Fix Vasicek zero-reversion limit

[jewonj0620]

The small-a branch is still needed for numerical stability, but returning 0.0 is not the correct limit.

Let tau = T - t. In the Vasicek model,

$$ P(t,T,r) = A(t,T)\exp(-B(t,T)r), $$

with

$$ B(t,T) = \frac{1-\exp(-a\tau)}{a}. $$

Using the Taylor expansion of exp(-a*tau) around a = 0,

$$ B(t,T)= \tau - \frac{a\tau^2}{2} + \frac{a^2\tau^3}{6} + O(a^3), $$

so (B(t,T) \to \tau).

For the A(t,T) term, QuantLib uses

$$ \log A(t,T)=\left(b + \frac{\lambda\sigma}{a} - \frac{\sigma^2}{2a^2}\right) (B(t,T)-\tau) -\frac{\sigma^2 B(t,T)^2}{4a}. $$

Substituting the expansion of (B(t,T)),

$$ B(t,T)-\tau= -\frac{a\tau^2}{2} +\frac{a^2\tau^3}{6} +O(a^3), $$

and

$$ B(t,T)^2 =\tau^2-a\tau^3+O(a^2). $$

Therefore,

$$ \log A(t,T) \to -\frac{1}{2}\lambda\sigma\tau^2 +\frac{\sigma^2\tau^3}{6}. $$

Hence,

$$ A(t,T) \to \exp\left( -\frac{1}{2}\lambda\sigma\tau^2 +\frac{\sigma^2\tau^3}{6}\right). $$

Therefore, the small-a branch should return this finite limit instead of 0.0.

[coveralls]

coverage: 74.641% (+0.003%) from 74.638% — jewonj0620:feature/fix-vasicek-zero-reversion-limit into lballabio:master