Continuous trading makes the hedged portfolio riskless, anchoring it to the riskless rate.

Evidence

Because it offers returns that are certain, the hedged portfolio can therefore earn only the riskless rate of interest, “not for the reason of [the capital] asset pricing model but … to avoid arbitrage, or money machine”: a way of generating certain profits with no net investment (Merton interview). For Merton, then, the “key to the Black-Scholes analysis” was an assumption Black and Scholes did not initially make: continuous trading, the capacity to adjust a portfolio at all times and instantaneously. “Only in the instantaneous limit are the warrant price and stock price perfectly correlated, which is what is required to form the `perfect’ hedge.” (Merton 1972a, p. 38) Black and Scholes were not initially convinced of the correctness of Merton’s approach. — Donald MacKenzie, An Engine, Not a Camera, 600

Categorization

• DEFINE: Clarifies the arbitrage/money-machine reasoning that anchors Black-Scholes, and pinpoints continuous trading as the load-bearing distinct assumption Merton added.
• XREF: Connects to the performativity thesis: Merton’s continuous-trading assumption is a mathematical idealization that Black himself doubted, yet it became canonical in derivatives practice. — connects to: Continuous dynamic hedging can eliminate all risk, not just systematic risk. — Donald MacKenzie, An Engine, Not a Camera: How Financial…, 599; the risk-neutral valuation insight lets options be priced by discounting at the riskless rate — Donald MacKenzie, An Engine, Not a Camera: How Financial…, 583; Option prices equal the expected payoff discounted at the riskless rate — Donald MacKenzie, An Engine, Not a Camera: How Financial…, 1284

Broader Connections

• the risk-neutral valuation insight lets options be priced by discounting at the riskless rate #corroborates (0.82) — Donald MacKenzie, An Engine, Not a Camera, 583
• Option prices equal the expected payoff discounted at the riskless rate #expands (0.80) — Donald MacKenzie, An Engine, Not a Camera, 1284