Quadrinomial trees with stochastic volatility to value real options.
| Autor | Marin-Sanchez, Freddy H. |
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Introduction
Authors such as Trigeorgis (1996), Mun (2002) and Damodaran (2019) recognize there is a difference between the theoretical concepts of valuation of capital assets and their empirical approximation, a discrepancy that is reflected in a gap not deeply explained and with sufficient academic rigour between the market value and its theoretical estimate. Therefore, different methodological alternatives have been proposed to mitigate this difference, but few consider key factors that generate value, such as those that try to measure and estimate uncertainty through the use of volatility, which usually has been moderately studied and ends up being relevant at the moment of making any type of valuation on a capital asset (Dixit and Pindyck, 1994; Valencia Herrera and Martinez Gandara, 2009; Alvarez Echeverria etal., 2012).
In academia, the use of discounted cash flow (DCF) as a method used to value capital assets is common, but there are numerous criticisms about its use because it does not include elements such as contingent events, risk present in the cash flows and volatility (Trigeorgis, 1996). To overcome some of these problems, the real options approach (ROA) was developed, which complements the DCF and allows it to include volatility as a fundamental parameter to quantify the risk and collect some elements associated with uncertainty (Keswani and Shackleton, 2006; Sabet and Heaney, 2017). The ROA model was derived from its simile in the theory of financial options by modelling the value of assets as a call or put options, considering that it is applied to investments in assets or real markets, although there is not a tradable market (Cobb and Charnes, 2004).
Since the Black and Scholes's (1973) seminal work, which is considered generally used as a method for estimating the value of European call and put options, multiple models and extensions have appeared that allow the valuation of other types of options in different contexts, using of different estimation techniques, such as closed analytical solutions, finite difference method, Adomian decomposition method and numerical method through multiplicative and additive binomial trees, but the problem of considering volatility constant persists. Recently, some models have appeared to include stochastic volatility and were created, fundamentally, to avoid considering this fixed variable in terms of evaluation time horizon. Motivated by this empirical evidence, several authors, such as Hull and White (1987), Scott (1987), Scott (1987), Chesney and Scott (1989), Stein and Stein (1991) and Heston (1993), have proposed models that involve stochastic volatility as a parsimonious extension of the Black-Scholes model (Black and Scholes, 1973). Amongst previously mentioned models, the GARCH-diffusion type proposed by Drost and Werker (1996), Duan (1997) and Duan (1996) offers the first approximation between a GARCH process and a stochastic volatility model, supported by multiple research theoretical (Ritchken and Trevor, 1999; Barone-Adesi et al, 2005; Christoffersen et al, 2010; Chourdakis and Dotsis, 2011) as well as empirical (Figa-Talamanca, 2009; Plienpanich et al, 2009; Wu et al, 2012,2014,2018). This paper proposes and motivates the inclusion of stochastic volatility in the ROA model, considering a rigorous mathematical development from a GARCH-diffusion stochastic differential equation (SDE) that has a numerical solution using multiplicative trees; besides, a prior estimate of a GARCH (1,1) process is required to allow an adequate real options valuation in real-world and risk-neutral situations.
This paper is organized as follows: Section 2 summarizes the stochastic volatility models and describes the GARCH-diffusion model used for the development of this research. Section 3 presents, in summary, the multiplicative quadrinomial tree method to assess real options. Section 4 offers a brief description of real options valuation, specifically, its use according to the method proposed. In Section 5, we present a series of numerical experiments and some examples. Finally, Section 6 is the discussion and conclusions.
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Literature review
One of the strongest assumptions considered in the Black-Scholes model (Black and Scholes, 1973) is that volatility ([sigma]) is constant; many studies have demonstrated that logarithm of returns on asset prices have leptokurtosis and conditional variance that changes randomly as a function of time (Grajales Correa and Perez Ramirez, 2008) and the assumption of conceiving a normal distribution does not apply (Fernandez Castano, 2007), additionally, implied volatilities are considered non-constant and differ between exercise prices and time to maturity. For these reasons, several extensions have been proposed in the literature in which volatility is considered a function of time as well as the price of the underlying asset, whereupon the linearity of drift and diffusion components of the asset price are maintained but incorporate a second equation that allows modelling the variance behaviour of [S.sub.t].
2.1 Stochastic volatility models
A model with stochastic volatility describes its change over time and generalizes the Black-Scholes model, defined in a given filtered probability space ([ohm], ,f, ,[f.sub.t], P), which generally satisfies a system of SDEs (Hull and White, 1987; Scott, 1987; Wiggins, 1987; Chesney and Scott, 1989; Nelson, 1990a, b; Stein and Stein, 1991; Heston, 1993; Hilliard and Schwartz, 1996; Drost and Werker, 1996; Duan, 1996, 1997; Wilmott, 1998; Ritchken and Trevor, 1999; Barone-Adesi et al, 2005; Chang and Fu, 2009; Figa-Talamanca, 2009; Moretto et al, 2010; Christoffersen et al, 2010; Chourdakis and Dotsis, 2011; Wu et al, 2012; Wu et al, 2014; Wu and Zhou, 2016; Peng and Peng, 2016; Wu et al, 2018; Wu et al, 2020, see Table 1). We considered an SDE system described as follows:
[Please download the PDF to view the mathematical expression] (1)
where [micro] is constant and volatility [[sigma].sub.t] is considered as a dynamic variable in the price [S.sub.t]. Studies like Wu et al (2018) showed that the implied volatility, captured from historical data from the market, takes a relevant role in fitting the prices of the options as Kim and Ryu (2015) explained as well. Functions f and g correspond to the tendency and diffusion of volatility, respectively. The model incorporates two sources of randomness [W.sub.t] and [W.sub.t] that correspond to Wiener s standard processes with a correlation coefficient p. The price process {[S.sub.t], 0
The GARCH-diffusion model was introduced by Wong in 1964, but its popularity only grew following the works of Nelson (1990a, b). An important condition was discovered by Christoffersen et al. (2010) who demonstrated empirically, through the use of realized volatilities on the S&P500 returns with an option's data panel, that the Heston model was poorly specified because, in the diffusion model presented by the author, volatility was found in the square root instead of being considered linear; these conclusions were reaffirmed by Chourdakis and Dotsis (2011) although they also suggested that the model should consider a nonlinear drift against a linear one.
Recent studies have indicated that this model allows a better description of the behaviour and dynamics of financial series than other types of models, such as Heston (1993), Ait-Sahalia and Kimmel (2007), Jones (2003) and Wu et al. (2018).
Also, it has been used as a good model for adjusting financial option data (Christoffersen et al, 2010; Chourdakis and Dotsis, 2011; Kaeck and Alexander, 2012; Wu et al, 2012, 2014, 2018,2020). The most recent applied research related to this model is summarized in Table 2.
In general terms, this type of model is usually characterized by not having a closed solution and is in the class of non-affine models; also, their solutions must be achieved with simulation, numerical methods (Wu et al, 2012) or the use of integrals in SDEs (BaroneAdesi et al, 2005; Florescu and Viens, 2008; Vellekoop and Nieuwenhuis, 2009). The system of traditional equations presented by this model has the following functional structure (Barone-Adesi et al, 2005):
[Please download the PDF to view the mathematical expression] (2)
where [c.sub.1], [c.sub.2] and [c.sub.3] are positive constants; [micro] is the tendency parameter; [c.sub.2] is the mean-reversion rate; ([c.sub.1])/([c.sub.2]) is the mean long-term variance and [c.sub.3] models the random behaviour of volatility and corresponds usually to the volatility of volatility. For its part, [W.sub.t](1) and [W.sub.t](2) correspond to Wiener processes with a correlation coefficient [rho].
An alternative way to present the structural form of equation (2) is an SDE; following (Wu et al., 2012):
[Please download the PDF to view the mathematical expression] (3)
[dV.sub.t] = a([theta] - [V.sub.t])dt + [sigma][V.sub.t][dW.sub.t(2)] (4)
where parameters a, 8 and a are constant and are equivalent to the mean-reversion speed, the mean long-term volatility or tendency and the volatility of volatility, respectively. Consequently, [W.sub.t] and [W.sub.t] correspond to independent one-dimensional standard Wiener motion processes.
This paper focuses on using equations (3) and (4) to summarize the numerical method for a multiplicative quadrinomial tree that includes stochastic volatility, which will allow the valuation of a derivative instrument, such as real options, when the volatility of the underlying asset is stochastic. Hence, we use the equivalence proposed by Hull (2003) and described by others (Pareja-Vasseur et al, 2020) that relates the conditional variance from GARCH (1,1) and the stochastic variance from differential equation proposed in (4), i.e. [V.sub.t] is equivalent to [h.sup.2.sub.t], with [Please download the PDF to view the mathematical expression]. These set of parameters of the conditional variance could be estimated by the maximum...
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