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      "statement": "# Black and Scholes: pricing options by a hedged portfolio\n\n## A right to buy is different from an obligation\n\nAn option can give its holder the right to buy an asset at a specified price under specified conditions. In a simple expiry example, a call with an exercise price of one hundred has a payoff of twenty if the asset finishes at one hundred and twenty, and zero if it finishes at eighty. That payoff is not profit: acquiring the option can cost money. These invented values introduce the contract, not a trading recommendation.\n\n## Why compare an option with a portfolio?\n\nA replicating portfolio combines other assets to produce the same modeled payoff. If two freely tradable positions have identical future outcomes but different prices, one could try to exploit the difference. The no-arbitrage argument rules out such a sure gain within its assumptions. It does not assert that all uncertain investments earn the same return or that a real market always behaves exactly like an idealized mathematical model.\n\n## Try a two-outcome teaching model\n\nSuppose an asset now costs one hundred and will finish at either one hundred and twenty or eighty. Ignore interest and transaction costs in this invented one-period model. Holding half a share and borrowing forty gives final values of twenty or zero, matching the call payoff above. Its initial cost is fifty minus forty, or ten. This illustrates replication; it is a small discrete example, not the continuous-time Black–Scholes formula.\n\n## What the original contribution established\n\nBlack and Scholes derived an option-valuation relation from a continuously adjusted hedged portfolio under explicit assumptions about markets and the price process. The mechanism is the key: remove the modeled risk in a way that constrains the price. Continuous rebalancing, the treatment of volatility and trading frictions matter when interpreting the result. Merton’s separate 1973 contribution and later extensions retain their own attribution. Neither a formula nor this WRS score guarantees a profitable trade.\n\n## Check a model without risking money\n\nAsk your agent to verify the two possible final portfolio values, then add a hypothetical transaction fee and explain why the previous exact comparison changes. Keep payoff, purchase cost and profit in different columns. A useful result is a transparent assumptions check with reproducible arithmetic. Do not use the toy price as a quote for any real option or silently extend two possible outcomes to the continuous model.\n\n## Scientific scope\n\nThis records the Black–Scholes paper, not guaranteed trading profit or universal market truth. Merton’s separate 1973 paper and extensions remain separate.\n\n## Sources for this explanation\n\n- [Black and Scholes, The Pricing of Options and Corporate Liabilities (1973)](https://www.journals.uchicago.edu/doi/10.1086/260062): Source for the recorded historical contribution and its scope. Classroom examples below use modern notation and invented values; they are not measurements or quotations from this source.\n",
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        "title": "A right to buy is different from an obligation",
        "text": "An option can give its holder the right to buy an asset at a specified price under specified conditions. In a simple expiry example, a call with an exercise price of one hundred has a payoff of twenty if the asset finishes at one hundred and twenty, and zero if it finishes at eighty. That payoff is not profit: acquiring the option can cost money. These invented values introduce the contract, not a trading recommendation."
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        "text": "A replicating portfolio combines other assets to produce the same modeled payoff. If two freely tradable positions have identical future outcomes but different prices, one could try to exploit the difference. The no-arbitrage argument rules out such a sure gain within its assumptions. It does not assert that all uncertain investments earn the same return or that a real market always behaves exactly like an idealized mathematical model."
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        "text": "Suppose an asset now costs one hundred and will finish at either one hundred and twenty or eighty. Ignore interest and transaction costs in this invented one-period model. Holding half a share and borrowing forty gives final values of twenty or zero, matching the call payoff above. Its initial cost is fifty minus forty, or ten. This illustrates replication; it is a small discrete example, not the continuous-time Black–Scholes formula."
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