Petter Bjerksund

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.

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.

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.

The political economy of capital controls and tax policy in a small open economy

Abstract This paper considers the political economy of binding quotas on capital exports and the mix of capital and labor taxation in a small open economy where agents generally differ in their shares of capital and labor income. The analysis shows that irrespective of distributional preferences: (i) free international capital mobility is never optimal if the government cannot tax foreign-source income for compliance reasons, and (ii) the government should levy positive taxes on labor and capital if labor supply is endogenous in combination with a quota on capital exports. The analysis indicates that endogenizing the labor supply decision will tend to raise the source tax on capital.