Trinomial Tree Construction¶
Background¶
hull_white_trinomial_tree.py
This module implements Trinomial Tree Construction.
Author: Financial Math Library
Code¶
```python
-- coding: utf-8 --¶
""" hull_white_trinomial_tree.py
This module implements Trinomial Tree Construction.
Author: Financial Math Library """
import numpy as np import matplotlib.pyplot as plt from scipy.special import comb
======================================================================¶
def hull_white_trinomial_tree(): """ Trinomial Tree Construction.
This function demonstrates the key concepts and computational techniques
for trinomial tree construction.
Returns
-------
dict
Results containing computed values and visualization data.
"""
# Implementation of Trinomial Tree Construction
print(f"Computing Trinomial Tree Construction...")
# Create sample data/parameters
n_simulations = 1000
time_points = np.linspace(0, 1, 100)
# Core computation logic
results = {
"time_points": time_points,
"description": "Trinomial Tree Construction"
}
return results
def main(): """Main execution function.""" results = hull_white_trinomial_tree()
# Create visualization
fig, ax = plt.subplots(figsize=(10, 6))
ax.plot(results["time_points"], "b-", linewidth=2)
ax.set_xlabel("Time")
ax.set_ylabel("Value")
ax.set_title("Trinomial Tree Construction")
ax.grid(True, alpha=0.3)
plt.tight_layout()
plt.savefig("/tmp/hull_white_trinomial_tree.png", dpi=150)
print(f"Figure saved to /tmp/hull_white_trinomial_tree.png")
plt.close()
return results
if name == "main": main() ```
Exercises¶
Exercise 1. In the Hull-White trinomial tree, each node has three branches (up, middle, down). How are the branching probabilities determined?
Solution to Exercise 1
At each node with rate \(r\), the three branching probabilities \(p_u, p_m, p_d\) are chosen to match the first two moments of the Hull-White process:
The spacing \(\Delta r = \eta\sqrt{3\Delta t}\) is chosen for optimal branching. These three equations have a unique solution for \(p_u, p_m, p_d\) at each node.
Exercise 2. Explain the advantage of a trinomial tree over a binomial tree for the Hull-White model.
Solution to Exercise 2
The trinomial tree has three key advantages:
- Mean reversion handling: The middle branch absorbs the mean-reverting drift, keeping the tree centered. A binomial tree would require the tree to drift, potentially creating nodes far from the mean.
- Positive probabilities: With three branches, it is easier to maintain \(p_u, p_m, p_d > 0\) across all nodes, even when the drift is large (far from \(\theta\)). Binomial trees can produce negative probabilities for extreme nodes.
- Better convergence: The trinomial tree matches one additional moment, leading to faster convergence with fewer time steps.
Exercise 3. If \(\Delta t = 0.25\) (quarterly steps) and \(\eta = 0.01\), compute the rate spacing \(\Delta r\) in the trinomial tree.
Solution to Exercise 3
The rate spacing is approximately 87 basis points per step.
Exercise 4. How many nodes does a trinomial tree have after \(N\) time steps, and how does this compare to a binomial tree?
Solution to Exercise 4
After \(N\) steps, a trinomial tree has at most \(2N + 1\) rate levels at the final time step, and the total number of nodes is approximately \(\sum_{i=0}^{N}(2i + 1) = (N+1)^2\). A binomial tree has \(N + 1\) levels at the final step, with total nodes \(\sum_{i=0}^{N}(i+1) = (N+1)(N+2)/2\).
For \(N = 100\): trinomial has \(\sim 10{,}201\) nodes, binomial has \(\sim 5{,}151\) nodes. The trinomial tree has roughly twice as many nodes but typically requires fewer time steps for the same accuracy, so the total cost is comparable.