Derivatives Pricing under Habit Formation and Catching-up with the Joneses


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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.

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Derivatives Pricing under Habit Formation and Catching-up with the Joneses

  1. 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 Caballé 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. 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. 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 (Galí, 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. 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 theindividual’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 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. 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 individual’s own consumption and is the average level ofconsumption in the economy.In (3.1) is the deterministic discount factor, is the coefficient of relative riskaversion, is the price of a stock on tree in period , is the dividend paid by thestock on tree in period and is the number of stocks on tree in period . Notethat utility is not longer time separable. Besides the fact that it depends on a benchmarklevel of consumption that is exogenous to the individual, consumption choice todayinfluences the future reference level of habit through . This specificationparameterizes two features of models with habits: 4
  6. 6. 1. Habit intensity: the parameters and index the importance of internal andexternal habit in the utility function, respectively. If then we are back tothe case of the standard power utility model. If and then the agentonly takes into account the average level of consumption in the economy. Ifand then only past levels of own consumption matter. Values of are notadmissible because the steady-state utility will be decreasing in consumption.2. Habit duration: the parameters and index the memory of the internal andexternal habit, respectively. If then only last period’s consumption is important.If then the larger is the further back in time the individual looks whenchoosing .The Euler EquationThe 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) 5
  7. 7. 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 pricingIn this section we present the analytical expressions used to price three types ofderivative securities: a one period forward contract, a one period call option and a oneperiod put option. In order to price these assets we needed to compute the value of theunderlying 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 futurecash 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 ofconsumption 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 6
  8. 8. (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 pricethe derivative securities of interest.For the forward contract we have that its cost is zero by definition, which leads to thefollowing 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 wehave (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 thisframework.The options considered are plain vanilla ones that are not protected for dividendpayments 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 doesnot matter whether we are pricing a European or an American option. This follows fromthe fact that the only possible date at which a holder of an American option can exercise 7
  9. 9. it is at t+1, which is its maturity and thus the American option will behave exactly as aEuropean option.3.3 MethodologyIt is common in the literature to approximate the solution to non-linear, dynamic,stochastic, general equilibrium models using linear methods. Linear approximationmethods are useful to characterize certain aspects of the dynamic properties ofcomplicated models. In particular, if the support of the shocks driving aggregatefluctuations is small and an interior stationary solution exists, first-order approximationsprovide adequate answers to questions such as local existence and determinacy ofequilibrium and the size of the second moments of endogenous variables.Evaluating the utility function using linear approximation of the policy functions, somesecond 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 agentsto be risk neutral, which does not hold under our assumptions. With this in mind,second order Taylor approximation allows to contemplate features of concave utilitiesand risk averse behavior, crucial features of models with externalities for asset pricing.We solve for deviations around steady state using second order approximation with theSchmitt-Grohe-Uribe Toolkit.2 Thus, in general, a correct second-order approximationof the equilibrium welfare function requires a second-order approximation to the policyfunction (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, wecompute these prices with numerical integration methods using the CompEconToolbox.3 With one of its functions we produce quadrature points and quadratureweights so to, using as many points as specified, reconstruct the normal distribution ofthe shocks, characterized with specified mean and variance. The set of quadrature pointsdiscretizes the range of possible values that the shock could take, making possible,having the policy functions, to compute all states and controls variables for eachquadrature point. For our purposes, using the policy functions, we compute the impliedprice of the share and the discount factor for next period associated to each quadrature2 S. Schmitt-Grohe, M. Uribe/Journal of Economic Dynamics & Control 28 (2004)3 Mario J. Miranda & Paul L. Fackler/MIT Press (2002) 8
  10. 10. point. Combining these with a strike price and discounting next period payoff, weobtain the associated price of any of the three derivatives contemplated for eachquadrature point. Weighting such prices with the quadrature weights we obtain the priceof a call, a put and a forward expiring next period.4. Quantitative Results4.1 Parameter valuesIn order to study the effects of internal and external habits on the prices of derivativesecurities we proceed in the three steps for each habit: (1) the effects due to the habitthat is not of interest is deactivated, by setting its intensity parameter to zero, in order tomake sure that all the effects on the prices are due to only one type of habit; (2) theintensity 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 intensityparameter up to 0.9 step (2) is repeated. With this not only we are able to analyze theeffect of each of the parameters but also how interactions between them affect the pricesof plain vanilla derivatives.The other parameters of the model that are still to be set are the coefficient of relativerisk aversion, , the deterministic discount factor, , the mean for the process of output, , the variance of the output shock, , and the autocorrelation of output, . The indexof relative risk aversion and the deterministic discount factor were chosen from theexistent 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 butsince Abel, 1990, provides values close to 1.00 when there are only internal habits wechose a middle point in order to analyze both habits with an utility specification thatwould be consistent with the two of them) and 0.98 (Fuhrer, 2000), respectively. We setthe autocorrelation of output to 0.9 in order to capture the high persistence of the levelof output. The value of was set to zero in order to normalize output at the steady stateto be equal to 1, while was set to 0.50. In order to price options on the stock we setthe strike price to be 50, a value close to the steady state price of the stock. 9
  11. 11. 4.2 Derivatives PricesUnder internal habits the reference level of consumption is the individual’s pastconsumption. The latter is more important as the weight attached to it, , increases andis more backwards looking as the duration of the habit, , increases. Under externalhabits the average aggregate consumption in the economy is the reference level ofconsumption and it enters the investor’s utility as a negative externality. The extent towhich it influences the investor’s consumption decision and, thus, his current utilitydepends on the intensity of the habit, , and its duration, . Figure 1 presents theresponses of the derivatives prices to changes in the habit parameters, under the twospecifications. Figure 1. Call, Put and Forward prices under internal and external habits4.2.1 Forward ContractA forward contract is an agreement between two parties to transact the underlying assetat a future date at a predetermined price. This asset is usually traded in the over thecounter market. 10
  12. 12. a) Internal habitsAs it can be seen in Figure 1 (Panel 5) and Figure 2, the forward price of a stock is anincreasing function of , the duration of internal habits. Given that habits affect themarginal utility of consumption in a way such that high levels of consumption todaydecrease future utility, individuals would rather have a constant level of consumptionthan to have it fluctuating around this constant every period. Such a pattern is similar towhat 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 thathouseholds are willing to pay in order to get rid of the uncertainty about future states. Itis worth noting that when the intensity parameter is high the price of the forwardcontract decreases when is above a certain threshold and such movement is linked tothe behaviour of the expected stock price. Figure 2. Forward price when changing the duration of the internal habitsAs the intensity of internal habits increase, gets bigger, individuals care more aboutkeeping 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 stockincreases (Figure 1, Panel 5, and Figure 3). Figure 3. Forward price when changing the intensity of the internal habits 11
  13. 13. b) External habitsIn the presence of external habits the reaction of the forward price with respect tomovements in , the duration of external habits, is similar to the one that was observedfor changes in . The main difference between the forward price reaction to andis that it is much more sensible to the former. This happens because each individual hasmuch better information about his own consumption and thus can better assess theeffect 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 habitsDifferently from what happened to the duration parameters, now the reaction of theforward price to an increase in , the intensity of external habits, does not resemble theone obtained for its internal habits’ counterpart (Figure 1). When habits are external thehousehold takes them as given, so as increases the weight of a term that thehousehold cannot control is getting bigger. This movement in does not affect the rateat which individuals will discount future payoffs but, since it resembles risk aversion ina sense that individuals dislike consumption volatility it makes individuals require ahigher risk premium to hold assets that have payoffs that are negatively correlated withmarginal utility.4 Since the stock is one of these assets, when the intensity of externalhabits increases, its price today decreases and so does its expected price for the nextperiods. 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 todifferent levels of habit intensity. Similarly to the non monotonous behaviour observedwith extreme values of , when is associated with high values of the forward4 Recall that we are departing from the steady state, so and only assets that maturein one period are being considered. 12
  14. 14. price reacts in a non standard way, following an increase in the expected future stockprice that will be further detailed bellow. Figure 5. Forward price when changing the intensity of the external habits4.2.2 Call OptionA 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 canalso take advantage from a potential market decrease. Taking the strike as given, it willbe useful for explanations below to recall that the price of the call depends positively onthe expected future price of the underlying and on the discount factor as it can be seenEquation 3.19.a) Internal habitsThe parameter reflects the scope of the memories that the agent has about hisprevious consumption standards. Figure 1 (Panel 1) and Figure 6 show that price of anext period call monotonically increases with . 13
  15. 15. 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 priceHaving a standard of living of high consumption, an increase of will make the agentbuy stocks, so to ensure a similar high level of consumption in periods to come, in morefuture periods. On the one hand, this makes expected future price of the stock, ,increase with . On the other side, this reduces consumption, increasing its marginalutility and reducing the discount factor. Overall, thus, the price of the callmonotonically increases with .The parameter reflects the strength of the internal habit. From Figure 7, it can beobserved how an increase in , for the 9 different levels of used to discretize therange of its admissible values, translates into prices of the call. 14
  16. 16. 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 priceMore relevant is the internal habit of the agent (higher ), more he will behave as morerisk 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 theagent increases his current marginal utility, bringing down the discount factor. The factthat the discount factor decreases, pushing downwards the price of the call is offset by achange in the expected price of the stock of a higher magnitude. Overall, independentlyon 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 ofsmallest risk aversion (low and ), it can be observed how, oppositely, expectedprice of shares decreases to increases in . Behaving more risk aversely, the agentrequires a higher risk premium, which, since dividends follow a given process, will beobtained by a fall in prices (falling more current price that the expected one). Under lowrisk aversion attitude in investors, expected price, , will decrease with andthe price of the call will also decrease.b) External habitsIt is very interesting to see that under external habits, the fact of departing from steadystate, 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 movementswhen only having Catching Up With the Jones utility are driven by expected futureprice, . The 3D figures appearing below illustrate the effects of and in theexpected price, . 15
  17. 17. The parameter reflects the scope of the memories that the agent has about previousconsumption standards of other agents. From Figure 1 (Panel 2) and Figure 8, it can beobserved 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 priceHaving the rest of the agents a standard of living of high consumption, an increase ofwill make the agent be willing to buy shares so to ensure not to be worse than agents inthe 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 anincrease of translates into prices for the 9 different levels of used to discretize therange of admissible values of . 16
  18. 18. Figure 9. The price of a call option when changing the duration of the external habitsAs 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 decreasewith . Overall, prices of a call decrease with . Only combined with high (highrisk aversion behavior with sufficiently long scope external habit memories) willgenerate 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 ofthe call will be driven by the expected price of the underlying, which behaving as justdetailed, will explain the non-monotonic response of the price of the call to increases in .4.2.3 Put OptionA put option can be seen as an insurance tool against a fall in the price of the underlyingasset. The buyer of such an option can hedge his downside price risk and still benefitfrom potential price gains if the market increases.a) Internal habitsThe longer the memory of the investor, the longer he is interested in keeping hisstandard of living. Therefore, he will want to own shares on the tree in the future andthis foreseen increase in demand will push the expected price of the stocks up. The priceof a put option is negatively correlated to the price of the stock at maturity (Equation3.20). Given that, as the duration of the habit increases, the expected stock priceincreases, pushing down the price of the put option (Figure 1, Panel 3, Figure 6 andFigure 10). 17
  19. 19. Figure 10. The price of a put option when changing the duration of the internal habitsWhen analyzing the variation in the intensity of the internal habit, the price of the putoption does not exhibit a monotonic response anymore (Figure 1, Panel 3). For highlevels of , the increase of reduces the price of the option. This links to the previousexplanation with the addition that in this case not only a high level of currentconsumption sets a high life standard for many subsequent periods, but also as theinvestor attaches more weight to his habits the movements previously identified of thestock price and of the discount factor are more pronounced (Figure 7). This makes theput price fall faster as the habit intensity increases than when duration exhibits the samepattern.For lower values of however, the response of the put price to changes in is not aswell 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 withlow intensity and duration as being close to the most basic case of a risk averse investorwho would prefer to keep his consumption path constant. Therefore, he will demand ahigh risk premium for holding the stock which is a risky asset. This translates into adecrease in the price of the share in the following periods and, therefore, in an increaseof the price of a put option. However, for values of bigger than 0.5 the price of theput option decreases. In this case, the investor attaches a lot of weight to his habit so anydeviation will have to be compensated by a strong movement in consumption in thenear future. To protect against this event the investor would want to hold shares to beable to consume the dividend in in the case of a positive deviation in or to keepopen the possibility of selling them at a future time to compensate the fall inconsumption that could be generated by a bad state. The increase in the demand ofshares pushes up the expected price of the stock and drives down the price of the putoption (Figure 11). 18
  20. 20. Figure 11. The price of a put option when changing the intensity of the internal habitsb) External habitsA high level of average consumption in the economy negatively affects the investor’scurrent utility so he will have incentives to increase his current consumption toneutralize, or at least diminish this effect. The higher is the memory of the externalhabit, , for longer is the investor exposed to the risk of having to adjust hisconsumption level to a negative externality. Therefore, he has a higher incentive to ownshares on the tree in the subsequent periods, which will increase the expected price ofthe stocks, and, in consequence, decrease the price of a put option on thestocks (Figure 1, Panel 4, Figure 8 and Figure 12). Recall that in the presence ofexternal habits the discount factor does not play any role in the variation of the price ofthe put. Figure 12. The price of a put option when changing the duration of the external habitsOn the other hand, the higher is the bigger is the weight the investor attaches to thepast aggregate average consumption. This increases his risk aversion and therefore hedemands a higher risk premium for holding a share, which translates into a decrease inthe price of the shares and into an increase in the price of a put option. For a highintensity of the habit combined with a high duration the investor will increase hisdemand of the shares, pushing the price of the stocks up and thus, the price of the putdown (Figure 1, Panel 4, Figure 9 and Figure 13). 19
  21. 21. Figure 13. The price of a put option when changing the intensity of the external habits5. ConclusionWe have analyzed the relationship of the price of derivative securities such as forwardcontracts and options with the duration and the intensity of the investors’ habits. Inparticular, we started from the asset pricing model proposed by Lucas (1978) and weassumed that the investors’ utility depends not only on the current level of consumptionbut also on their stock of habit, internal and external.We have solved for the equilibrium allocation and prices by performing a second orderapproximation of the policy functions. This way we have ensured the fact that the riskaversion is preserved. We used the price of the shares to recover the prices of theforward contract, the call and the put option. To study the effect of habit parameters onthe prices of the derivatives we considered separately two cases: the one in which theinvestor has internal habits and the one in which he has external habits and we analyzedthe sensitivity of the prices with respect to a specific grid of possible values ofparameters.We have shown that, on average, there is a monotonic relationship between the durationof the habits, both internal and external, and the price of the derivative securities. Inparticular, a longer memory increases the forward price and the price of a call anddecreases the price of a put. For the case of the intensity of the habits however, theprices of the securities considered respond differently to changes in the intensity underdifferent values of the duration and under different specifications of the habit. Obtainingexact solutions using projection methods would be a way to obtain further reliableresponses to parameters characterizing the habits. 20
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