American Finance Association Empirical Testing of Real Option-Pricing Models Author(s):

This research is the first to examine the empirical predictions of a real option-pricing model using a large sample of market prices. We find empirical support for a model that incorporates the option to wait to develop land. The option model has explanatory power for predicting transactions prices over and above the intrinsic value. Market prices reflect a premium for the option to wait to invest that has a mean value of 6% in our sample. We also estimate implied standard deviations for individual commercial property prices ranging from 18 to 28% per year. DESPITE EXTENSIVE TESTING OF option-pricing models for financial assets, virtually no research has addressed the empirical implications of option-based valuation models for real assets.' This research is the first effort that examines the empirical predictions of a real option-pricing model using a large sample of market prices. Real options that have been valued in the academic literature include capital investments and natural resources, as well as urban land. The model we consider incorporates the option to wait to invest in the valuation of urban land. This paper provides empirical information about the option-based value of land, relative to its intrinsic value and its market price. Using data on 2700 land transactions in Seattle, we find a mean option (time) premium of 6% of the theoretical land value. This premium ranges from 1% to 30% in various subsamples. We define the "option premium" as the difference between the intrinsic value and the option model value, divided by the option model value.2 We also find that the option model has explana* Department of Finance, University of Illinois at Urbana-Champaign. The author thanks Peter Berck, Peter Colwell, Robert Edelstein, Bjorn Flesaker, Steven Grenadier, Hayne Leland, Jay Ritter, Anthony Sanders, Rene Stulz, Sheridan Titman, Nancy Wallace, Joseph Williams, and seminar participants at the University of Illinois, the 1992 Western Finance Association Meetings, and the Norwegian School of Management, and two anonymous referees for useful comments. 1 The exception is the Paddock, Siegel, and Smith (1988) empirical study discussed below. Theoretical real option-pricing models include Titman (1985), Brennan and Schwartz (1985), McDonald and Siegel (1985, 1986), Majd and Pindyck (1987), Morck, Schwartz, and Stangeland (1989), Brennan (1990), Gibson and Schwartz (1990), Triantis and Hodder (1990), and Williams (1991a). 2 The mean of 6% is an unweighted average across all sample observations. 621 622 The Journal of Finance tory power over and above the intrinsic value for predicting transaction prices. Therefore, to the extent that it is possible to coordinate and time an investment, valuation models should account for the option to wait. We believe that the premia for more speculative properties might be much larger than the values given here.3 In addition, we estimate implied standard deviations of individual commercial real estate asset prices in the range of 18 to 28% per year. This result is a contribution in itself, as a lack of repeat sales data for this class of assets makes it difficult to estimate the price variance directly. The implied variances we estimate are equivalent to Black-Scholes implied volatilities from stock options. Previous research that evaluates the prices obtained from a real option model is limited to Paddock, Siegel, and Smith (1988). Paddock et al. develop an option-based model that values offshore petroleum leases as a function of the market price of oil. For 21 tracts, they compare the prices computed from this model both to the government's discounted cash flow model (which uses the same underlying data inputs) and to industry bids (both highest and geometric mean). The Paddock et al. and government models give very highly correlated values, and neither comes close to predicting industry bids. The highest industry bid would generally correspond to the market price, providing a comparison between real option values and transaction prices. While these high bids are more than twice either the option-based or government valuations, the mean industry bid is within the range (either above or below depending on the assigned value for gas) of the alternative valuations. As the authors point out, due to a "winner's curse," the high bid may exceed the true expected tract value. In Section I we discuss our model that prices land, incorporating the option to wait to develop. The option value is a function of the building developed on the site and development costs. In Section II, the marginal prices of each building's characteristics are estimated using hedonic estimation on a separate sample of 3200 developed properties. Using these results, we calculate the value of an optimally scaled building for each of 2700 undeveloped properties. In Section III we evaluate the theoretical land values given by the option-based model relative to the intrinsic values and to market prices. We present conclusions in Section IV. I. The Model The model we consider is a fairly general, infinite horizon, continuous time model that in form most closely resembles Williams (1991a), but also tests 3 The finding of small but consistently positive premia seems reasonable given that Seattle experienced moderate growth during the sample period (1976 to 1979). However, the exact figures obtained for Seattle may not be representative of the overall economy, given the city's dependence on a single industry (aerospace). Empirical Testing of Real Option-Pricing Models 623 the implications of Titman (1985). The principal features of previous optimal timing models are incorporated. Through ownership of an undeveloped or underdeveloped property, the landholder holds a perpetual option to construct an optimal size building at an optimal time, subject to zoning restrictions. The cost of development is, X=f+qyxl, (1) where f represents fixed costs, q is the square footage of the building, y is the cost elasticity of scale, and x1 is the development cost per square foot. Development costs are assumed to follow a geometric Brownian motion with a constant drift, ax and a constant variance, o0X2, dX/X= ax dt + ax dzx. (2) We assume that the price P of the underlying asset, the building, is observable. The implications of this assumption are discussed in Section JJJ.B. P is given by P = q 'e, where e is a function of other attributes of the property and 4 is the price elasticity of scale.4 The complete formulation and estimation of P are discussed in Section III. A. P follows a geometric Brownian motion with constant drift, ap, and constant variance, j2 dP/P = (apx2) dt + op dzp, (3) where x2 are payouts to the developed property and p dt is the constant correlation between dzx and dzp. We require that the cost elasticity of scale, y, exceed the price elasticity of scale, 4.' 4We allow for 4) < 1, giving a concave relationship between price and building size in the option model, which is consistent with the relationship estimated in the hedonic function described in Section III.A. This concave relationship may exist for several reasons. Colwell (1992) argues that the cost functions are concave because the exterior walls increase less than proportionally to the floorspace. Concavity in the industry offer curves requires, in turn, that the lower envelope of the offer curves, and hence the hedonic function, must be concave. For the building as a whole, the marginal cost curve must intersect the marginal price curve from below in order to obtain an interior solution for the optimal building size. In the market for commercial space, there might be a downward sloping demand curve for a given location, and it is likely that as the building size grows, the prime rentable space decreases as a proportion of the total space (e.g., more interior offices). In the market for residential space doubling an apartment's size does not normally double the rent, since it would still only suit one family and have one kitchen. 5 Williams' (1991a) model assumes that unit development costs (x1) and unit cash inflows to the developed property (x2) are the underlying stochastic variables, both following lognormal processes. The data that we have provides information about building prices, but not about the rents generated by the building. Therefore, we alter the Williams' model with the assumption that the building price and total construction costs, both dependent on scale, are the state variables. Williams solves for price as a linear function of unit cash inflows (P = rrqx2, rr constant), and he assumes that total development cost is a linear function of unit development cost (X = x1q7). Therefore the two derivations are formally equivalent. However, we are not required to make the assumption, as Williams does, that price is a linear function of scale and therefore that development costs must exhibit decreasing returns to scale (in order to obtain an interior solution to the optimal scale problem). Instead, with our formulation we require only that the scale cost parameter exceed the price elasticity of scale, thus allowing for either increasing or decreasing returns to scale. 624 The Journal of Finance We also make the following assumptions. There is a known riskless instantaneous interest rate, i, which is constant through time and equal for borrowers and lenders. Land owners are price takers, giving a partial equilibrium model.6 The investment is irreversible, i.e., once the investor has built on the property, it no longer has any optimal timing value. f8P is the income to the undeveloped or underdeveloped property.7 Finally, we assume that there is an equilibrium in the economy in which contingent claims on the pair of processes for the development costs and building price, (X, P) are uniquely priced.8 We will represent the corresponding pricing operator by taking the expectation of future cash flows under the risk-adjusted probability measure and discounting at the risk-free rate. This is carried out by changing the drifts of X and P, ax and ap , respectively, to VX-_ Ax o-x and vp (aop-x2)-APop-, where Ax and Ap are constant parameters representing the excess mean return per unit of standard deviation. We can then express the value of the undeveloped property, V(X, P), as the solution to the fundamental valuation equation: O 0O.5uXXVxx +oAxpXPV + .5o2P2V + v XV, + vPV -iV+f3P (4) subject to the appropriate boundary conditions. Making a change of variables, z P/X and W(z) V(X, P)/X, we obtain: 0-OO.5wo2z2W" + (v)z + + ( i)W + 8z, where c 2 = o ox 22pox + 2 To solve this differential equation, we assume that there is a ratio of the building price to development costs, z, at which it is optimal to build. The investor exercises optimally at this "hurdle ratio," z*, giving the "smoothpasting" condition. The Appendix provides a more detailed solution and description of these conditions. 6 The model assumes that an individual's decision to develop has no impact on the market price of buildings. The empirical implications of this assumption are discussed in Section III.B. Williams (1991b) models a Nash equilibrium among developers. 7 The payouts to the undeveloped property are thereby assumed to be proportional to the developed property value. 8 This assumption is sometimes derived as a consequence of no arbitrage opportunities in an economy in which there exist tradeable securities whose prices are perfectly correlated with P and X. See, e.g., Titman (1985) and Brennan and Schwartz (1985). Given the nature of the underlying processes, we find it more palatable not to explicitly rely on a hedging argument. An equilibrium similar to the one we assume was explicitly derived by Rubinstein (1976) and applied by Milne and Turnbull (1991). An intermediate solution assumes that the component of the risk in (P, X) that is priced in equilibrium can be dynamically spanned by tradeable securities. Empirical Testing of Real Option-Pricing Models 625 The solution is given as follows: V(P,X) =X(AzX + k), (6) where, A = (z* 1-k)(z*)-J, z* =j(l +k)l(j-1), k = f8z/(i vx)X j = w (.5w9+ Up+ [2(.25w2 vV + 2i) + (v)2]) z = P/X The intrinsic value of the option can be found by taking the limit of (6) as the variance w goes to zero. This result is given by, VI(X,P) =P-X, z21+k VI(X, P) = ,SPI(i -vx), z < 1 + k (7) If the ratio z P/X exceeds 1 + k, the landowner will build immediately. Otherwise he will hold the land for the income it generates.9 For tractability, the optimal scale or building square footage, q*, is determined by the initial values of P and X, and is therefore the same for both the option-based value, V(P, X), and the intrinsic value, VI(P, X). This assumption understates the value of the option, as we discuss further in Section III.B. q* is found by maximizing the value of the undeveloped property, V(q) = P(q) X(q) = qe(f + q7'x1), over q. The solution is, q = ( q*<5 q* q* > q 6, (8) where 8 is the maximum size permitted by the zoning regulations. Our empirical work examines the option-based value given by (6), compared to the intrinsic value (7) and compared to market prices. The building is assumed to be built to the optimal scale in (8), and the optimal time to build is when the ratio of building price to development costs exceeds z*, in (6). II. The Data The primary data set consists of a large number of real estate transactions within the city of Seattle.10 All properties are within the city limits and are zoned for investment purposes: business, commercial, industrial, or lowand 9This "hurdle ratio," 1 + k, corresponds to the ratio z* in the option-based model. It is found by taking the limit of z* as w -O 0 (j 1). 10 The source of the data is the Real Estate Monitor Corporation. 626 The Journal of Finance high-density residential.1" The data cover the second half of 1976 through the end of 1979 and include the characteristics of 3200 transactions of developed properties (developed to a reasonable approximation of the permitted zoning) and 2700 transactions of unimproved land parcels. The data on the developed properties are used in the hedonic estimation of the potential building values. The unimproved parcels represent the real options, the land which the owner has the option to develop. The cost function is given by (1). These costs are estimated using the Marshall Valuation Service. This service provides indexes of per-square-foot construction costs for various types and qualities of buildings and assigns multipliers for adjusting these unit costs to particular years and localities.'2 Estimation of the cost scale parameter y is described in Section III.C. III. Empirical Results A. Estimates of Building Values Land is valued as an option, for which the underlying asset is the building that potentially would be built on that site. The price of this building is not observable, and thus must be estimated. The method we employ is hedonic estimation. Hedonic theory focuses on markets in which a generic commodity can embody varying amounts of each of a vector of characteristics or attributes Z. A hedonic price function p(Z) specifies how the market price of a commodity varies as these characteristics vary. Rosen (1974) provides a theoretical framework in which p(Z) emerges as the equilibrium price arising from bids and offers of the suppliers and demanders of the good. The distribution of the quantity, as a function of Z, that is supplied and consumed is also endogenously determined. We separate the sample into years (1976-1977, 1978, and 1979) and into five zoning categories (commercial, business, industrial, low-density residential, and high-density residential), to improve the predictive power of the coefficients.13 For each subsample, we regress the log price for an improved 1 The Seattle Zoning Code classifies these zoning categories cumulatively. The lowest (i.e. most inclusive) zoning category is industrial. Commercial includes nonretail business and light manufacturing. The purpose of business zoning is to provide for retail and office uses. The lowest residential use (high rise and mixed use) is not included in our sample because there were few data points and a large amount of heterogeneity. Therefore, what we term high-density residential is actually medium density (midrise). Low-density residential includes duplexes and triplexes. 12 We calculate the 1977 to 1979 square foot costs for building types according to the purpose of each zoning category (apartment, store, office, warehouse, and industrial building) for an average quality C and a good quality B building, and chose values at the middle of each range. These costs range from $23 per square foot to $34 per square foot. We assume fixed costs of $10,000. 13 A Wald test of the restrictions that the parameters are the same across zoning categories is statistically significant in nearly all cases. Therefore, we separated these groups in this first estimation, and also in the tests of the land valuation model. Empirical Testing of Real Option-Pricing Models 627 property on its characteristics, logPi = c + dlogqi + i logLSFi + a1HTi + a2HT 2 + a3AGEi + b'Li + d'Qi + ei (9) The independent variables included are the log of square footage of the building (q), the log of the lot (LSF), and the height and age of the building. L is a vector of six dummy variables, obtained by combining groups of census tracts in the city. These are abbreviated n (north), w (west), ce (central east), cw (central west), se (southeast), and sw (southwest). Q is a vector of dummy variables representing the quarter in which the property was sold, with the last quarter of each subsample omitted. This functional form is used because its Box-Cox transformation gives the highest log likelihood, lowest standard error, and highest R2. From these equations we estimate the coefficients to be used to determine the potential building value on an undeveloped plot of land. The results from each of these regressions are presented in Tables I to V. The fit of the regressions is good for all zoning categories and all years, with R2 ranging from 80.2 to 95.6. Each of the price elasticities of size is less than one and significantly greater than zero at conventional levels. The elasticity of lot size is fairly large, in most cases greater than 0.5. The elasticity of building size ranges approximately between 0.3 and 0.5. This situation is reversed for low-density residential properties, for which the elasticity of the building size is much higher than that of the lot. The estimates vary across years; the variation might reflect that each data set consists of the differing properties that have sold each year. However, because of the unknown time series properties of this data, it is possible that the standard errors are understated. Therefore, we cannot reject the hypothesis that the true elasticities are constant across time. For business and commercial properties, the downtown area (central west, cw) is the eliminated locational dummy variable. For all other subsamples the north is eliminated due to the lack of data points in the downtown area. From these estimates it is clear that Seattle is not a monocentric city, and that a simple variable measuring the distance from the city center would not capture important locational features. In particular, the area east of downtown (central east, ce) is a location that has negative price effects. North and west generally add to value, while central west is not always the most attractive location.14 When the coefficient for age is statistically significant, age has a negative impact. For the height levels that make up most of the sample, the effect of height on value is increasing at a decreasing rate, although the coefficients on these variables often are only marginally significant. 14 Our goal was to estimate coefficient that serve as predictors of marginal prices. We separated the city into as many regions as possible, given the number of data points in each region. 628 The Journal of Finance Table I Hedonic Estimation for Business Properties in Seattle logPi = c + klogqi + f log LSFi + ajHT, + a2HT,2 + a3AGEj + b'L, + d'Qi + ei For each property, i, P is the price; q and LSF are the building and lot sizes, respectively, as measured in square feet. Height (HT) is measured in stories and age in years. L and Q are dummy variables representing the location and quarter in which the sale took place. The last quarter of each time period and Central West are omitted variables. N is the sample size. The regression is estimated separately for each of the three time periods. 1976-1977 1978 1979 R-squared 0.922 0.843 0.878 Std. Error 0.312 0.457 0.426 F-statistic 156.11 67.31 82.86 N 215 177 164 Variable Coeff. Std. Error Coeff. Std. Error Coeff. Std. Error Constant -0.741 0.463 3.210 0.575 1.152 0.527 log(q) 0.565 0.039 0.495 0.058 0.314 0.057 log(LSF) 0.875 0.063 0.507 0.067 0.935 0.061 HT 0.126 0.053 0.163 0.066 0.255 0.094 HT2 -0.011 0.006 -0.006 0.004 -0.019 0.011 AGE -0.003 0.001 -0.005 0.002 -0.003 0.002 Locations North -0.158 0.089 -0.074 0.147 -0.185 0.134 West -0.101 0.094 0.070 0.158 -0.262 0.148 Central east -0.355 0.100 -0.434 0.172 -0.276 0.146 Central west 0 0 0 Southeast -0.340 0.094 -0.117 0.172 -0.198 0.159 Southwest -0.415 0.105 0.511 0.166 -0.332 0.155 Quarters 76-3 -0.202 0.078 78-1 -0.258 0.097 79-1 -0.190 0.143 76-4 --0.075 0.065 78-2 -0.167 0.098 79-2 -0.136 0.147 77-1 -0.067 0.081 78-3 -0.007 0.112 79-3 -0.120 0.148 77-2 0.045 0.070 77-3 -0.039 0.084 In order to predict the building value that corresponds to a particular plot of land, we need to estimate the size and height of the building. From Section I, the building price is given by P(q) = qte where the building size, q, and the price elasticity of scale, 4, are the same as in the hedonic equation (9), and ? is a function of other attributes of the property. We assume that the investor develops the optimally sized building, q = q*, as given by equation (8). We base the height estimates on existing property heights, so that the estimated height, HT, of a given property is equal to the average height for the relevant zoning and location. We then conclude that the estimated value of a building developed on each of the 2700 land transactions is given by, Pi = qp*bLSFi7exp{c + a1HTiiii Ti 2 + b'Li + d'Qi + e (10) Empirical Testing of Real Option-Pricing Models 629 Table II Hedonic Estimation for Commercial Properties in Seattle logPi = c + 4logq,J + frlog LSF, + a1HT, + a2HTi2 + a3AGEi + b'L, + d'Q, ?ei For each property, i, P is the price; q and LSF are the building and log sizes, respectively, as measured in square feet. Height (HT) is measured in stories and age in years. L and Q are dummy variables representing the location and quarter in which the sale took place. The last quarter of each time period and Central West are omitted variables. N is the sample size. The regression is estimated separately for each of the three time periods. 1976-1977 1978 1979 R-squared 0.885 0.856 0.893 Std. Error 0.336 0.397 0.326 F-statistic 108.9 54.26 75.05 N 229 133 131 Variable Coeff. Std. Error Coeff. Std. Error Coeff. Std. Error Constant 1.887 0.366 2.854 0.476 0.853 0.469 log(q) 0.419 0.036 0.501 0.064 0.415 0.050 log(LSF) 0.702 0.038 0.513 0.067 0.804 0.061 HT 0.051 0.068 0.151 0.076 0.209 0.059 HT2 0.002 0.011 -0.004 0.005 -0.014 0.004 AGE -0.006 0.001 -0.004 0.002 -0.000 0.002 Locations North -0.119 0.090 0.080 0.146 -0.055 0.103 West 0.144 0.085 0.151 0.154 -0.064 0.102 Central east -0.208 0.100 -0.267 0.160 0.075 0.127 Central west 0 0 0 0 0 Southeast -0.283 0.134 0.090 0.261 0.065 0.185 Southwest -0.325 0.104 -0.199 0.178 -0.285 0.116 Quarters 76-3 -0.148 0.079 78-1 -0.228 0.092 79-1 -0.092 0.098 76-4 -0.086 0.078 78-2 -0.122 0.105 79-2 -0.026 0.092 77-1 -0.079 0.073 78-3 -0.008 0.104 79-3 0.059 0.094 77-2 -0.084 0.087 77-3 0.017 0.067 where the L, LSF, and Q represent the actual location, size, and date sold of each parcel. B. Observation Errors There are a number of sources of error in our estimation. Consistent with the assumption of an exogenously given building price process, the estimated building values do not account for a possible depression in prices due to additions to supply. This would tend to overstate the value of the building if the true price process reflects a downward sloping demand curve. The intrinsic value of the land would tend to be overstated and the option premium correspondingly understated. However, the sample consists of many scattered, mostly small, heterogeneous lots in a market which at the time 630 The Journal of Finance Table III Hedonic Estimation for Industrial Properties in Seattle logPi = c + 4logqi + qilogLSFi + a,HTi + a2HTi2 + a3AGEi + b'Li + d'Qi +ei For each property, i, P is the price; q and LSF are thebuilding and lot sizes, respectively, as measured in square feet. Height (HT) is measured in stories and age in years. L and Q are dummy variables representing the location and quarter in which the sale took place. The last quarter of each time period and North are omitted variables. N is the sample size. The regression is estimated separately for each of the three time periods. 1976-1977 1978 1979 R-squared 0.824 0.948 0.956 Std. Error 0.464 0.318 0.252 F-statistic 22.0 26.8 45.5 N 75 33 41 Variable Coeff. Std. Error Coeff. Std. Error Coeff. Std. Error Constant 2.740 0.907 2.866 0.923 1.989 0.512 log(q) 0.332 0.085 0.373 0.105 0.289 0.058 log(LSF) 0.728 0.082 0.681 0.110 0.789 0.054 HT2 -0.329 0.422 -0.177 0.283 0.171 0.202 HT2 0.085 0.108 0.037 0.042 0.013 0.030 AGE 0.000 0.003 -0.004 0.004 0.001 0.003 Locations North 0 0 0 West -0.224 0.222 -0.079 0.372 -0.230 0.234 Central east * -0.144 0.372 -0.651 0.293 Central west * 0.225 0.422 1.011 0.316 Southeast -0.491 0.225 -0.102 0.186 -0.330 0.159 Southwest -0.288 0.149 -0.108 0.176 -0.277 0.147 Quarters 76-3 -0.590 0.278 78-1 -0.548 0.302 79-1 -0.353 0.189 76-4 -0.460 0.259 78-2 -0.431 0.241 79-2 -0.262 0.178 77-1 -0.412 0.253 78-3 -0.308 0.233 79-3 -0.169 0.195 77-2 0.283 0.263 77-3 -0.107 0.258 * No transactions in this region. was neither saturated nor underdeveloped. The supply of land available for development was fairly limited. The building on each of these lots probably does not have much impact on the price. Therefore, the potential for substantially overstating the building value due to the assumption of exogeneity is much smaller than for a large tract of similar properties. In addition, we may tend to overstate the true price because we observe prices only for those undeveloped or developed properties that sell. However, we also assume that the observable characteristics of existing buildings have the same marginal prices as the new buildings, and we only control for the depreciation of the existing stock in a simple way. This assumption would tend to understate the true price. Note that we do not assume that the new building is developed to the maximum density. We assume only that its value Empirical Testing of Real Option-Pricing Models 631 Table IV Hedonic Estimation for Low-Density Residential Properties in Seattle logPi =c + logq, + qilogLSF ?+a,HT ?+a2HT12 +a3AGE, +b'L, +d'Qi +ei For each property, i, P is the price; q and LSF are the building and lot sizes, respectively, as measured in square feet. Height (HT) is measured in stories and age in years. L and Q are dummy variables representing the location and quarter in which the sale took place. The last quarter of each time period and North are omitted variables. N is the sample size. The regression is estimated separately for each of the three time periods. 1977 1978 1979 R-squared 0.820 0.815 0.856 Std. Error 0.288 0.282 0.254 F-Statistic 112.2 112.7 121.6 N 361 319 259 Variable Coeff. Std. Error Coeff. Std. Error Coeff. Std. Error Constant 2.560 0.400 3.611 0.350 0.948 0.306 log(q) 0.800 0.036 0.622 0.374 0.637 0.034 log(LSF) 0.279 0.044 0.378 0.044 0.631 0.042 HT 0.195 0.114 -0.152 0.159 0.022 0.041 HT2 -0.054 0.031 0.051 0.046 -0.001 0.002 AGE -0.006 0.001 0.005 0.001 -0.001 0.001 Locations North 0 0 0 West 0.127 0.047 0.040 0.052 0.046 0.048 Central east -0.292 0.047 0.467 0.049 0.064 0.053 Central west * * * Southeast -0.073 0.044 -0.072 0.051 -0.097 0.056 Southwest -0.399 0.057 -0.358 0.052 0.072 0.050 Quarters 76-3 -0.359 0.060 78-1 -0.333 0.045 79-1 -0.287 0.061 76-4 -0.309 0.061 78-2 -0.213 0.046 79-2 -0.243 0.045 77-1 -0.241 0.048 78-3 -0.117 0.047 79-3 -0.010 0.044 77-2 -0.235 0.048 77-3 -0.191 0.047 * No transactions in this region. is estimated based on values of other buildings in the same zoning classification, which generally are not developed to the maximum density.'5 In sum, the building value estimation introduces several potential biases, the net impact of which is diffilcult to gauge. 15 Because the zoning is cumulative, we attempted a search over all possible building types for a given zoning category, e.g., allowing an industrial parcel to be developed commercially. The prices obtained were unreasonably high, indicating that the industrial parcel could not command the same marginal prices as a commercially zoned parcel. The fact that the land was zoned as industrial conveys information about its location and potential. Moreover, the marginal prices we estimate for the industrial parcel are based only on other industrially zoned lots, which are not necessarily developed to the maximum density. 632 The Journal of Finance Table V Hedonic Estimation for High-Density Residential Properties in Seattle logPi = c + 4logqi + qflogLSFi + a,HTi + a2HTi2 + a3AGEz + b'Li +d'Qz +ei For each property, i, P is the price; q and LSF are the building and lot sizes, respectively, as measured in square feet. Height (HT) is measured in stories and age in years. L and Q are dummy variables representing the location and quarter in which the sale took place. The last quarter of each time period and North are omitted variables. N is the sample size. The regression is estimated separately for each of the three time periods. 1977 1978 1979 R-squared 0.859 0.848 0.802 Std. Error 0.379 0.353 0.350 F-statistic 161.3 119.4 54.8 N 413 269 175 Variable Coeff. Std. Error Coeff. Std. Error Coeff. Std. Error Constant 3.181 0.274 2.637 0.401 2.472 0.492 log(q) 0.595 0.045 0.772 0.050 0.548 0.063 log(LSF) 0.407 0.051 0.315 0.062 0.593 0.071 HT 0.090 0.069 0.149 0.110 0.015 0.122 HT2 -0.004 0.010 -0.031 0.020 -0.002 0.023 AGE 0.007 0.001 -0.003 0.001 -0.003 0.001 Locations North 0 0 0 West 0.118 0.058 * * Central east -0.159 0.053 0.018 0.070 0.021 0.094 Central west 0.156 0.391 -0.178 0.055 -0.080 0.693 Southeast 0.056 0.073 0.152 0.089 0.007 0.126 Southwest -0.282 0.073 -0.203 0.094 0.068 0.136 Quarters 76-1 -0.296 0.069 78-1 -0.249 0.064 79-1 -0.221 0.123 76-2 -0.252 0.069 78-2 -0.189 0.064 79-2 -0.273 0.126 77-1 -0.254 0.068 78-3 -0.001 0.064 79-3 -0.174 0.124 77-2 -0.212 0.058 77-3 -0.140 0.059 * No transactions in this region. The state variable to the option model is the ratio z of the building price to development costs. Both these values are likely to be observed with error. Since z centers into the model in a nonlinear way, the expected value of the land is affected by the estimation error. To be specific, if we assume that the estimation error is normal and multiplicative, i.e., if ln z^ = ln z + ? and ? is distributed N(O, ure2), then, E[W(z^)] = AzIexp( j2or2/2) + k > W(z). (11) The presence of this bias means that we are more likely to reject the null hypothesis, and therefore that our tests tend to be more conservative than Empirical Testing of Real Option-Pricing Models 633 stated. In addition, in our regressions of theoretical prices on actual prices, the presence of errors-in-variables problems biases the slope coefficient downward and the intercept upward.16 We assumed that the building price is observable. For many reasons, including building delay, the value is usually not observable. As was shown in Flesaker (1991), this uncertainty can lead to errors of omission, in which the option is not exercised when it should be, or errors of commission, in which the option is exercised when it should not be, and uniformly lowers the value of the option itself. However, if the developer retains the option to alter the building plans during the building process, the option value may possibly increase. C. Results from Tests of the Land Valuation Model In this section we discuss the results of the estimation and specification tests. As with the hedonic estimation, we break the sample into the five data classes organized by zoning category, for which the parameters are assumed constant across each class. We evaluate real option values, (6), relative to market prices and to the intrinsic values, (7). In order to calculate the model prices, we must make assumptions about several parameters. We assume that the risk-adjusted drift parameters vp = vx = 0.03 and the interest rate i = 0.08. The model does not appear to be very sensitive to these assumptions. We find, however, that the theoretical values are extremely sensitive to assumptions regarding the values of the development cost scale parameter, y, and the payout to the underdeveloped property, ,3. Since we lack information about the true values of these variables, we estimate values that minimize the pricing errors in our sample. We estimate a value for ,3 ranging from 0.3% to 3% of the developed building value. Our priors were that y should be less than but close to one, giving some economies of scale. The values we estimate range from 0.9 to 1. The prices we calculate are extremely sensitive to y, primarily through its function in determining the optimal building size. Only a small fraction of the sample was developed to the maximum density. Because both the intrinsic value and option-based value models depend in the same way on the building value, development costs, and land payout, the estimates of y and ,B, the optimal building size q*, and the height assumptions should affect the option model and intrinsic value equivalently.17 That is, for positive variance, reasonable changes in these values do not alter the theoretically positive difference between the option model price and intrinsic value. Table VI shows variance estimates that are "implied" from the real option model. The parameter we estimate is wo 2, given by (5), which is the variance of the developed property value and development costs. The standard errors 16 See Theil (1971), for example. 17 In the intrinsic value case, the building is either developed immediately, a function of the factors which affect P and X, or held as income-producing land, a function of P and ,3. 634 The eJournal of Finance Table VI Variance Estimates Implied from the Option Model We estimate Var(P, X), the variance of developed property value, P, and development costs, X, from the option model (equation (6)), which incorporates the option to wait to develop land. The standard error is of this estimate. The variance and standard deviation of developed property value, P, are calculated assuming a 5% annual standard deviation of development costs and zero covariance. Standard Standard n Var(P, X) Error Var(P) Deviation (P) Business 1977 76 0.0369 0.0030 0.0344 0.1855 1978 64 0.0616 0.0053 0.0591 0.2431 1979 48 0.0571 0.0046 0.0546 0.2337 Commercial 1977 102 0.0503 0.0024 0.0478 0.2186 1978 90 0.0533 0.0032 0.0508 0.2254 1979 73 0.0526 0.0024 0.0501 0.2238 Industrial 1977 62 0.0525 0.0037 0.0500 0.2236 1978 43 0.0813 0.0038 0.0788 0.2807 1979 25 0.0658 0.0073 0.0633 0.2516 Low-density residential 1977 490 0.0720 0.0011 0.0695 0.2636 1978 401 0.0577 0.0017 0.0552 0.2348 1979 340 0.0488 0.0016 0.0463 0.2152 High-density residential 1977 224 0.0475 0.0014 0.0450 0.2121 1978 336 0.0647 0.0031 0.0622 0.2494 1979 360 0.0699 0.0022 0.0674 0.2595 of the estimates are very low. We then present values for the variance and standard deviations of the developed property value only, assuming the p = 0 and o= 0.05. We find annual standard deviations ranging from 18.55% to 28.07%, with no significant differences among the different property types.18 We can reject that the variance is constant, but we do not find any uniform movement up or down in the variance estimates across the years. Because these figures represent the annual standard deviation of individual properties, based on actual prices rather than on appraisals, it is difficult to find comparable numbers in the literature. The closest we find comes from a recent study by Case and Shiller (1989) of repeat sales; the study reports a 15% annual standard deviation of individual housing prices, from 1970 to 1986, in Atlanta, Chicago, Dallas, and San Francisco-Oakland. Titman and Torous (1989) estimate an implied property value standard deviation of 18 Clearly, different cities and different sample periods would generate different results. The results obtained in this study provide an indication of how the model fares empirically, but do not purport to be representative. Emipirical Testing of Real Option-Pricing Models 635 15.5% using their commercial mortgage-pricing model.'9 There is a fundamental inconsistency in utilizing a model that assumes constant variance to estimate implied variances that are allowed to change. This issue has been addressed in several papers. Preliminary findings of Sheik and Vora (1990) show that, under certain circumstances that allow for changing variance (such as a constant elasticity of variance diffusion process), implied volatilities measure the average volatility of the underlying stocks' returns fairly accurately. Based on our assumptions and estimates, most properties would not be developed if the investor correctly accounts for the option to wait. For most properties the current ratio of building price to development costs, z, is still less than the optimal development ratio z*. However, most properties would be developed in the intrinsic value case, where a null variance is assumed. We lack information about the actual development of these properties,20 and therefore cannot test whether the developer follows an optimal strategy. In Table VII, we present summary statistics for the option-based model price and the intrinsic value. As expected, the former exceeds the latter in every subsample, and in some cases by a substantial margin. Based on these values, we calculate the option (time) premium as the mean percentage difference bewteen the option model price and the intrinsic value. These premia range from 1% to 30%, with a mean of 6%.21 In support of the theory, they are consistently positive. There is no reason for the estimated option premium to be constant across the sample, since properties bought for current development should have a premium of zero, while more "speculative" transactions could have premia approaching 100% of the value. We consider that these numbers represent a lower bound on the option premium for land, since our sample consists of urban land during a period of expansion in a city with tight growth controls.22 Some larger figures appear, especially for industrial properties.23 When the industrial properties are excluded, the average premium is 5%. As previously discussed, our assumptions and 19 By comparison, Cox and Rubinstein (1985) report individual stock standard deviations ranging from 17% (for utilities) to 68% (for Winnebago) during the 1980 to 1984 sample period. 20 From 1977 to 1979 Seattle experienced a steady increase in building permits for all property types, but building activity subsequently dropped off. No trend can be seen in our data which reflects the statistics. 21 This mean is calculated across all observations, i.e., E15 1(N, opL)/N, where for each subsample i, Ni is the number of observations and opI is the option premium, and N is the total number of observations (2734). 22 We would expect higher premia in locations where very little building is currently taking place, indicating that the value in the land is mostly as an option to build far out in the future. The overall expansion in Seattle during this period indicates that many options are "in the money." 23 The premia for the industrial properties are not driven by just a few outliers. The industrial properties have by far the widest range of sizes and prices compared to the other zoning categories. The land prices and lot sizes are also the largest. Given the small sample size and heterogeneous data, it is quite possible that the building values fitted from the hedonic regressions are less representative of the sample of vacant lots than for the other categories. It is also possible that the industrial properties in 1977 and 1978 do have a larger option premium. 636 The Journal of Finance Table VII Summary Statistics and Option Premia Average land values given by the option model (equation (6)), which incorporates the option to wait to invest, and the intrinsic value (equation (7)) which does not value this option. The option premium for each parcel is defined as: (option model price intrinsic value)/option model price. We present the mean option premium for each subsample (not the option premium of the average values). Sample sizes are as given in Table VI. Option Model ($) Intrinsic Value ($) Option Premium Business 1977 30,550 29,090 0.0377 1978 115,092 112,578 0.0222 1979 84,773 74,998 0.0449 Commercial 1977 144,237 136,844 0.0518 1978 180,221 177,735 0.0095 1979 184,477 171,792 0.0256 Industrial 1977 146,670 122,124 0.298