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 Caballé

Abstract

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.

Keywords: derivative securities, internal habit, external habit

1

1. Introduction

The 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 this

specification to characterize numerically the relationship between the habit parameters

and 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 of

assets. However, under the standard power utility specification the model fails to

explain important facts about stock returns. In response to this, the extensive asset

pricing 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, individuals

derive utility from the comparison of their current level of consumption with the one of

the previous periods. When choosing a level of consumption they implicitly set a

standard of living for the subsequent periods. Under the latter, individuals derive utility

from the comparison of their own consumption with the average level of past

consumption in the economy such that any increase in the average consumption is

perceived as a negative externality.

The literature uses two ways of introducing habits in the utility function of the

individuals. One is the additive manner under which habits play the role of a minimum

level 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 the

coefficient of relative risk aversion. The other is the multiplicative manner under which

utility depends on the current level of consumption relative to a reference level

determined 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 the

event the individual is confronted to a consumption below the habit, and we calibrate

2

the parameters of the model such that we guarantee that the utility is always increasing

in consumption and concave.

The remainder of this paper is organized as follows. Section 2 surveys briefly the asset

pricing literature of models with habits, Section 3 describes the model, the derivatives

pricing logic and the methodology used and Section 4 presents the quantitative results.

Finally, Section 5 concludes the paper.

2. Literature Review

The asset pricing model developed by Lucas (1978) establishes a link between the

financial markets and the real side of the economy, represented by consumption. The

model was designed to price any financial asset in the setup of a rational expectations

economy. However, it has been shown that under the power utility specification, the

model fails to explain important facts about stock returns such as the high equity

premium, the high volatility of returns and the countercyclical variation in the equity

premium (Mehra and Prescott, 1985).

In response to these failures, financial economists have considered alternative models of

preferences. One prominent line of research is the one taking into account the social

nature of portfolio decisions which arises from the presence of consumption

externalities: agents have preferences defined on their own consumption, as well as on

the average consumption in the economy (Galí, 1994). There are two approaches in

modeling these externalities: one is called the internal-habit formation model, as

proposed by Constantinides (1990), for example, in which habit depends upon the

agent's own consumption and the agent takes this into account when choosing future

levels 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 habit

depends upon the average level of consumption in the economy that is unaffected by

any individual agent's own decisions.1

Under internal habits, on one side, individuals derive utility from the comparison of the

current level of consumption with the one of the previous periods. This has

1 Abel (1990, 1999) calls it catching up with the Joneses

3

consequences on the optimization problem faced by consumers because when they

choose their current consumption they implicitly select a standard of living for the

future periods. On the other side, under external habits, the individuals derive utility

from the comparison of their own consumption with the average level of consumption

in the economy. The spillovers of others’ consumption could increase or decrease the

individual’s 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 attempt

to explain the equity premium puzzle by authors like Abel (1990, 1999), Campbell and

Cochrane (1999). Even if this line of research performs better than the standard power

utility model in explaining empirical facts, most of the times it has to rely on high

coefficients of relative risk aversion. Yogo (2008) develops a standard model with

external habit formation that is able to account for the empirical facts appealing to low

risk aversion by introducing a utility function that evaluates gains and losses in

consumption relative to the habit.

Boldrin et al. (1995) argue that in comparing the efficiency of the two types of models it

is important to distinguish between the relative risk aversion of the investor and the

measure of the curvature of his utility function. While in external-habit formation

models the two are identical, in internal-habit formation models one needs to

disentangle between them. Therefore, in the case of external habits specifications,

accounting for the equity premium by increasing the curvature of the utility function

also leads to counterfactually high levels of risk aversion, while in the case of internal

habits it is possible to induce a high curvature without using such high values of the

relative risk aversion (Constantinides, 1990).

It can be seen that the extensive literature on asset pricing lends credence to the

presence of habit formation. Even if such a specification does not fully explain all asset

pricing 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 of

models that incorporate habit formation to value derivative securities and analyze how

the features of habits, such as duration and intensity, affect the prices of these securities.

4

3. The Model

3.1 Preferences

The economy is populated by a continuum of identical infinitely lived agents that

maximize expected life-time utility. As in the Lucas (1978) fruit-tree model, there is no

exogenous endowment and the output produced by fruit-trees is completely perishable.

At every period the individual consumer chooses how many shares on tree to

purchase, , and, thus, consumption, , such that he maximizes the following:

(3.1)

subject to:

and

,

where

, is the stock of internal habits and the

stock of external habits defined, like in Fuhrer (2000), as follows:

, (3.2)

and

, (3.3)

where is the individual’s own consumption and is the average level of

consumption in the economy.

In (3.1) is the deterministic discount factor, is the coefficient of relative risk

aversion, is the price of a stock on tree in period , is the dividend paid by the

stock on tree in period and is the number of stocks on tree in period . Note

that utility is not longer time separable. Besides the fact that it depends on a benchmark

level of consumption that is exogenous to the individual, consumption choice today

influences the future reference level of habit through . This specification

parameterizes two features of models with habits:

5

1. Habit intensity: the parameters and index the importance of internal and

external habit in the utility function, respectively. If then we are back to

the case of the standard power utility model. If and then the agent

only takes into account the average level of consumption in the economy. If

and then only past levels of own consumption matter. Values of are not

admissible because the steady-state utility will be decreasing in consumption.

2. Habit duration: the parameters and index the memory of the internal and

external habit, respectively. If then only last period’s consumption is important.

If then the larger is the further back in time the individual looks when

choosing .

The Euler Equation

The Lagrangian associated to the consumer’s problem is the following:

, (3.4)

where is the Lagrange multiplier. The first order conditions with respect to ,

and are the following:

, (3.5)

where

(3.6)

and

(3.7)

By combining (3.7) and (3.8) and defining

, we obtain the Euler Equation:

(3.8)

where is defined as follows:

(3.9)

6

Note that:

,

and

. Therefore,

derivative with respect to consumption becomes

(3.10)

which can be rewritten in a compact way as

(3.11)

Defining

(3.12)

as in Fuhrer (2000), the marginal utility of consumption can be rewritten as

(3.13)

3.2 Derivatives pricing

In this section we present the analytical expressions used to price three types of

derivative securities: a one period forward contract, a one period call option and a one

period put option. In order to price these assets we needed to compute the value of the

underlying asset, shares of the goods producing trees that represent the whole economy.

After that we compute the price of the assets today by discounting their expected future

cash flows with the stochastic discount factor,

.

The price of the stock is derived directly from the Euler condition, from which we have:

(3.14)

where is the price of a share of tree n in period t, is the marginal utility of

consumption in period t and is the dividend paid by tree n in period t+1.

In equilibrium, all output is consumed in the period in which it is produced so that

. Assume that the whole output is produced by a single fruit-tree ( )

and that its stochastic process

is

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(3.15)

where .

It follows that the pricing equation can be rewritten as

(3.16)

where is the ex-dividend price in period . Equation (3.16) can now be used to price

the derivative securities of interest.

For the forward contract we have that its cost is zero by definition, which leads to the

following equation describing a long position in the forward contract

(3.17)

Rearranging equation (3.16) to solve for , the forward price of a share of the tree we

have

(3.18)

Even though we have called this the forward price of a share and not the futures price,

since only one period assets are being considered they would be exactly the same in this

framework.

The options considered are plain vanilla ones that are not protected for dividend

payments and have strike . In this case, the pricing equation of the call can be written as

(3.19)

Conversely, for the put option we have

(3.20)

It is worth pointing out that since the options considered mature in one period it does

not matter whether we are pricing a European or an American option. This follows from

the fact that the only possible date at which a holder of an American option can exercise

8

it is at t+1, which is its maturity and thus the American option will behave exactly as a

European option.

3.3 Methodology

It is common in the literature to approximate the solution to non-linear, dynamic,

stochastic, general equilibrium models using linear methods. Linear approximation

methods are useful to characterize certain aspects of the dynamic properties of

complicated models. In particular, if the support of the shocks driving aggregate

fluctuations is small and an interior stationary solution exists, first-order approximations

provide adequate answers to questions such as local existence and determinacy of

equilibrium and the size of the second moments of endogenous variables.

Evaluating the utility function using linear approximation of the policy functions, some

second and higher-order terms of the utility function are ignored while others are not.

Linearizing equations using a first order approximation would mean we assume agents

to be risk neutral, which does not hold under our assumptions. With this in mind,

second order Taylor approximation allows to contemplate features of concave utilities

and risk averse behavior, crucial features of models with externalities for asset pricing.

We solve for deviations around steady state using second order approximation with the

Schmitt-Grohe-Uribe Toolkit.2 Thus, in general, a correct second-order approximation

of the equilibrium welfare function requires a second-order approximation to the policy

function (Schmitt-Grohe and Uribe, 2004).

Due to the expressions of the price of puts and calls being non-differentiable functions,

thus not being able to be approximated with the previously mentioned approach, we

compute these prices with numerical integration methods using the CompEcon

Toolbox.3 With one of its functions we produce quadrature points and quadrature

weights so to, using as many points as specified, reconstruct the normal distribution of

the shocks, characterized with specified mean and variance. The set of quadrature points

discretizes the range of possible values that the shock could take, making possible,

having the policy functions, to compute all states and controls variables for each

quadrature point. For our purposes, using the policy functions, we compute the implied

price of the share and the discount factor for next period associated to each quadrature

2 S. Schmitt-Grohe, M. Uribe/Journal of Economic Dynamics & Control 28 (2004)

3 Mario J. Miranda & Paul L. Fackler/MIT Press (2002)

9

point. Combining these with a strike price and discounting next period payoff, we

obtain the associated price of any of the three derivatives contemplated for each

quadrature point. Weighting such prices with the quadrature weights we obtain the price

of a call, a put and a forward expiring next period.

4. Quantitative Results

4.1 Parameter values

In order to study the effects of internal and external habits on the prices of derivative

securities we proceed in the three steps for each habit: (1) the effects due to the habit

that is not of interest is deactivated, by setting its intensity parameter to zero, in order to

make sure that all the effects on the prices are due to only one type of habit; (2) the

intensity parameter of the habit that is activated is set to 0.1 and then different values for

, the duration parameter, from 0 to 0.9 are chosen; (3) for values of the intensity

parameter up to 0.9 step (2) is repeated. With this not only we are able to analyze the

effect of each of the parameters but also how interactions between them affect the prices

of plain vanilla derivatives.

The other parameters of the model that are still to be set are the coefficient of relative

risk aversion, , the deterministic discount factor, , the mean for the process of output,

, the variance of the output shock, , and the autocorrelation of output, . The index

of relative risk aversion and the deterministic discount factor were chosen from the

existent literature and are set to 1.50 (Campbell and Cochrane, 1999, and Fuhrer, 2000,

report a coefficient of risk aversion of 2.00 in the presence of external habits only but

since Abel, 1990, provides values close to 1.00 when there are only internal habits we

chose a middle point in order to analyze both habits with an utility specification that

would be consistent with the two of them) and 0.98 (Fuhrer, 2000), respectively. We set

the autocorrelation of output to 0.9 in order to capture the high persistence of the level

of output. The value of was set to zero in order to normalize output at the steady state

to be equal to 1, while was set to 0.50. In order to price options on the stock we set

the strike price to be 50, a value close to the steady state price of the stock.

10

4.2 Derivatives Prices

Under internal habits the reference level of consumption is the individual’s past

consumption. The latter is more important as the weight attached to it, , increases and

is more backwards looking as the duration of the habit, , increases. Under external

habits the average aggregate consumption in the economy is the reference level of

consumption and it enters the investor’s utility as a negative externality. The extent to

which it influences the investor’s consumption decision and, thus, his current utility

depends on the intensity of the habit, , and its duration, . Figure 1 presents the

responses of the derivatives prices to changes in the habit parameters, under the two

specifications.

Figure 1. Call, Put and Forward prices under internal and external habits

4.2.1 Forward Contract

A forward contract is an agreement between two parties to transact the underlying asset

at a future date at a predetermined price. This asset is usually traded in the over the

counter market.

11

a) Internal habits

As it can be seen in Figure 1 (Panel 5) and Figure 2, the forward price of a stock is an

increasing function of , the duration of internal habits. Given that habits affect the

marginal utility of consumption in a way such that high levels of consumption today

decrease future utility, individuals would rather have a constant level of consumption

than to have it fluctuating around this constant every period. Such a pattern is similar to

what would have been observed if the individual had a higher degree of risk aversion.

With this in mind one can see the increase in the forward price as a premium that

households are willing to pay in order to get rid of the uncertainty about future states. It

is worth noting that when the intensity parameter is high the price of the forward

contract decreases when is above a certain threshold and such movement is linked to

the behaviour of the expected stock price.

Figure 2. Forward price when changing the duration of the internal habits

As the intensity of internal habits increase, gets bigger, individuals care more about

keeping their level of consumption constant across time, resembling risk aversion and,

like what happened when the duration of habits increased, the forward price of the stock

increases (Figure 1, Panel 5, and Figure 3).

Figure 3. Forward price when changing the intensity of the internal habits

12

b) External habits

In the presence of external habits the reaction of the forward price with respect to

movements in , the duration of external habits, is similar to the one that was observed

for changes in . The main difference between the forward price reaction to and

is that it is much more sensible to the former. This happens because each individual has

much better information about his own consumption and thus can better assess the

effect that increasing consumption today will have on subsequent periods’ utility

(Figure 1, Panel 6, and Figure 4).

Figure 4. Forward price when changing the duration of the external habits

Differently from what happened to the duration parameters, now the reaction of the

forward price to an increase in , the intensity of external habits, does not resemble the

one obtained for its internal habits’ counterpart (Figure 1). When habits are external the

household takes them as given, so as increases the weight of a term that the

household cannot control is getting bigger. This movement in does not affect the rate

at which individuals will discount future payoffs but, since it resembles risk aversion in

a sense that individuals dislike consumption volatility it makes individuals require a

higher risk premium to hold assets that have payoffs that are negatively correlated with

marginal utility.4 Since the stock is one of these assets, when the intensity of external

habits increases, its price today decreases and so does its expected price for the next

periods. This happens in order to increase the return on all periods and not only today.

Because the expected price in the following period decreases, so will the forward price,

as can be seen in equation (3.18). Figure 5 captures the reaction of the forward price to

different levels of habit intensity. Similarly to the non monotonous behaviour observed

with extreme values of , when is associated with high values of the forward

4 Recall that we are departing from the steady state, so

and only assets that mature

in one period are being considered.

13

price reacts in a non standard way, following an increase in the expected future stock

price that will be further detailed bellow.

Figure 5. Forward price when changing the intensity of the external habits

4.2.2 Call Option

A call option is used to hedge against an increase in the price of the underlying asset.

The buyer of such an option not only is protected against the raise of the price but can

also take advantage from a potential market decrease. Taking the strike as given, it will

be useful for explanations below to recall that the price of the call depends positively on

the expected future price of the underlying and on the discount factor as it can be seen

Equation 3.19.

a) Internal habits

The parameter reflects the scope of the memories that the agent has about his

previous consumption standards. Figure 1 (Panel 1) and Figure 6 show that price of a

next period call monotonically increases with .

14

Figure 6. The price of a call option when changing the duration of the internal habits

and the reactions of the discount factor and the expected stock price

Having a standard of living of high consumption, an increase of will make the agent

buy stocks, so to ensure a similar high level of consumption in periods to come, in more

future periods. On the one hand, this makes expected future price of the stock, ,

increase with . On the other side, this reduces consumption, increasing its marginal

utility and reducing the discount factor. Overall, thus, the price of the call

monotonically increases with .

The parameter reflects the strength of the internal habit. From Figure 7, it can be

observed how an increase in , for the 9 different levels of used to discretize the

range of its admissible values, translates into prices of the call.

15

Figure 7. The price of a call option when changing the intensity of the internal habits

And the reaction of the discount factor and the expected stock price

More relevant is the internal habit of the agent (higher ), more he will behave as more

risk averse. Applying the same logic than with high , he is willing to buy more shares.

On the one side this will bring the expected price of the stock up. On the other hand the

agent increases his current marginal utility, bringing down the discount factor. The fact

that the discount factor decreases, pushing downwards the price of the call is offset by a

change in the expected price of the stock of a higher magnitude. Overall, independently

on the value of the parameter , high values of increase the price of a call option.

Besides, in the parameter configuration where the agent has an attitude towards risk of

smallest risk aversion (low and ), it can be observed how, oppositely, expected

price of shares decreases to increases in . Behaving more risk aversely, the agent

requires a higher risk premium, which, since dividends follow a given process, will be

obtained by a fall in prices (falling more current price that the expected one). Under low

risk aversion attitude in investors, expected price, , will decrease with and

the price of the call will also decrease.

b) External habits

It is very interesting to see that under external habits, the fact of departing from steady

state, as argued in the forward section, the external habit cancels at the discount factor,

being the latter simply constant across and . In this sense, all price movements

when only having Catching Up With the Jones utility are driven by expected future

price, . The 3D figures appearing below illustrate the effects of and in the

expected price, .

16

The parameter reflects the scope of the memories that the agent has about previous

consumption standards of other agents. From Figure 1 (Panel 2) and Figure 8, it can be

observed that prices of a call monotonically increases as well with .

Figure 8. The price of a call option when changing the duration of the external habits

and the reactions of the discount factor and the expected stock price

Having the rest of the agents a standard of living of high consumption, an increase of

will make the agent be willing to buy shares so to ensure not to be worse than agents in

the economy in periods to come, in more future periods. This makes expected price,

, increase with . Overall, prices of a call monotonically increase as well with .

The parameter reflects the strength of the external habit. Figure 9 shows how an

increase of translates into prices for the 9 different levels of used to discretize the

range of admissible values of .

17

Figure 9. The price of a call option when changing the duration of the external habits

As detailed before, higher will be , more the agent will behave as more risk averse.

As explained for the case of internal habits the expected price, , will decrease

with . Overall, prices of a call decrease with . Only combined with high (high

risk aversion behavior with sufficiently long scope external habit memories) will

generate to the agent the will to buy shares today and in the subsequent periods,

producing the observed increase in the expected price of the shares. Again, the price of

the call will be driven by the expected price of the underlying, which behaving as just

detailed, will explain the non-monotonic response of the price of the call to increases in .

4.2.3 Put Option

A put option can be seen as an insurance tool against a fall in the price of the underlying

asset. The buyer of such an option can hedge his downside price risk and still benefit

from potential price gains if the market increases.

a) Internal habits

The longer the memory of the investor, the longer he is interested in keeping his

standard of living. Therefore, he will want to own shares on the tree in the future and

this foreseen increase in demand will push the expected price of the stocks up. The price

of a put option is negatively correlated to the price of the stock at maturity (Equation

3.20). Given that, as the duration of the habit increases, the expected stock price

increases, pushing down the price of the put option (Figure 1, Panel 3, Figure 6 and

Figure 10).

18

Figure 10. The price of a put option when changing the duration of the internal habits

When analyzing the variation in the intensity of the internal habit, the price of the put

option does not exhibit a monotonic response anymore (Figure 1, Panel 3). For high

levels of , the increase of reduces the price of the option. This links to the previous

explanation with the addition that in this case not only a high level of current

consumption sets a high life standard for many subsequent periods, but also as the

investor attaches more weight to his habits the movements previously identified of the

stock price and of the discount factor are more pronounced (Figure 7). This makes the

put price fall faster as the habit intensity increases than when duration exhibits the same

pattern.

For lower values of however, the response of the put price to changes in is not as

well defined as before. For low levels of duration and intensity, as the latter increases,

the price of the put also increases. One can perceive the presence of internal habits with

low intensity and duration as being close to the most basic case of a risk averse investor

who would prefer to keep his consumption path constant. Therefore, he will demand a

high risk premium for holding the stock which is a risky asset. This translates into a

decrease in the price of the share in the following periods and, therefore, in an increase

of the price of a put option. However, for values of bigger than 0.5 the price of the

put option decreases. In this case, the investor attaches a lot of weight to his habit so any

deviation will have to be compensated by a strong movement in consumption in the

near future. To protect against this event the investor would want to hold shares to be

able to consume the dividend in in the case of a positive deviation in or to keep

open the possibility of selling them at a future time to compensate the fall in

consumption that could be generated by a bad state. The increase in the demand of

shares pushes up the expected price of the stock and drives down the price of the put

option (Figure 11).

19

Figure 11. The price of a put option when changing the intensity of the internal habits

b) External habits

A high level of average consumption in the economy negatively affects the investor’s

current utility so he will have incentives to increase his current consumption to

neutralize, or at least diminish this effect. The higher is the memory of the external

habit, , for longer is the investor exposed to the risk of having to adjust his

consumption level to a negative externality. Therefore, he has a higher incentive to own

shares on the tree in the subsequent periods, which will increase the expected price of

the stocks, and, in consequence, decrease the price of a put option on the

stocks (Figure 1, Panel 4, Figure 8 and Figure 12). Recall that in the presence of

external habits the discount factor does not play any role in the variation of the price of

the put.

Figure 12. The price of a put option when changing the duration of the external habits

On the other hand, the higher is the bigger is the weight the investor attaches to the

past aggregate average consumption. This increases his risk aversion and therefore he

demands a higher risk premium for holding a share, which translates into a decrease in

the price of the shares and into an increase in the price of a put option. For a high

intensity of the habit combined with a high duration the investor will increase his

demand of the shares, pushing the price of the stocks up and thus, the price of the put

down (Figure 1, Panel 4, Figure 9 and Figure 13).

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Figure 13. The price of a put option when changing the intensity of the external habits

5. Conclusion

We have analyzed the relationship of the price of derivative securities such as forward

contracts and options with the duration and the intensity of the investors’ habits. In

particular, we started from the asset pricing model proposed by Lucas (1978) and we

assumed that the investors’ utility depends not only on the current level of consumption

but also on their stock of habit, internal and external.

We have solved for the equilibrium allocation and prices by performing a second order

approximation of the policy functions. This way we have ensured the fact that the risk

aversion is preserved. We used the price of the shares to recover the prices of the

forward contract, the call and the put option. To study the effect of habit parameters on

the prices of the derivatives we considered separately two cases: the one in which the

investor has internal habits and the one in which he has external habits and we analyzed

the sensitivity of the prices with respect to a specific grid of possible values of

parameters.

We have shown that, on average, there is a monotonic relationship between the duration

of the habits, both internal and external, and the price of the derivative securities. In

particular, a longer memory increases the forward price and the price of a call and

decreases the price of a put. For the case of the intensity of the habits however, the

prices of the securities considered respond differently to changes in the intensity under

different values of the duration and under different specifications of the habit. Obtaining

exact solutions using projection methods would be a way to obtain further reliable

responses to parameters characterizing the habits.

21

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