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A Recombining Binomial Tree for Valuing Real Options With Complex Structures

In this paper, we allow the (risk-neutral) drift and the volatility of the underlying asset vary as an arbitrary (continuous) function of the underlying asset value and time. We develop a lattice technique for valuing derivative assets (including real options) that have values dependent on the underlying asset. The prior literature shows how to do this for specific functional forms for the drift and volatility, such as for a lognormal diffusion. These papers do this by transforming the diffusion to a simple diffusion, such as an additive normal diffusion and generating a lattice with constant parameters. We allow the lattice jumps to vary to accommodate the arbitrary changes in the volatility. We accomplish this by defining a lattice and probabilities on the lattice that achieve the correct volatility in the limit as the step size vanishes even though the drift may be incorrect in the limit. Then we consider a change of measure to correct the drift, in the spirit of a discrete-time version of Girsanov’s theorem. The revised process converges to the desired underlying process.