In a previous paper it was shown that a Markov-functional model with log-normally distributed rates in the terminal measure displays nonanalytic behaviour as a function of the volatility, which is similar to a phase transition in condensed matter physics. More precisely, certain expectation values have discontinuous derivatives with respect to the volatility at a certain critical value of the volatility. Here we discuss the implications of these results for the pricing of interest rates derivatives. We point out the presence of nonanalyticity effects in other quantities of the model, focusing on the properties of the Libor probability distribution function in a measure in which it is simply related to caplet prices. We show that the moments of this distribution function have nonanalytic dependence on the volatility , which are also similar to a phase transition. We study in some detail the pricing of caplets on Libor rates and Libor payments in arrears, and show that the convexity adjustment for the latter is also nonanalytic in the model volatility .
We consider an interest rate model with log-normally distributed rates in the terminal measure in discrete time. Such models are used in financial practice as parametric versions of the Markov functional model, or as approximations to the log-normal Libor market model. We show that the model has two distinct regimes, at high and low volatilities, with different qualitative behavior. The two regimes are separated by a sharp transition, which is similar to a phase transition in condensed matter physics. We study the behavior of the model in the large volatility phase, and discuss the implications of the phase transition for the pricing of interest rate derivatives. In the large volatility phase, certain expectation values and convexity adjustments have an explosive behavior. For sufficiently low volatilities the caplet smile is log-normal to a very good approximation, while in the large volatility phase the model develops a non-trivial caplet skew. The phenomenon discussed here imposes thus an upper limit on the volatilities for which the model behaves as intended .
R
In a previous paper it was shown that a Markov-functional
We consider an interest rate
R
displays nonanalytic behaviour as a function of the volatility,
in discrete time. Such models are used in financial practice as parametric versions of the Markov functional model, or as approximations to the log-normal Libor market model. We show that the model has two distinct regimes, at high and low volatilities, with different qualitative behavior. The two regimes are separated by a sharp transition,
R
More precisely, certain expectation values have discontinuous derivatives with respect to the volatility at a certain critical value of the volatility. Here we
We study the behavior of the model in the large volatility phase, and
R
these results
the phase transition
R
rates derivatives. We point out the presence of nonanalyticity effects in other quantities of the model, focusing on the properties of the Libor probability distribution function in a measure in which it is simply related to caplet prices. We show that the moments of this distribution function have nonanalytic dependence on the volatility , which are also similar to a phase transition. We study in some detail the pricing of caplets on Libor rates and Libor payments in arrears, and show that the convexity adjustment for
rate derivatives. In the large volatility phase, certain expectation values and convexity adjustments have an explosive behavior. For sufficiently low volatilities the caplet smile is log-normal to a very good approximation, while in the large volatility phase
R
latter is also nonanalytic in the model volatility
model develops a non-trivial caplet skew. The phenomenon discussed here imposes thus an upper limit on the volatilities for which the model behaves as intended