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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
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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
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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
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3. The Model
�� � ������� , � , � �∞
�=0
�� + ��,� ��,�+1 = ���,� + ��,� ���,� s.t.:
�� ��� , � , � � = 11 − � � ��
� � �1−�
�� ��� , � , � � = 11 − � � ��
��1 ��2 �1−�
where
� = �1�−1 + �1 − �1���−1
� = �2�−1 + �1 − �2���−1
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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 �
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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, �* �
'� = �� = ��
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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
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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
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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
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4. Quantitative Results
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4. Quantitative Results
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4. Quantitative Results
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4. Quantitative Results
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4. Quantitative Results
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4. Quantitative Results
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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
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The End
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4. Quantitative Results
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Internal Habits: duration
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Internal Habits: intensity
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4. Quantitative Results
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External Habits: duration
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External Habits: intensity