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