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

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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--")