

Market Risk, Credit Risk, Operational Risk, … ; Model Risk perhaps?

Risk Management Process ขั้นตอนแบ่งออกเป็น: Identify/พสิจูน์ทราบ Measure/วัด(ประมาณ) Mitigate/ลด-ละ-เลิก Report/รายงาน(ทบทวน)

Risk Measure ตามแต่กรณี เช่น: Value-at-Risk (VaR) quantile, qtarget' Pr(Return £ qtarget) = very small percentage a Expected Shortfall (ES) exceedance, E[VaR – Return | Return < VaR] Probability of Default (PD) parameter, such as due to bankruptcy, Pr(Equity £ qtreshold)

Risk Aggregation ตามแต่กรณี เช่น: Stock portfolio: variance-covar. matrix S, wt. vector w fi quadratic form s2(w) = w'Sw

Bond portfolio: (additive) duration μ ∂BondPrice/∂DiscountRate, convexity μ ∂2Price/∂2Rate

Loan portfolio: default correlation via Gaussian Copula, i.e. symmetric, linear, not tail-heavy?

Operational risk exposure: compound Poisson process + Extreme Value Theory (EVT)

ˇThe (Some) Mathematics of (in) Banking & Finance Ito calculus – how do we do ‘Financial Engineering’? Financial Derivatives: In the simplest form: given (observable) S0 and (contractual) CT = max {S0 – K, 0}, C0 = ?

Black & Scholes (1972): 1. continuous hedge ratio delta Dt = ∂Ct/∂St

2. riskless portfolio Pt = Ct – DtSt should ‘grow’ like a money market account: Bt = B0e r t

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

4. Ito’s Lemma fi expression for dCt

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

6. Apply Green’s function and the boundary condition fi the famous Black-Scholes formula!

From which: 7. Note: entire edifice singly parameterised by the volatility parameter s 8. Note: with Bt as numeraire, (Ct – DtSt)/Bt 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) fi Equivalent Martingale Measure (EMM)

ˇThe (Some) Mathematics of (in) Banking & Finance Network modelling – way forward for framing the ‘Financial Stability’ mandate Background: Central Banking Mandate = Monetary Stability + Financial Stability + Infrastructural Stability

Financial Stability = Microprudential Stability + Macroprudential Stability

Analysis: (i) Radiation Disease ~ exposure dose times length of exposure

(ii) Vector-Borne Diseases ~ exposure vs. individual’s immunity

(iii) Sexually Transmitted Disease (STD) ~ connectivity matrix

Network Models: (iii) Agent-driven vs. (ii) Dynamic vs. (i) Static: Bonacich, Phillip (1987), “Power and Centrality: A Family of Measures”, The American Journal of

Sociology, vol. 92, no. 5, pp. 1170–1182, [http://www.comhealth.ch/pdf/Bonacich_1987.pdf].

Brin, S. & Page, L. (1998), “The Anatomy of a Large–Scale Hypertextual Web Search Engine”, Computer Networks and ISDN Systems, vol. 30, pp. 107–117.

Nacaskul, Poomjai (2010), “Systemic Import Analysis (SIA) – Application of Entropic Eigenvector Centrality (EEC) Criterion for a Priori Ranking of Financial Institutions in Terms of Regulatory-Supervisory Concern, with Demonstrations on Stylised Small Network Topologies and Connectivity Weights”, SSRN Working Paper Series, [http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1618693].




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ˇ4+4=8 (QM/FE) Critical (Knowledge Gap) Areas [fe1] Volatility Surface & Path Dependency Pricing ‘exotics’? Why Black-Scholes formula is both wrong and immortal.

[fe2] Term Structure & Market Models How FX/IR products remain the bulk of global derivatives/hedging activities.

[fe3] Default Correlation & Copula Functionals From pricing CDO, to estimating credit VaR, to modelling economic capital?

[fe4] Levy Process & Stochastic Dynamics How to capture continuous diffusion as well as jump phenomena seamlessly.

[qm1] Optimisation Algorithms & Network Modelling Bank capitalisation as optimisation problem? Is Lehman systemic? Who next?

[qm2] Nonparametric, Semiparametric & Bayesian Inference What to with missing (historical default) data? What about human judgement?

[qm3] Extremal, Chaotic & Cybernetic Stability Are financial crises 'rare’ events? Will bubbles form/burst every 7-15 years?

[qm4] Game Theory & Information Asymmetry Level playing field equals fair game? What would make credit bureau work?

ˇ4+4=8 (QM/FE) Critical (Knowledge Gap) AreasPrudential

Regulation –

Prudential

Supervision –

Prudential

Supervision –

Prudential

Regulation –

Competition

Regulation –

Conduct

Supervision –

Capital

Adequacy

Risk

Model

Derivatives

Book

Financial

Stability

Optimal

Landscape

Transparent &

Fair

qm1 Optm. Algorithms

& Network Modelling

qm2 Extremal, Chaotic

& Cybernetic Stability

qm3 Non/Semiparam.

& Bayesian Inference

qm4 Game Theory

& Information Asymmetry

fe1 Volatility Surface

& Path Dependency

fe2 Term Structure

& Market Models

fe3 Default Correlation

& Copula Functionals

fe4 Levy Process

& Stochastic Dynamics

ˇ[qm2] Non-, Semiparametric & Bayesian Inference Parametric: How? Hypothesise the relationship in terms of function y = fmodel-specific(x), (e.g. Is it governed by a power law?

Can 2 linear factors account for most of the variations observed?), then estimate based on the data.

Best route to pursue if and when possible; must take care not to impose too restrictive assumptions.

Nonparametric: How? Use the data to parameterise the fitting function itself, i.e. y = fempirical-kernel(x), where = {x,y}.

Closest as we can to the “let the data speak” ideal; avoid degenerating into vacuous “averaging game”.

Semiparametric (my preferred interpretation/classification): How? Attribute fundamental properties (e.g. Is the mapping continuous? Are there discontinuities?), then minimise

the fitting error by performing search over the prescribed parametric/function space {f ; f }generic

Gives up some explanatory power hopefully in exchange for realism, robustness, and generalisation.

This way, NN would be considered nonparametric, while ANN/SVM would be considered semiparametric.

Semiparametric, as per Hardle, et al. (2004): How? “Semiparametric models combine components of parametric and nonparametric models, keeping the easy

interpretability of the former and retaining some of the flexibility of the latter.”

Achieves “reduction of dimensionality as well as allowance for partly parametric modeling.”

Bayesian: How? Bootstrap/embellish the state of information as expert knowledge/empirical evidence warrant.

Close to the way humans reason; best summarised as “parametric in form, and yet nonparametric in spirit”!

ˇ[qm3] Extremal, Chaotic & Cybernetic Stability

[qm4] Game Theory & Information Asymmetry Extreme Value Theory (EVT): Just as the Central Limit Theorem (CLT) is concerned with distribution of sample averages, EVT deals with distribution of sample

extrema …

Chaos Theory: System behaviour is deterministic (non-stochastic), yet practically unpredictable: “behavior of certain dynamical systems … highly

sensitive to initial conditions … the butterfly effect … an exponential growth of perturbations in the initial conditions ... appears to be random … even though these systems are deterministic” [http://en.wikipedia.org/wiki/Chaos_theory].

A complex system is “composed of interconnected parts that as a whole exhibit ... properties (behavior ...) not obvious from the properties of the individual parts” [http://en.wikipedia.org/wiki/Complex_system]

Cybernetics: “interdisciplinary study of … complex systems, … control mechanisms and feedback principles”

[http://en.wikipedia.org/wiki/Cybernetics]; a science of government, governance, and governorship.

A radical proposal: from “Business Management as Commercial Entity Cybernetics” to “Central Banking as Cybernetics of the Monetary-Financial System” and perhaps “Government as Cybernetics of the State”

Game Theory: Theoretical-analytical framework for analysing the action-counteraction decision-making strategies amongst a finite number of

self-interested (utility maximising) agents.

A radical proposal: from “regulating/supervising banks’ competition (with other banks)” to “creating an economic game that promotes competitive, socially responsible, and welfare enhancing behaviours”

Information Asymmetry: Theoretical-analytical framework for analysing the economic behaviours between two sets of transaction counterparties when

each commands different quantity, quality, type, and/or source of information.

A radical proposal: from “regulating/supervising banks’ conduct (with the customers)” to “incentivising/optimising/counterbalancing/hedging/mitigating information asymmetry”




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ˇQuantitative Models & Financial Engineering @ BOT[http://www.bot.or.th/English/FinancialInstitutions/New_Publications/QMFE] Quantitative/Research Papers:

2010 “Systemic Import Analysis (SIA) – Application of Entropic Eigenvector Centrality (EEC) Criterion for a Priori Ranking of Financial Institutions in Terms of Regulatory-Supervisory Concern, with Demonstrations on Stylised Small Network Topologies and Connectivity Weights”, SSRN Working Paper Series, [http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1618693].

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/sol3/papers.cfm?abstract_id=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/sol3/papers.cfm?abstract_id=1618692].

Lectures/Qualitative Papers: 2010 “Financial Modelling with Copula Functions”, Lecture Notes,

[http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1726313].

2010 “Toward a Framework for Macroprudential Regulation and Supervision of Systemically Important Financial Institutions (SIFI)”, SSRN Working Paper Series, [http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1730068].

2010 “The Global Financial (nee US Subprime Mortgage) Crisis – 12 Contemplations from 3 Perspectives”, SSRN Working Paper Series, [http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1677890].

2010 “The Case for All Asset Investing: What You Need to Know about Relative Numeraire Risk (RNR)”, Institutional Investor’s Americas Sovereign Funds Roundtable, 8th-9th March, Coral Gables, FL, USA, [www.iimemberships.com/dl/usi/ASFR%202010%20Summary.pdf] (summary) & [www.euromoney.com/videos/asfr/03082010-0814-1034.htm?LS=EMS396453] (video highlight).

ˇQuantitative Models & Financial Engineering @ BOT[http://www.bot.or.th/English/FinancialInstitutions/New_Publications/QMFE]

ˇIdeal Math Applications & Applied Mathematicians Ideal Math Applications The ‘downstream’ path (i.e. new problem mapped onto an old class, recognised in a

different application…) vs. ‘upstream’ path (i.e. new problem identifies methodological gaps, motivates basic research areas…)

Ideal Applied Mathematicians Solid background in PURE mathematics ‘There is nothing more practical than a good theory.’ Kurt Lewin (1945)

Mastering ‘the art of assuming & the science of NOT …’ Suppose you were given a choice: guaranteed optimality over convex search space (not

supported by reality) vs. heuristic covergence over more realistic search terrain.

Think APPLICABILITY, not EXPERTISE! ‘If all you have is a hammer, everything looks like a nail.’ Abraham Maslow (1966)

Think PRACTICALITY, not SHORTCUT! ‘Everything should be rendered as simple as possible, but no simpler.’

Can ‘TALK the WALK’ ‘You do not really understand something unless you can explain it to your grandmother.’

ˇWhat am I looking for right now? (a help ad!) Quantitative Models Network research ideas: Simulating dynamics of financial contagion in a global-network financial economy.

Endowing financial network nodes with agent decision making.

Copula research ideas: Control over asymmetry, heavy-tailedness, nonlinearity in dependency modelling?

Higher-dimensional ‘Gaussian Slug’ copula?

Financial Engineering Unification issues: Risk Management: toward a modular representation of market/credit/operational risk.

Financial Engineering: consistency in terms of pricing vs. hedging vs. calibration.

Policy issues: Risk Management: operational risk involving extreme losses, multiply-sourced data.

Financial Engineering: applicability of market models vis-à-vis Thai financial markets.




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