A reduced-form model for lease contract valuation with embedded options

The valuation of the lease contract with a structural approach was first introduced by Grenadier (1995). Some of the other papers using this approach are Ambrose et al. (2002), Buetow and Albert (1998), Clapham (2003), etc. Structural-form models are not efficient in pricing lease and mortgage-related contracts (see Ambrose and Yildirim 2008, Liao et al. 2008, Chernov et al. 2018, Cananà et al. 2023), because it requires strong assumptions about latent state variables, such as the lessee's asset value process and its related parameters.

Since we don’t access to the data, we look for a commercial lease agreement from sec.gov, covering the clauses on the cancelation, default, and purchase including rent adjustment. Ignoring one or two of these clauses would cause miscalculation of the initial fair lease rate (see https://​www.​sec.​gov/​Archives/​edgar/​data/​1098009/​0001002334130000​62/​elmonteleaseforf​iling.​htm.

Grenadier (1995) advocates that adjustable-rate leases may provide the lessor a hedge against unexpected inflation or cost fluctuations.

Ambrose and Yildirim (2008) note that the Vasicek model allows the possibility of negative interest rates; however, this model captures the mean-reverting property of the short interest rate. The Vasicek model is also a Gaussian process that provides simple analytic results and enables us to obtain reduced-form solutions.

In practice, multiple adjustment time points are common in adjustable-rate lease contracts. To simplify the derivation and illustration of our model, we set a single adjustment date, T₁. The resulting model can be extended to the multiple adjustment time points case. In our numerical example, T₁ is assumed as the middle point in \(\left[ {t,T} \right]\), \(i.e. \, T_{1} = \frac{t + T}{2}\). However, to keep the time of adjustment flexible, we do not place any restrictions on T₁.

In the literature of derivatives pricing, the market price of risk is specified to eliminate the drift terms in the stochastic process under the measure Q. However, as in Ambrose and Yildirim (2008), in this paper we analyze the impact of the market price of risk of each variable on the lease term structure. Therefore, we use the notations of the market price of risk in the model and thus the drift terms under the physical measure appear in the process. Recently, real estate related derivatives became popular in the market. The trading of these derivatives provides investors instruments to hedge real-estate service-flow risk and leased asset price risk and be free of arbitrage opportunity. A more detailed introduction of real estate derivatives can be referred to Van Bragt et al. (2015).

The reference index maybe CPI, prevailing market rent, or another exogenous index as defined in lease contracts.

Notice that when the lessee defaults or exercises the cancellation option between T₁ and T, the rent and service-flow recovery rates will not be the same as those between t and T₁, since they depend on different levels of rental payments.

Liao et al. (2008) note that the specifications in Eq. (9) imply that the hazard rates can be negative with a positive probability. However, Duffee (1999) suggests that this problem can be largely ignored if the model accurately prices the relevant instruments. Furthermore, such assumptions facilitate the model's pricing procedure and bring economic intuition into the hazard rates.

The detailed derivations of the proposition are given in the Internet Appendix and available upon request.

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One may use the historical recovery rates of the events in which the lessee defaults, exercises the cancellation option, or purchases an option for implementation.

We have a rich model of analytic solution; we can also analyze multiple things such as the impact of the interest rate volatility, the market price of risk, the recovery rate, and the correlations on the lease contract prices. All these comparative statistics are put in the Internet Appendix. This Appendix is available upon request.