Identifying Incomplete Markets and Unattainable Claims
Summary
The discussion explains that counting Brownian drivers and risky assets alone does not establish market incompleteness. In the example, the two assets depend on three named Brownian motions, but their exposures can be rewritten using only two independent Brownian sources. Redundant representation therefore does not create an unhedgeable risk; the Black–Scholes model would otherwise appear incomplete simply by expressing it with extra Brownian motions.
The answer contrasts this with stochastic volatility models, where volatility introduces a distinct source of randomness that cannot be reduced to the stock-price driver. Claims whose values depend on the volatility drift may then be unattainable under the model's traded instruments. The discussion gives a conceptual explanation rather than a procedure for constructing a particular claim, and its broad statement about attainability in Heston should be read in the context of the model assumptions and available hedging assets.
Key ideas
- The number of named Brownian motions does not by itself determine whether a market is incomplete.
- Redundant Brownian representations can make a complete model appear to have more sources of risk than it actually has.
- Stochastic volatility can introduce a separate risk source that traded assets do not span.
- Claims sensitive to an unspanned volatility drift may be unattainable without additional instruments.
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# Non attainable claim - Incomplete market
# Non attainable claim - Incomplete market
I am wondering whether there is a standard procedure to find a non attainable (i.e. non replicable) asset in an incomplete market. As an example, let us have the following market ($B = (B^1, B^2, B^3)$ is a $\mathbb{P}$-SBM): $$dX_t^0 = 0 \\ dX_t^1 = dt + dB_t^1 + dB_t^2 - dB_t^3 \\ dX_t^2 = 5dt - dB_t^1 + dB_t^2 + dB_t^3$$ This is clearly incomplete (number of risky assets is less than dimension of the underlying brownian motion), which means a non attainable claim exists. How do I find such a claim? What should I look for?
## Answer by Andrea (score 1)
https://quant.stackexchange.com/a/81173
This is clearly not incomplete.
You have simply expressed it in a complicated way, only 2 BMs are necessary. If your statement were true, then even the BS model would be incomplete as you could express it (redundantly) with multiple BMs.
A model which is really incomplete is a stochastic volatility model (e.g. Heston) since it is not possible to reduce it to a single source of randomness.
To answer your question for the Heston model: no claim is attainable.
Or better, the cause is related to a change of measure, which for BMs is the drift.
Any drift in your model must be constrained: for $S$ it must be $r$ due to the forward argument. You need something for the vol too.
So, if a claim's price depends on the vol drift, then it is incomplete.
And how do you complete it? You pick any such claim with an exogenous price and calibrate the drift.
This is only a theoretical consideration since one has to calibrate anyway all the other parameters as well at the same time.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.