Contrasting two approaches in real options valuation: contingent claims versus dynamic programming

This paper compares two well-known approaches for valuing a risky investment using real options theory: contingent claims (CC) with risk neutral valuation and dynamic programming (DP) using a constant risk adjusted discount rate. Both approaches have been used in valuing forest assets. A proof is presented which shows that, except under certain restrictive assumptions; DP using a constant discount rate and CC will not yield the same answers for investment value. A few special cases are considered for which CC and DP with a constant discount rate are consistent with each other. An optimal tree harvesting example is presented to illustrate that the values obtained using the two approaches can differ whcn we depart from these special cases to a more realistic scenariio. Further, the implied risk adjusted discount rate calculated from CC is found to vary with the stochastic state variable and stand age. We conclude that for real options problems the CC approach should be used.optimal tree harvesting, real options, contingent claims, dynamic programming

On the Economics of Ramping Rate Restrictions at Hydro Power Plants: Balancing Profitability and Environmental Costs

This paper examines the impact of ramping rate restrictions imposed on hydro operations to protect aquatic ecosystems. A dynamic optimization model of the profit maximizing decisions of a hydro operator is solved for various restrictions on water flow, using data for a representative hydro operation in Ontario. Profits are negatively affected, but for a range of restrictions the impact is not large. Ramping restrictions cause a redistribution of hydro production over a given day, which can result in an increase in total hydro power produced. This affects the need for power from other sources with consequent environmental impacts.

Regime switching in stochastic models of commodity prices: An application to an optimal tree harvesting problem

This paper investigates a regime switching model of stochastic lumber prices in the context of an optimal tree harvesting problem. Using lumber derivatives prices, two lumber price models are calibrated: a regime switching model and a single regime model. In the regime switching model, the lumber price can be in one of two regimes in which different mean reverting price processes prevail. An optimal tree harvesting problem is specified in terms of a linear complementarity problem which is solved using a fully implicit finite difference, fully-coupled, numerical approach. The land value and critical harvesting prices are found to be significantly different depending on which price model is used. The regime switching model shows promise as a parsimonious model of timber prices that can be incorporated into forestry investment problems.optimal tree harvesting, regime switching, calibration, lumber derivatives prices, fully implicit finite difference approach

The Impact of Stochastic Convenience Yield on Long-term Forestry Investment Decisions

This paper investigates whether convenience yield is an important factor in determining optimal decisions for a forestry investment. The Kalman filter method is used to estimate three different models of lumber prices: a mean reverting model, a simple geometric Brownian motion and the two-factor price model due to Schwartz (1997). In the latter model there are two correlated stochastic factors: spot price and convenience yield. The two-factor model is shown to provide a reasonable fit of the term structure of lumber futures prices. The impact of convenience yield on a forestry investment decision is examined using the Schwartz (1997) long-term model which transforms the two-factor price model into a single factor model with a composite price. Using the long-term model an optimal harvesting problem is analyzed, which requires the numerical solution of an impulse control problem formulated as a Hamilton-Jacobi-Bellman Variational Inequality. We compare the results for the long-term model to those from single-factor mean reverting and geometric Brownian motion models. The inclusion of convenience yield through the long-term model is found to have a significant impact on land value and optimal harvesting decisions.

On the option to invest in pollution control under a regime of tradable emissions allowances

Optimal decisions of a firm facing the option of retrofitting its plant to reduce pollution and thereby eliminate the need to purchase emissions allowances are analysed. The decision is treated as a real option with the price of pollution permits following a known stochastic process. The model is formulated as a set of one-dimensional partial differential equations. At discrete points in time, the firm owner makes optimal decisions about the retrofit, including whether to mothball temporarily. The model is used to analyse a firm's decision to instal a scrubber as a result of the 1990 U.S. Clean Air Act.

Hedges and Trees: Incorporating Fire Risk into Optimal Decisions in Forestry Using a No-Arbitrage Approach

This paper investigates the impact of including the risk of fire in an optimal tree harvesting model at the stand level, assuming timber prices follow a mean-reverting stochastic process. The relevant partial differential equation is derived under different assumptions about hedging the risk of fire. The assumption that fire risk is fully diversifiable is contrasted with the assumption that it can be hedged with another asset. It is conjectured that the risk-neutral probability of fire exceeds the historical probability of fire, which will affect forest land valuation. An empirical example is presented for two different silvicultural regimes

CLIMATE GAMES: WHO’S ON FIRST? WHAT’S ON SECOND?

We study four different climate change games and compare with the outcome of choices by a Social Planner. In a dynamic setting, two players choose levels of carbon emissions. Rising atmospheric carbon stocks increase average global temperature which damages player utilities. Temperature is modelled as a stochastic differential equation. We contrast the results of a Stackelberg game with a game in which both players act as leaders (a Leader-Leader, or Trumpian game). We also examine an Interleaved game where there is a significant time interval between player decisions. Finally we examine a game where a Nash equilibrium is chosen if it exists, and otherwise a Stackelberg game is played. One or both players may be better off in these alternative games compared to the Stackelberg game, depending on state variables. We conclude that it is important to consider alternate game structures in examining strategic interactions in pollution games. We also demonstrate that the Stackelberg game is the limit of the Interleaved game as the time between decisions goes to zero.Nous étudions quatre jeux différents sur le changement climatique et les comparons aux résultats des choix d’un Planificateur Social. Dans un contexte dynamique, deux joueurs choisissent des niveaux d’émissions de carbone. L’augmentation des stocks de carbone dans l’atmosphère augmente la température moyenne mondiale, ce qui nuit aux services publics des joueurs. La température est modélisée comme une équation différentielle stochastique. Nous contrastons les résultats d’un jeu à la Stackelberg avec un jeu dans lequel les deux joueurs jouent le rôle de meneur (un jeu Leader-Leader, ou Trumpian). Nous examinons également un jeu Entrelacé dans lequel il existe un intervalle de temps important entre les décisions des joueurs. Enfin, nous examinons un jeu dans lequel un équilibre de Nash est choisi s’il existe, et un jeu de Stackelberg est joué dans le cas contraire. Un seul ou les deux joueurs peuvent terminer dans une meilleure position avec ces jeux alternatifs par rapport au jeu à la Stackelberg, dépendamment des variables d’état. Nous concluons qu’il est important d’envisager d’autres structures de jeu lors de l’examen des interactions stratégiques dans les jeux portant sur pollution. Nous démontrons également que le jeu de Stackelberg constitue une limite du jeu Entrelacé lorsque le temps entre les décisions tend vers zéro

Hedges and Trees: Incorporating Fire Risk into Optimal Decisions in Forestry Using a No-Arbitrage Approach

This paper investigates the impact of including the risk of fire in an optimal tree harvesting model at the stand level, assuming timber prices follow a mean-reverting stochastic process. The relevant partial differential equation is derived under different assumptions about hedging the risk of fire. The assumption that fire risk is fully diversifiable is contrasted with the assumption that it can be hedged with another asset. It is conjectured that the risk-neutral probability of fire exceeds the historical probability of fire, which will affect forest land valuation. An empirical example is presented for two different silvicultural regimes.fire risk, forest value, hedging, jumps, no-arbitrage, optimal harvesting, Poisson process, real options, Resource /Energy Economics and Policy,

"Real options in harvesting decision on publicly owned forest lands"

This paper extends the literature on optimal tree harvesting assuming stochastic prices. With volatile prices, the value of a stand of trees is increased when harvesting dates are flexible, depending on wood volume and product prices of the day. Flexibility adds value because a forest owner can delay harvesting when prices are depressed, or can harvest earlier than planned if there is a uptick in prices. The stand owner thus has a natural hedge against price volatility. Regulatory policy in some jurisdictions has reduced the flexibility of firms harvesting on public lands by imposing allowable cut restrictions. This paper develops a two factor real options model of the harvesting decision over infinite rotations with mean reverting stochastic prices. The model is used to examine a proposed investment in intensive forest management in Ontario's boreal forests. The value of a representative stand in the Romeo Malette forest is estimated assuming complete harvesting flexibility. This value is then compared to the value when regulations dictate a window of time during which harvesting must occur.optimal harvesting, real options, allowable cut restrictions, intensive forest management, Markov decision process, linear complementarity problem