Skip to content
Stochastic Calculus / Discrete Martingales and the Gambling Blueprint

Overview

Every option-pricing formula of the form Vt=e−r(T−t)EQ[VT∣Ft]V_t = e^{-r(T-t)}\mathbb{E}^{\mathbb{Q}}[V_T \mid \mathcal{F}_t] rests on a single mathematical spine: the martingale. A martingale is a process whose conditional expected future value is always the current value — no predictable trend, no arbitrage opportunity. In continuous-time pricing this condition appears via Girsanov's theorem, Itô's formula, and the fundamental theorem of asset pricing. But the cleanest route into all three is the discrete theory developed here.
The topic opens by rebuilding conditional expectation on a more rigorous footing — as a Radon-Nikodym derivative and as the best L2L^2 predictor with respect to a σ-algebra. This precision is not pedantry: path-dependent payoffs, stochastic volatility models, and change-of-measure arguments all require CE to be a random variable adapted to a filtration, not merely a number. Once that foundation is laid, martingales follow as processes where consecutive conditional expectations are always equal, and the entire theory — optional stopping, convergence, quadratic variation, and discrete stochastic integration — unwinds from that single definition.

By the end of this topic you should be able to:

  • State the measure-theoretic definition of conditional expectation and prove its core properties from that definition.
  • Define a martingale, submartingale, and supermartingale with respect to a filtration, and verify or falsify the property for concrete processes.
  • Apply the optional stopping theorem accurately, citing the conditions (bounded stopping time or uniform integrability) that make it valid.
  • Identify the Doob decomposition of a submartingale and state the martingale convergence theorem.
  • Construct a discrete stochastic integral (trading strategy P&L) and prove it is a martingale when the integrand is predictable.
  • Apply Doob's L2L^2 maximal inequality and explain why it is the discrete precursor to the Itô isometry.

In this lesson

  1. Contents
  2. The big picture

The full lesson is for members.

Read the full lesson

New to SelfMadeQuant.com? See membership options or explore the curriculum.