This method causes the object to forward all notifications, even when not calculated. The default behavior is to forward the first notification received, and discard the others until recalculated; the rationale is that observers were already notified, and don't need further notification until they recalculate, at which point this object would be recalculated too. After recalculation, this object would again forward the first notification received.
warning Forwarding all notifications will cause a performance hit, and should be used only when discarding notifications cause an incorrect behavior.
This method performs all needed calculations by calling
the performCalculations method.
warning Objects cache the results of the previous
calculation. Such results will be returned upon
later invocations of
calculate. When the results depend
on arguments which could change between
invocations, the lazy object must register itself
as observer of such objects for the calculations
to be performed again when they change.
warning Should this method be redefined in derived classes, LazyObject.calculate() should be called in the overriding method.
Single correlation parameter
This method constrains the object to return the presently cached results on successive invocations, even if arguments upon which they depend should change.
Integral over the density $ \rho(m) $ of M and a one-dimensional function $ f $ of conditional probabilities related to the input vector of probabilities p:
$$ \int_{-\infty}^\infty,dm,\rho(m), f (\hat p_1, \hat p_2, \dots, \hat p_N), \qquad \hat p_i (m) = F_Z \left( \frac{F_Y^{-1}(p_i)-a,m}{\sqrt{1-a^2}} \right) $$
Integral over the density $ \rho(m) $ of M and a multi-dimensional function $ f $ of conditional probabilities related to the input vector of probabilities p:
$$ \int_{-\infty}^\infty,dm,\rho(m), f (\hat p_1, \hat p_2, \dots, \hat p_N), \qquad \hat p_i = F_Z \left( \frac{F_Y^{-1}(p_i)-a,m}{\sqrt{1-a^2}}\right) $$
This method must implement any calculations which must be (re)done in order to calculate the desired results.
This method force the recalculation of any results which
would otherwise be cached. It is not declared as
const since it needs to call the
non-const notifyObservers method.
note Explicit invocation of this method is not necessary if the object registered itself as observer with the structures on which such results depend. It is strongly advised to follow this policy when possible.
This method reverts the effect of the freeze
method, thus re-enabling recalculations.
Observer interface
Abstract base class for one-factor copula models
Reference: John Hull and Alan White, The Perfect Copula, June 2006
Let $Q_i(t)$ be the cumulative probability of default of counterparty i before time t.
In a one-factor model, consider random variables $$ Y_i = a_i,M+\sqrt{1-a_i^2}:Z_i $$ where $M$ and $Z_i$ have independent zero-mean unit-variance distributions and $-1\leq a_i \leq 1$. The correlation between $Y_i$ and $Y_j$ is then $a_i a_j$.
Let $F_Y(y)$ be the cumulative distribution function of $Y_i$. $y$ is mapped to $t$ such that percentiles match, i.e. $F_Y(y)=Q_i(t)$ or $y=F_Y^{-1}(Q_i(t))$.
Now let $F_Z(z)$ be the cumulated distribution function of $Z_i$. For given realization of $M$, this determines the distribution of $y$: $$ Prob ,(Y_i < y|M) = F_Z \left( \frac{y-a_i,M}{\sqrt{1-a_i^2}}\right) \qquad \mbox{or} \qquad Prob ,(t_i < t|M) = F_Z \left( \frac{F_Y^{-1}(Q_i(t))-a_i,M} {\sqrt{1-a_i^2}} \right) $$
The distribution functions of $ M, Z_i $ are specified in derived classes. The distribution function of $ Y $ is then given by the convolution $$ F_Y(y) = Prob,(Y<y) = \int_{-\infty}^\infty,\int_{-\infty}^{\infty}: D_Z(z),D_M(m) \quad \Theta \left(y - a,m - \sqrt{1-a^2},z\right),dm,dz, \qquad \Theta (x) = \left{ \begin{array}{ll} 1 & x \geq 0 \ 0 & x < 0 \end{array}\right. $$ where $ D_Z(z) $ and $ D_M(m) $ are the probability densities of $ Z$ and $ M, $ respectively.
This convolution can also be written $$ F(y) = Prob ,(Y < y) = \int_{-\infty}^\infty D_M(m),dm: \int_{-\infty}^{g(y,a,m)} D_Z(z),dz, \qquad g(y,a,m) = \frac{y - a\cdot m}{\sqrt{1-a^2}}, \qquad a < 1 $$
or
$$ F(y) = Prob ,(Y < y) = \int_{-\infty}^\infty D_Z(z),dz: \int_{-\infty}^{h(y,a,z)} D_M(m),dm, \qquad h(y,a,z) = \frac{y - \sqrt{1 - a^2}\cdot z}{a}, \qquad a > 0. $$
In general, $ F_Y(y) $ needs to be computed numerically.
Improve on simple Euler integration