# market completion in stochastic volatility model

Hi all,

Consider a stochastic volatility model. As there are two sources of risk and one asset only, this is an imcomplete 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?

Many thanks in advance.

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I agree with Fred that many more things have to be specified (the stock process, the pricing concept, what you mean exactly by completing and where you take the prices from, which kind of trading strategies you consider...). But instead bothering with the technicalities in the SV framework, I think it might you help much more to think first about much easier incomplete market models, lets say a one-period trinomial model. This will help you to get an idea whats going on here. Btw. I am not sure if this is the right kind of questions for MO. –  Stephan Sturm Aug 3 '12 at 23:30

## 2 Answers

The question is incomplete itself! What kind of derivative prices are you interested in ? Second, do assume the presence of a numeraire to compute the prices after completion of the market ? Otherwise, the answer is trivially yes since the unit of account chnages the prices (consider for instance the case you introduce some $V_1$ and $V_2=2\times V_1$. Details of the question need to be fixed...

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Assuming there are no arbitrage opportunities, the price of a derivative depends on the prices of other derivatives available in the market. If you introduce a derivative without giving it a price, or by giving it a price that is already theoretically determined by your initial setup, then you do not affect the prices of other derivatives.

To take a simpler example, suppose we have two sources of risk, two (biased) coin tosses $\omega_1$ and $\omega_2$ and the only tradable asset is $S=\$1_{H(\omega_1)} 1_{H(\omega_2)}$i.e. we get 1 dollar if both coin tosses are heads and 0 otherwise. Now we can complete the market by introducing two options having payoffs $$V_1=h S +\1_{H(\omega_2)}$$ $$V_2=\1_{H(\omega_1)}$$ where$h$is a huge number, Assume there are no arbitrage opportunities and the price of the option$V_1$is$\$1$. Then $S=\$1$must be very unlikely and so the original asset$S\$ is now essentially worthless. But if we don't assume the price of any introduced option to be given then introducing the option has no effect.

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