LO 14.3: Assess the efficacy o f time-dependent volatility models.

LO 14.3: Assess the efficacy o f time-dependent volatility models.
Time-dependent volatility models add flexibility to models of future short-term rates. This is particularly useful for pricing multi-period derivatives like interest rate caps and floors. Each cap and floor is made up of single period caplets and floorlets (essentially interest rate calls and puts). The payoff to each caplet or floorlet is based on the strike rate and the current short-term rate over the next period. Hence, the pricing of the cap and floor will depend critically on the forecast of cr(t) at several future dates.
It is impossible to describe the general behavior of the standard deviation over the relevant horizon because it will depend on the deterministic model chosen. However, there are some parallels between Model 3 and the mean-reverting drift (Vasicek) model. Specifically, if the initial volatility for both models is equal and the decay rate is the same as the mean reversion rate, then the standard deviations of the terminal distributions are exactly the same. Similarly, if the time-dependent drift in Model 3 is equal to the average interest rate path in the Vasicek model, then the two terminal distributions are identical, an even stronger observation than having the same terminal standard deviation.
There are still important differences between these models. First, Model 3 will experience a parallel shift in the yield curve from a change in the short-term rate. Second, the purpose of the model drives the choice of the model. If the model is needed to price options on fixed income instruments, then volatility dependent models are preferred to interpolate between
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Topic 14 Cross Reference to GARP Assigned Reading – Tuckman, Chapter 10
observed market prices. On the other hand, if the model is needed to value or hedge fixed income securities or options, then there is a rationale for choosing mean reversion models.
One criticism of time-dependent volatility models is that the market forecasts short-term volatility far out into the future, which is not likely. A compromise is to forecast volatility approaching a constant value (in Model 3, the volatility approaches 0). A point in favor of the mean reversion models is the downward-sloping volatility term structure.
C o x -In g e r s o l l -R o s s (CIR) a n d L o g n o r m a l M o d e l s