naginterfaces.library.pde.dim1_​blackscholes_​means

naginterfaces.library.pde.dim1_blackscholes_means(t0, tmat, td, phid)

dim1_blackscholes_means computes average values of a continuous function of time over the remaining life of an option. It is used together with dim1_blackscholes_closed() to value options with time-dependent arguments.

For full information please refer to the NAG Library document for d03ne

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/d03/d03nef.html

Parameters

t0float

The current time $t_0$.

tmatfloat

The maturity time $T$.

tdfloat, array-like, shape $\left(\text{ntd}\right)$

The discrete times at which $\varphi$ is specified.

phidfloat, array-like, shape $\left(\text{ntd}\right)$

$phid[\text{i}-1]$ must contain the value of $\varphi$ at time $td[\text{i}-1]$, for $\text{i}=1,2,\dots ,\text{ntd}$.

Returns

phiavfloat, ndarray, shape $\left(3\right)$

$phiav\left[0\right]$ contains the value of $\varphi$ interpolated to $t_0$, $phiav\left[1\right]$ contains the first-order average $\hat{\varphi }$ and $phiav\left[2\right]$ contains the second-order average $\overline{\varphi }$, where:

$$\begin{array}{cc}\hat{\varphi }=\frac{1}{T-t_0}{\int }_{t_0}^T\varphi \left(\zeta \right)d\zeta \text{,}& \overline{\varphi }={\left(\frac{1}{T-t_0}{\int }_{t_0}^T{\varphi }^2\left(\zeta \right)d\zeta \right)}^{1/2}\text{.}\end{array}$$

Raises

NagValueError

(errno $1$)

On entry, $td\left[0\right]\le td[⟨value⟩]$.

Constraint: $td\left[0\right]<td\left[1\right]<\cdots <td[\text{ntd}-1]$.

(errno $1$)

On entry, $\text{ntd}=⟨value⟩$.

Constraint: $\text{ntd}\ge 2$.

(errno $1$)

On entry, $tmat=⟨value⟩$, $td\left[0\right]=⟨value⟩$ and $td[\text{ntd}-1]=⟨value⟩$.

Constraint: $td\left[0\right]\le tmat\le td[\text{ntd}-1]$.

(errno $1$)

On entry, $t0=⟨value⟩$, $td\left[0\right]=⟨value⟩$ and $td[\text{ntd}-1]=⟨value⟩$.

Constraint: $td\left[0\right]\le t0\le td[\text{ntd}-1]$.

(errno $2$)

Unexpected failure in internal call to spline function.

Notes

dim1_blackscholes_means computes the quantities

$$\begin{array}{ccc}\varphi \left(t_0\right)\text{,}& \hat{\varphi }=\frac{1}{T-t_0}{\int }_{t_0}^T\varphi \left(\zeta \right)d\zeta \text{,}& \overline{\varphi }={\left(\frac{1}{T-t_0}{\int }_{t_0}^T{\varphi }^2\left(\zeta \right)d\zeta \right)}^{1/2}\end{array}$$

from a given set of values $phid$ of a continuous time-dependent function $\varphi \left(t\right)$ at a set of discrete points $td$, where $t_0$ is the current time and $T$ is the maturity time. Thus $\hat{\varphi }$ and $\overline{\varphi }$ are first and second order averages of $\varphi$ over the remaining life of an option.

The function may be used in conjunction with dim1_blackscholes_closed() in order to value an option in the case where the risk-free interest rate $r$, the continuous dividend $q$, or the stock volatility $\sigma$ is time-dependent and is described by values at a set of discrete times (see Use with dim1_blackscholes_closed).