The Pricing of Embedded Options in Real Estate Lease Contracts

Leases and rental agreements often have options attached or embedded in them. These options sometimes depend on a number of economic variables such as the Consumer Price Index (CPI), a real estate index and/or the value of real estate underlying the agreement. The evaluation of these options often involves the solution or approximation to a partial differential equation (PDE). This study analyzes the appropriate PDEs which model the situation where the lessee is granted an option to either purchase the property or to renew the lease at a price (rent) indexed to the CPI or some other readily measured economic variable. The PDEs that result from the usual contingent claim asset-pricing framework are derived and numerically solved using the finite difference method with absorbing boundaries. The value of an embedded option to renew a five year lease on class A office space in each of the twenty-five markets for which the National Real Estate Index reports quarterly rental data is estimated. An evaluation of the model’s ‘‘Greeks’’ confirm that the model conforms to financial intuition which provides support for the accuracy of the estimates. Introduction The explicit valuation of embedded options in financial contracts, such as the conversion option of convertible bonds or the put option held by a mortgagor, has received substantial attention in the contingent claims pricing literature. However, in real estate, little attention has been given to the pricing of explicit options, which now appear in almost all commercial lease contracts. The literature which does address lease option pricing is largely descriptive and, where pricing models are offered, tends to be simplistic to the point of inapplicability. Though lease options have not been explicitly priced, there is little doubt that these contract contingencies have value, which is necessarily reflected in the contractural rental stream. Even seemingly benign options, such as an option to renew at market rent, must have both a value to the lessee and impose a cost on the lessor. It would seem, given the extent to which option components are negotiated into leases, that both property/asset managers as well as tenants would have substantial interest in a formal and quantitative method of incorporating the option value into the income stream. The ability to explicitly identify the value of embedded options would make the negotiation of their inclusion a more straightforward and exact process. *Department of Finance, James Madison University, Harrisonburg, VA 22807. **Department of Finance, James Madison University, Harrisonburg, VA 22807 or [email protected]. 254 JOURNAL OF REAL ESTATE RESEARCH VOLUME 15, NUMBER 3, 1998 This study examines a number of different options that are common in commercial leases and develops pricing models that yield efficient prices for these options. First there is a brief discussion of various lease options and the stochastic properties of the underlying processes are explored. Then the Partial Differential Equations (PDEs) that model the dynamics of the options are derived and numerically approximated. A discussion of the prices and properties of the pricing model is then offered in order to demonstrate the efficacy of the models. A discussion of the applicability of the models to a broader range of lease options concludes the study. Lease Options Though there is a large variety of lease options, this study focuses on two types that are most common, and potentially, the most valuable. The first type is the option to renew a lease at the end of the initial lease period and the second is the option to purchase the leased space upon expiration of the lease. Though these two options are substantially different in terms of the right conveyed, their pricing is remarkably similar. For the traditional option, an exercise, or strike price, is defined at the time the option is written and the option’s value is largely a function of the dynamics of the underlying market price. In lease options, however, this is frequently not the case. Most commonly, the strike price is defined as a function of the underlying’s market price, or related to the cumulative value of some index at the time of expiration. For example, lease renewal options commonly define renewal rent in one of three ways. First, the renewal rent could simply be defined as the market rent at the time the lease expires. Second, the renewal rent could be defined to be a fixed percentage of market rent at expiration, with 90% to 95% being the most commonly used percentages. Finally, renewal rent could be current rent grossed up by the cumulative change in some index, such as the Consumer Price Index (CPI). Similarly, an option to purchase the leased property could define the exercise price to be market value at expiration, some defined percentage of market value or current market value grossed up by the cumulative change in some index. The first possibility, renewal or purchase at market, is the least interesting of the three alternatives and the one to which standard option pricing methodology does not apply. The option will, by definition, be at-the-money or have an intrinsic value of zero at expiration. It, therefore, cannot take on value in the traditional sense. Yet, it does indeed have a value for which bounds can be identified but its price is a function of the negotiating skills and positions of the two parties. The value to the lessee is simply the present value of the combined cost of relocating and the locational goodwill established during the initial tenure. This amount is the most that the lessee will pay for the option and represents the upper bound on its price. The cost to the lessor is the present value of the opportunity cost imposed by the obligation to release the property to the lessee. This opportunity cost could take on various forms but would consist of at least the forgone ability to redirect the property theme or use (i.e., medical building, legal building, etc.), the inability to attract a large block user because of the THE PRICING OF EMBEDDED OPTIONS IN REAL ESTATE LEASE CONTRACTS 255 existing tenant and a restructuring of lease terms to shift risks from the lessor to the lessee. The present value of the perceived opportunity cost would be the lower bound of the option price. As indicated, the exact price between these two bounds cannot be mathematically defined because it is a function of the negotiating skills or positions of the two parties. Of somewhat greater interest is an option where the exercise price is a defined percentage of market at the expiration of the lease. By definition, the option will be in-the-money at expiration by the defined fractional amount of the then current market rent, or price, depending on whether the option is an option to renew or an option to purchase. The value of this option is simply the present value of the defined fractional amount of expected market rent or price. The typical pricing problem for a call option requires a model that will determine the probability that the market price will exceed the exercise price at expiration and also identify the expected market price, given that it is greater than the exercise price. In this case, the exercise price is defined to be less than the market price at expiration no matter what its value, so the in-the-money probability is one. Therefore, an appropriate pricing model must only identify the expected value of the market price at the expiration of the lease term. In order to identify the expected future value of market rent or price, a stochastic process must be assumed. Since the value of income-producing real estate is a direct function of the expected rental stream, then it is easily assumed that both rent and price follow the same stochastic process. A standard assumption for investment assets is that their market prices follow Geometric Brownian Motion (GBM). However, other stochastic processes are possible and some of these possibilities are explored later. The assumption of GBM implies that real estate prices and rents have a lognormal distribution and by the properties of a lognormal distribution the expected value of real estate prices or rents (R) at expiration time T is: 2 mT1s T / 2 E(R ) 5 R e , (1) T 0 where m is the constant expected rate of return on real estate and s is the standard deviation of returns. The value of the call option (O) held by the lessee is then, 2 (m2r)T1s T / 2 O 5 (1 2 p)R e , (2) 0 where p is the fractional proportion of market price or rent and r is the risk-free interest rate. It is obvious from Equation (2) that the value of the option to renew or purchase at some fraction of market will vary between geographical markets according to the level and volatility of current prices or rents. For example, the option would be less valuable when attached to a lease on office space in the Houston or Denver markets than it would for office space in Manhattan due to the difference in rent levels. It would also be relatively less valuable when attached to a lease in a stable market, 256 JOURNAL OF REAL ESTATE RESEARCH VOLUME 15, NUMBER 3, 1998 such as the northern New Jersey office market, than it would be in a volatile market, such as the Boston office market. Lease options where the exercise price is a direct function of the market price at time of expiration are common and they are not difficult to price, given appropriate data from which an expected return and volatility can be estimated. Of greater interest, and certainly degree of difficulty, is the pricing of purchase and renewal options that are tied to an index, such as the CPI. Such options are commonplace in lease contracts, but there seems to have been no attempt to develop pricing models for lease options with this feature. The reason for this void is likely due to the complexity that the dynamic strike price adds to the pricing problem. When both the asset price and the strike price follow a stochastic process the Partial Differential Equation (PDE) that models the situation is significantly more complex and more difficult to solve than when only the asset price is stochastic. In the following section, a number of PDEs that, depending on the assumed stochastic process, model the situation where the lessee is granted the right to renew the underlying lease at a rent indexed to the CPI are presented. Note that market price can be substituted for market rent in each of these equations and the right to purchase the property at a price indexed to the CPI will be modeled instead. Also, note that any index can be used as long as it exhibits similar stochastic properties to the CPI. Since it is impossible to identify the analytic solutions to the PDEs, a numerical approach will be employed to obtain approximations to the equations. A number of studies, Brennan and Schwartz (1977), Geske and Shastri (1985), Courtadon (1982), Hull and White (1990) and Hilliard (1994) have demonstrated the usefulness of the finite difference method (FDM) for approximating the solution of a PDE where the analytic solution cannot be identified. Buetow and Sochacki (1995) use a modified version of the FDM to evaluate problems similar to those addressed in this study. The FDM is used here to approximate the PDEs that are developed for pricing the lease option with a dynamic strike price. FDM allows various dynamics to be easily incorporated into the problem and several possible stochastic processes can be used. For example, GBM can be used if it accurately represents the dynamics of the variable. Alternatively, the dynamics can be modeled by a mean reverting process (MRP) if the variable follows a trend, but experiences short term disturbances. In this study, both possibilities, as well as combinations of the two, are presented. For a solution to exist using the traditional FDM, pre-determined boundary conditions are required. When the state variables (market rent and the CPI) are extremely volatile, the solutions to these standard contingent claim finite difference equations (FDEs) are unreliable. However, this problem can be eliminated by using the absorbing boundaries (AB) technique (Sochacki, Kubichek, George, Fletcher and Smithson, 1987), which is employed in this study. The results show that the empirical properties of the variables involved substantially impact the value of the renewal option. For example, as the market rent and the CPI THE PRICING OF EMBEDDED OPTIONS IN REAL ESTATE LEASE CONTRACTS 257 become more closely related, the value of the option decreases and increases as the relationship diminishes. Several other relationships are found between the option value and the two variables and it would be expected that real estate property and portfolio managers will find these relationships to be useful in negotiating property leases. The Model Let O(R,X,t) denote the option which gives the lessee the right to renew the lease at an indexed rent (X) at expiration t 5 T. This would be the classical Black-Scholes European call option if X were fixed. However, the options addressed here have a stochastic X resulting in dynamic boundary conditions. Due to the dynamic boundary conditions, an analytic solution is not known. The development of the model begins by letting X follow a mean reverting process defined as: gX dX 5 k (m 2 X)dt 1 s X dZ , (3) x x x X where kx is the speed of adjustment parameter, mx is the long-run mean return of X, sx is the volatility of the returns on X, gX is the volatility exponent of X, and dZx is the standard Wiener process. The square root mean-reverting process is defined when gX 5 .5. Mean-reverting processes are appropriate for positive economic variables that tend toward a long-run mean (with or without a trend) but experience short-term disturbances. Consequently, it is often used to model interest rates (Cox, Ingersoll and Ross, 1985) and the CPI, which is why an MRP model is included in addition to the GBM model. Care must be taken when choosing the process to describe the dynamics of R, since the process must allow for R . X. Let the dynamics of R be expressed as follows: gR dR 5 k (m 2 R)dt 1 s R dZ , (4) R R R R where the R subscript denotes the same variables defined for X to be operating on R. The values of gR, kR, gX and kx must allow for the possibility of R . X. For example, if kR , kx and gR , gX then R . X for some period of time following the departure from the mean (i.e., the reversion back to the mean will be slower for R than for X, thus allowing O(R,X) . 0). Several combinations result in positive option values. The alternative case would be for R and X to follow GBM with the stochastic process of X defined as: dX 5 m Xdt 1 s XdZ , (5) x x x and the stochastic process of R as: dR 5 m Rdt 1 s RdZ , (6) R R R where the variables are the same as above. Again, R . X must be possible. 258 JOURNAL OF REAL ESTATE RESEARCH VOLUME 15, NUMBER 3, 1998 Combinations of these processes are also possible. R can follow a GBM and X an MRP; or R an MRP and X a GBM. The dynamics chosen are dictated by the properties of the option being valued. The PDEs Using the usual no arbitrage assumption and a variant of the riskless-hedge portfolio, the following PDE is derived when R and X are assumed to follow the stochastic process expressed by the stochastic differential equations (SDEs) in Equations (3) and (4): 2 2g 2 2g R x s R s X R x g g R x O 1 O 1 r s s R X O 1 rRO 1 rXO 2 O 2 rO 5 0, RR xx Rx R x Rx R x t 2 2 (7) where t 5 T 2 t and is the time to expiration of the option. The subscripts on O represent partial derivatives. Similarly, when both R and X follow the SDEs expressed by Equations (5) and (6) respectively, the PDE is: 2 2 2 2 R s X s R x O 1 O 1 r s s RXO 1 rRO 1 rXO 2 O 2 rO 5 0. (8) RR xx Rx R x Rx R x t 2 2 Equation (8) is similar to Stulz (1982), except that the boundary conditions differ considerably. This difference makes the use of risk-neutral valuation an impossibility. Equation (9) represents the PDE when R follows a GBM (Equation (6)) and X an MRP (Equation (3)): 2 2 2 2gx R s s X R x gX O 1 O 1 r s s RX O 1 rRO 1 rXO 2 O 2 rO 5 0. (9) RR xx Rx R x Rx R x t 2 2 Equation (10) is the PDE when R follows an MRP (Equation (4)) and X follows a GBM (Equation (5)): 2 2g 2 2 R s R X s R x gR O 1 O 1 r s s R XO 1 rRO 1 rXO 2 O 2 rO 5 0. (10) RR xx Rx R x Rx R x t 2 2 Four PDEs (Equations (7)–(10)) have been identified that, when appropriately solved, yield the value of the option to renew the lease at a rent indexed to the CPI. If R is assumed to be the market price of the asset, instead of market rent, the same equations can be solved for the value of an option to purchase the leased space at a price indexed to the CPI. Analytic solutions for these PDEs are not known because of the dynamics of the equations and the boundary conditions. The FDM with absorbing boundaries will be used here to approximate the solutions. Since the dynamics of the strike and market THE PRICING OF EMBEDDED OPTIONS IN REAL ESTATE LEASE CONTRACTS 259 Exhibit 1 Boundary Conditions for Equation 8