A General Approach to Real Option Valuation with ... - ICMA Centre

ICMA Centre Discussion Papers in Finance DP2012-04The displaced logarithmic utility may be written(U(w) =λw 0 log 1+ w − w )0, for w>(1 − λ)w 0 , (17)λw 0where w includes initial wealth w 0 ,sow − w 0 is the P&L from the investment. Note that(17) is a standardized from of logarithmic utility for which U(w 0 )=0,U ′ (w 0 )=1andtheasymptote to −∞ is at (1 − λ)w 0 . Also, −U ′ (w)/U ′′ (w) =λw 0 + w − w 0 ,soatw = w 0 thelocal coefficient of absolute risk tolerance is λw 0 and λ is the local coefficient of relative risktolerance. 11 When λ = 1, local relative risk aversion is 1, independent of wealth, so in thiscase the utility (17) is said to have the constant relative risk aversion (CRRA) property.The power utility is best written in terms of risk aversion γ = λ −1 ,asU(w) =−γ|1 − γ| −1 ( ww 0) 1−γ, for w/w 0 > 0ifγ>1. (18)Since −U ′ (w)/U ′′ (w) =λw, λ is the local coefficient of relative risk tolerance. It is a constant,so the power utility has the CRRA property. Absolute risk tolerance increases linearly w, butfor (18) the sensitivity of local risk tolerance to changes in w is λ, whereas it is 1 for (17).Thus, when λw 0 (w1.Exponential, logarithmic and power utilities are all special cases of the general HARAutility, which has two parameters determining the local coefficient of absolute risk toleranceand its sensitivity to wealth. That is, HARA utility functions have a risk tolerance λ thatincreases linearly with wealth at the rate η, and are defined as:U(w) =−[1 +ηλw 0(w − w 0 )] 1−η−1 (1 − η) −1 , for w>(1 − η −1 λ)w 0 . (19)When η = 0 we have the exponential utility, η = 1 corresponds to the displaced logarithmicutility, η =0.5 gives the hyperbolic utility and η = λ gives the power utility.4 Numerical ResultsRecall that RNV principal yields a unique price for an option, and this reference price impliesthat the option could be tradable on a secondary market. By contrast, the option value for a11 Relative risk tolerance is expressed as a percentage of wealth, not in dollar terms. So if, say, λ =0.4 thedecision maker is willing to take a gamble with approximately equal probability of winning 40% or losing 20%of his wealth, but he would not bet on a 50:50 chance (approximately) of winning x% or losing x%/2 for anyx>0.4.Copyright c○ 2012 Alexander and Chen 14