SABR LIBOR Market Model

Abstract

The extensive analysis carried out in Chapter 5 has shown how well the SABR model is able to capture the volatility dynamics characterizing the vanilla interest rate markets. Its major limitation is however the impossibility of modelling more than one forward rate at a time. This deficiency makes the SABR model impractical in the valuation of exotic interest rate derivatives, since these have payoffs which depend on a combination of several forward rates. While evaluating these payoffs, it is critical to simulate all relevant forward rates under a single measure to avoid arbitrage in the pricing model. Obviously, we have already achieved this by simulating multiple forward LIBOR rates in a market model as shown in Chapter 6. The market models are calibrated to the hedging vanilla instruments, one per expiry, whose payoff references the LIBOR forward rates or swap rates desired. As some of the complex products, such as range accruals or ratchets, depend on several strikes per expiry, we realize that these models suffer from an important shortcoming — they can only be calibrated to one strike per expiry and therefore cannot model the volatility smile characterizing today’s markets. Various attempts to overcome this limitation have been pursued by academics and practitioners, following two main directions.