## Discrete Time Mean Reversion Models

I recently valued a complex derivative where WTI oil was the underlying price. The valuation required a Monte Carlo simulation. When the valuation was reviewed by a major accounting firm, the values produced by their specialists deviated from mine more than expected. After a cooperative exchange, I identified the primary source of the difference. The reviewers’ model produced average future spot prices that were not equal to the market forward price. This violates the “no-arbitrage” requirement of pricing models. The source of the error was the omission of “drift-adjustment” terms (DATs).

The application of Ito’s lemma gives rise to the DAT in mean-reversion models, just as it does in geometric Brownian motion models. This has been recognized in the continuous-time literature on mean-reversion models.[1] However, I wanted to refer the reviewers to a published reference that would explain how to calculate the DATs required in a discrete time model. If such a reference exists, I was not able to find it. Therefore, I prepared this note.

After completing the derivation of DATs for a Monte Carlo simulation, I wondered how these results would apply to building a lattice of prices for a mean-reverting process. I reviewed Hull’s[2] presentation of the building of a trinomial lattice of prices for a mean-reverting process. His method involves a search process to identify the DATs. The derivation of the DATs for the Monte Carlo implementation provides the basis for the analytical calculation of the DATs required by lattices. I use Hull’s example to demonstrate how to build a lattice without the search process. For details please see the PDF file.

1. Eduardo S. Schwartz. “The Stochastic Behavior of Commodity Prices: Implications for Valuation and Hedging”, Journal of Finance, Vol. 52, No. 3, p. 926. (I want to thank Andrew Lyasoff for suggesting this reference.)

2. John C. Hull. Options Futures and Other Derivatives, 8th Edition, 2012, Prentice-Hall, New York, N. Y.