In mathematical finance, a replicating portfolio for a given asset or series of cash flows is a portfolio of assets with the same properties. This is meant in two distinct senses: static replication, where the portfolio has the same cash flows as the reference asset, and dynamic replication, where the portfolio does not have the same cash flows, but has the same "Greeks" as the reference asset, meaning that for small changes to underlying market parameters, the price of the asset and the price of the portfolio change in the same way. Dynamic replication requires continual adjustment, as the asset and portfolio are only assumed to behave similarly at a single point. Given an asset or liability, an offsetting replicating portfolio is called a static hedge or dynamic hedge, and constructing such a portfolio is called static hedging or dynamic hedging. The notion of a replicating portfolio is fundamental to rational pricing, which assumes that market prices are arbitrage-free – concretely, arbitrage opportunities are exploited by constructing a replicating portfolio. In practice, replicating portfolios are seldom, if ever, exact replications. Most significantly, unless they are claims against the same counterparties, there is credit risk. Further, dynamic replication is invariably imperfect, since actual price movements are not infinitesimal – they may in fact be large – and transaction costs to change the hedge are not zero.
Applications
Derivatives pricing
Dynamic replication is fundamental to the Black–Scholes model of derivatives pricing, which assumes that derivatives can be replicated by portfolios of other securities, and thus their prices determined. See explication under Rational pricing #The replicating portfolio. In limited cases static replication is sufficient, notably in put–call parity. An important technical detail is how cash is treated. Most often one considers a self-financing portfolio, where any required cash is borrowed, and excess cash is loaned.
Insurance
In the valuation of a life insurance company, the actuary considers a series of future uncertain cashflows and attempts to put a value on these cashflows. There are many ways of calculating such a value, but these approaches are often arbitrary in that the interest rate chosen for discounting is itself rather arbitrarily chosen. One possible approach, and one that is gaining increasing attention, is the use of replicating portfolios or hedge portfolios. The theory is that we can choose a portfolio of assets whose cashflows are identical to the magnitude and the timing of the cashflows to be valued. For example, suppose your cash flows over a 7-year period are, respectively, $2, $2, $2, $50, $2, $2, $102. You could buy a $100 seven-year bond with a 2% dividend, and a four-year zero-coupon bond with a maturity value of 48. The market price of those two instruments might be $145 - and therefore the value of the cashflows is also taken to be $145. Such a construction, which requires only fixed-income securities, is even possible for participating contracts. The proof relies on a fixed point argument. It should be clear that the advantages of a replicating portfolio approach include:
Valuing options and guarantees can require complex nested stochastic calculations. Replicating portfolios can be set up to replicate such options and guarantees. It may be easier to value the replicating portfolio than to value the underlying feature. For example, bonds and equities can be used to replicate a call option. The call option can then be easily valued as the value of the bond/equity portfolio, hence not requiring one to value the call option directly. For additional information on economic valuations and replicating portfolios can be found here:
