Fixed Income Securities: Valuation, Risk, and Risk Management

PART III

TERM STRUCTURE MODELS: CONTINUOUS TIME

In this third part of the book we discuss one of the most applied tools used to price and hedge fixed income instruments, namely, continuous time methodologies. In a nutshell, we consider the case in which underlying variables, such as interest rates or yields, move at a high frequency, such as daily and even intra-daily. Why do we want to do that? For at least three reasons:

1. Realism: Market variables do move at a high frequency. That is, if a bank sells a derivative product and it has to hedge its variation by using some underlying security, the bank will need to mark-to-market and rehedge the position very frequently, typically daily. While trees are excellent devices, a model that takes into account the high frequency nature of trading is welcome, as it offers immediacy to the trading necessities.
2. Simplicity: A number of analytical tools are available to analyze fixed income instruments and their derivatives when the trading interval time is small (when, in fact, it converges to zero).
3. Analytical Formulas: For many fixed income instruments and derivatives, we will obtain analytical formulas for their prices or hedge ratios. These analytical formulas enourmously influence the speed at which trades can be implemented. The ability to react swiftly to market changes is an important tool in modern markets.