Owen Graduate School of Management 401 21 ^{st} Avenue SouthNashville, TN 37203 | nick.bollen@owen.vanderbilt.edu office: 615.343.5029 fax: 615.343.7177 |

**Research Story 1: Option pricing in regime-switching models**

When I was a PhD student at Duke University, I worked with Bob Whaley and focused on option pricing. Duke had recently hired Stephen Gray out of Stanford. Steve's research was on estimating parameters of regime-switching models for describing the evolution of important economic variables such as interest rates and exchange rates. The main idea is that the underlying stochastic process undergoes discrete shifts as some latent state, like monetary policy, itself changes. The hard part is that the state is unobservable. It dawned on me that since options on interest rates and exchange rates are quite important, an option pricing model that explicitly allows for regime switches could be useful.

I began with the simplest possible regime-switching model: two normal distributions can govern the underlying variable at any point in time. In other words, the mean and variance parameters can discretely switch between two values. My first impulse was to construct a lattice to represent the possible paths that a variable governed by such a regime-switching process might follow. The standard binomial lattice is consistent with a normal distribution (thanks to the DeMoivre-Laplace theorem). Hence, I pictured pasting two binomial lattices together like this:

The inner two branches correspond to the distribution with lower variance. This set-up works. However, when implementing the lattice, the topology quicky became unwieldy. The reason is that four branches emanate from each node, and in general the branches do not recombine. This means that as the number of time steps in the lattice increases, the number of nodes at each time step increase exponentially.

My "Eureka" idea for my dissertation, which subsequently led to a number of publications, was to substitute a trinomial lattice for one of the binomial lattices. The extra degree of freedom provided by the third branch allows me to stretch out one of the lattices so that all branches are evenly spaced, while at the same time choosing branch probabilities so that the original mean and variance of the underlying normal distribution are preserved:

To illustrate the benefit of this modification, I created the following figure for my 1998 *Journal of Derivatives* paper,
Valuing Options in Regime-Switching Models. The figure on the left shows the lattice using the original binomials. After 3 time steps there are 16 nodes. After *t*
time steps there are (*t* + 1)^{2} nodes. The figure on the right shows the "pentanomial" lattice. After 3 time steps there are 13 nodes. After *t* time steps there
are 4*t* + 1 nodes. After 50 time steps, there are 2,601 nodes in the original lattice and only 201 nodes in the pentanomial:

I used the pentanomial lattice to value foreign currency options in my 2000 *Journal of Econometrics* paper Regime-switching in foreign
Exchange Rates: Evidence from Currency Option Prices and my 2003 *Journal of International Money and Finance* paper
The Performance of Alternative Valuation Models in the OTC Currency Options Market. Then, I started thinking about real options and came up with the
idea that a regime-switching model would be an appropriate way to reflect a product life cycle. In other words, if demand for a product is the underlying
variable, then one can view different stages of the product life cycle as different regimes. Real options to alter capacity, for example, should
reflect the random switches between the life cycle regimes. I found the following data regarding sales of successive generations of DRAM (dynamic random access memory) chips to
show the classic bell-shaped product life cycle:

This resulted in my 1999 *Management Science* paper
Real Options and Product Life Cycles.