Unit 4 of 14

Discrete arbitrage pricing

  • 8 lessons
  • Videos 25 to 32

In a finite market, prices follow from replication and the absence of arbitrage. This unit develops one-period and multiperiod models, pricing measures and the fundamental theorem of asset pricing.

Replication is a linear system across states, and its dual gives state prices and a pricing kernel. Binomial trees price claims by backward induction with a new hedge at each node. Risk-neutral probabilities are tied to a numeraire and differ from the physical probabilities used for forecasting, and discounted prices are martingales under a pricing measure. The finite fundamental theorem is proved with a separating hyperplane. Incomplete markets give price intervals, American options lead to optimal stopping and the Snell envelope, and bid-ask costs give buyer and seller bounds.

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.

Replication, state prices and binomial trees

  1. One period replication and state prices

    How one-period replication becomes a linear system in the payoff matrix, how state prices solve the dual pricing system, and how a pricing kernel writes the same prices with physical probabilities.

    Video 25

  2. Binomial trees and backward induction

    How a binomial tree prices a claim by one-period replication at every node, why it needs the local no-arbitrage condition, how the hedge ratio changes by node, and which scaling rule gives a diffusion limit.

    Video 26

Pricing measures, martingales and the fundamental theorem

  1. Risk neutral probabilities and physical probabilities

    How physical probabilities for forecasting differ from risk-neutral probabilities for pricing, what equivalent measures and the likelihood ratio are, and why a pricing measure names its numeraire, here the cash account.

    Video 27

  2. Martingales and discrete conditional pricing

    How a filtration models information, why trading strategies must be predictable, and why discounted prices, conditional values and cumulative gains with dividends are martingales under a pricing measure.

    Video 28

  3. The fundamental theorem of asset pricing

    What the finite fundamental theorem of asset pricing says, why no arbitrage holds exactly when an equivalent martingale measure exists, and how a separating hyperplane and the full-support condition enter the proof.

    Video 29

Incompleteness, early exercise and transaction costs

  1. Market completeness and nonunique prices

    Which claims are attainable, why a finite market is complete exactly when its payoff matrix has full row rank and a unique pricing measure, and how incompleteness gives an open price interval bounded by superhedging.

    Video 30

  2. American options and optimal stopping

    How admissible exercise times are stopping times, how the Snell envelope compares immediate exercise with the continuation value, and where the exercise boundary and the early-exercise premium come from.

    Video 31

  3. Transaction costs and no arbitrage price intervals

    How proportional bid-ask costs turn one replication price into an interval, how superreplication with covering hedges gives seller and buyer price bounds, and how shadow prices certify no arbitrage inside them.

    Video 32