In this chapter we discuss when it is necessary to use stochastic rates to price a derivative and what effects they have on the prices, giving the conditional trigger swap TARN, convertible bond and exchangeable bond as examples.
All options depend on interest rates through the discounting of future payments. If we treat interest rates as stochastic, then the money market account (often used as our numeraire) becomes stochastic. So apart from the explicit hybrid products, where we receive payments based on both interest rate market observables (LIBOR and CMS rates) and equities, we may also need to consider stochastic interest rates when pricing options where the value of some equity/index affects the time when we receive some payments. A good example of this is the target redemption note (see section 4.3).
Whether or not we choose to price a particular option with stochastic rates will depend on what risks we think are significant and against which we need to hedge ourselves. Interest rates tend to be less volatile than equities, with typical short-rate volatilities being around a few percent, whereas equity volatilities may be of the order 10% to 100%. Often, the effect of stochastic rates will be swamped by the effects of the more volatile equities, and it will not be necessary to use a more CPU-intensive two-factor model.
Stochastic LIBOR and CMS rates The most obvious effect of stochastic interest rates ...