Derivative Prices Depend on How an Incomplete Market Is Completed

Summary

The document considers a stochastic volatility model with more sources of risk than traded assets, so the market is incomplete. It asks whether derivative prices depend on the additional derivative selected to complete the market. The answer says they do: incompleteness leaves insufficient market constraints to uniquely determine all derivative prices, and the price assigned to the added instrument supplies an important constraint.

The explanation uses a simple interpolation analogy: a polynomial cannot be uniquely determined from too few observed points. It also notes that a derivative used to hedge volatility risk must have a known price to value the hedged position. The discussion is conceptual and does not derive a pricing measure, show a numerical example, or specify conditions under which a particular completion is preferred. It therefore establishes price dependence in principle without comparing alternative hedging instruments or calibration methods.

Key ideas

• A market with more risk sources than traded assets is incomplete.
• In an incomplete market, available prices do not uniquely determine every derivative price.
• The chosen instrument used to complete the market adds a constraint that can affect pricing.
• A derivative used to hedge volatility risk must have an assigned price for the hedged position to be valued.

Tags

# market completion under stochastic volatility model


# market completion under stochastic volatility model












Consider a stochastic volatility model. As there are two sources of risk and one asset only, this is an incomplete market. One can complete the market by considering a derivative V1 used to hedge the volatility risk. My question is: Do derivatives prices depends on the derivative V1 one chooses to complete the market? And if no, why?

## Answer by joelhoro (score 2)

https://quant.stackexchange.com/a/3909

Of course. The whole point of pricing derivatives is that it is an interpolation exercise. Saying that the market is incomplete means there are not enough constraints to determine the price of all derivatives. It's a bit as if you had a polynomial of degree 2 but you gave only 2 points rather than 3.

So yes, the price of your V1 will be vital because if you need to use it to hedge your position then for sure you need to know how much it costs...

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