Illustrates an example of complete market with volatility

Question

Illustrates an example of complete market with volatility?

Answer 1

Take volatility as an illustration. As long as we contain a lognormal equity random walk, continuous hedging, no transaction costs and perfectly divisible assets and constant volatility after that we have a complete market. If such volatility is a known time-dependent function after that the market is yet complete. This is even still complete when the volatility is an identified function of stock price and time. But immediately that volatility becomes random after that the market is no longer complete. It is because now there are more states of the world than there are linearly independent securities. Actually, we don’t know what volatility will be in the future therefore markets are incomplete.