The link between caplet and swaption volatilities in a Brace–Gatarek–Musiela/Jamshidian framework: approximate solutions and empirical evidence

We present an approximation for the volatility of European swaptions in a forward rate-based Brace–Gatarek–Musiela/Jamshidian framework (Brace, Gatarek and Musiela, 1997; Jamshidian, 1997) that enables us to calculate prices for swaptions without the need for Monte Carlo simulations. Also, we explain the mechanism behind the remarkable accuracy of these approximate prices. For cases in which the yield curve varies noticeably as a function of maturity, a second, and even more accurate, formula is derived