# LO 11.4: Distinguish between true and risk-neutral probabilities, and apply this

LO 11.4: Distinguish between true and risk-neutral probabilities, and apply this difference to interest rate drift.
Using the 0.5 probabilities for up and down states as shown in the previous example may not produce an expected discounted value that exactly matches the market price of the bond. This is because the 0.5 probabilities are the assumed true probabilities of price movements. In order to equate the discounted value using a binomial tree and the market price, we need to use what is known as risk-neutral probabilities. Any difference between the risk-neutral and true probabilities is referred to as the interest rate drift.
U s i n g t h e R i s k -N e u t r a l In t e r e s t R a t e T r e e
There are actually two ways to compute bond and bond derivative values using a binomial model. These techniques are referred to as risk-neutral pricing.
The first method is to start with spot and forward rates derived from the current yield curve and then adjust the interest rates on the paths of the tree so that the value derived from the model is equal to the current market price of an on-the-run bond (i.e., the tree is created to be arbitrage free). This is the method we used in the previous example. Once the interest rate tree is derived for an on-the-run bond, we can use it to price derivative securities on the bond by calculating the expected discounted value at each node using the real-world probabilities.
The second method is to take the rates on the tree as given and then adjust the
probabilities so that the value of the bond derived from the model is equal to its current market price. Once we derive these risk-neutral probabilities, we can use them to price derivative securities on the bond by once again calculating the expected discounted value at each node using the risk-neutral probabilities and working backward through the tree.
The value of the derivative is the same under either method.
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Topic 11 Cross Reference to GARP Assigned Reading – Tuckman, Chapter 7