New Developments in Interest Rate Simulation Processes

by Michael Bykhovsky President and CEO, Applied Financial Technology

Latest innovation by Applied Financial Technology resolves a common error in the implementation of the BGM process to be truly arbitrage free and calibrates a portfolio of caps and swaptions in seconds.

It almost goes without saying, when valuing MBS and other securities whose cash flows are path dependent, one needs to
model a series of possible future interest rate scenarios. While the number of scenarios required to realize acceptable results
may be debatable, all agree that the underlying process used to generate the scenarios must be extremely reliable, practical to
use and mathematically sound. Without that, analysis of the securities is little more than a house built on sand.

What does it mean for an interest rate process (IRP) to be “reliable, practical and sound”? Over the past twenty-five years
there have been numerous attempts to develop a best of class IRP. These efforts have met with mixed success.

Essential Criteria

To begin, let’s review the essential criteria for any valid generator of future interest rate realizations. There are five conditions
that must be met:

• Must be completely arbitrage free
  
  There cannot be a combination of short/long positions securities and derivatives that will on average make or lose
  money at any point in time in the future. If this condition is violated, the IRP introduces a bias in the calculation that
  is impossible to account for, and thus, renders all calculations suspect. This is a primary condition, but surprisingly it
  has been violated in some IRPs.
• Must be consistent with the term structure of interest rates and both short and long structures of volatilities, at- and out-of-the-money
  
  This is a critical condition that is difficult to satisfy and is almost always violated. If the IRP does not price caps and swaptions correctly (i.e. their expected value being equal to their current market price), then one can be assured that it will not price correctly the cap and the prepayment option of a capped floater or of an ARM.
• Must use a tree implementation for a straightforward
  evaluation of American style options
  
  We must always use the same IRP to value MBS positions and their hedging instruments. For example, if the IRP used to calculate callable debentures differs from the one used to calculate the MBS that they fund, then one is essentially comparing apples and oranges and the resulting hedge ratios cannot be trusted. Surprisingly, using different IRPs for different instruments is a very common technique, probably because heretofore there have been few, if any, IRPs that satisfy all of these conditions.
• Must be a log-normal process
  
  This is because out-of-the-money options trade at a volatility structure closer to the log-normal than any other one. “Log-normal” means that the expected change in interest rates is proportional to the level of interest rates, while “normal” means that the expected change in interest rates is not related to the level of interest rates. One of the more unpleasant outcomes (from a modeller’s viewpoint) of a “normal” interest rates process is the possibility of negative interest rates in the simulation, leading to all sorts of mathematical difficulties or ad-hoc corrections.
  
  Also, prices of an interest rate derivative (IRD) are generally quoted in percent of Black volatility. Black volatility measure is a log-normal measure. There is generally a volatility “smile” – out-of-the-money options trade at greater Black volatility than at-the-money ones. A normal volatility based IRP would lead to a volatility “well” – a dramatic difference between-at-the-money implied volatilities and out-of-the-money ones. If one were to fit to these, the process would end up being essentially log-normal. (I’ll leave it to the reader to try this for him/herself and see the results.)
• Must complete calculations in a reasonable amount of time – preferably not involving a super computer
  
  Moore’s law continues to help here, but I have yet to see a trader or fund manager who didn’t want faster results.

As noted above, most efforts have fallen short – at least until recently. Here are some examples:

• CIR, BDT, Black-Karazinsky type processes cannot fit the complex volatilities term structures that we see today.
• HJM tree implementation becomes computationally untenable beyond 8-9 time nodes.
• HJM (or its more common implementation – BGM) simulation implementation requires many thousands of paths to start converging, a super computer to calculate even a simple CMO OAS and it does not allow for tree implementations.

AFT’s First IRP

After years of research, in 1996 AFT published a two-factor model that adequately met all of the conditions outlined above.

• First factor captures the volatility of short interest rates
• Second factor captures the volatility of the slope of the yield curve
• Acceptable calibration to an arbitrary volatilities term structure as well as an interest rates term structure typically in a range of five to fifteen minutes
• Implemented by a two dimensional tree structure (three dimensional if time dimension is included)
• Process is a log-normal and mean reverting (Note: mean reversion is a function of time and can be set by the user to be anything, including zero.)
• Volatilities are time-varying and “out-of-the-money” varying with user supplied correlations

Mathematical Details of AFT’s First IRP

Define short rate: r = a(t)e^x

It can be shown that is the excess slope of the yield curve for a given time on a given node versus the average expected slope.

It can also be shown that the correlation between changes in the slope and changes in the short rate is:

AFT’s New, Revolutionary IRP

In Q2 2006, AFT completed development of an interest rates process that advances the state of the art remarkably. AFT’s new IRP satisfies all of the conditions outlined above, fits more completely than ever, and allows for calibration to hundreds of caps and swaptions in a fraction of the time.

This is the first IRP of its kind, resulting from several years of practical experience and research by AFT. A major paper detailing the underlying methodology, “Analytic Backward Induction of Option Cash Flows: A New Application Paradigm for the Markovian Interest Rate Models”, has recently been published in the International Journal of Theoretical and Applied Finance (IJTAF) The paper was authored by Dr. Junwu Gan, AFT’s Director of Research and Development, and is available from AFT.

In his paper Dr. Gan demonstrates that the standard implementation of the BGM process is not arbitragefree. (Given fairly wide utilization of this approach, the conclusion should be a source of concern for many market participants.) AFT’s new IRP - itself a version of the BGM process – is strictly arbitrage-free.

Some of the other features of the new process are outlined below:

• Calibrates to at-the-money and out-of-the-money caps and swaptions simultaneously and rapidly.
  
  In our testing, it takes less than a minute to calibrate to hundreds of options. This rapid calibration to all caps and swaptions allows for
• Partial vega (or partial volatility sensitivity) calculations – sensitivity of a portfolio to the change of the market quote of single cap or swaption or a set of caps/swaptions
• Rapid arbitrage opportunity identification – allows the arbitrage of volatility pricing between IRD market and the MBS markets
• Leverages a tree implementation
  
  The process is based on the quantitative relation between the Monte Carlo sampled LIBOR market model and its related short-rate model. There, an explicit and heuristic relation model to the short-rate (tree implementation) exists: every model volatility parameter function has a corresponding interpretation in terms of the related short-rate model.

Based on the tree implementation of the process, AFT has a complete set of calculator libraries for non-path- dependent securities that:

• Values swaps, swaptions, caps/floors, all bonds in existence (bullet, sinking, optional sinks, putable, callable, capped floaters, etc.)
• Provides an interface for valuation of any IRD where exercised and unexercised values can be defined

Based on the Monte Carlo implementation of the process, AFT has a complete set of calculator libraries for path-dependent securities that:

• Values all MBS as well as synthetic MBS derivatives
• Provides an interface for valuation of any IRD where cash-flows based on a path of interest rates can be generated

• Satisfies the Markovian constraint along the implied yield curve, i.e., the price of the underlying bonds is history independent in the small volatility limit
• Introduces a CEV (constant elasticity of variance) model with a “smile” to fit the three-dimensional market volatility manifold (maturity, tenor, and strike price).

Acceptance of the new AFT IRP is growing rapidly. The model and its related calibration software have already been successfully implemented in several analytical systems with more organizations expressing interest or in evaluation mode.

For more information contact Kevin Williams at 603.401.1400.