Derivatives Pricing under HabitFormation and Catching-up with           the Joneses   Corina Boar   Rodrigo Gaze Antoni Ta...
1. Motivation• Standard power utility models fail to explain  important empirical facts• The introduction of habits improv...
2. Literature Review• Lucas (1978):  – Asset pricing in a dynamic setup  – Can be used to price any kind of security• Abel...
3. The Model                ∞                               ,       ,                =0s.t.:   +   ,       , +1   =       ...
3. The ModelEuler Equation        ,            ′                    =       ,   +1 +      , +1                            ...
3. The Model• Output is perishable and produced by one  single tree and evolves according to:        ln   = 1−        ln +...
3. The Model• Forward contract:                                          +1                              ′                ...
3. The Model• Second-order approximation• Gaussian quadrature  – Discretizes the normal distribution of the output    shoc...
4. Quantitative Results• Parameter values:             Parameter             Value                σ                  1.50 ...
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 der...
The End
4. Quantitative Results
Internal Habits: duration
Internal Habits: intensity
4. Quantitative Results
External Habits: duration
External Habits: intensity
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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. Derivatives Pricing under HabitFormation and Catching-up with the Joneses Corina Boar Rodrigo Gaze Antoni Targa Advisor: Prof. Jordi Caballé
  2. 2. 1. Motivation• Standard power utility models fail to explain important empirical facts• The introduction of habits improves their performance• 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
  3. 3. 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
  4. 4. 3. The Model ∞ , , =0s.t.: + , , +1 = , + , , 1− 1 , , = 1− 1 2where = 1 −1 + 1− 1 −1 = 2 −1 + 1− 2 −1
  5. 5. 3. The ModelEuler Equation , ′ = , +1 + , +1 ′ +1where ′ = 1− − 1 1− 1− 1 +1 +1 ≡ 1 +1 + 1 +1
  6. 6. 3. The Model• Output is perishable and produced by one single tree and evolves according to: ln = 1− ln + ln −1 + where ~ 0,• In equilibrium we have: = =
  7. 7. 3. The Model• Forward contract: +1 ′ +1 = ′ ′ +1 ′• Call option: +1 ′ = ′ +1 − ,0• Put option: +1 ′ = ′ − +1 , 0
  8. 8. 3. The Model• Second-order approximation• Gaussian quadrature – Discretizes the normal distribution of the output shock• 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
  9. 9. 4. Quantitative Results• Parameter values: Parameter Value σ 1.50 β 0.98 μ 1.00 σε 0.50 φ 0.90 X 50.00• Activate one habit at a time• Start from a low γi and loop over all possible values for ρi
  10. 10. 4. Quantitative Results
  11. 11. 4. Quantitative Results
  12. 12. 4. Quantitative Results
  13. 13. 4. Quantitative Results
  14. 14. 4. Quantitative Results
  15. 15. 4. Quantitative Results
  16. 16. 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 however, the prices of the securities considered respond differently – Under certain values for the duration parameter the relationship is no longer monotonous
  17. 17. The End
  18. 18. 4. Quantitative Results
  19. 19. Internal Habits: duration
  20. 20. Internal Habits: intensity
  21. 21. 4. Quantitative Results
  22. 22. External Habits: duration
  23. 23. External Habits: intensity

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