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Bjørn Kjos-Hanssen
Bjørn Kjos-Hanssen
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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)}$$S=\\\$1_{H(\omega_1)} 1_{H(\omega_2)}$ i.e. we get 1 dollar if the firstboth coin toss istosses are heads and 0 if it is tailsotherwise.

Now we can complete the market by introducing an option that has payofftwo options having payoffs $$ V=h \\\$1_{H(\omega_1)} +\\\$1_{H(\omega_2)} $$$$ V_1=h S +\\\$1_{H(\omega_2)} $$ $$ V_2=\\\$1_{H(\omega_1)} $$ where $h$ is a huge number.,

Assume there are no arbitrage opportunities and the price of the option $V$$V_1$ is $\\\$1$. Then $H(\omega_1)$$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 theany introduced option to be given then introducing the option has no effect.

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 $S=\\\$1_{H(\omega_1)}$ i.e. we get 1 dollar if the first coin toss is heads and 0 if it is tails.

Now we can complete the market by introducing an option that has payoff $$ V=h \\\$1_{H(\omega_1)} +\\\$1_{H(\omega_2)} $$ where $h$ is a huge number.

Assume there are no arbitrage opportunities and the price of the option $V$ is $\\\$1$. Then $H(\omega_1)$ must be very unlikely and so the original asset $S$ is now essentially worthless. But if we don't assume the price of the introduced option to be given then introducing the option has no effect.

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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Bjørn Kjos-Hanssen
Bjørn Kjos-Hanssen
  • 24.8k
  • 3
  • 58
  • 114

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 $S=\\\$1_{H(\omega_1)}$ i.e. we get 1 dollar if the first coin toss is heads and 0 if it is tails.

Now we can complete the market by introducing an option that has payoff $$ V=h \\\$1_{H(\omega_1)} +\\\$1_{H(\omega_2)} $$ where $h$ is a huge number.

Assume there are no arbitrage opportunities and the price of the option $V$ is $\\\$1$. Then $H(\omega_1)$ must be very unlikely and so the original asset $S$ is now essentially worthless. But if we don't assume the price of the introduced option to be given then introducing the option has no effect.