Discount factor convergence

The convergence of the stochastic discount factors generated by the Hull-White model.

The charts below examine the convergence of the discount factors for various combinations of \(\sigma\) and \(a\), first by changing \(\sigma\) and secondly by changing \(a\). As Balaraman’s study shows, the convergence gets worse as \(\sigma/a\) gets larger than 1, and gets better as \(\sigma/a\) gets smaller than 1.

See also

• HullWhite in BasicHullWhite

• Overview of BasicHullWhite notebook in the economic library

• 
• 

import modelx as mx
import matplotlib.pyplot as plt

HW = mx.read_model("BasicHullWhite").HullWhite

fig, axs = plt.subplots(2, 2, sharex=True, sharey=True)
fig.suptitle(r"$a=$" + str(HW.a))
for sigma, (h, v) in zip([0.05, 0.075, 0.1, 0.125], [(0, 0), (0, 1), (1, 0), (1, 1)]):
    HW.sigma = sigma
    axs[h, v].set_title(r"$\sigma=$" + str(sigma) +  r", $\sigma/a=$" + "%.2f" % (sigma/HW.a))
    axs[h, v].plot(range(HW.step_size+1), [HW.mkt_zcb(i) for i in range(HW.step_size+1)], "b-")
    axs[h, v].plot(range(HW.step_size+1), HW.mean_disc_factor(), "r--")

fig, axs = plt.subplots(2, 2, sharex=True, sharey=True)
fig.suptitle(r"$\sigma=$" + str(HW.sigma))
HW.sigma = 0.1
for a, (h, v) in zip([0.05, 0.1, 0.15, 0.2], [(0, 0), (0, 1), (1, 0), (1, 1)]):
    HW.a = a
    axs[h, v].set_title(r"$a=$" + str(a) +  r", $\sigma/a=$" + "%.2f" % (HW.sigma/HW.a))
    axs[h, v].plot(range(HW.step_size+1), [HW.mkt_zcb(i) for i in range(HW.step_size+1)], "b-")
    axs[h, v].plot(range(HW.step_size+1), HW.mean_disc_factor(), "r--")