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In our last article on Hedging the sale of a Call Option with a Two-State Tree we showed that there was one unique price for a call option on an underlying stock, in a world with two-future states.

Risk Neutral Pricing of a Call Option with a Two-State Tree | QuantStart

This was guaranteed by the principle of no arbitrage. The most surprising consequence of the argument was that the probability of the stock going up or down did not factor into the discussion. It will be necessary to read the prior articles on the Binomial Trees in order to familiarise yourself with the example of the stock and option before proceeding.

Click for Part 1 and Part 2. We will use a probability argument for this particular technique, which is known as risk neutral pricing.

How does this affect the expected value of the stock in tomorrow's world? Note that this is a risk free price because we are still setting interest rates to zero and can synthesise the stock using zero-coupon bonds worth , as before.

Rational pricing - Wikipedia

It is very important to realise that we have assumed any purchasers of this stock are risk neutral and do not need to be compensated for taking on the extra risk associated with a stock that can take on two differing values.

In reality, this is not likely to be the case. The fact that we can hedge the entire portfolio has removed the diversifiable risk and thus eliminated the premium usually associated with holding this risk.

This would in fact lead to an arbitrage opportunity. In order to see this, we could borrow money at the risk free rate which is currently zero! Tomorrow when the bond matures we could sell the stock, which is guaranteed to increase in value, and then pay back the bond, leaving us with a risk-free profit.

Thus the probabilities in this argument are only present in order to allow both world states to occur. So what is actually happening here and why does this method work?

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Risk Neutral Pricing of a Call Option with a Two-State Tree. By Michael Halls-Moore on September 25th, In our last article on Hedging the sale of a Call Option with a Two-State Tree we showed that there was one unique price for a call option on an underlying stock, in a world with two-future states.

Next we will consider a third method of pricing an option, that of replication.