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Financial products continue to become increasingly sophisticated, and more sophisticated models need to be developed to keep up with innovation. In the interest rate world, the Libor Market Model (LMM) has developed and evolved to meet these needs.

Just what is the Libor Market Model? The stock definition we often see states the LMM differs from models that came before because it is based on the movement of Libor rates as they are quoted by traders in the market, as opposed to continuously compounded rates that are the favorite of academics. This is actually a common misconception: models like Black Derman Toy (BDT), Ho-Lee and others can be fairly easily derived and implemented with money market rates, and semi-annual compounding. So this is really nothing special.

Others say the Libor Market Model is the first to price instruments consistent with both caplet and swaption volatilities. This, too, is not completely true, though it is partially correct, depending on the assumptions made about the stationarity of the interest rate process (which is just a fancy way of saying the relationship between short and long dated vols stays the same over time).

But the really big innovation in the Libor Market Model is that it can handle yield curve shifts with multiple factors. In other words, unlike the one-factor models that came before it, the LMM allows the yield curve to have non-parallel shifts. Changes in slope, changes in curvature, and more, can occur, and lead to a much richer set of possible future scenarios to use for pricing. Now, the Heath-Jarrow-Morton (HJM) model also handled non-parallel shifts, but it did require volatility structures which were not considered as realistic as the LMM provides. And it has been said that HJM and LMM are much the same, with one a special case of the other.

But HJM and,later, LMM, have always had one very big problem: they require Monte Carlo simulation to price anything interesting (they do offer a formula for pricing an option on a zero coupon bond, though Black-Scholes will work fine for that since that's how volatilities are quoted anyway). Monte Carlo simulation is inaccurate, and s l o w. It is also only truly appropriate for pricing European options, which are relatively rare (while some heuristic attempts have been made to use simulation to price American options, even proponents of the technique admit it is biased. We also believe it is based on faulty mathematics).

In nearly every market, the preferred method for pricing non-European options is to build trees. When there is one factor, that means binomial trees. When there are multiple factors, that means multinomial trees.

But building a tree isn't enough: for all practical purposes, it has to be a recombining tree. In a recombining binomial tree, an up jump followed by a down jump leads to the same node as a down jump followed by an up jump. So one node becomes two, and two become three. In a bushy binomial tree, one node becomes two, and two nodes become four. That extra node doesn't look like much at first, but after 10 periods the recombining tree has 11 nodes, and the bushy tree has 1024. And after 30 periods, the bushy tree has over a billion.

HJM and LMM result in bushy trees. The world's top quants have tried to get past this problem, but the best they've been able to come up with is Monte Carlo simulation. Which is slow, inaccurate, and notoriously difficult to use for calculating sensitivities and hedges.

But now, for the first time, an LMM model is available which builds strongly recombining trees (we call them Tanenbaum Trees, after Rich Tanenbaum, the founder of Savvysoft, and the developer of the model). Not just binomal trees, but multinomial trees, with as many factors as the user requires. This makes valuations incredibly fast, and accurate. Even for American and Bermudan options.

The trees in Savvysoft's LMM models are strongly recombining. This means that most of the nodes in the tree can be reached from multiple points earlier in the tree. There are some implementations in the literature of weakly recombining trees, where just a few of the nodes recombine, but these do not generate nearly the same speed and acuracy of Savvysoft's model.

Savvysoft has also developed several other innovations inside the Tanenbaum Tree model. First, the interest rate process is a blend of a lognormal and a normal, with the user choosing the blending ratio between the two. A ratio of 1 is a pure lognormal, a 0 is a pure normal, with any number in between allowed. This can be used inside the model to handle vol smiles and skew.

In addition, Savvysoft offers a function to convert a three dimensional swaption volatility surface into a vol grid appropriate for the option being valued. This method has been shown to most accurately match the market for off-the-run swaptions, far superior to models which attempt to create best fit models of the skew which are good on average and correct none of the time.

Savvysoft has also created a function which, given a vol grid, calculates the implied correlation between different points on the yield curve. Correlation is an important input into any multi-factor model, and Savvysoft's function removes the guesswork, and replaces it with a reproducible and auditable calculation.

Savvysoft offers the tree-based LMM as part of its TOPS suite of derivative's pricing models, as well as OmniPricer, which allows users to price any derivatives structure by simply specifying a payoff function.

Only Savvysoft offers the tools you need to manage all the types of swaps you trade today, and in the future. And only TOPS lets you price and manage the risk of every type of derivative, in every market, backed by award-winning support from a team led by Rich Tanenbaum.

TOPS features at a glance:

• Measure derivatives across all asset classes, including cross-asset products like structured notes
• Full coverage of every type of derivative in each market, including all the latest structures like TARNs, Snowballs and Snowblades
• Handle an infinite variety of variations on each structure, with amortization/accretion, step-up/down coupons, strike schedules, and more
• Calculate prices as well as every relevant Greek
• Price instruments in Excel with handy dialogs, or with our exclusive Template Builder, or plug TOPS into any in-house or third party system
• Price instruments that haven't even been invented yet with OmniPricer, which lets you specify the derivative's payoff function in just a few cells of an Excel spreadsheet
• Backed by "always on" world-class support that users rated 98/100

Only TOPS covers ALL your trades, now, and in the future, to ensure you get to see ALL your risk. And if you ever need help, Savvysoft's support team is there to guide you every step of the way.

To schedule a FREE Demo of TOPS, click here now.