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What is the time value of a stream of investments?

Part of: Probabilistic methods, simulation and stochastic differential equations Mathematical economics

Published online by Cambridge University Press:  14 July 2016

Ragnar Norberg  and
Mogens Steffensen
Ragnar Norberg*
Affiliation:
London School of Economics
Mogens Steffensen*
Affiliation:
University of Copenhagen
*
∗Postal address: Department of Statistics, London School of Economics, Houghton Street, London WC2A 2AE, UK. Email address: r.norberg@lse.ac.uk
∗∗Postal address: Laboratory of Actuarial Mathematics, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark. Email address: mogens@math.ku.dk
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Abstract

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The titular question is investigated for fairly general semimartingale investment and asset price processes. A discrete-time consideration suggests a stochastic differential equation and an integral expression for the time value in the continuous-time framework. It is shown that the two are equivalent if the jump part of the price process converges. The integral expression, which is the answer to the titular question, is the sum of all investments accumulated with returns on the asset (a stochastic integral) plus a term that accounts for the possible covariation between the two processes. The arbitrage-free price of the time value is the expected value of the sum (i.e. integral) of all investments discounted with the locally risk-free asset.

Keywords

Cash flowcompounding assetdiscountingstochastic exponential

MSC classification

Secondary: 91B30: Risk theory, insurance 65C30: Stochastic differential and integral equations
Type
Short Communications
Information
Journal of Applied Probability , Volume 42 , Issue 3 , September 2005 , pp. 861 - 866
Copyright
© Applied Probability Trust 2005 

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