## Monte Carlo Challenges

Option values are often computed using a Monte-Carlo simulation (MC). In this case however, using traditional MCs is technically challenging. Under the various default scenarios, recovery depends on future cashflow scenarios, each of which requires another MC simulation. In other words, the valuation estimation requires many nested MC runs, making the computation taxing. This situation is similar to pricing American options, where the calculations not only depend on the current scenario but on all future ones as well.

An alternative method is the so-called 'least squared MC', which uses a surrogate function (usually a polynomial function) to replace the second level of MC simulations. However, this approach is still computationally challenging, especially when calibrating the forward-looking volatility in the final stage of the investment.

Hence, we use a third option for model calibration, which relies on so-called tree-type approximation:

## Tree-type approximation

Using binomial trees to price options is also a standard method, and one that is suitable for various option styles. It requires a tree of the possible scenarios and a corresponding probability for each scenario at each time step. Since computations are required only for a limited number of the scenarios, rather than all the possible ones, it is much faster than the MC method.

While the binomial tree approach only generates two possible future scenarios at each step, it requires fine-grained scenarios to compute the default probability (PD) in our case. Given the CFADS forecast from the CFADS model CFADS(t), we compute the forecast CFADS movement in the log space as:

m_{t} = Ln\left( \text{CFADS}\left( t + 1 \right) + 1 \right) - Ln\left( \text{CFADS}\left( t \right) + 1 \right) |

As the traditional binomial tree, we discretize the continuous CFADS space as the n_{t} possible value \text{CFAD}S_{i}\left( t \right) for the i-th scenario at the step t. As shown on the figure below:

The transition probability between the i-th node at step t and the j-th node at step t+1 is
p_{i,j}\left( t \right) and follows the below

conditions:

w_{i,j}(t) = Ln\left( \text{CFAD}S_{j}\left( t + 1 \right) + 1 \right) - Ln\left(CFADS_{i}\left( t \right) + 1 \right)\\ \sum_{j}^{}{p_{i,j}\left( t \right)\ w_{i,j}(t)} = m_{t}\\ \sum_{j}^{}{p_{i,j}\left( t \right)\left( w_{i,j}(t) \right)^{2}} = \sigma^{2}\\ \sum_{j}^{}{p_{i,j}\left( t \right) = 1} |

where the \sigma is the CFADS movement volatility \sigma_{CFADS}.

Assuming the log of CFADS follow a Brownian motion, the transition probability can be approximated by:

p_{i,j}\left( t \right) = Z\exp\left( - \frac{\left( w_{i,j}\left( t \right) - m_{t} \right)^{2}}{2{\tilde{\sigma}}^{2}} \right)\\ Z = \frac{1}{\sum_{j}^{}{\exp\left( - \frac{\left( w_{i,j}\left( t \right) - m_{t} \right)^{2}}{2{\tilde{\sigma}}^{2}} \right)}} |

where {\tilde{\sigma}}^{2} is the variance should be found to satisfy the conditions of mean and variance of a Brownian motion.

In reality, CFADS movement do not follow a normal distribution and have larger kurtosis. However, such differences would be mitigated when we use a calibrated \sigma instead of the realised value, as is typically done in equity option pricing.

We use the approach to compute the CFADS scenarios and their transition probability, which is used in the calculation of equity and debt value.

Technical notes

- The number of the node n_{t} in each step is arbitrarily set. More nodes increase the PD estimation granularity and transition probabilities will also become closer to a normal distribution. The trade-off is computation time. We use a large number of nodes at t=1 n_{t} = 100 with the coverage of 3-sigma.
- p_{i,j}\left( t \right) approaches a normal distribution when n_{t} becomes very large number and the CFADS nodes cover enough range of \sigma^{2}.
- As the nature of Brownian motion, each node at the step t must has at least one node at t+1 whose w_{i,j}\left( t \right) - m_{t} > 0 and one node with w_{i,j}\left( t \right) - m_{t} < 0.

**Transition probability of the scenarios at
t and
t+1**