HullWhite¶

The main Space in the BasicHullWhite model.

Mathematical notations are defined consistent with those in Brigo and Mercurio (2001, 2nd Ed. 2006)

See also

Damiano Brigo, Fabio Mercurio (2001, 2nd Ed. 2006). Interest Rate Models — Theory and Practice with Smile, Inflation and Credit

scen_size¶: Number of scenarios. 1000 by default.

np¶: numpy module.

step_size¶: Number of time steps. 360 by default.

time_len¶: Simulation length in years. 30 by default.

a¶: Parameter \(a\) in the Hull-White stochastic differential equation. 0.1 by default.

sigma¶: Parameter \(\sigma\) in the Hull-White stochastic differential equation. 0.1 by default.

seed1¶: Seed number to generate random numbers. 1234 by default. See std_norm_rand().

seed2¶: Seed for the second random numbers used for accum_short_rate2().

Cells

`A_t_T`(i, j)	\(A(t_i, t_j)\) in \(P(t_i, t_j)\)
`B_t_T`(i, j)	\(B(t_i, t_j)\) in \(P(t_i, t_j)\)
`E_rt`()	The expected values of \(r(t_i)\) at time 0 for all \(t_i\).
`E_rt_s`(i, j)	Conditional expected values of \(r(t_j)\)
`P_t_T`(i, j)	The price at \(t_i\) of a zero-coupon bond paying off 1 at \(t_j\)
`V_t_T`(i, j)	The variance of \(\int_{t_j}^{t_i} r(u)du\|\mathcal{F}_{t_i}\)
`Var_rt`()	The variance of \(r(t_i)\) at time 0 for all \(t_i\).
`Var_rt_s`(i, j)	The variance of \(r(t_j)\) conditional on \(\mathcal{F}_{t_i}\)
`accum_short_rate`(i)	Accumulated short rates.
`accum_short_rate2`(i)	Alternative implementation of accumulated short rates.
`alpha`(i)	\(\alpha(t_i)\)
`disc_factor`(i)	Discount factors
`disc_factor_paths`()	Discount factor scenarios.
`mean_disc_factor`()	Discount factor means
`mean_short_rate`()	The means of generated short rates
`mkt_fwd`(i)	The initial instantaneous forward rate for `t_(i)`.
`mkt_zcb`(i)	The initial price of zero coupon bond
`short_rate`(i)	Stochastic short rates at `t_(i)`
`short_rate_paths`()	Short rate paths.
`std_norm_rand`([seed])	Random numbers from the standard normal distribution.
`t_`(i)	time index \(t_i\)
`var_short_rate`()	Variance of generated short rates

A_t_T(i, j)[source]¶

\(A(t_i, t_j)\) in \(P(t_i, t_j)\)

See P_t_T().

B_t_T(i, j)[source]¶

\(B(t_i, t_j)\) in \(P(t_i, t_j)\)

See P_t_T().

E_rt()[source]¶

The expected values of \(r(t_i)\) at time 0 for all \(t_i\).

Returns, in a numpy array, the expected values of \(r(t_i)\) for all \(t_i\). Calculated as \(E\{r(t_i) | \mathcal{F}_{0}\}\).

See also

E_rt_s()

E_rt_s(i, j)[source]¶

Conditional expected values of \(r(t_j)\)

Returns, in a numpy array, \(E\{r(t_j) | \mathcal{F}_{i}\}\), the expected values of \(r(t_j)\) conditional on \(\mathcal{F}_{i}\) for all scenarios. \(E\{r(t) | \mathcal{F}_{s}\}\) is calculated as:

\[r(s)e^{-a(t-s)} + \alpha(t) - \alpha(s)e^{-a(t-s)}\]

where \(\alpha(t)\) is calculated by alpha().

P_t_T(i, j)[source]¶

The price at \(t_i\) of a zero-coupon bond paying off 1 at \(t_j\)

This formula corresponds to \(P(t, T)\) in Brigo and Mercurio, which is defined as:

\[P(t, T)=A(t, T)e^{-B(t, T)r(t)}\]

where \(t_i\) and \(t_j\) substitute for \(t\) and \(T\).

See also

A_t_T() for \(A(t, T)\)
B_t_T() for \(B(t, T)\)
short_rate() for \(r(t)\)
Brigo and Mercurio (2001, 2nd Ed. 2006). Interest Rate Models — Theory and Practice with Smile, Inflation and Credit

V_t_T(i, j)[source]¶

The variance of \(\int_{t_j}^{t_i} r(u)du|\mathcal{F}_{t_i}\)

This formula corresponds to \(V(t, T)\) in Brigo and Mercurio, which is defined as:

\[V(t, T)=\frac{\sigma^2}{a^2}\left[T-t+\frac{2}{a}e^{-a(T-t)}-\frac{1}{2a}e^{-2a(T-t)}-\frac{3}{2a}\right]\]

where \(t_i\) and \(t_j\) substitute for \(t\) and \(T\).

See also

sigma for \(\sigma\)
a for \(a\)
Brigo and Mercurio (2001, 2nd Ed. 2006). Interest Rate Models — Theory and Practice with Smile, Inflation and Credit

Var_rt()[source]¶

The variance of \(r(t_i)\) at time 0 for all \(t_i\).

Returns, in a numpy array, the variance of \(r(t_i)\) for all \(t_i\). Calculated as \(Var\{r(t_i) | \mathcal{F}_{0}\}\).

See also

Var_rt_s()

Var_rt_s(i, j)[source]¶

The variance of \(r(t_j)\) conditional on \(\mathcal{F}_{t_i}\)

\(Var\{r(t_{j}) | \mathcal{F}_{t_i}\}\), the variance of \(r(t_j)\) conditional on \(\mathcal{F}_{t_i}\), calculated as:

\[Var\{ r(t) | \mathcal{F}_s \} = \frac{\sigma^2}{2a} (1 - e^{-2a(t-s)})\]

See also

a
sigma

accum_short_rate(i)[source]¶

Accumulated short rates.

a descrete approximation to the integral \(\int_0^{t_i}r(t)dt\), calculated as \(\sum_{j=1}^{i}r(t_{j-1})(t_j-t_{j-1})\)

See also

disc_factor()

accum_short_rate2(i)[source]¶

Alternative implementation of accumulated short rates.

An alternative approach to simulate \(Y(t_i)=\int_0^{t_i}r(t)dt\) by using the fact that \(Y(t_i)\) follows a normal distribution, and by simulating the joint distribution of \((r(t_i), Y(t_i))\), as suggested in Glasserman (2003).

See also

accum_short_rate()
seed2
Paul Glasserman (2003). Monte Carlo Methods in Financial Engineering

alpha(i)[source]¶

\(\alpha(t_i)\)

Returns, in a numpy array, \(\alpha(t_i)\) for all scenarios. \(\alpha\) appears in the expression of \(E\{r(t) | \mathcal{F}_{s}\}\) and is defined as:

\[\alpha(t) = f^M(0, t) + \frac{\sigma^2} {2a^2}(1-e^{-at})^2\]

See also

E_rt_s()

disc_factor(i)[source]¶

Discount factors

Returns, in a numpy array, the discount factors for cashflows at \(t_i\) for all scenarios. Defined as:

np.exp(-accum_short_rate(i))

See also

accum_short_rate

disc_factor_paths()[source]¶

Discount factor scenarios.

Returns, as a 2D numpy array, the simulated discount factors for all scenarios.

See also

disc_factor()

mean_disc_factor()[source]¶

Discount factor means

Returns, as a numpy array, the mean of discount factors of all scenarios for each \(t_i\).

See also

disc_factor()

mean_short_rate()[source]¶

The means of generated short rates

Returns, as a numpy array, the means of short rates of all scenarios for all \(t_i\). This should converge to the theoretical variances calculated by E_rt().

See also

short_rate()
E_rt()

mkt_fwd(i)[source]¶

The initial instantaneous forward rate for t_(i).

By default, returns 0.05 for all i.

mkt_zcb(i)[source]¶

The initial price of zero coupon bond

The initial price of the unit zero coupon bond maturing at t_(i).

If i=0 returns 1. Otherwise, defined as:

mkt_zcb(i-1) * np.exp(-mkt_fwd(i-1)*dt)

where dt = t_(i) - t_(i-1).

See also

t_
mkt_fwd

short_rate(i)[source]¶

Stochastic short rates at t_(i)

Returns, in a numpy array, simulated stochastic short rates at t_(i) for all scenarios.

For i=0, defined as mkt_fwd(0).

For i>0, defined as \(r(t_i) = E\{r(t_i) | \mathcal{F}_{i-1}\} + \sqrt{Var\{ r(t_i) | \mathcal{F}_{i-1} \}} * Z\),

where \(E\{r(t_i) | \mathcal{F}_{i-1}\}\), the expected value of \(r(t_i)\) conditional on \(\mathcal{F}_{i-1}\) is calculated by E_rt_s(i-1, i), \(Var\{ r(t_i) | \mathcal{F}_{i-1} \}\) the variance of \(r(t_i)\) conditional on \(\mathcal{F}_{i-1}\) is calculated by Var_rt_s(i-1, i), and \(Z\), a random number drawn from \(\mathcal{N}(0, 1)\) a standard normal distribution calculated by std_norm_rand().

See also

scen_size
mkt_fwd()
E_rt_s()
Var_rt_s()
std_norm_rand()
seed1

short_rate_paths()[source]¶

Short rate paths.

Returns, as a 2D numpy array, the simulated short rate paths for all scenarios.

See also

short_rate()

std_norm_rand(seed=1234)[source]¶

Random numbers from the standard normal distribution.

Returns a numpy array shaped scen_size x step_size. The elements are random numbers drawn from the standard normal distribution.

t_(i)¶: time index \(t_i\)

var_short_rate()[source]¶

Variance of generated short rates

Returns, as a vector in a numpy array, the variances of the generated short rates for all \(t_i\). This should converge to the theoretical variances calculated by Var_rt().

See also

short_rate()
Var_rt()

The BasicHullWhite Model

Overview of BasicHullWhite