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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 Jonesesthe Joneses
Corina Boar Rodrigo Gaze Antoni Targa
Advisor: Prof. Jordi Caballé
1. Motivation
• Standard power utility models fail to explain
important empirical facts
• The introduction of habits improves their
performanceperformance
• The effects of habits on stock prices and bond
prices have been already widely studied but
work on how derivatives prices respond to
them is scarce
2. Literature Review
• Lucas (1978):
– Asset pricing in a dynamic setup
– Can be used to price any kind of security
• Abel (1990, 1999) and Campbell and Cochrane
(1999)
– Attempt to explain the equity premium puzzle by
adding habits to the utility function
3. The Model
�� � ������� , � , � �∞
�=0
�� + ��,� ��,�+1 = ���,� + ��,� ���,� s.t.:
�� ��� , � , � � = 11 − � � ��
� � �1−�
�� ��� , � , � � = 11 − � � ��
��1 ��2 �1−�
where
� = �1�−1 + �1 − �1���−1
� = �2�−1 + �1 − �2���−1
3. The Model
��,� �′ ��� � = ��� !���,�+1 + ��,�+1��′ ���+1�"
�′ ��� � = �1 − �� ���� − �1�1 − ���1 − �1�#�
Euler Equation
where
�′ ��� � = �1 − �� ���� − �1�1 − ���1 − �1�#�
#� ≡ ��1�� �#�+1 + ��+1���+1��1�+1 �
3. The Model
• Output is perishable and produced by one
single tree and evolves according to:
ln '� = �1 − (� ln ) + ( ln '�−1 + *�
where
• In equilibrium we have:
ln '� = �1 − (� ln ) + ( ln '�−1 + *�
*�~,�0, �* �
'� = �� = ��
3. The Model
• Forward contract:
-� = �� .� �′�'�+1��′�'�� ��+1/�� .� �′�'�+1��′�'�� /
• Call option:
• Put option:
-� = � �'���� .� � �'�+1��′�'�� /
�011� = ��� 2�′�'�+1��′�'� � 3045��+1 − #, 067
�8�� = ��� 2�′�'�+1��′�'� � 3045# − ��+1, 067
3. The Model
• Second-order approximation
• Gaussian quadrature
– Discretizes the normal distribution of the output shockshock
• Maps states today into states next period
• Maps states into controls
• Allows us to recover the expected stock price and discount factor and therefore derivatives’ prices
4. Quantitative Results
• Parameter values:Parameter Value
σ 1.50
β 0.98
μ 1.00
• Activate one habit at a time
• Start from a low γi and loop over all possible values for ρi
σε 0.50
φ 0.90
X 50.00
4. Quantitative Results
4. Quantitative Results
4. Quantitative Results
4. Quantitative Results
4. Quantitative Results
4. Quantitative Results
5. Conclusion
• On average, there is a monotonic relationship
between the duration of the habits and the
price of the derivative securities
• For the case of the intensity of the habits • For the case of the intensity of the habits
however, the prices of the securities
considered respond differently
– Under certain values for the duration parameter
the relationship is no longer monotonous
The End
4. Quantitative Results
Internal Habits: duration
Internal Habits: intensity
4. Quantitative Results
External Habits: duration
External Habits: intensity
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