Unit 6 of 14

Continuous time derivative pricing

  • 8 lessons
  • Videos 41 to 48

The Black-Scholes model prices options by replication in continuous time. This unit derives its pricing equation, formulas, Greeks and hedging error, and extends it to digital and American options.

Geometric Brownian motion has an exact lognormal solution. A self-financing delta hedge under frictionless assumptions gives the Black-Scholes PDE, and the Feynman-Kac formula writes its solution as a discounted expectation in a stated solution class. Gaussian integrals give the call and put formulas and put-call parity, and their derivatives are the Greeks. Discrete hedging leaves an error driven by gamma and realized variance. A smooth call curve recovers digital prices and the state-price density, and early exercise leads to a free-boundary problem, solved exactly for the perpetual put.

Every lesson will be one video. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.

The Black-Scholes pricing equation

  1. Geometric Brownian motion and exact distributions

    Solves geometric Brownian motion exactly with Itô's formula and derives its lognormal transition density, all power moments, the gap between mean and median, and exact sampling compared with Euler steps.

    Video 41

  2. Black Scholes replication and the pricing PDE

    Derives the Black-Scholes PDE from a self-financing delta hedge under frictionless assumptions, and explains the replication argument, why the drift drops out, and the terminal and boundary conditions.

    Video 42

  3. Feynman Kac and discounted expectations

    States and proves the Feynman-Kac formula, which writes a classical solution of a backward terminal value problem as a discounted conditional expectation, with uniqueness in a controlled-growth class.

    Video 43

Black-Scholes formulas, Greeks and hedging

  1. The Black Scholes formula and put call parity

    Derives the Black-Scholes call and put prices from truncated normal integrals, defines d1 and d2, proves put-call parity, and explains the in-the-money probability and the expiry and zero-volatility limits.

    Video 44

  2. Greeks and differential identities

    Derives the Black-Scholes delta, gamma, vega, theta and rho from one density identity, states their units and conventions, and checks the vega-gamma identity and the pricing PDE in a worked example.

    Video 45

  3. Discrete delta hedging and hedging error

    Derives the leading residual of a delta hedge rebalanced on a finite grid, shows how gamma exposure turns realized versus model variance into hedging error, and explains the role of trading frequency and costs.

    Video 46

Digital and American options

  1. Digital options and distribution recovery

    How a cash-or-nothing digital price is minus the strike derivative of the call price, how the Breeden-Litzenberger formula reads the state-price density from call price curvature, and what a probability atom changes.

    Video 47

  2. Early exercise in continuous time

    How the American value becomes a continuous-time optimal stopping problem, how the variational inequality and its free boundary describe it, and how value matching and smooth pasting solve the perpetual American put.

    Video 48