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reza
Senior Member

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Tue Jan 07, 03 09:13 PM

just to get the discussion going:
a complete market is a market where any derivative product can be dynamically replicated via cash and the underlying asset.
needless to say the Black-Scholes set of assumptions corresponds to a complete market.
this is why the market price of the risk does not appear in their PDE.

under the assumption of (non degenerate) stochastic volatility, this property is lost
some (Peter Carr) call the markets complete even here, assuming we can include a vanilla option in the replication portfolio

I know there is long discussion on the Technical Forum ... so I let others talk

Vincent
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Wed Jan 08, 03 04:18 AM

Some complements for reza's points: complete market is perfectly hedgeable. Incomplete market is not perfectly hedgeable. In point of probabilistic view, there exists unique martingale measure for complete market, many infinitely martingale measure for incomplete. Incomplete market is due to non-tradable assets, if banks are willing to trade these asset, the incomplete market will become complete. e.g stochastic vol model, if vol swap is tradable, then sv becomes complete.

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Vincent

Marsden
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Wed Jan 15, 03 09:52 PM

I think of a complete market as being one in which any desired profile -- whether in payoff outcome or in matching of derivatives/greeks -- can be achieved through available securities. And an incomplete market is one in which there are profiles that cannot quite be matched. Reza's definition requires the ability to costlessly and instantaneously rebalance, but I don't think this is really a necessary condition for a complete market.

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"Socialism collapsed because it did not allow prices to tell the economic truth. Capitalism may collapse because it does not allow prices to tell the ecological truth." -- Oystein Dahle, former Vice President of Esso

Thu Jan 16, 03 02:02 PM

When you say 'any desired profile', this includes not only the underlying price but the time-dimension... Once you admit that, you should expect that costless and instantaneous rebalancing is equivalent. The mathematical nitty-gritty may be somewhat conditional on the underlying process. I'm far from expert on this subject so I'm not exactly sure if things like jump-diffusion processes can have complete markets under this definition (i.e. limit orders). I'm also a little confused about whether a complete market with some underlying processes must include infinitely many underlyings so that portfolio strategies can produce any desired profile (so that you could try to take the risk out of the jumps through diversification and other kinds of CAPM-style tricks).

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young
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Fri Jan 17, 03 01:42 PM

There exist complete capital markets; that is, there exist n securities with known linearly independent payoff vector and known values. These n securities form the basis for the n-dimensional outcome space. In complete market, we can make any other cash flow with linear combinations of price from the control securities.

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Fri Jan 17, 03 02:41 PM

I challenge the board to identify a single complete market.

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David
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Fri Jan 17, 03 04:18 PM

In complete markets, information is equally distributed between all precipitants. It is also highly transparent ? the reliability of the information is very high with no error transcription of data that occur in incomplete markets.

The dissemination of real and best existing prices, bid-ask spread as well as the trades executed is visible to all precipitants at al levels. This is in my humble opinion an ideal complete market. Otherwise, it is an incomplete market.

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Johnny
Senior Member

Fri Jan 17, 03 06:02 PM

I agree with Young's definition of market completeness in terms of the number of linearly independent securities being equal to the number of possible states of the world. This definition is frequently used in the economics literature. A simple example is a binomial tree in a world with one (stochastic) share and one risk free bond, where there are only two possible states of the world at any time.

I do not agree with David's characterisation of complete markets as requiring all agents to have (a) the same information where (b) this information is "highly transparent". Although I agree that there are many models which assume both market completeness and homogeneous information, they are nonetheless quite separate assumptions. I could easily construct a model which assumed incomplete markets and homogeneous perfect information - for example, see much of the rational expectations literature.

Marsden
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Sun Jan 19, 03 10:13 AM

kr:

Suppose that at time T two separate portfolios have the exact same payout under all outcomes. Then for any time 0<t<T they must have the same value. Thus, the time dimension profile is generally matched implicitly, and need not have its own accounting.

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"Socialism collapsed because it did not allow prices to tell the economic truth. Capitalism may collapse because it does not allow prices to tell the ecological truth." -- Oystein Dahle, former Vice President of Esso

Tue Jan 21, 03 07:44 PM

Yes, in theory this is true, but probably the single largest hole in the theory is that most instruments are not tradeable 'on demand', and the risk that you can't trade when you want to in quantity has a serious impact on pricing and hedging. The qualitative phenomenon is that good liquidity is correlated with good market behavior, and this lack of completeness is an issue of first-magnitude. To take the first example that comes to mind, the interplay between portfolio insurance and the 1987 crash falls squarely into this category.

I am acutely aware of this fact because I play the 'cash is king' game every day of the week to distressed companies, offering oxygen at a not-too-extortionate price when the existing bank group is pulling the plug. To take it from yet another angle, if market vol were more of a storable commodity, then turbulence and liquidity potholes would be less of a problem. But knowing how to do that is a kind of alchemy, which is introduced to complete the market at a price.

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Johnny
Senior Member

Tue Jan 21, 03 08:15 PM

"I challenge the board to identify a single complete market. "

I challenge the board to identify a single ideal gas, or frictionless surface, or rigid body, or perfect electrical resistor, or perfect vacuum, or ...

Alan
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Sun Jan 26, 03 02:21 AM

Quote
Originally posted by: kr
I challenge the board to identify a single complete market.

In the U.S., most states have a lottery. These are complete -- every possible outcome is available for purchase.

rcohen
Senior Member

Posts: 251
Joined: Nov 2001

Mon Jan 27, 03 01:59 PM

Quote
Originally posted by: Johnny

I challenge the board to identify a single ideal gas, or frictionless surface, or rigid body, or perfect electrical resistor, or perfect vacuum, or ...

Johnny,

You don't need to challenge the board for an answer to these questions. You could easily challenge the whole world.

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Ockham's razor: "All other things being equal, the simplest solution is generally the right one."

Mon Jan 27, 03 02:20 PM

Even in the lottery you can not easily buy derivatives that provide non-lottery payouts based on the numbers that are drawn... i.e. pay me $0.40 if at least one '6' is drawn. Not that it would be that interesting, but I bet people would still buy them - it would make the whole thing a lot more like roulette, which is popular for a reason well beyond any reason I can discern.

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mrbadguy
Senior Member

Wed Feb 26, 03 09:30 PM

With reference to financial market, options and futures make market dinamically complete because, in theoretical terms, any option payoff could be replicated by trading underlying asset and related riskless bonds. Studies on options show that they make the market statically complete, because every trader could hedge any stock portfolio with correspondent extended option portfolio. This is an ideal environment, in fact officially quoted and OTC instruments sometimes cannot cover all ranges of strikes and expiries, matching correct volatilities of underlyings; the real quest are ways of reaching this completeness.

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mrbadguy