Why Replicating an Option Requires Initial Capital
Summary
The document clarifies the relationship between option prices and dynamic replication in the frictionless Black–Scholes framework. Frictionless trading means rebalancing a self-financing replicating portfolio has no transaction costs or imposed limits; it does not mean the portfolio starts with zero value. By no arbitrage, the option price equals the initial value of a portfolio that replicates its payoff.
The key mathematical distinction is that a discounted self-financing portfolio is a martingale with constant expected value under the risk-neutral measure, not necessarily an asset with zero expected value. A payoff that is positive at maturity, such as a call or put, generally requires initial funding or borrowing to construct its replication. The explanation is conceptual rather than a derivation of the Black–Scholes formula, and it relies on the idealized assumptions of frictionless markets and successful replication.
Key ideas
- Frictionless assumptions remove transaction costs for rebalancing a self-financing portfolio.
- A replicating portfolio can still require capital at inception.
- No arbitrage equates the derivative price with the initial value of its replicating portfolio.
- A martingale has constant expected value, which need not be zero.
- Options with positive terminal payoffs illustrate why replication can have nonzero initial value.
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# Intuitive understanding of Black-Scholes pricing
# Intuitive understanding of Black-Scholes pricing
The Black-Scholes formula entails market completeness, so the price of an option is only the cost associated with dynamically hedging the option.
Where does this cost come from? I don't see how sustaining a replicating portfolio in a frictionless market can cost anything.
## Answer by AFK (score 4, accepted)
https://quant.stackexchange.com/a/12654
I think you misinterpreted what you read.
The whole point of the frictionless market assumption is that you can forget about any cost or any bound on volumes or latency associated with transactions used to rebalance a self-financed portfolio. So you are right when you say that sustaining a replicating portfolio doesn't cost anything.
This implies that the price of the option has to be equal to the initial value of any replicating portfolio (and by absence of arbitrage all replicating portoflios have the same initial value).
From a mathematical point of vue, your mistake comes from assuming that any martingale (like the actualized value of a self-financing portfolio $\widetilde{\Pi}_t := e^{-\int_0^t r_s ds} \Pi_t$, for $t\leq T$, under the risk neutral probability) has zero expected value when actually it just has constant expectated value. So if $\Pi$ is a replicating portfolio, the value of your derivative is equal to $\Pi_0$ and it can be non zero.
Take any simple example. If your terminal payoff is almost surely positive (like a call or put option), you will need some money (or to borrow some) to start your replication strategy. The replication of a bond with fixed coupons rates or the floating leg of a swap is quite enligntening imho.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.