Why Black–Scholes Option Value Equals the Initial Replication Cost
Summary
The document explains the link between an option’s Black–Scholes value and the cost of replicating its payoff. It asks whether continuous rebalancing creates a loss from buying the underlying after it rises and selling after it falls, even when trading is frictionless and time steps become infinitesimal.
The answer clarifies that, in the model, the hedge’s cost is paid upfront to establish the replicating portfolio. Subsequent trades are financed by borrowing or investing at the risk-free rate, so the portfolio is self-financing and incurs no additional trading cost. At expiry, it matches the derivative payoff; therefore, its initial value equals the option price. This addresses the apparent rebalancing cost under ideal Black–Scholes assumptions, but does not analyze how transaction costs, discrete trading, or other market frictions would change the result.
Key ideas
- Black–Scholes prices an option by the initial cost of a replicating portfolio.
- In a frictionless continuous-time model, the hedge is self-financing after it is established.
- Purchases of the underlying are financed through borrowing, while sale proceeds earn the risk-free rate.
- The replicating portfolio matches the option payoff at maturity.
- The explanation depends on idealized assumptions and does not quantify real-world trading frictions.
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Full text
# The source of "Cost of hedging" in the Black Scholes model # The source of "Cost of hedging" in the Black Scholes model I am trying to get some intuition for the fact that a Black-Scholes price for an option is equal to the cost of replicating the option. Say the interest is 0. The option is obviously still worth something, which must be the cost of hedging the option. From what I have read, this cost comes from the fact that you always rebalance your portfolio after the stock has moved, so you incur a small loss from "buying high, selling low". But this is not completely clear when considering the fact that we are trading in continuous time and in a frictionless market. Does it mean that even in the limit, for infinitesimal time steps, the stock moves faster than the trading strategy? In that case, which assumptions in the BS-model create this effect, and on the other hand what assumptions would make the cost of hedging equal 0? ## Answer by pbr142 (score 1, accepted) https://quant.stackexchange.com/a/17877 In the BS model, with friction-free markets in continuous time, the cost of the hedging portfolio is the initial cost of setting up the portfolio. There are no costs over time as the hedging portfolio is self-financing: any purchase of the underlying is paid for by borrowing money and any selling of the underlying is invested at the risk-free rate. At maturity of the derivative, the hedging portfolio will have exactly the same value as the payoff of the derivative. Hence, the cost today of setting up the hedging portfolio has to be the same as the price of the derivative.
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