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Asset valuation operators with diffusion processes

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Heaney and Garman develop a linear valuation operator which prices risky income streams when arbitrage profits are precluded. Both study the case where the states of nature are presumed to follow a diffusion process over the real line; each developing a differential equation involving the values (prices) of assets, as a function of the underlying states and time, dividends to these assets and the valuation operator. It is shown that the differences in the developments of these two equations - arising partially from different definitions of diffusion processes - are more apparent than real. These differences in derivation are only changes in the order that the steps are performed, not the application of different assumptions. Further, Heaney's differential equation, which governs the valuation operator for all times and states, is shown to hold only when a certain consistency condition is satisfied. Requiring this condition to be satisfied restricts the class of accepted no-arbitrage economies, but allows the valuation operator to be obtained from Heaney's equation. Lastly the effect of barriers to the diffusion process is investigated.
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