A note on embedded lease options

Abstract Buetow and Albert (1998) discuss options embedded in lease contracts. They present a pricing framework, calibrate it using data from the National Real Estate Index and apply it using a numerical method known as the finite difference method with absorbing boundaries. This note extends the analysis. Analytic solutions are presented and some of the findings are discussed. The framework developed by Grenadier is used to compare indexed renewal options for different lease lengths.

Introduction

Buetow and Albert (1998) discuss the pricing of options embedded in real estate lease contracts. They focus on renewal options where the strike price is either tied to a price index or is a fraction of the prevailing lease price. The authors argue that correctly valuing such options is of importance to practitioners, but conclude that the complexity of their numerical method precludes its use to individual properties. The message is repeated in Albert and Buetow (2000) and the methodology is also referred to in a real estate case study (Albert, Frankfort and Hobson, 2000).

This study extends the analysis by avoiding the use of numerical methods, such as the finite difference approach with absorbing boundaries in Buetow and Albert (1998). The analytical derivatives of the indexed option are compared to the results earlier inferred numerically. There is a discussion of how the various parameters enter into the pricing formulas of the options. In addition, the results are related to the framework of Grenadier (1995, 1996), which enables a systematic comparison of the indexed option for different lease lengths.

Framework

Buetow and Albert (1998) value options using continuous time arbitrage pricing theory. The value of a contingent claim can often be expressed in terms of a so-called risk neutral expectation (i.e., where drifts have been adjusted to reflect the risk aversion prevailing in the market). The risk neutral pricing approach is traditionally motivated by arbitrage arguments in a frictionless market (Bjork, 1998), but this is not always a realistic assumption for real estate. As argued by Grenadier (1995), risk neutral dynamics can also be inferred from general equilibrium arguments. In this approach, the price is seen as that which would tend to prevail in general equilibrium. This makes it possible to motivate the use risk neutral pricing even in the presence of market imperfections.

In the following, notation, E^sup Q^^sub t^[[middot]] denotes the risk neutral expectation at time t. The standard Black and Scholes (1973) call option formula, C([middot]) = C(A^sub t^, K, [sigma], T - t, r) will also sometimes be used. This denotes the call option price as a function of the price of the underlying, strike price, volatility, time to maturity and interest rate.

Valuing the Options

Buetow and Albert (1998) consider the value at a European call option to enter into a ? period lease at a future time T. Denoting the lease price and strike price by R(T,[tau]) and K respectively, the payoff of the option becomes max(R(T, [tau]) - K, 0). In the following, the notation of Buetow and Albert is used, where the lease length is assumed to be [tau] = 5:

Two different types of strike prices are considered:

* The lease price prevailing at origin at time 0, adjusted by the change in a price index. The strike price becomes K = R^sub 0^X^sub T^, if the index is normalized to 1 at origin.

* A fraction p of the prevailing market lease price at time T, or K = pR^sub T^. This type of option is always in the money by construction and has the simple payoff (1 - p)R^sub T^.

In the following the more interesting indexed option is discussed first, followed by an analysis of second option.

Indexed Strike Price

The value of the indexed option can be expressed as a risk neutral expectation:

This general expression is valid for any stochastic processes. The case when the risk neutral processes are correlated geometric Brownian motions is:

The driving Wiener processes have correlation coefficient [rho].

A similar pricing problem was studied by Fischer (1978) and the value of the option is as follows at any time up to maturity:

N([middot]) denotes the cumulative standard normal distribution.

A Different Viewpoint

By rewriting the payoff of the option, it becomes clearer how the pricing formula can be derived and some intuition can be provided. With notation Z^sub T^. = R^sub T^/X^sub T^, the option price in Equation (2) can be expressed as:

This representation relates to methods known as reduction of the state space in differential equation theory or change of numeraire in probabilistic theory (Bjork, 1998). The intuition is that while the strike price is stochastic in nominal terms, it is fixed in relation to the index X^sub t^. As a result, it is possible to price the option in terms of the index ("real terms") using the Black-Scholes formula, and then convert back to the current price level:

The following definitions apply:

This is the same result as in Equation (5), but expressed using the Black-Scholes formula. Here, Z^sub t^ may be interpreted as the real lease price and r as the real interest rate. Further, at origin the option is at the money and has value:

By indexing the option, its value changes through two effects (Fischer, 1978): adjusting the interest rate and adjusting volatility.

First, a positive (negative) risk neutral drift of the price index lowers the adjusted interest rate, reducing (increasing) the value of a call option. Second, higher (lower) volatility increases (decreases) the value of the call option. Volatility is changed by the index itself as well as through the covariance between the two processes, as can be seen from Equation (12). The first effect increases the value, while the second can go either way depending on the sign of the correlation coefficient. The more positive the correlation coefficient is, the lower the value of the option.

One would probably believe that indexing tends to reduce the value of the option: a generally increasing price level means, on average, a higher strike price at maturity. However, the indexed option could be more valuable as both the interest and volatility effects may, in general, go either way.

Greeks

By differentiating the value of the option with respect to its parameters, the so-called "greeks" are obtained. Buetow and Albert (1998) inferred the signs using numerical simulations. The analytical expressions are given in Exhibit 1, and the signs conform to their results with one exception. They report that the price of the option is always an increasing function of the volatility in the lease price process. However, the sign is ambiguous for the same reason as discussed above for the volatility of the price index. That is, there is a direct volatility effect and a covariance effect, where the latter may go either way. More specifically, the derivative of the option with respect to the volatility of the lease price can be written as [kappa]^sub R^ = M([sigma]^sub R^ - [rho][sigma]^sub X^) where M is a complicated expression that is always positive. Since the correlation coefficient can be at most equal to 1, a necessary condition for the derivative to be negative is that volatility of the lease price is smaller than that of the price index. Further, the option's reaction to changes in the volatility of the two stochastic processes is completely symmetric, i.e., the derivative with respect to volatility in the price index can be written as K^sub X^ = M([sigma]^sub X^ - [rho][sigma]^sub R^), with the same definition of M as before. As a result, the option can be decreasing in the volatility of at most one of the two processes at the same time.

Method of Buetow and Albert

Buetow and Albert (1998) price the payoff max(R^sub T^ - X^sub T^R^sub 0^, 0) using the pricing PDE (partial differential equation). Risk neutral pricing and the PDE approach are equivalent, and the pricing formula presented above may be viewed alternatively as the risk neutral expectation or as the analytic solution to the pricing PDE. Buetow and Albert use a numerical method known as finite differences with absorbing boundaries. Looking at the PDE one sees that the authors have used the drift r, which is thus the postulated risk neutral drift of the index and lease price processes.

Regarding the lease price, Buetow and Albert (1998) make the following statement: "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." It is well known that the risk neutral drift of a traded asset is equal to the short interest rate less any dividend yield (e.g., Bjork, 1998). Since the asset pays out a dividend (the value of the lease service flow), its risk neutral drift must be less than the risk free rate. When considering a stochastic process that is not a traded asset (e.g., a price index), then the risk neutral drift is not defined by the interest rate in general. In order to find the risk neutral drift in this case, one must infer its value from traded contracts that depend on the price index (such as real bonds), or resort to some theoretical equilibrium model (such as the capital asset pricing model).

Buetow and Albert (1998) price indexed renewal options by calibrating volatility using data from the National Real Estate Index. Their results would also be affected by the choice of risk neutral drifts for the lease price and index.

Numerical Algorithms

For more complicated derivatives or stochastic processes it may be difficult to obtain simple analytic results. Although one could then consider the numerical method of Buetow and Albert (1998), a Monte Carlo valuation is also feasible. This proceeds as follows: (1) Find the joint risk neutral distribution of the lease price and the index at time T. (2) Generate an outcome from the joint distribution. (3) Compute the option payoff for this outcome and discount back to obtain the present value. (4) Repeat steps 2 and 3 a large number of times, for instance 10000 times. (5) The average in step 4 is the estimated price of the call option.

The above can be used to evaluate the case when the initial value of the index is specified as a lower bound, which is suggested as a task for possible future work by Buetow and Albert (1998). The option's price then becomes:

O^sub t^ = e^sup -r(T-t)^E^sup Q^^sub t^ [max(R^sub T^ - R^sub 0^ max(X^sub T^, 1), 0)]. (15)

This is the same expression as in Equation (2), except that X^sub T^ has been replaced by max(X^sub T^, 1). In Exhibit 2 the ratio between the indexed option with and without lower bound is given (for parameter values see below the exhibit). As might be expected, the impact of the lower bound is less significant, the higher the risk neutral drift of the price index. Higher volatility of the price index will tend to be offset by the fact that the lower bound cuts off the positive potential of the volatility.

Strike Price as a Fraction of Market Price

Next, the option is considered where strike price is equal to a fraction p of the market price prevailing at the time of maturity. The drift of the lease price is the same as previously [i.e., it follows Equation (3)]. Thus, the value of the option at origin is given by:

O^sub 0^ = e^sup -rT^ (1 - p)E^sup Q^^sub 0^[R^sub T^] = (1 - p)R^sub 0^ e^sup ([mu]-r)T^. (16)

This contradicts the result reported by Buetow and Albert (1998):

O^sub 0^ = e^sup -rT^ (1 - p)E[R^sub T^] = (1 - p)R^sub 0^ e^sup ([mu]-r)T+[sigma]^sup 2^T/2^. (17)

First, there is no information regarding under which measure the expectation is taken. For pricing purposes, it is the risk neutral measure that is appropriate. Second, a variance term appears in the result. The authors therefore conclude that the option would be "less valuable when attached to a lease in stable market, such as the northern New Jersey office market, than it would be in a volatile market, such as the Boston office market." However, since the value of the option is a fraction of the risk neutral expectation of the lease price, volatility does not matter all else being equal. In a standard option on the other hand, the holder only has upside potential and therefore volatility increases the expected payoff.

Renewal Options for Different Lease Lengths

The aim of this section is to relate the previous results to the lease literature in finance. This will give the same type of valuation formula as before for the indexed renewal option, but in addition makes it possible to systematically compare the effect of the renewal option across lease lengths. It is of interest to analyze how the value of the indexed renewal option depends on the length of the lease. Also, since the work by Buetow and Albert (1998) is one of few continuous time real estate papers with an empirical section on lease option pricing, this helps to establish a link between the theoretical and empirical work.

Several earlier papers consider lease options. Grenadier (1995) presents a general discussion based on the premise that the price of a lease with an option should be equal to the value of its two parts, namely the pure lease and the option. Among other things, he specifically compares the value of a lease with and without a nominal renewal option for different lease lengths (illustrated in his Figure 3). The renewal option becomes more valuable for longer lease lengths with an increasing rent level. Grenadier (1995, 2002) also discusses tying lease payments to a price index. This arrangement does not include an option, but highlights the importance of indexing. Beardsley, Hendershott and Ward (2000) use a Monte Carlo approach to value multiple indexed renewal options. They find that the value to repeatedly renew a lease at an indexed price can become significant, especially if real rents are increasing. Ambrose, Hendershott and Klosek (2002) further analyze an upward-only adjusted lease common in the United Kingdom and many Commonwealth countries. In this case, it is the lessor, rather than the lessee, that has an option to increase the lease rate at certain points in time to the currently prevailing lease rate. This section uses an analytic expression for the indexed renewal option to analyze its impact on the lease price for varying contract lengths.

The approach used follows the seminal work of Grenadier (1995, 1996), which is set in continuous time and abstracts from transaction costs. Further, he assumes that in equilibrium, the present value of lease payments should equal the present value of the service flow from the leased asset, which is also the case in Miller and Upton (1976), McConnell and Schallenheim (1983) and Schallenheim and McConnell (1985). Grenadier (1995, 2002) proceeds to derive models with endogenously determined supply sides and uses those models to consider the term structure of lease rates and many different leasing arrangements. Grenadier (1996) considers the effect of credit risk (i.e., a risky lessee). In that paper, exogenously specified real estate dynamics are used, and that is also the approach followed here.

Parameterization

In the following an exogenous short rent (similar to a dividend yield for a stock) is assumed to follow a geometric Brownian motion (e.g., Grenadier, 1996; Beardsley, Hendershott and Ward 2000; Ambrose, Hendershott and Klosek, 2002; and Stanton and Wallace, 2002). This gives a well known and simple expression for the lease price for different contract lengths. It is then straightforward to derive results for renewal options with indexed strike prices.

The short rent (V^sub t^) is thus defined by the following dynamics:

dV^sub t^ = [mu]V^sub t^dt + [sigma]V^sub t^dW^sub t^. (18)

Following Grenadier (1995), the equilibrium price of the real estate asset (H^sub t^) is assumed equal to the present value of the short rent stream:

It is thus necessary that the interest rate is lager than the drift of the short rent for the real estate asset to have a finite price. Further, introduce notation for the difference between the interest rate and the drift in the short rent, [delta] = r - [mu]. Since V^sub t^ = [delta]H^sub t^, it is possible to interpret [delta] as a yield or payout ratio. The risk neutral dynamics of the real estate asset price, which is the same as for the short rent, can now be written:

dH^sub t^ = (r - [delta])H^sub t^dt + [sigma]H^sub t^dW^sub t^. (20)

This rewriting is convenient because it expresses risk neutral dynamics in terms of observable quantities (real estate price, payout ratio and volatility). It has been frequently used in the real options literature (e.g., Dixit and Pindyck, 1994). A short rent that follows geometric Brownian motion thus implies constant payout ratio and volatility. This may be empirically less plausible for lease lengths where business cycle dynamics are likely to be important.

The price R(t, [tau]) for leasing over a [tau] period is equal to the present value of the corresponding short rent stream (Grenadier 1995, 1996):

Typically, payments are made in periodical installments throughout the lease and not as a lump sum at the beginning of the lease. The continuous lease rate paid throughout the lease is just the annuitized value of the lease price:

The lease price R(t, [tau]) observed at different times t is not a traded asset, but in fact a different asset for each t. However with a constant payout ratio, the lease price is just a constant times the asset price and therefore the two have the same dynamics. Now consider a call option that expires at T to rent over a [tau] time period at indexed price K. Since the lease price follows a geometric Brownian motion, the same type of Black-Scholes analysis as before is valid. If the index continues to follow the geometric Brownian motion of Equation (2) and be normalized to 1 at origin, the price at any time prior to maturity is:

Indexed Renewal Option

In order to employ the above, a [tau] year lease with an indexed option to renew for a further [tau] years is considered. The option will be valued at origin (i.e., at time 0). As also noted by Beardsley, Hendershott and (2000), there are two possible approaches to handle this:

* The lessee makes a separate payment to cover the cost of the option and leases at the standard price R(0, [tau]). The strike price in the renewal option is R(0, [tau]) adjusted by the index.

* The cost of the renewal option is embedded into the lease price. The strike price in the renewal option is equal to this new lease price adjusted by the increase in the price index.

In the first case, the indexed option need only be evaluated at origin and its value is a fraction of the standard lease price, R(0, [tau])C(e^sup -[delta]r^ 1, [sigma], [tau], r ). In the second case, the premium for the renewal option must be incorporated in the lease price. Since the price for the contract including the option must be equal to its two parts, the following equation is obtained:

If it is the rate including the option that is known, the above reasoning can be used in reverse to solve for the pure lease rate without an option.

Exhibit 3 shows the rate as a function of lease length (parameter values are given below the graph). The graph shows the term structure with (upper line) and without (lower line) an embedded renewal option. With no indexed renewal option, the term structure is downward sloping in this case (i.e., the risk neutral drift of the lease price is negative). However, with the option the term structure becomes hump shaped. That is a result of two counteracting forces: As the lease period increases, effective volatility becomes larger, which drives up the price of the option. On the other hand, the expected future lease price decreases, which reduces the value of the option. The first effect dominates the second one for short horizons, widening the gap between the lease with and without the renewal option. For longer maturities, however, the lease with a renewal option slowly converges to the one without.

The above analysis for specific contract types and dynamics is a special case of the general discussion in Grenadier (1995). In particular, the lease price with an option must always be at least as high as the price without an option.

Conclusion

Renewal options of many types are common in real world leasing arrangements. They are therefore an important phenomenon and as a result, tractable methods could potentially be very useful. Buetow and Albert (1998) present work involving just that, and here an attempt to extend the analysis has been made. The pricing was implemented in a manner that involved little more than the Black-Scholes formula. Further work could try to apply more flexible lease option models, perhaps by drawing on the theoretical lease valuation literature that has recently been developed.

Valuable comments from three anonymous referees are gratefully acknowledged.

References

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Eric Clapham, Stockholm School of Economics, S-113 83 Stockholm, SWEDEN or Eric.Clapham@hhs.se.

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