period and the asset value at the end of the period. As transacting at the maximum value assumes perfect market timing, the option price is an upper bound for the value of marketability. This upper bound is an increasing function of the length of marketability restriction as well as asset volatility. This property is intuitive since the longer it takes to sell an asset and the more volatile its price, the higher is the opportunity cost of not being able to trade. Longstaff mentions that this upper bound can also be viewed as the maximum amount that any investor would be willing to pay to obtain immediacy in liquidating an asset position. He finally assesses whether the bound is consistent with the empirical studies of Pratt (1989) and Silber (1991) who estimate the value of the lack of marketability of restricted stock and private equity. He finds that the model could actually provide a tight bound, representing a useful approximation of the value of marketability. Its closed form solution is

D_{max }

=

2+

σ^{2}T

2

N(d) +

r

σ^{2}T 2π

e

2 8

T

1,

(20)

where T is the length of the marketability period, σ is the standard deviation of the asset under consideration, N(·) is the cumulative normal distribution function and d = ^{√}σ^{2}T /2.

(20) provides a discount while k_{2 }represents a surcharge. The reference value for the verification of k_{2 }is

k 0 2 =

1 1 D_{max }

1.

(21)

To apply the Longstaff model to verify k_{2}, values for the marketability period and for the volatility of property are required. According to market practice, it takes about six months to buy or sell a property as described above. Volatility of property indexes needs to be assessed with caution, as property prices

usually exhibit inertia. returns by a correction coefficient. We measure in 1983 as well as over

Geltner and Miller (2001) propose to adjust standard deviation of property factor equal to 1/(1 AR(1)) where AR(1) is the first order autoregressive standard deviation and autocorrelation for the Halifax HPI since its inception

the mentioned market cycle since 1991 and over the last ten years.

# Table 2

summarizes the results. While plain time, the adjusted standard deviation

standard deviation increases turns out to be very stable.

and

autocorrelation

decreases

over

[Insert Table 2 about here]

For an illiquidity period of six months and an adjusted volatility level over the considered market

12