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IBBC CAPITAL ACADEMY / 2027

One-step binomial replication

Independent IBBC Capital draft content for self-study. Not affiliated with or endorsed by CFA Institute. New lessons await subject-matter review. No CFA designation or official PSM credit is awarded.

Learning goals

  • Explain state payoffs using mechanisms and limitations.
  • Explain replicating hedge using mechanisms and limitations.
  • Explain risk-neutral weights using mechanisms and limitations.

Draft · approximately 25 minutes, excluding further revision.

State payoffs

A one-step tree specifies current spot and up/down ending prices; derivative payoffs must be calculated in each state before pricing today.

Delta equals the difference between derivative state payoffs divided by the difference between underlying state prices; it hedges across the two modeled states.

Replicating hedge

Combine delta units of underlying with borrowing or lending so both terminal payoffs match; the replicating portfolio cost gives the derivative value.

Risk-neutral up probability is [(1+r)−d]/(u−d) for a no-income one-period asset; it is a pricing weight rather than a claim about actual investor beliefs.

Risk-neutral weights

A valid no-arbitrage tree requires d<1+r<u, yielding weights between zero and one; otherwise the tree or financing assumptions need repair.

Discount the risk-neutral expected payoff at the compatible risk-free rate, not the stock’s subjective expected return; model simplicity still leaves specification risk.

Formula and assumptions

q=[(1+r)−d]/(u−d); V0=[qVu+(1−q)Vd]/(1+r)

Original applied case: One-step binomial replication

  1. Spot 100, up 120, down 80, risk-free gross 1.05, call strike 100; one period and no dividends.
  2. Call payoffs 20 and 0; q=(1.05−0.8)/(1.2−0.8)=0.625.
  3. Value=0.625×20/1.05=11.9048; delta=(20−0)/(120−80)=0.5.

Errors to catch

  • Do not assume: Price an option before specifying state payoffs.
  • Do not assume: Delta equals the option premium divided by strike.
  • Do not assume: Replication only needs to match the up state.

Original knowledge check

Answer every question before checking. Review explanations after submitting; retry starts a new attempt.

Sources and coverage status

These original lessons are drafts. Public outlines are used to check scope; the official learning-outcome text and proprietary question banks are not reproduced here. Follow the source for exam-year requirements and updates.