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TERM STRUCTURE MODELS: TREES

In this second part of the book we move one step forward in the understanding of fixed income instruments. In particular, we introduce the notion of no arbitrage, and the basics of term structure modeling. To advance the topic, let’s consider the following example.

Example

Consider a trader in a prominent investment bank. By using a bootstrap methodology or any of the other methodologies discussed in Chapter 2, the trader has estimated the current discount function Z (0, T). Recall that Z (0, T) gives the value today (0) of one dollar at time T.

If a client asks the trader to quote the price of 10%, 5-year, T-bond, the trader has all the information needed. The price can be computed from

Suppose now that the client asks the trader to quote the price of a 10%, 5-year, callable T-bond, that is, such that the Treasury has the option to buy it back at par at some date in the future.

• How can the trader compute the value of such a bond?
• How can the trader effectively hedge it?

Methodology 1 (Naive)

Naively we can follow this reasoning:

1. We have data on interest rates, so we can use past data on interest rates to forecast future interest rates.
2. The Treasury will exercise the options when interest rates are low in the future, as low interest rates imply high bond prices.
3. By forecasting future interest rates, we can forecast the future cash flows of ...

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