Why Dynamic Stock Hedging Can Replicate an Option’s Greeks

Summary

The document explains why a replicating strategy made from cash and the underlying stock can reproduce an option’s behavior even though cash and stock do not themselves have the option’s full set of Greeks. A static holding in those assets provides delta exposure, while a continuing rule for changing the stock position as prices move creates the additional effects associated with gamma. The strategy’s response to volatility changes can likewise reflect the expected future rebalancing activity, which is the sense in which it can replicate vega.

The explanation emphasizes that replication belongs to a trading strategy over time, not to the Greeks of a frozen portfolio at one instant. It offers a conceptual account rather than equations, a worked hedge, or empirical evidence. Its scope is therefore limited: it does not specify the assumptions required for exact replication, quantify transaction costs, or discuss how discrete rebalancing and model error affect the result.

Key ideas

• A static stock and cash portfolio does not have an option’s full set of Greeks.
• Dynamic changes to the stock holding can reproduce option-like exposures over time.
• Adjusting the hedge as the underlying price changes creates a gamma-like effect.
• Vega replication is attributed to volatility changing the expected future hedging activity.
• The document gives an intuition rather than a quantified replication method.

Tags

# Replicating an option


# Replicating an option












When we replicate a portfolio of cash and stock for a call option, shouldn't the replicating portfolio's greeks be equal to options greeks?

Is that true? If it is, how is it that a portfolio of cash and stock has same vega as the option, since the vega of stock and cash is 0. What am I missing here?

Do the weights on cash and stock change in such a way with time such as to mimic the greeks, but at any single point the greeks of the portfolio aren't equal to options?

## Answer by Attack68 (score 3, accepted)

https://quant.stackexchange.com/a/40218

The pricing of options is married with the concept of a hedging strategy that replicates the effect of the option. If you can only long or short a stock that will not replicate the greeks, it only creates delta. It is the commitment to the strategy that achieves it.

For example if the price goes up and you are committed to buying more to increase your delta then you have simulated gamma. And if volatility increases then your expected activity under your hedging strategy also increases hence you have simulated vega, but it doesn't inherently exist in your portfolio of just owning a stock.

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.