Finance & Insurance

Portfolio · Pricing · Risk · ALM · Credit · Rates · Execution · Real Options · Actuarial

Finance sits at the intersection of mathematical finance, operations research, financial economics, and actuarial science. Unlike domains where OR is applied to engineering systems, finance OR confronts decisions under inherent and irreducible uncertainty — stochastic processes, Itô calculus, and martingales are not an extension but the foundation. Since Markowitz (1952), Black–Scholes–Merton (1973), and the coherent-risk axioms of Artzner et al. (1999), the field has grown into ten distinct problem families spanning pricing, optimisation, risk measurement, credit, rates, asset–liability management, execution, real options, actuarial, and systemic networks.

Educational Purpose

This page describes mathematical models used in finance and insurance for educational and research purposes. It is not investment advice, financial planning, tax guidance, or regulatory counsel. Numerical examples are illustrative. Real financial decisions require licensed professional advice; historical model performance never implies future results.

Why finance OR matters

Four anchor facts on the scale of the field

$128 T

global professionally managed assets (2023) — the direct field of portfolio-optimisation theory originating with Markowitz (1952).

BCG Global Asset Management 2024 · bcg.com

1997

year Merton & Scholes received the Nobel Prize in Economics for the options-pricing framework that still underlies most derivative valuations today.

Nobel Committee · nobelprize.org

Basel III

regulation requires banks to hold capital calibrated to Expected Shortfall (CVaR) at 97.5% — a direct adoption of Rockafellar & Uryasev (2000) into global financial policy.

BCBS Standards · bis.org

$40 T+

insured assets worldwide require asset–liability management under long-horizon stochastic programming, a stream founded by Ziemba, Mulvey, Dempster and colleagues.

Swiss Re Institute, sigma 2023 · swissre.com

The finance-OR landscape

Four lenses on the same ten problem families

Two fundamental divides — Optimisation vs Pricing, Discrete vs Continuous-time. Click any family to jump to its catalog.

Optimisation-flavoured Pricing-flavoured Hybrid / both

Buy-side

Asset managers, hedge funds, pensions, endowments — firms that take investment risk on behalf of principals.

Markowitz Mean-Variance CVaR Portfolio (coming soon) Robust Portfolio (coming soon) Value at Risk & ES (coming soon)

Sell-side

Dealers, broker-dealers, market makers — firms that price instruments and provide liquidity.

Black-Scholes-Merton (coming soon) Binomial Option Pricing (coming soon) Monte Carlo Option Pricing (coming soon) Short-Rate Models (coming soon) Optimal Execution (coming soon)

Corporate

CFOs, treasurers, project teams — non-financial firms making investment and capital-structure decisions.

Real Options Valuation (coming soon) Merton Structural Credit (coming soon)

Actuarial & Regulatory

Insurers, pensions, regulators, central banks — institutions concerned with long-horizon liabilities and system stability.

Asset-Liability Management (coming soon) Insurance Ratemaking (coming soon) Systemic Risk & Clearing (coming soon)

Equity

Stocks, indices, equity derivatives.

Markowitz CVaR Portfolio Robust Portfolio Optimal Execution

Fixed Income

Bonds, rate derivatives, yield curves.

Short-Rate Models Merton Credit ALM

Derivatives

Options, futures, swaps, exotics.

Black-Scholes Binomial Monte Carlo

FX & Commodities

Currency pairs, metals, grains, energy.

Market Timing (grain, cross-domain) Energy Portfolio (cross-domain)

Alternatives

Private equity, real estate, infrastructure, crypto.

Real Options Project Portfolio (cross-domain)

Insurance

Ratemaking, reserving, catastrophe bonds.

Insurance Ratemaking ALM (Pension)

Market structure

Clearinghouses, payment systems, contagion.

Systemic Risk

High-frequency

Seconds – milliseconds

Optimal Execution

Operational

Hours – days

Option Pricing Rate Curves VaR / ES

Tactical

Weeks – quarters

Portfolio Rebalancing CVaR / Robust

Strategic

Years – decades

ALM Real Options Insurance Ratemaking Systemic Risk

All Finance & Insurance Applications

Ten problem families · fourteen sub-applications

01 Portfolio Optimisation Seminal anchor: Markowitz (1952) · Cornuejols & Tütüncü (2007)

Static and multi-period allocation of capital across risky assets under uncertainty. Quadratic programming, second-order cone programming, stochastic programming, and robust optimisation all originate in or take canonical form here.

Portfolio Equity Tactical QP

Markowitz Mean-Variance Portfolio

Markowitz (1952), J. Finance. The founding quadratic program of modern portfolio theory: minimise portfolio variance subject to a target return.

View application →

Portfolio Multi-asset Tactical LP (stochastic) Coming soon

CVaR Portfolio Optimisation

Rockafellar & Uryasev (2000). Replace variance with the coherent tail-loss measure; reduces to a linear programme under discrete scenarios.

Coming soon

Portfolio Multi-asset Tactical SOCP (robust) Coming soon

Robust Portfolio Optimisation

Goldfarb & Iyengar (2003); Ben-Tal & Nemirovski (1998). Ellipsoidal uncertainty on expected returns; reformulates as second-order cone programme.

Coming soon

02 Derivative Pricing Seminal anchors: Black & Scholes (1973) · Cox-Ross-Rubinstein (1979) · Shreve (2004)

Valuing contingent claims under no-arbitrage. Pricing uses the risk-neutral measure rather than the physical measure used for portfolio optimisation. Three canonical computational approaches: PDE (Black-Scholes), lattice (binomial trees), and Monte Carlo simulation.

Pricing Derivatives Operational PDE Coming soon

Black-Scholes-Merton Option Pricing

Black & Scholes (1973); Merton (1973). Closed-form European option price under geometric Brownian motion; interactive Greeks.

Coming soon

Pricing Derivatives Operational DP / lattice Coming soon

Binomial Option Pricing

Cox, Ross & Rubinstein (1979). Recombining binomial tree; handles American options directly; converges to Black-Scholes as time steps increase.

Coming soon

Pricing Derivatives Operational Monte Carlo Coming soon

Monte Carlo Option Pricing

Boyle (1977); Glasserman (2004); Longstaff & Schwartz (2001) for American options. Path-dependent and exotic payoffs.

Coming soon

03 Risk Measurement & Management Seminal anchor: Artzner, Delbaen, Eber & Heath (1999)

Quantifying loss risk. VaR is a quantile of the loss distribution; CVaR (Expected Shortfall) is the average loss beyond VaR. CVaR is coherent in the sense of Artzner et al.; VaR is not — the reason Basel III (2019) moved to an ES-based capital rule.

Risk Multi-asset Tactical Simulation + LP Coming soon

Value at Risk & Expected Shortfall

Parametric, historical, and Monte Carlo estimation, with side-by-side comparison of VaR and CVaR on the loss distribution.

Coming soon

04 Interest-Rate & Term-Structure Modelling Seminal anchors: Vasicek (1977) · Cox-Ingersoll-Ross (1985) · Heath-Jarrow-Morton (1992)

Modelling the dynamics of interest rates and the zero-coupon curve under the pricing measure. Short-rate models are one-factor SDEs; HJM and LIBOR market models characterise the full forward-rate curve.

Rates Fixed Income Operational SDE Coming soon

Vasicek / CIR Short-Rate Models

Mean-reverting one-factor SDEs for the short rate. Vasicek (Gaussian) vs Cox-Ingersoll-Ross (non-negative, square-root diffusion).

Coming soon

05 Credit Risk Seminal anchors: Merton (1974) · Jarrow & Turnbull (1995) · Duffie & Singleton (2003)

Modelling default. Structural models view equity as a call option on firm assets (Merton 1974); reduced-form models treat the default time as the first jump of a point process with a hazard rate.

Credit Fixed Income Strategic SDE + implicit eq. Coming soon

Merton Structural Credit Model

Merton (1974), J. Finance. Default as firm-asset value breaching a debt threshold; closed-form credit spread. Reduced-form alternative included.

Coming soon

06 Asset-Liability Management Seminal anchors: Ziemba & Mulvey (1998) · Consigli & Dempster (1998)

Long-horizon matching of assets to stochastic liabilities (pensions, insurance, banks) via multi-stage stochastic programming over scenario trees.

ALM Multi-asset Strategic Multi-stage SP Coming soon

Asset-Liability Management (Pension)

Pension-fund ALM over a scenario tree; funding-ratio evolution; illustrative benchmark inspired by the Russell-Yasuda Kasai and InnoALM families.

Coming soon

07 Trading & Execution Seminal anchor: Almgren & Chriss (2001)

Scheduling order execution under temporary and permanent market impact, optimally trading off expected execution cost against variance of the realised cost.

Execution Equity HFT / operational Stochastic control Coming soon

Optimal Execution (Almgren-Chriss)

Almgren & Chriss (2001), J. Risk. Mean-variance execution frontier; closed-form exponential trading schedule under linear impact.

Coming soon

08 Corporate Finance & Real Options Seminal anchors: Myers (1977) · Dixit & Pindyck (1994)

Firm-level capital investment under uncertainty with flexibility (defer, expand, abandon, switch). Extends classical NPV by pricing the option value of managerial flexibility.

Real options Alternatives Strategic DP / binomial Coming soon

Real Options Valuation

Dixit & Pindyck (1994). Value the option to defer, expand, or abandon an investment under stochastic cash flows; binomial-lattice valuation.

Coming soon

09 Actuarial & Insurance Seminal anchors: Bühlmann (1970) · Klugman-Panjer-Willmot (2019)

Ratemaking, loss reserving, ruin theory, and catastrophe-bond pricing. Distinct from corporate-finance derivatives because the risk is idiosyncratic to the insurance pool and diversification is the dominant loss-absorption channel.

Actuarial Insurance Strategic Credibility / regression Coming soon

Insurance Ratemaking & Reserving

Bühlmann credibility premium combined with chain-ladder reserving; illustrative loss triangles.

Coming soon

10 Financial Networks & Systemic Risk Seminal anchors: Eisenberg & Noe (2001) · Glasserman & Young (2016)

Optimal clearing, contagion propagation, and systemic-risk amplification in interconnected financial systems. A foundational problem is the Eisenberg-Noe fixed-point clearing vector.

Networks Market structure Strategic Fixed-point / LP Coming soon

Eisenberg-Noe Clearing & Systemic Risk

Eisenberg & Noe (2001), Management Science. Clearing vector as fixed point of a monotone mapping; interactive bank-network contagion.

Coming soon

Key references

Seminal papers and canonical textbooks per family

Markowitz, H. (1952).

Portfolio Selection.

The Journal of Finance, 7(1), 77–91. doi:10.1111/j.1540-6261.1952.tb01525.x Family 1

Rockafellar, R. T. & Uryasev, S. (2000).

Optimization of Conditional Value-at-Risk.

Journal of Risk, 2(3), 21–41. doi:10.21314/JOR.2000.038 Family 1, 3

Ben-Tal, A. & Nemirovski, A. (1998).

Robust Convex Optimization.

Mathematics of Operations Research, 23(4), 769–805. doi:10.1287/moor.23.4.769 Family 1

Goldfarb, D. & Iyengar, G. (2003).

Robust Portfolio Selection Problems.

Mathematics of Operations Research, 28(1), 1–38. doi:10.1287/moor.28.1.1.14260 Family 1

Black, F. & Scholes, M. (1973).

The Pricing of Options and Corporate Liabilities.

Journal of Political Economy, 81(3), 637–654. doi:10.1086/260062 Family 2

Merton, R. C. (1973).

Theory of Rational Option Pricing.

The Bell Journal of Economics and Management Science, 4(1), 141–183. doi:10.2307/3003143 Family 2

Cox, J. C., Ross, S. A. & Rubinstein, M. (1979).

Option Pricing: A Simplified Approach.

Journal of Financial Economics, 7(3), 229–263. doi:10.1016/0304-405X(79)90015-1 Family 2

Artzner, P., Delbaen, F., Eber, J.-M. & Heath, D. (1999).

Coherent Measures of Risk.

Mathematical Finance, 9(3), 203–228. doi:10.1111/1467-9965.00068 Family 3

Vasicek, O. (1977).

An equilibrium characterization of the term structure.

Journal of Financial Economics, 5(2), 177–188. doi:10.1016/0304-405X(77)90016-2 Family 4

Heath, D., Jarrow, R. & Morton, A. (1992).

Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claims Valuation.

Econometrica, 60(1), 77–105. doi:10.2307/2951677 Family 4

Merton, R. C. (1974).

On the Pricing of Corporate Debt: The Risk Structure of Interest Rates.

The Journal of Finance, 29(2), 449–470. doi:10.1111/j.1540-6261.1974.tb03058.x Family 5

Jarrow, R. A. & Turnbull, S. M. (1995).

Pricing Derivatives on Financial Securities Subject to Credit Risk.

The Journal of Finance, 50(1), 53–85. doi:10.1111/j.1540-6261.1995.tb05167.x Family 5

Ziemba, W. T. & Mulvey, J. M. (eds.) (1998).

Worldwide Asset and Liability Modeling.

Cambridge University Press. ISBN 978-0-521-57108-3. Family 6

Almgren, R. & Chriss, N. (2001).

Optimal Execution of Portfolio Transactions.

Journal of Risk, 3, 5–40. doi:10.21314/JOR.2001.041 Family 7

Dixit, A. K. & Pindyck, R. S. (1994).

Investment under Uncertainty.

Princeton University Press. ISBN 978-0-691-03410-2. Family 8

Klugman, S. A., Panjer, H. H. & Willmot, G. E. (2019).

Loss Models: From Data to Decisions, 5th ed.

Wiley. ISBN 978-1-119-52378-9. Family 9

Eisenberg, L. & Noe, T. H. (2001).

Systemic Risk in Financial Systems.

Management Science, 47(2), 236–249. doi:10.1287/mnsc.47.2.236.9835 Family 10

Cornuejols, G. & Tütüncü, R. (2007).

Optimization Methods in Finance.

Cambridge University Press. ISBN 978-0-521-86170-2. Textbook

Shreve, S. E. (2004).

Stochastic Calculus for Finance II: Continuous-Time Models.

Springer. ISBN 978-0-387-40101-0. Textbook

Hull, J. C. (2021).

Options, Futures, and Other Derivatives, 11th ed.

Pearson. ISBN 978-0-13-693997-9. Textbook

Explore Related Domains

Stochastic programming and robust optimisation appear across the site. See how related techniques apply in healthcare, energy, agriculture, and logistics.

Stochastic family Continuous family All domains

Reminder — educational purpose only

Every model, illustration, numerical example, and interactive widget on this page and its sub-applications is provided for educational and research purposes. They are not investment, tax, legal, accounting, or regulatory advice. All model assumptions (geometric Brownian motion, Gaussian returns, stationarity, known parameters, continuous trading, no transaction costs) are violated in real markets; model outputs cannot be interpreted as forecasts. Historical model performance never implies future results. For real financial decisions, consult licensed professionals.