We analyze the prices of derivative securities in response to the changes in the parameters characterizing investors’ internal and external habits. Using a multiplicative specification for preferences, we solve for the equilibrium allocation with a second order approximation of the policy function. We recover the prices of the derivatives and we characterize their response to changes in the duration and the intensity of internal and external habits separately. We show that there is a monotonic relation between the duration parameter and the forward and options’ price under both types of habits. The effect of the intensity parameter however, depends of the level on the duration and on the particular habit that is analyzed.
Text of Derivatives Pricing under Habit Formation and Catching-up with the Joneses
1. Barcelona Graduate School of Economics MSc in Macroeconomic Policy and Financial Markets Derivatives Pricing under Habit Formation and Catching-up with the Joneses Corina BOAR Rodrigo GAZE Antoni TARGA Advisor: Prof. Jordi Caball AbstractWe analyze the prices of derivative securities in response to the changes in the parameterscharacterizing investors internal and external habits. Using a multiplicative specification forpreferences, we solve for the equilibrium allocation with a second order approximation of the policyfunction. We recover the prices of the derivatives and we characterize their response to changes inthe duration and the intensity of internal and external habits separately. We show that there is amonotonic relation between the duration parameter and the forward and options price under bothtypes of habits. The effect of the intensity parameter however, depends of the level on the durationand on the particular habit that is analyzed.Keywords: derivative securities, internal habit, external habit
2. 1. IntroductionThe purpose of this paper is to extend the asset pricing model proposed by Lucas (1978)by introducing habits in the utility function of the representative investor and use thisspecification to characterize numerically the relationship between the habit parametersand the prices of derivative securities such as forward contracts, call and put options.The consumption capital asset pricing model (CCAPM) is suitable to price all kinds ofassets. However, under the standard power utility specification the model fails toexplain important facts about stock returns. In response to this, the extensive assetpricing literature has introduced habit formation in the preferences of the investors.There are two approaches to model these alternative specifications of preferences:internal-habit formation and external-habit formation. Under the former, individualsderive utility from the comparison of their current level of consumption with the one ofthe previous periods. When choosing a level of consumption they implicitly set astandard of living for the subsequent periods. Under the latter, individuals derive utilityfrom the comparison of their own consumption with the average level of pastconsumption in the economy such that any increase in the average consumption isperceived as a negative externality.The literature uses two ways of introducing habits in the utility function of theindividuals. One is the additive manner under which habits play the role of a minimumlevel of consumption and the utility has the following functional form: , (1.1)where is the current level of consumption, is the stock of habit and is thecoefficient of relative risk aversion. The other is the multiplicative manner under whichutility depends on the current level of consumption relative to a reference leveldetermined by habits and it has the following functional form: (1.2)In this paper we adopt the latter specification to avoid the utility to be negative in theevent the individual is confronted to a consumption below the habit, and we calibrate 1
3. the parameters of the model such that we guarantee that the utility is always increasingin consumption and concave.The remainder of this paper is organized as follows. Section 2 surveys briefly the assetpricing literature of models with habits, Section 3 describes the model, the derivativespricing logic and the methodology used and Section 4 presents the quantitative results.Finally, Section 5 concludes the paper.2. Literature ReviewThe asset pricing model developed by Lucas (1978) establishes a link between thefinancial markets and the real side of the economy, represented by consumption. Themodel was designed to price any financial asset in the setup of a rational expectationseconomy. However, it has been shown that under the power utility specification, themodel fails to explain important facts about stock returns such as the high equitypremium, the high volatility of returns and the countercyclical variation in the equitypremium (Mehra and Prescott, 1985).In response to these failures, financial economists have considered alternative models ofpreferences. One prominent line of research is the one taking into account the socialnature of portfolio decisions which arises from the presence of consumptionexternalities: agents have preferences defined on their own consumption, as well as onthe average consumption in the economy (Gal, 1994). There are two approaches inmodeling these externalities: one is called the internal-habit formation model, asproposed by Constantinides (1990), for example, in which habit depends upon theagents own consumption and the agent takes this into account when choosing futurelevels of consumption. The other approach is called the external-habit formation model,as suggested by Abel (1990, 1999) and Campbell and Cochrane (1999), in which habitdepends upon the average level of consumption in the economy that is unaffected byany individual agents own decisions.1Under internal habits, on one side, individuals derive utility from the comparison of thecurrent level of consumption with the one of the previous periods. This has1 Abel (1990, 1999) calls it catching up with the Joneses 2
4. consequences on the optimization problem faced by consumers because when theychoose their current consumption they implicitly select a standard of living for thefuture periods. On the other side, under external habits, the individuals derive utilityfrom the comparison of their own consumption with the average level of consumptionin the economy. The spillovers of others consumption could increase or decrease theindividuals marginal utility of own habit-adjusted consumption (Alonso-Carrera et al.,2006).Models with habit formation have been used in the asset pricing literature in an attemptto explain the equity premium puzzle by authors like Abel (1990, 1999), Campbell andCochrane (1999). Even if this line of research performs better than the standard powerutility model in explaining empirical facts, most of the times it has to rely on highcoefficients of relative risk aversion. Yogo (2008) develops a standard model withexternal habit formation that is able to account for the empirical facts appealing to lowrisk aversion by introducing a utility function that evaluates gains and losses inconsumption relative to the habit.Boldrin et al. (1995) argue that in comparing the efficiency of the two types of models itis important to distinguish between the relative risk aversion of the investor and themeasure of the curvature of his utility function. While in external-habit formationmodels the two are identical, in internal-habit formation models one needs todisentangle between them. Therefore, in the case of external habits specifications,accounting for the equity premium by increasing the curvature of the utility functionalso leads to counterfactually high levels of risk aversion, while in the case of internalhabits it is possible to induce a high curvature without using such high values of therelative risk aversion (Constantinides, 1990).It can be seen that the extensive literature on asset pricing lends credence to thepresence of habit formation. Even if such a specification does not fully explain all assetpricing anomalies, it is widely agreed that it fits the data better than standard time-separable utility models. The main contribution of this paper is extending the usage ofmodels that incorporate habit formation to value derivative securities and analyze howthe features of habits, such as duration and intensity, affect the prices of these securities. 3
5. 3. The Model3.1 PreferencesThe economy is populated by a continuum of identical infinitely lived agents thatmaximize expected life-time utility. As in the Lucas (1978) fruit-tree model, there is noexogenous endowment and the output produced by fruit-trees is completely perishable.At every period the individual consumer chooses how many shares on tree topurchase, , and, thus, consumption, , such that he maximizes the following: (3.1)subject to:and ,where , is the stock of internal habits and thestock of external habits defined, like in Fuhrer (2000), as follows: , (3.2)and , (3.3)where is the individuals own consumption and is the average level ofconsumption in the economy.In (3.1) is the deterministic discount factor, is the coefficient of relativ