Simulation of Gourieroux & Jasiak (2025)'s stochastic-tree bubble process:
a positive stationary submartingale generated by a binomial tree with
stochastic branching intensity (a random-coefficient autoregression, as
opposed to Cox-Ross-Rubinstein's deterministic branches). Blanchard &
Watson (1982)'s bubble (sim_blan) is the special case of
constant intensity.
Usage
sim_tree(
n,
a = 0.95,
eta = 1,
mu = -1,
rho = 0.7,
sigma = 4,
y0 = eta/(1 - a),
seed = NULL
)Arguments
- n
A positive integer specifying the length of the simulated output series.
- a
A scalar in (0, 1) (note: \(1/a > 1\) is the growth rate).
- eta
A positive scalar setting the price floor
eta / (1 - a).- mu, rho, sigma
Parameters of the latent Gaussian AR(1) intensity process (
rhoin (-1, 1),sigma > 0).- y0
Starting value. Defaults to the price floor
eta / (1 - a).- seed
An object specifying if and how the random number generator (rng) should be initialized. Either NULL or an integer will be used in a call to
set.seedbefore simulation. If set, the value is saved as "seed" attribute of the returned value. The default, NULL, will not change rng state, and return .Random.seed as the "seed" attribute. Results are reproducible across the parallel and non-parallel option when the same seed is used.
Details
The stochastic intensity is \(p_t = \Phi(X_t)\), with \(X_t\) a latent stationary Gaussian AR(1): $$X_t = \mu + \rho(X_{t-1}-\mu) + \sigma\sqrt{1-\rho^2}u_t,\quad u_t \sim iid\,N(0,1)$$ Given \(p_t\), draw \(Z_t \sim Bernoulli(p_t)\) and set (the paper's eq. 5-6): $$Y_t = \xi_{1t}Y_{t-1}+\epsilon_t,\quad \xi_{1t}=\frac{1}{a}\frac{Z_t}{p_t},\quad \epsilon_t=\frac{\eta}{1-a}\left(1-\xi_{1t}\right)+\frac{\eta}{a}\frac{1-Z_t}{1-p_t}$$ \(a>1\) controls the growth rate in a branch's active phase, \(\eta>0\) sets the price floor \(\eta/(1-a)\) (Corollary 1 in the source: \(Y_t \ge \eta/(1-a)\)), \(\rho\) controls persistence of the bubble-growth phase, and \(\sigma\) controls the frequency of bubbles. The process has no finite mean (the source's Proposition 3) – occasional very large values are a feature of the model, not a bug.
Default parameters (\(\mu=-1,\eta=1,a=0.95,\sigma=4,\rho=0.7\)) reproduce the source's own illustrative example (Section 2.3, Figure 2).
References
Gourieroux, C. & Jasiak, J. (2025). "A Stochastic Tree for Bubble Asset Modelling and Pricing." JTSA, 46(5), 932-944.
