Identifier: C4T2L3 | Revision: 2026-10-26 | Planned study: 10–15 minutes, excluding extended practice.

Decision and objectives

Choose whether the method addresses this lesson’s task before placing an order. Objectives: 1. Replicate a two-state call with stock and borrowing. 2. Calculate the one-step option value. 3. Explain why replication weights are not forecasts.

Prerequisite: C4T2L2.

All numbers, quotes, contract terms and dates in the case are hypothetical teaching assumptions, not a listed series or offer. Unless varied below, options are European, expire together in 90 calendar days, are cash settled in TRY, and use 100 underlying units per contract. Premiums are TRY per underlying unit, paid at entry; signed quantities identify long (+) and short (−) legs. Assume no distributions and zero financing for the isolated payoff, no holiday adjustment, and final cash on the next business day. Actual tax is excluded, not assumed zero.

Mechanics

A one-step binomial model replaces all possible price paths with two terminal states. It is a teaching model for replication: choose stock units and a cash account that reproduce the option in both states. The model does not claim the asset can only take two prices in practice. Replication requires trading and financing at the assumed terms, with no transaction costs or constraints. Those assumptions are tested after solving the arithmetic, rather than silently treated as available financing.

Price and return drivers

Hedge units per option unit = (up payoff − down payoff)/(up asset − down asset). Determine the terminal cash account from either state, then discount at the supplied one-period factor. Alternatively, a pricing weight q = (growth factor − down factor)/(up factor − down factor) produces the same discounted value. This weight arises from equal cash flows, not from a prediction of the up-state frequency. If the growth factor falls outside the two price factors, the assumed model violates its own basic no-arbitrage setup.

Worked case

Override expiry to one year. S100 moves to 120 or 80; K100 call pays 20 or 0. One-period cash growth is 1.05. Hedge ratio = 20/40 = 0.5 share per option unit. A half-share is worth 60 in the up state and 40 in the down state; borrowing 40 repayable at expiry leaves 20 or 0. Borrowing today is 40/1.05 = 38.095238. Replicating cost = 0.5 × 100 − 38.095238 = 11.904762 per unit, or TRY 1190.4762 for multiplier 100. Pricing weight is (1.05 − 0.8)/(1.2 − 0.8) = 0.625; 0.625 × 20/1.05 gives the same result.

Adverse case

The terminal scenarios S80, S100 and S120 give a long call net of the theoretical entry value and TRY 20 charges: −1210.4762, −1210.4762 and +789.5238. The middle state was not in the two-state replication model; at S100 the fixed hedge would produce 0.5 × 100 − 40 = 10 while the call pays zero. That TRY 10 per-unit mismatch exposes model risk. Before expiry, a real price jump and changing volatility need new valuation inputs; the original two-state value is not a standing exit quote.

Practical use

Confirm whether the assumed borrowing and fractional hedge can actually be implemented. The tree is an arithmetic teaching model; its cost does not establish market depth.

For any actual domestic implementation, verify the exact issue/contract, authorized intermediary, account permission and executable size. Include spread, commissions, passed-through fees, funding/borrow, custody and exercise/exit costs without double-counting embedded charges. Recheck income year, residency/status, issue/acquisition dates and effective tax law; all case results remain before actual tax.

Decision boundaries

Use the tree to understand how funding enters a contingent claim and why a hedge can replicate within a model. Do not infer that a retail account can borrow at the model cash rate or rebalance continuously. A model price without costs may fall inside a wide spread and produce no profitable implementation. Decline a trade justified only by a fragile two-state forecast. More steps refine a model, but do not eliminate gaps, transaction costs or wrong assumptions.

Practice and references

Exercise: S50 moves to 60 or 40; K50 call pays 10 or 0; growth factor 1.02. Solution: Hedge = 10/20 = 0.5. Terminal borrowing = 20; present borrowing = 20/1.02 = 19.607843. Call value = 25 − 19.607843 = 5.392157 per unit. The pricing weight is (1.02 − 0.8)/0.4 = 0.55; 0.55 × 10/1.02 confirms the same value.

Takeaway: Two-state replication values only the declared model states; its pricing weight is not a real-world forecast. Reference links: product card, formula; glossary: Replication; Binomial model. Next: C4T2L4. Topic test.

Sources: Borsa Istanbul: Options (retrieved 2026-10-26; General information and basic concepts). Borsa Istanbul: Settlement Method (retrieved 2026-10-26; Settlement by contract family). Options Industry Council: Black-Scholes Formula (retrieved 2026-10-26; Inputs and pricing-model limitations). GIB: Provisional Article67 Guide for Individuals (retrieved 2026-10-26; Sections1,3–9 and19; publication introduction). Revision 2026-10-26; recheck changing terms before publication.

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