Multifactor model

Hobbs BF (1980) A comparison of weighting methods in power plant siting. Decision Sciences 11 725-737 Hodder JE, Dincer MC (1986) A Multifactor Model for International Plant Location and Financing under Uncertainty. Computers Operations Research 13 601-609... 

While this approach would work in practice, this would only be for a single security portfolio it would be unwieldy and inaccurate for valuing a number of securities. As banks and market makers must value many hundreds of cash and off-balance sheet instruments, another approach is required. This other approach was considered in this chapter and involves describing the dynamics of the bond price process in the form of a term structure model. Under this situation, a multifactor model may be more suitable, particularly when used to value options. 

Two-factor models were based on a second source of random shocks. Two factor models were developed by Brennan and Schwartz, Fong and Vasicek, and Longstaff and Schwartz. However, Hogan " proved that the solution to the Brennan and Schwartz model explodes, that is reaches infinity in a finite amount of time with positive probability. The Brennan and Schwartz model shows that adding more factors may cause unseen problems. More complex multifactor models are described by Rebonato, and by Brigo and Mercurio. 

Heath, Jarrow, and Morton (HJM) derived one-factor and multifactor models for movements of the forward rates of interest. The models were complex enough to match the current observable term structure of forward rate and by equivalence the spot rates. Ritchken and Sankara-subramanian provide necessary and sufficient conditions for the HJM models with one source of error and two-state variables such that the ex post forward premium and the integrated variance factor are sufficient... 

When short rates are modelled with single-factor models, Jamshidian proved that an option on a coupon bond can be priced by valuing a portfolio of options on discount bonds. This approach does not work in multifactor models as proved by El Karoui and Rochet." ... 

Cnrrency risk is potentially a large source of risk for global investors that can be handled with a multifactor model with one factor per cnrrency. Yet, special attention has to be paid in forecasting the variances... 

Another form of the variance-covariance model incorporates a multifactor approach. Instead of looking at covariances between individual bonds the multifactor model aggregates information into common factors. These so-called principal factors can be obtained either by regres-... 

While this also uses a variance-covariance matrix much like the full covariance method, the actual matrix is much more condensed. As an example, the matrix used in a 20-factor model would have a size of (20 X 20) 400 cells, which is moderate compared with the one-million-cell matrix mentioned previously for the full variance-covariance model. The advantages of using a multifactor model are that it easily allows for mapping a new issue into past data for similar bonds by looking at its descriptive characteristics, and it can be inverted for use in a portfolio optimizer without too much effort. The multifactor model is also more tolerant to pricing errors in individual securities since prices are averaged within each factor bucket. 

For the following example, we utilize the Barclays Capital Portfolio Analytics System XQA, which incorporates the aforementioned multifactor model. Again, this model incorporates factors that include points on the yield curve as well as factors related to credit spreads. We took the yield curve data in the sterling model from gilts and for the euro model from Bunds. The credit spread factors consist of buckets by sector and rating, among other factors. 

Our sterling multifactor model consists of 32 factors reflecting changes in yield curve and credit spreads. We obtained historical monthly... 

EXHIBIT 25.8 Multifactor Model—Daily Returns versus ex ante Daily Tracking... 

The approach is similar to the historical simulation method, except that it creates the hypothetical changes in prices by random draws from a stochastic process. It consists of simulating various outcomes of a state variable (or more than one in case of multifactor models), whose distribution has to be assumed, and pricing the portfolio with each of the results. A state variable is the factor underlying the price of the asset that we want to estimate. It could be specified as a macroeconomic variable, the short-term interest rate or the stock price, depending on the economic problem. 

In multifactor models, it is also important that the random variables generated have the desired correlation. ... 

The approach described in Heath-Jarrow-Morton (1992) represents a radical departure from earlier interest rate models. The previous models take the short rate as the single or (in two- and multifactor models) key state variable in describing interest rate dynamics. The specification of the state variables is the fundamental issue in applying multifactor models. In the HJM model, the entire term structure and not just the short rate is taken to be the state variable. Chapter 3 explained that the term structure can be defined in terms of default-free zero-coupon bond prices or yields, spot rates, or forward rates. The HJM approach uses forward rates. 

Some of the newer models refer to parameters that are difficult to observe or measure direcdy. In practice, this limits their application much as B-S is limited. Usually the problem has to do with calibratii the model properly, which is crucial to implementing it. Galibration entails inputtii actual market data to create the parameters for calculating prices. A model for calculating the prices of options in the U.S. market, for example, would use U.S. dollar money market, futures, and swap rates to build the zero-coupon yield curve. Multifactor models in the mold of Heath-Jarrow-Morton employ the correlation coefficients between forward rates and the term structure to calculate the volatility inputs for their price calculations. 

Hodder, J.E. and Dincer, M.C. (1986) A multifactor model for international plant location and financing under uncertainty. Computers and Operations Research, 13 (5), 601-609. 

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