What is Meant by 'Complete' and 'Incomplete' Markets?

Short answer

A complete market is one in which a derivative product can be artificially made from more basic instruments, such as cash and the underlying asset. This usually involves dynamically rebalancing a portfolio of the simpler instruments, according to some formula or algorithm, to replicate the more complicated product, the derivative. Obviously, an incomplete market is one in which you can't replicate the option with simpler instruments.

Example

The classic example is replicating an equity option, a call, say, by continuously buying or selling the equity so that you always hold the amount

in the stock, where

Long answer

A slightly more mathematical, yet still quite easily understood, description is to say that a complete market is one for which there exist the same number of linearly independent securities as there are states of the world in the future.

Consider, for example, the binomial model in which there are two states of the world at the next time step, and there are also two securities, cash and the stock. That is a complete market. Now, after two time steps there will be three possible states of the world, assuming the binomial model recombines so that an up-down move gets you to the same place as down-up. You might think that you therefore need three securities for a complete market. This is not the case because after the first time step you get to change the quantity of stock you are holding; this is where the dynamic part of the replication comes in.

In the equity world the two most popular models for equity prices are the lognormal, with a constant volatility, and the binomial. Both of these result in complete markets, you can replicate other contracts in these worlds.

In a complete market you can replicate derivatives with the simpler instruments. But you can also turn this on its head so that you can hedge the derivative with the underlying instruments to make a risk-free instrument. In the binomial model you can replicate an option from stock and cash, or you can hedge the option with the stock to make cash. Same idea, same equations, just move terms to be on different sides of the 'equals' sign.

As well as resulting in replication of derivatives, or the ability to hedge them, complete markets also have a nice mathematical property. Think of the binomial model. In this model you specify the probability of the stock rising (and hence falling because the probabilities must add to 1). It turns out that this probability does not affect the price of the option. This is a simple consequence of complete markets, since you can hedge the option with the stock you don't care whether the stock rises or falls, and so you don't care what the probabilities are. People can therefore disagree on the probability of a stock rising or falling but still agree on the value of an option, as long as they share the same view on the stock's volatility.

In probabilistic terms we say that in a complete market there exists a unique martingale measure, but for an incomplete market there is no unique martingale measure. The interpretation of this is that even though options are risky instruments in complete markets we don't have to specify our own degree of risk aversion in order to price them.

Enough of complete markets: where can we find incomplete markets? The answer is 'everywhere.' In practice, all markets are incomplete because of real-world effects that violate the assumptions of the simple models.

Take volatility as an example. As long as we have a lognormal equity random walk, no transaction costs, continuous hedging, perfectly divisible assets,..., and constant volatility then we have a complete market. If that volatility is a known time-dependent function then the market is still complete. It is even still complete if the volatility is a known function of stock price and time. But as soon as that volatility becomes random then the market is no longer complete. This is because there are now more states of the world than there are linearly independent securities. In reality, we don't know what volatility will be in the future so markets are incomplete.

We also get incomplete markets if the underlying follows a jump-diffusion process. Again more possible states than there are underlying securities.

Another common reason for getting incompleteness is if the underlying or one of the variables governing the behaviour of the underlying is random. Options on terrorist acts cannot be hedged since terrorist acts aren't traded (to my knowledge at least).

We still have to price contracts even in incomplete markets, so what can we do? There are two main ideas here. One is to price the actuarial way, the other is to try to make all option prices consistent with each other.

The actuarial way is to look at pricing in some average sense. Even if you can't hedge the risk from each option it doesn't necessarily matter in the long run. Because in that long run you will have made many hundreds or thousands of option trades, so all that really matters is what the average price of each contract should be, even if it is risky. To some extent this relies on results from the Central Limit Theorem. This is called the actuarial approach because it is how the insurance business works. You can't hedge the lifespan of individual policyholders but you can figure out what will happen to hundreds of thousands of them on average using actuarial tables.

The other way of pricing is to make options consistent with each other. This is commonly used when we have stochastic volatility models, for example, and is also often seen in fixed-income derivatives pricing. Let's work with the stochastic volatility model to get inspiration. Suppose we have a lognormal random walk with stochastic volatility. This means we have two sources of randomness (stock and volatility) but only one quantity with which to hedge (stock). That's like saying that there are more states of the world than underlying securities, hence incompleteness. Well, we know we can hedge the stock price risk with the stock, leaving us with only one source of risk that we can't get rid of. That's like saying there is one extra degree of freedom in states of the world than there are securities. Whenever you have risk that you can't get rid of you have to ask how that risk should be valued. The more risk the more return you expect to make in excess of the risk-free rate. This introduces the idea of the market price of risk. Technically in this case it introduces the market price of volatility risk. This measures the excess expected return in relation to unhedgeable risk. Now all options on this stock with the random volatility have the same sort of unhedgeable risk, some may have more or less risk than others but they are all exposed to volatility risk.

The end result is a pricing model which explicitly contains this market price of risk parameter. This ensures that the prices of all options are consistent with each other via this 'universal' parameter. Another interpretation is that you price options in terms of the prices of other options.

References and Further Reading

Joshi, M 2003 The Concepts and Practice of Mathematical Finance. Cambridge University Press

Merton, RC 1976 Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics 3 125-144

Wilmott, P 2006 Paul Wilmott on Quantitative Finance, second edition. John Wiley & Sons Ltd

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