Stochastic Calculus & Derivative Pricing Theory

  1. F1

    Probability Foundations

    The formal setting. Probability space, sigma-algebra, measure and expectation.

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      Topic 01

      State Price Density

      Topic 02

      Risk-Neutral Probability

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  2. F2

    Stochastic Processes and Jumps

    Randomness through time. Martingale, filtration, stopping time, Markov, and jumps.

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      Topic 01

      Brownian Motion

      Topic 02

      Geometric Brownian Motion

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      Topic 01

      Quadratic Variation

      Topic 02

      Ordinary Variation

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      Topic 01

      Markov Process

      Topic 02

      Martingale: Two Different Promises

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  3. F3

    Ito Calculus

    Integrating against randomness, and why ordinary calculus fails.

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      Topic 01

      The Ito Integral

      Topic 02

      the Riemann Integral: Where They Part

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      Topic 01

      Ito Calculus

      Topic 02

      Ordinary Calculus: What Changes and Why

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  4. F4

    Stochastic Differential Equations

    Modelling change with a random term. Drift, diffusion, solutions and discretisation.

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      Topic 01

      Stochastic Differential Equation

      Topic 02

      Ordinary Differential Equation

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  5. F5

    Pricing Theory and No-Arbitrage

    The argument. No-arbitrage, the fundamental theorems, measure change and the pricing kernel.

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      Topic 01

      Replication

      Topic 02

      Hedging: An Argument and an Activity

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      Topic 01

      Complete Market

      Topic 02

      Incomplete Market: One Price or a Range

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  6. F6

    Option Pricing Theory

    The canonical application. Black-Scholes, the binomial model, delta hedging and boundaries.

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      Topic 01

      European and American Options: Why Early Exercise Changes Everything

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  7. F7

    Volatility Models

    Relaxing the constant volatility assumption. Local, stochastic, Heston, SABR.

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      Topic 01

      Vasicek Model

      Topic 02

      CIR Model: Two Ways to Hold a Rate

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      Topic 01

      Volatility Smile

      Topic 02

      Skew

      Topic 03

      Surface: Reading the Shape

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  8. F8

    Interest Rate Models

    Why rates need their own models. Mean reversion, short rate, the discount curve.

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      Topic 01

      Short-Rate Model

      Topic 02

      Market Model: What Each Takes as Given

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