Real option value and random jumps: application of a simulation model

Abstract

This article studies how sensitive real option valuations are to incorrect assumptions about the stochastic process followed by the state variables. We design a valuation model which combines Monte Carlo simulation and dynamic programming and provides an appropriate framework to evaluate the effect of estimation errors on both the value of real options and their critical frontier. Although the model is flexible enough to value American-type options contingent on a wide range of stochastic processes, we focus on the analysis of the effect of stochastic jumps. We apply our model to the valuation of an investment in the car parts industry documented in previous literature. Our results clearly show that underestimating this type of jumps might lead to substantial misjudgements in a firm's decision-making processes. For instance, it may lead to profitable projects being rejected when jump diffusion is low, or negative expanded net present value projects being accepted.

Acknowledgements

The authors benefited from the comments of Esther del Brío, Rosa Mayoral, Alberto de Miguel, and participants at the XIII ACEDE Conference. They especially thank Mark Taylor (editor) and anonymous referees for useful suggestions. Financial support from Ministerio de Educación y Ciencia and FEDER (ref. SEJ2007-67382/ECON), and Junta de Castilla y León (ref: VA05204), is also acknowledged. Any errors are the responsibility of the authors.

Notes

¹In addition to its theoretical superiority, a number of papers have found empirical insights regarding the relevance of real option in market values. Such is the case of Paddock et al. (1988), Berger et al. (1996), Danbolt et al. (2002) or Andrés-Alonso et al. (2005), among others.

²Our model's advantage decreases with the number of early exercise opportunities, since determining each critical value requires simulating new paths for the state variable, and hence the model costs – in terms of time and computing resources – grow exponentially.

³This does not reduce the advance of the ROV approach over traditional DCF models, but highlights interest in analysing the value to postpone the exercise of real options when possible. See Vandenbroucke (1999) for a comparison of the NPV approach and ROV approximations.

⁴A good example of this view is the second edition of Hull's handbook on financial options (Hull, 1993), which on page 334 explains that ‘one of the drawbacks of the Monte Carlo approach is that it may only be used for European style derivatives’. In this same sense, Hull and White (1993) postulate that ‘Monte Carlo simulation cannot deal with early exercise since there is no way of knowing whether this is optimal when a specific price is reached at a given moment’.

⁵Some authors cite Bossaerts’ (1989) working paper as the earliest reference to the analysis of early exercise of American options through simulation.

⁶This procedure entails certain significant drawbacks, such as the need to store all the simulated paths – a time-consuming exercise – as well as the complexity linked to the sorting process when dealing with multiple sources of uncertainty.

⁷Unlike binomial and trinomial trees, the values that appear at each node are placed in the order in which they are generated and not following a hierarchic order.

⁸The equidistance (τ) of the exercise dates, derived from this formula, is assumed for clarification and explanation purposes only and does not condition the model in any way. Logically, T ^ο  < T.

⁹Hence, the mean growth rate caused by the discrete jumps is λ k.

¹⁰This expression derives from applying the risk-neutral valuation approach to the ‘twin’ financial asset, which is perfectly correlated to the state variable.

¹¹Following Merton (1976), we assume the risk associated to the discontinuous jump of the state variable to be diversifiable. The risk-neutral simulation would then show a continuous modified drift, r − δ, rather than the initial α. This is the equivalent of subtracting from the continuous drift the risk premium of the corresponding asset (Trigeorgis, 1996: p. 102).

¹²The simulation of the number of discrete jumps at a time interval, Δt, is obtained from applying the Monte Carlo method to the accumulated probability function P[q ≤ X].

¹³The simulation may begin at any moment and for any value of the state variable (Grant et al., 1996). However, when dealing with American options, whose optimal exercise at each moment depends on future expectations, the first critical value to be calculated must correspond to expiration.

¹⁴In order to simplify the analysis, we have not considered the car-makers’ control of the value chain and its implications for the option valuation. For a detailed study, see Azofra et al. (op. cit.).

¹⁵The parameter λ = 0.2 implies that, on average, only one discrete jump will occur during the five-year life span of the underlying project. We feel it is more interesting to show the valuation results assuming a highly volatile and low frequency process of random jumps, as opposed to multiple smaller jumps, which may prove hard to distinguish from continuous evolution itself. On the choice of parameters see http://www.puc-rio.br/marco.ind/stoch-a.html#jump-dif

¹⁶The technique of antithetical variates consists of generating two symmetrical observations at zero for each of the random simulations of the normal distribution.

¹⁷Logically, the 0% level of the volatility of the discontinuous jump corresponds to the pure geometric Brownian process.

¹⁸The estimation of the total volatility for the mixed process was performed using the expression obtained in Navas (2003), who amends the one initially obtained in Merton (1976).

¹⁹In fact, the reader can note that this negative relation disappears in cases of zero jump volatility and α values equal to 0 and 7%.

²⁰It should be noted that, as the project presents a finite life span, approaching the option expiration implies less time to compensate for the cost of additional investment through the higher cash flows arising from expansion.