# Approximation versus repricing **BSM-06 · Black–Scholes & Greeks · Core** **Lesson question:** Why must both methods receive the same scenario? Original fictional examples; educational writing, not live market data, orders or production approval. ## By the end - Explain the mechanism in the lesson question. - Reproduce the worked comparison and recognize its failure case. **Prerequisites:** BSM-01 A Greek approximation estimates local P/L; full repricing evaluates the same accepted new state under the model. Their difference is informative only when the input shocks, dates and units match. Comparing different applied scenarios produces a software inconsistency disguised as model error. ## Understand the mechanism A common approximation includes Delta dS + 0.5 Gamma dS² + Theta dt + Vega dSigma + Rho dr + Epsilon dq. Here dt is positive elapsed calendar years. Additional cross terms are omitted, so combined large moves can diverge. Full repricing is still exact only within its stated model and numerical accuracy, not exact future market P/L. ## Walk through the process Validate requested shocks first. Reject invalid resulting spot or negative volatility, or explicitly report a shared adjusted scenario. If elapsed time reaches expiry, use exact intrinsic for value and stop beyond the contract's life. Apply the same validated state to the approximation and repricer. State which terms are included. ## Read before applying All numerical market states are hypothetical. The calculations illustrate a European continuous-yield model, not a quote feed. Financial-model validation, instrument suitability and executable pricing are separate questions. ## A worked example Starting at the common call, apply spot +0.50, IV +1 point and one calendar day elapsed. Delta contribution is 0.263782308; Gamma 0.006934859; Theta -0.051508870; Vega 0.113997676 per unit. Estimated new value is 3.310780128; BSM revaluation at S=100.50, sigma=.26 and T=29/365 gives 3.308087339. Estimate minus revaluation is +0.002692789. The graph uses dollars per unit for every contribution. A request for ten elapsed days on a seven-day option must be rejected or explicitly limited to seven for both methods. Charging ten days of Theta while pricing seven is not an honest approximation comparison. ## Calculation popup: Scenario contributions ```text dV≈Delta dS+0.5 Gamma dS²+Theta dt+Vega dSigma ``` Use one shared scenario: dS=.50, dSigma=.01, dt=1/365. 1. Delta contribution=0.263782308201 2. Gamma contribution=0.006934858616 3. Theta contribution=-0.051508869653 4. Vega contribution=0.113997675882 5. Estimated=3.310780128268; repriced=3.308087339302. **Result:** Difference +0.002692788966. ## Graph: Approximation versus repricing—worked comparison **Horizontal:** Delta / Gamma / Theta / Vega contribution **Vertical:** Model P/L per option unit (USD) Fictional data from the written example. The chart organizes the comparison; its shape is not a forecast. Complete numerical data are embedded in `scripts/black-scholes-greeks/BSM-06.js` and displayed as an accessible table in the HTML lesson. ## What would invalidate the reading? Different scenario clamps or time horizons can make the error meaningless. Large shocks also require more than a low-order Taylor expansion. ## Check your understanding ### What is the essential comparison requirement? **Answer:** Both methods use the same accepted shocks and model conventions. **Why:** The remaining difference can then be studied as approximation error. ### What is the critical failure to avoid? **Answer:** Different scenario clamps or time horizons can make the error meaningless. Large shocks also require more than a low-order Taylor expansion. **Why:** Explain this risk before treating the example as a decision rule. A correct calculation does not guarantee a favorable market outcome. **Critical risk check.** ## Takeaway Return to the question: Why must both methods receive the same scenario? State the observed facts, the assumptions and the failure condition separately. ## References and provenance The cited references support definitions and conventions. The teaching prose, tickets and numerical scenarios are original; they are not attributed as historical market observations. - [OIC Greeks](https://www.optionseducation.org/advancedconcepts/understanding-options-greeks) ## Plain-English quick read ### Start small Try spot +0.25, no IV change and no time passing. Then increase the spot move and compare the approximation error. ### Change volatility separately Reset the spot shock to zero, then change IV by one percentage point. Only after that combine spot and IV. ### Use identical shocks Both methods receive the accepted input changes. A request for 30 days passing on a 7-day option is rejected, not quietly floored in just one method. ### Interpret the gap The gap is approximation error within the same model. Even perfect model agreement does not prove an executable live-market result. ## Try it in the working calculator 1. Set spot change +0.25, IV/time/rate/yield changes zero. 2. Compare local estimate and full-model repricing. 3. Increase the spot shock, then add +1 IV point and inspect Vanna/Vomma contributions. The working lab uses fictional shared inputs; the printed worked example above stays fixed. Open it from the lesson's same-page Live Greeks Lab control. The engine prices European calls and puts with continuous yield and calendar days/365.