Identifier: C4T2L4 | 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. Read a conditional model-output table. 2. Reproduce valuation from supplied components. 3. Identify features that invalidate the basic model.

Prerequisite: C4T2L3.

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

Black–Scholes is a model for conditional option value, not a market-price guarantee. The basic European form uses current asset price, strike, time in years, volatility, rate and a distribution assumption. It assumes an idealized continuous price process and frictionless replication. Discrete dividends, early exercise, barriers, changing volatility, jumps and funding constraints require appropriate treatment. A familiar formula does not make an unsupported input reliable. Model output should carry its valuation time and quotation unit.

Price and return drivers

For an accessible calculation, write call value as discounted asset component × supplied N1 minus discounted strike component × supplied N2. N1 and N2 are model-generated normal-distribution weights; learners need not compute an integral. They are not two real-world state probabilities. For zero distributions and rates, the components are S and K. A basic zero-rate at-the-money call must increase when the supplied volatility or time increases, all else equal; an executable bid may still differ from model value.

Worked case

Original model outputs, no live quote: European call S100 K100, T0.25 years, volatility 20%, rate0, distribution0. Supplied N1 = 0.5199388058 and N2 = 0.4800611942. Value = 100 × N1 − 100 × N2 = 3.9878 per unit. One multiplier-100 contract has model value TRY 398.78. Change only volatility to 30%: N1 0.5298926441, N2 0.4701073559, value 5.9785. Change only time to 0.5 years at 20%: N1 0.5281859889, N2 0.4718140111, value 5.6372. At rate5% with T0.25 and volatility20%, supplied discounted strike 98.75778005 and weights 0.5694601832,0.5298926441 give value 4.6150. Adding a continuous distribution yield3% gives discounted asset99.25280548 and weights0.5398278373,0.5, value4.2005.

Adverse case

If the investor pays an executable ask 4.30 rather than model 3.9878, entry premium is TRY 430. At expiry S90,100,110, net after TRY 20 charges is −450,−450,+550. Before expiry at the unchanged base inputs, an illustrative bid3.70 yields only TRY 370 and net loss80. It is incorrect to sell in the arithmetic at model value when only a lower bid is supplied. A jump-sensitive issue can warrant a different model even if its headline inputs appear similar.

Practical use

Keep model inputs/output beside the timestamped bid and ask. Use an appropriate exercise/feature model and avoid treating extra decimal places as executable precision.

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 table to explain attribution and sensitivity to assumptions. Do not report a calculated value to four decimals as evidence of executable precision. Store actual quote sides beside the model and test uncertain volatility, distributions and rates. Decline when payoff features cannot be represented by the chosen model. A validated calculator can recompute conditional values, but cannot validate an issuer promise, a live order book or an unavailable financing leg.

Practice and references

Exercise: a supplied model decomposition has discounted asset100, N1 0.56, discounted strike98, N2 0.52. Solution: Value = 56 − 50.96 = 5.04 per unit. For 2 contracts, multiplier100, model value is TRY 1008. A supplied purchase ask5.30 costs TRY 1060 before fees, TRY 52 above that conditional value. These rounded weights are an arithmetic exercise, not an independent calibrated model specification.

Takeaway: A Black–Scholes number is a conditional model value, so compare its inputs and limitations with the actual issue and quote. Reference links: product card, formula; glossary: Black Scholes model; Model risk. Next: C4T2L5. 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.

Shared reference guide