Gunnar Stensland

Closed-form approximation of American options

This paper presents a closed-form approximation of the value of the finite-lived American option where the underlying asset provides a constant pay-out rate. It is derived by imposing a restriction on the set of feasible exercise strategies, and thus represents a lower bound to the option value. The exercise strategy is to exercise the option the first time the price of the underlying asset hits a flat boundary. Our numerical results show that this lower bound is very close to the true option value.

Closed Form Spread Option Valuation

This paper considers the valuation of a spread call when asset prices are lognormal. The implicit strategy of the Kirk formula is to exercise if the price of the long asset exceeds a given power function of the price of the short asset. We derive a formula for the spread call value, conditional on following this feasible but non-optimal exercise strategy. Numerical investigations indicate that the lower bound produced by our formula is extremely accurate. The precision is much higher than the Kirk formula. Moreover, optimizing with respect to the strategy parameters (which corresponds to the Carmona-Durrleman procedure) yields only a marginal improvement of accuracy (if any).

Valuation and Risk Management in the Norwegian Electricity Market

The purpose of this paper is twofold: Firstly, we analyse the option value approximation of traded options in the presence of a volatility term structure. The options are identified as: (a) “European” (written on the forward price of a future flow delivery); and (b) “Asian”. Both types are in fact written on (arithmetic) price averages. Secondly, adopting a 3-factor model for market risk which is compatible with the valuation results, we discuss risk management in the electricity market within the Value at Risk concept. The analysis is illustrated by numerical cases from the Norwegian electricity derivatives market.

On optimal timing of investment when cost components are additive and follow geometric diffusions

Abstract The paper gives a partial characterization of the optimal stopping rule for a problem with reward function g(t, x) = e−rt(x1 − x2 − ··· − xn), where x1,…, xn follow geometric Brownian motions. The characterization takes the form of a simple and operational sufficient condition for when it is optimal to stop (invest). It is further shown that the optimal value function for the problem at hand is convex in the state variables, and that this fact has some interesting implications for comparative static results.

Optimal shutdown decisions in resource extraction

Abstract In this letter we give an analytic solution to the problem of when to stop production in resource extraction. The shutdown decision is irreversible. Both price and production follow geometric diffusion processes while extraction costs are fixed.

Gas Storage Valuation: Price Modelling v. Optimization Methods

In the literature, one approach is to analyse gas storage within a simple one-factor price dynamics framework that is solved to optimality. We follow an alternative approach, where the market is represented by a forward curve with daily granularity, the price uncertainty is represented by six factors, and where we impose a simple and intuitive storage strategy. Based on UK natural gas market price data, we obtain the gas storage value using our approach, and compare with results from a one-factor model as well as with perfect foresight. We find that our approach captures much more of the true flexibility value than the one-factor model.

Optimal sequencing of resource pools under uncertainty

Abstract This paper characterizes the optimal scheduling of extraction from two reserves under price and production uncertainty. An essential step in the analysis is the characterization of the optimal “switching date” and the associated “flexibility value”; i.e., the value of the option to switch production from the first to the second reserve.

An American call on the difference of two assets

Abstract We consider the finite-lived American call option written on the difference of two assets (the spread). For this problem, no closed form solution to the optimal exercise strategy or to the option value is known. The outline of this paper is to present and evaluate a simple closed form exercise strategy. Extensive numerical investigation indicates that the opportunity loss from following this simple exercise strategy is marginal. Thus, the option value corresponding to the simple strategy represents a very accurate approximation to the true option value.

On optimal control of income generating activities, and the value of flexible production design

Abstract The paper provides a closed form solution for the value of a firm which costlessly can change its mode of operation between two activities. The technology is such that production is zero from the inactive production line. In addition we assume that the production potential on any production line is idle when not operated, else the production follows a geometric Brownian motion. Although this framework is rather general, it is motivated by the problem to produce a natural resource located in two separate wells/ores, using a single (flexible) production unit. For what seems to be reasonable parameter values, the flexible unit is seen to achieve a surprisingly large value relative to the value achievable by a comprehensive unit, producing both alternatives simultaneously. In addition it is interesting to note that switching might occur even if the active production line gives the highest immediate income.

Simple rules to order irreversible income alternatives sequentially

Abstract We identify a simple sufficient condition for optimal sequential exploitation of a set of income generating alternatives. If these are identical except for differences in the standard deviation of income, the condition implies that the optimal ordering is obtained by always setting the most uncertain alternative up front. If the alternatives are identical except for differences in flow-fixed operating costs, then, depending on the magnitudes of cost differentials, the optimal sequential ordering may or may not be to produce in order of increasing costs.