A Mathematical Formulation of the Valuation of Short Sales and Put Options on Real Estate

Abstract

The purpose of this article is to construct mathematical valuation models of short sales and put options on real estate with declining values. For 1) farmland, the underlying assumption is that prices decline gradually. The first formulation is for the risk-averse investor, who takes minimum gains with the decline in real-estate prices. The investor’s sentiments are represented by the Arrow-Pratt coefficient of risk-aversion. The pricing distribution is for a Levy-Ito decomposition. The second formulation is for the moderate risk-taker whose price expectations are modeled by a Bessel function with a Levy-Ito decomposition as price function. This strategy offers modest gains. The third formulation is for the risk-taker with price expectations modeled by an exponential distribution and a pricing function with a Levy-Khintchine formula that has sharp gains. For 2) decaying shopping malls, prices decline initially, followed by upward movement as the decaying malls are repurposed into apartments and cultural centers. The risk-averse investor’s Arrow-Pratt coefficient of risk-aversion is subject to a Lebesgue integral of falling, then rising prices, to yield modest gains. The moderate risk-taker’s Bessel function is also subject to the Lebesgue integral with puts that are exercised to yield gains. The risk-taker’s exponential distribution is subject to a Lebesgue integral. However, the overconfidence of this investor results in prices increasing before put exercise, so that no exercise occurs. For 3) mines and quarries, a freely falling price distribution is used, as no recovery in prices is anticipated. Riemann integrals model the price distribution for each type of investor, as they show a vertical decline in prices. The short-sale strategy of the risk-averse investor and the put purchase strategy of the moderate risk-taker yield gains, while the put purchase strategy of the risk-taker results in losses. The paper advances knowledge by developing profit and loss strategies for three negatively priced investments.