A Brief Review Of LIBOR And Swap Market Models - Part 2

This week's Learning Curve covers pricing model application, namely calibration of market models to caps and swaptions, closed form solutions useful for calibration and pricing of Bermudan options with Monte Carlo in the context of the market models. Calibration To Caps & Swaptions

Calibration consists of finding model parameters that result in a close fit to a set of liquid instruments, in this case, caps and swaptions. This is a high dimensional root finding or minimization problem, which generally involves iterations and calls to cap and swaption valuation routines. It is computationally feasible only when the valuation is relatively fast.

The Black and Karansinski model is slow for this purpose. It is computationally difficulty and the solution introduces additional time-dependency in model parameters, such as the short-rate volatility or mean-reversion parameters. Short-rate models generally have too few parameters to enable a good stationary fit. One way to gain additional degrees of freedom is to increase the number of factors. For example, at least two state variables driving the short rate are needed to capture stationary parameters in a humped cap volatility curve. But increasing the number of factors in short-rate models means higher dimensional trees or grids, resulting in a substantial increase to computational cost. But in the market model there are more degrees of freedom available for a stationary fit. By modeling the volatilities of the entire yield curve, the market model is inherently high dimensional. Even a one-factor market model can capture the hump in the cap volatility curve.

The cap data essentially determines the volatilities of each forward LIBOR rate in a stationary market model. In the one-factor case, this does not leave much room for further calibration to European swaptions. In order to get a reasonable stationary fit to a market quoted swaption volatility grid; at least two factors in addition to forward-rate volatilities are required to parameterize forward-rate correlations. Increasing the number of factors in the market models is straightforward and comes at little expense to computational cost.

Fast valuation of caps and European swaptions is required for the purpose of calibration. But before discussing the solution offered by the market model, there is the illusive issue of whether one should calibrate to both caps and swaptions in all circumstances. The drawback of calibrating to both sets of data is that the quality of the fit deteriorates. The first column of the swaption volatility matrix, the short-tenor swaption, replicates the same information in cap volatilities. So unless these two sets of data are in line, one is likely to over-fit one and under-fit the other. Bermudan swaptions and CMSs are more sensitive to swaption volatilities than to cap volatilities. Market model calibration also provides a tool for determining arbitrage opportunities between caps and swaptions.

 

Closed-Form Solutions & Approximations

The success of market models in calibration is due to closed-form solutions and approximations for caps and swaptions. Less well known are similar formulas for related derivatives, such as LIBOR in arrears, or caps on them, CMS convexity adjustment, and CMS caps and spread options. The existence of these closed-form solutions and approximations and the ease with which they are derived is a major advantage of the market model.

The lognormal LIBOR market model leads to the standard Black-Scholes formula for caplets, and the lognormal swap market model leads to the standard Black-Scholes formula for a series of European swaptions with the same maturity. Unfortunately forward LIBOR rates and forward swap rates are not simultaneously lognormal. But, if one series is lognormal, we might expect other series to be approximately lognormal. This turns out to be experimentally the case, at least for reasonable maturities of less than 30 years and normal volatility levels. While at present there is no mathematical demonstration of this claim, it can be substantiated by numerical comparison with Monte-Carlo valuation, as reported by several authors.

The calibration thus proceeds as follows. The lognormal LIBOR market model is chosen, but one also uses the closed-form solutions provided by the lognormal swap market model as close approximations to swaption prices in the LIBOR model. The next step is to sensibly parameterize the covariance matrix of forward LIBOR and use non-linear least-square optimization to find best-fit parameters. This is computationally feasible even when calibrating to over 100 swaption volatilities because of closed-form approximation. There are different approaches to parameterization. It is natural to impose stationarity. But the decaying characteristic of quoted volatilities can also imply mean reversion on LIBOR and CMS spot rates. Similar approximations are available for CMS convexity adjustment, CMS caps, etc. Currently, the most widely used method for CMS convexity adjustment is a duration-based argument that treats a CMS flow as if it were a contract on the yield of a certain fixed bond. While this too may be a valid approximation, it requires exogenous specification of correlation between LIBOR and swap rates. In the market model however, such correlations can be determined indigenously as a consequence of calibration.

We should point out the lognormal market model has difficulty coping with the cap volatility smile. There are ad-hoc remedies, but a consistent approach requires dispensing with the lognormality assumption. Fortunately, the fundamentals of market model theory is applicable in full generality, and a few authors have developed alternative volatility specifications, such as constant elasticity value or quadratic, which give rise to skew or smile while admitting practical approximations for European swaptions. Extensions to jump-diffusion processes giving rise to a smile have also been considered.

 

Evaluating Bermudan & Path-Dependent Derivatives By Monte Carlo Simulation

The market model is theoretically elegant and a powerful calibration tool. It also provides closed-from approximations for several important derivatives. But the numerical valuation of derivatives is harder in the market model than short-rate models. While several approaches have been proposed, such as bushy trees, or approximation by low-dimensional Markovian systems and partial differential equations, only one of them, namely Monte-Carlo simulation, seems scalable to all structures and maturities.

There are complex path-dependent structures where, regardless of the model, Monte Carlo simulation is the method of choice or the only possible approach, these include periodic caps, or knock-out swaps. But such "paths by paths" Monte Carlo simulation cannot be applied to Bermudan swaptions. It is a disadvantage to the short rate models that low dimensional tree or grid implementation are not available for market models.

The good news is that due to recent advances it is possible to account for the early exercise of Bermudan swaptions using Monte Carlo simulation. There are at least five different approaches to date, and research is continuing. These methods proceed backward inductively as in trees, and utilizing the cross sectional path information some methods directly calculate conditional expectation prior to exercise, while other methods calculate optimal exercise levels by maximizing average payoff.

 

Conclusion

The key idea common to all these methods is to use cross-sectional path information. For ordinary path-dependent derivatives, one path is generated at a time, which values the derivative along the path and then dispenses with the path before repeating the procedure with another path. For Bermudan derivatives the values along different paths, have to be stored and compared with each other at each exercise date. These methods have consequently higher computational and storage costs than non-Bermudan Monte-Carlo valuations. Nevertheless, some of them have been found to be highly effective in practice.

This week's Learning Curve was written by Farshid Jamshidian, head of model research at Cygnifiin London.