Lognormal Forward Market Model (LFM) Volatility Function Approximation

In the lognormal forward Market model (LFM) framework, the specification for time-deterministic instantaneous volatility functions for state variable forward rates is required. In reality, only a discrete number of forward rates is observable in the market. For this reason, traders routinely construct time-deterministic volatility functions for these forward rates based on the tenor structure given by these rates. In any practical implementation, however, it is of considerable importance that volatility functions can also be evaluated for forward rates not matching the implied tenor structure. Following the deterministic arbitrage-free interpolation scheme introduced by Schlögl in (Advances in Finance and Stochastics: Essays in Honour of Dieter Sondermann. Springer, Berlin 2002) in the LFM, this paper, firstly, derives an approximate analytical formula for the volatility function of a forward rate not matching the original tenor structure. Secondly, the result is extended to a swap rate volatility function under the lognormal forward rate assumption. Finally, a modified Black’s market formula is derived for the cases of the payoff dates differing from the original tenor structure, employing the LFM convexity adjustment formula. Implications for Monte Carlo simulation of non-tenor rates are also discussed. Most importantly in this context, the present paper introduces a simulation method avoiding the unrealistic behaviour of implied volatilities for interpolated caplets documented in (Schlögl, Advances in Finance and Stochastics: Essays in Honour of Dieter Sondermann. Springer, Berlin 2002).