fraction of the LGD where protection starts
fraction of the LGD where protection ends
vector of basket nominal amounts
default basket represented by a vector of default term structures that allow computing single name default probabilities depending on time
one-factor copula
sold protection if set to true, purchased otherwise
schedule for premium payments
annual premium rate, e.g. 0.05 for 5% p.a.
day count convention for the premium rate
recovery rate as a fraction
premium as a tranche notional fraction
yield term structure handle
number of distribution buckets
time step for integrating over one premium period; if larger than premium period length, a single step is taken
returns the net present value of the instrument.
returns all additional result returned by the pricing engine.
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.
returns the error estimate on the NPV when available.
When a derived result structure is defined for an instrument, this method should be overridden to read from it. This is mandatory in case a pricing engine is used.
This method constrains the object to return the presently cached results on successive invocations, even if arguments upon which they depend should change.
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.
returns any additional result returned by the pricing engine.
set the pricing engine to be used.
warning calling this method will have no effects in case the performCalculation method was overridden in a derived class.
When a derived argument structure is defined for an instrument, this method should be overridden to fill it. This is mandatory in case a pricing engine is used.
This method reverts the effect of the freeze
method, thus re-enabling recalculations.
Observer interface
returns the date the net present value refers to.
collateralized debt obligation
The instrument prices a mezzanine CDO tranche with loss given default between attachment point $ D_1$ and detachment point $ D_2 > D_1 $.
For purchased protection, the instrument value is given by the difference of the protection value $ V_1 $ and premium value $ V_2 $,
$$ V = V_1 - V_2. $$
The protection leg is priced as follows:
Build the probability distribution for volume of defaults $ L $ (before recovery) or Loss Given Default $ LGD = (1-r),L $ at times/dates $ t_i, i=1, ..., N$ (premium schedule times with intermediate steps)
Determine the expected value $ E_i = E_{t_i},\left[Pay(LGD)\right] $ of the protection payoff $ Pay(LGD) $ at each time $ t_i$ where $$ Pay(L) = min (D_1, LGD) - min (D_2, LGD) = \left{ \begin{array}{lcl} \displaystyle 0 &;& LGD < D_1 \ \displaystyle LGD - D_1 &;& D_1 \leq LGD \leq D_2 \ \displaystyle D_2 - D_1 &;& LGD > D_2 \end{array} \right. $$
The protection value is then calculated as $$ V_1 :=: \sum_{i=1}^N (E_i - E_{i-1}) \cdot d_i $$ where $ d_i$ is the discount factor at time/date $ t_i $
The premium is paid on the protected notional amount, initially $ D_2 - D_1. $ This notional amount is reduced by the expected protection payments $ E_i $ at times $ t_i, $ so that the premium value is calculated as
$$ V_2 = m , \cdot \sum_{i=1}^N ,(D_2 - D_1 - E_i)
\cdot \Delta_{i-1,i}\,d_i$$
where $ m $ is the premium rate, $ \Delta_{i-1, i}$ is the day count fraction between date/time $ t_{i-1}$ and $ t_i.$
The construction of the portfolio loss distribution $ E_i $ is based on the probability bucketing algorithm described in
John Hull and Alan White, "Valuation of a CDO and nth to default CDS without Monte Carlo simulation", Journal of Derivatives 12, 2, 2004The pricing algorithm allows for varying notional amounts and default termstructures of the underlyings.
Investigate and fix cases $ E_{i+1} < E_i. $