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Replicate a one-step option with stock and cash

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Start with the idea

In a two-state world, two traded instruments can match two possible option payoffs.

Symbols, units & horizon
  • S₀,S_u,S_d: current and next-step stock prices, currency per share with S_d
  • H_u,H_d: option payoffs in the same currency per option covering one share
  • Δ: shares held per option
  • R>0: cash gross return over one step
  • B: present cash position
  • V₀: replicated current option value

When and why to use this

Understand replication before continuous-time pricing and hedge approximations.

In a two-state world, two traded instruments can match two possible option payoffs.

The stock hedge makes the difference between the up and down states match. A cash balance then fixes the common level. Risk-neutral pricing probabilities are derived from no-arbitrage financing; they are not the trader’s forecast of the next move.

The model assumes exactly two states, the ability to borrow/lend at the stated rate, and frictionless rebalancing. Check that the financing gross return lies between the down and up stock gross returns so the model itself admits no stock/bond arbitrage.

Δ=Hu−HdSu−Sd,B=Hd−ΔSdR,V0=ΔS0+B
Exact replication algebra in an assumed two-state market

Replicate a one-step option with stock and cash

  1. Subtract the two replication equations ΔS_u+BR=H_u and ΔS_d+BR=H_d.
  2. Solve for Δ, then substitute the down state to recover present cash B.
  3. Add current stock cost ΔS₀ and cash B.
Work it by hand

S₀=100, S_u=120, S_d=80, call strike 100, R=1. Payoffs 20 and 0 give Δ=.5, B=−40 and V₀=10.

Apply it in a strategy

  • Freeze inputs at the stated decision time and record their units.
  • Understand replication before continuous-time pricing and hedge approximations.
  • Recompute the example, then change the material assumption and explain the difference.

Research deliverable

Replicate a one-step option with stock and cash: produce the worked calculation, a timestamped input record and a written decision addressing this limitation: Real prices have more than two states and trading costs; exact one-step replication does not transfer automatically.

These are synthetic mechanics examples, not historical performance or paper replications. Module evidence and research boundaries record the 12 September 2026 review.

Python implementation

Self-contained teaching example. Python 3.10+; dependencies and input conventions are shown in the code and notation. Run in your own Python environment.

# Python 3.10+; standard library and NumPy only.
# Synthetic teaching inputs; conventions and units are defined in the notation above.
def binomial_replicate(spot,up,down,pay_up,pay_down,gross_rate):
    if not 0<down<spot<up or not down/spot<gross_rate<up/spot: raise ValueError('Invalid or arbitrage-inconsistent tree')
    delta=(pay_up-pay_down)/(up-down)
    cash=(pay_down-delta*down)/gross_rate
    return delta,cash,delta*spot+cash

assert binomial_replicate(100,120,80,20,0,1)==(.5,-40,10)
print(binomial_replicate(100,120,80,20,0,1))

Continue learning

Options: Payoffs, Replication & Hedge Accounting — all lessons
  1. Call and put payoffs versus profit
  2. A bull call spread caps gains and initial cost
  3. Put–call parity as identical terminal cash flows
  4. Replicate a one-step option with stock and cash
  5. Black–Scholes as a conditional benchmark
  6. Delta and gamma are local sensitivities
  7. Cash accounting for a discretely hedged option
  8. Early exercise compares immediate and continuation value

Quantitative finance and development glossary · Python resources and libraries · Research sources and limitations