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I'm currently reading the following lecture notes: http://www1.maths.leeds.ac.uk/~jitse/math2515/lecture04.pdf

On the second page, under the subsection titled "The Risk-Neutral World" it points out that the model previously used to value options contracts does not take into account the probability of prices going up or down. It then suggests that this may be because a stocks current price takes into account the probability of it going up (or down) in the future.

Can anyone explain, in simple terms, why this might be true?

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well, the current share price reflects fair value. So you'd expect it to be close to its expected price, but slightly below because of risk aversion and discounting. If it was very far off its expectation, it would either be over or under valued and people would trade accordingly.

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Mark has rightly pointed.

You may think like this, If there is high probability price would go up in future(in very short period), lets say 90% then investors would continue to buy until they no longer expect price to increase with such a high probability. Or until this strategy of buying shares with high probability of going up is not profitable(excess risk adjusted return).

Similarly, if there is very high probability that price will fall in future, then investors would continue to sell until they no longer expect price to fall beyond that level. Or until this strategy of selling shares with high probability of going down is not profitable(excess risk adjusted return).

So your question : is it realistic to assume that current price of stock takes into account probability of going up and down in the future? Answer is completely yes. After all price are not deterministic.

Lets come to your dilemma on pricing of option contract. Since derivatives derive their value from some underlying assets it is completely realistic to assume that current price of security is fair( only for actively traded) and hence does not require to take into account expected return on the underlying one.

I hope it would help you to understand.

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What the author is arguing is that the current price exactly at this instant takes into account all the views of the market participants as expressed by orders (which would be correct). Note that those views may change almost instantaneously generating orders thus causing price to change.

An example is you have an expected future price of 10 and the current price is 9, therefore you will allocate a certain amount of capital to this trade (say 1 lot - 100 shares). Let's assume that the next Ask is 9.10 @ 1 lot. So the price is now $9.10, given enough participants with views as you the price will move accordingly.

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