Replicating portfolio
Quantitative Finance glossary · Derivatives Pricing & Volatility Surfaces
Definition
Dynamically-rebalanced self-financing portfolio in the underlying (Δ_t shares) and risk-free bond that exactly matches an option's payoff at expiry. In a complete market the cost of replication equals the no-arbitrage price: V_0 = mathbbE^mathbbQ[e^-rT,payoff]. The constructive content of the BSM PDE — every option is, in continuous time, a synthetic position in the underlying. In incomplete markets (jumps, stochastic vol with two-sided unhedged risk) perfect replication fails and one resorts to super-replication, quadratic hedging, or utility-indifference pricing.
Replicating portfolio in plain English
Instead of buying an option, imagine you could brew up a homemade copy of it using just two ingredients — some shares of the stock and some cash in a savings account — and keep adjusting the recipe as the price moves so that, at the end, your mixture is worth exactly what the option would have paid. That's a replicating portfolio, and it's the deep idea behind option pricing: if you can perfectly copy the payoff, then the option must cost exactly what the copy costs, or someone earns free money. The Δ_t tells you how many shares to hold at each moment, and 'self-financing' means you never pour in or pull out extra cash — you just rebalance what's already there. This works cleanly in idealized 'complete' markets; when prices can suddenly jump or wobble in ways you can't fully offset, an exact copy is impossible and traders settle for the best imperfect hedge they can build.
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