LIBOR Market Model Lattice Approach

The LIBOR Market Model (LMM) is an interest rate model based on evolving LIBOR market forward rates under a risk-neutral forward probability measure. In contrast to models that evolve the instantaneous short rates (e.g., Hull-White, Black-Karasinski models) or instantaneous forward rates (e.g., Heath-Jarrow-Morton (HJM) model), which are not directly observable in the market, the objects modeled using the LMM are market observable quantities. The explicit modeling of market forward rates allows for a natural formula for interest rate option volatility that is consistent with the market practice of using the formula of Black for caps. It is generally considered to have more desirable theoretical calibration properties than short rate or instantaneous forward rate models.

In general, it is believed that Monte Carlo simulation is the only viable numerical method available for the LMM (see Piterbarg [2003]). The Monte Carlo simulation is computationally expensive, slowly converging, and notoriously difficult to use for calculating sensitivities and hedges. Another notable weakness is its inability to determine how far the solution is from optimality in any given problem.

We introduce a shifted forward measure that uses a variable substitution to shift the center of a forward rate distribution to zero. This ensures that the distribution is symmetric and can be represented by a relatively small number of discrete points. The shift transformation is the key to achieve high accuracy in relatively few discrete finite nodes. In addition, we present several fast and novel drift approximation approaches. Other concepts used in the model are probability distribution structure exploitation, numerical integration and the long jump technique (we only position nodes at times when decisions need to be made).

This model is actually quite useful for risk management because normally full-revaluations of an entire portfolio under hundreds of thousands of different future scenarios are required for a short time window (see FinPricing (2011)). Without an efficient algorithm, one cannot properly capture and manage the risk exposed by the portfolio.

For ease of illustration, we present the lattice model based on the Trapezoidal Rule integration. A better but slightly more complicated solution is to spline the payoff functions. The cubic spline of the option payoffs can achieve higher accuracy, especially for Greeks calculations, and higher speed. Although cubic spline takes some time, the lattice will require much fewer nodes (23 ~ 28 nodes are good enough) and can perform a much faster integration. In general, the spline method can provide a speedup factor around 3 ~ 5 times.

We present the lattice model based on the Trapezoidal Rule integration. A better but slightly more complicated solution is to spline the payoff functions. The cubic spline of the option payoffs can achieve higher accuracy, especially for Greeks calculations, and higher speed. Although cubic spline takes some time, the lattice will require much fewer nodes (23 ~ 28 nodes are good enough) and can perform a much faster integration. In general, the spline method can provide a speedup factor around 3 ~ 5 times.

• inarxiv LMM Valuation
• MyScienceWork lmm
• Authorea lmm
• Authorea lmm latex
• Zenodo pass through
• Zenodo digital adjustment
• Pubpub composite
• gitbook lmm
• Socarxiv lmm
• github inverse swap