Financial Mathematics

Author

John Robin Inston

Published

September 25, 2026

1 Financial Mathematics

Mathematical finance (also known as quantitative finance and mathematical finance) is a branch of applied mathematics concerned with mathematical modeling in the financial field. It lies at the intersection of various subjects including pure mathematics, statistics and data science.

1.2 Topics

  1. Market Models
  2. Derivatives Pricing
    • Arbitrage Theory
      • Risk-Neutral Measures
      • Stochastic Discount Factors
      • PDE Financial Models
    • Stock Derivatives
      • Forwards
      • Futures
      • Options
        • European Options
        • [[american-options]]
        • Exotic Options
          • Lookback Options
          • Asian Options
          • Barrier Options
          • Compound Options
          • Basket Options
    • Delta-Hedging
    • Option Greeks
  3. Interest Rate Models
    • Interest Rates
      • Yield Curve
      • LIBOR Rates
      • Instantaneous Rates
    • Short-Rate Models
      • Vasiček Model
      • CIR Model
      • Hull-White Model
      • Ho-Lee Model
    • Heath-Jarrow-Morton (HJM) Framework
    • Yield Curve Models
    • LIBOR Market Models
    • Bond Pricing and Term Structure Models
    • Bond Derivatives
  4. Risk-Management
    • Market, Credit and Operational Risk
    • Value-at-Risk (VaR) and Conditional VaR
    • Expected Shortfall
    • Credit Risk CR Models
      • Structural CR Models
      • Intensity Based CR Models
    • Copulas and Tail Dependence
    • Credit Default Swaps (CDS)
    • Stress Testing and Scenario Analysis
    • Basel and Regulatory Capital Models
  5. Portfolio Theory
    • Markowitz mean-variance optimization
    • CAPM (Capital Asset Pricing Model)
    • Efficient Frontier and Utility Maximization
    • Black-Litterman Model
    • Kelly Criterion and Growth-Optimal Portfolios
  6. Numerical Methods in Financial Mathematics
  7. Algorithmic and High-Frequency Trading (see PSTAT223C 2024)
    • Market Microstructure
    • Limit Order Books and Execution Methods
    • Alpha Generation and Signal Processes
    • Latency, Slippage and Execution Algorithms
  8. Statistical Methods
    • Time Series Analysis (see PSTAT274 2023)
      • ARIMA Model
      • GARCH Model
      • ARMA-GARCH Model
    • MLE and Bayesian Inference
    • Regression and Factor Models
    • Cointegration and Error Correction Models
  9. Statistical & Machine Learning Methods
    • Deep Hedging
    • Reinforcement Learning
    • Game Theory
    • Mean Field Games

1.2.1 2.3.2 Topics

Mathematics

Financial Models

1.3 1. Basic Concepts

  • Derivative: A financial contract whose value is derived from an underlying asset.

  • Underlying asset: The asset (e.g., stock, bond, index) on which the derivative is based.

  • Payoff: The amount the holder receives at maturity or exercise.

  • No-arbitrage principle: Core pricing concept that prevents riskless profit.

  • Replication: Constructing a portfolio that mimics the derivative’s payoff.

1.4 2. Risk-Neutral Valuation

  • Key idea: Price derivatives by taking the expectation under a risk-neutral measure.

  • Under risk-neutral measure QQ, discounted asset prices become martingales.

  • Fair price of derivative =EQ[e−rTΦ(ST)]=EQ[e−rTΦ(ST​)] where:

    • rr is the risk-free rate

    • Φ(ST)Φ(ST​) is the payoff function

1.5 3. The Black-Scholes Framework

  • Model Assumptions:

    • Asset follows geometric Brownian motion:

      dSt=μStdt+σStdWtdSt​=μSt​dt+σSt​dWt​

    • Constant interest rate rr, volatility σσ

    • No arbitrage, continuous trading, no transaction costs

  • Black-Scholes PDE:

    ∂V∂t+rS∂V∂S+12σ2S2∂2V∂S2−rV=0∂t∂V​+rS∂S∂V​+21​σ2S2∂S2∂2V​−rV=0

  • Black-Scholes formula for a European call:

    C(S,t)=SΦ(d1)−Ke−r(T−t)Φ(d2)C(S,t)=SΦ(d1​)−Ke−r(T−t)Φ(d2​)

    where d1=ln⁡(S/K)+(r+σ2/2)(T−t)σT−td1​=σT−t​ln(S/K)+(r+σ2/2)(T−t)​, and d2=d1−σT−td2​=d1​−σT−t​

1.6 4. Greeks (Sensitivity Measures)

Used for risk management and hedging:

  • Delta: sensitivity to underlying price

  • Gamma: sensitivity of Delta to underlying price

  • Theta: time decay

  • Vega: sensitivity to volatility

  • Rho: sensitivity to interest rate

1.7 5. Beyond Black-Scholes

For more realistic modeling:

  • Stochastic volatility models (e.g., Heston)

  • Jump-diffusion models (e.g., Merton)

  • Local volatility models

  • Monte Carlo methods for path-dependent options

  • Finite difference methods for solving PDEs numerically

1.8 6. Exotic Options

Options with more complex features than standard calls/puts:

  • Barrier options: payoffs depend on whether asset hits a barrier

  • Asian options: depend on average price

  • Lookback options: depend on maximum or minimum price

  • American options: exercisable at any time before maturity (require solving a free-boundary problem)

1.9 7. Interest Rate Derivatives

  • Instruments: swaps, caps/floors, swaptions

  • Models: Vasicek, Cox-Ingersoll-Ross (CIR), Heath-Jarrow-Morton (HJM), LIBOR Market Model

  • Pricing involves modeling the term structure of interest rates

1.10 8. Credit Derivatives

  • Includes credit default swaps (CDS), collateralized debt obligations (CDOs)

  • Pricing involves modeling default probabilities and loss given default

  • May require intensity-based or structural models

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