In hsloot/cvalr: Credit Derivative Valuation in R

General Assumptions

We consider $d$ entities with default times $\boldsymbol{\tau} = (\tau_1, \ldots, \tau_d)$ and average default counting process $$ C_t =\frac{1}{d} \sum_{i=1}^{d} 1_{{ \tau_i \leq t}} . $$

Furthermore, we assume deterministic discount factors $d_t$ and a recovery rate $R \in (0, 1)$.

Portfolio CDS

A portfolio CDS is a financial contract between two parties which serves as insurance for the buyer against the default of companies from a bond-portfolio. The insurance buyer pays a periodic spread $s \Delta t N$ and receives the recovered fraction $(1-R) \Delta L$ in case of default.

If accounting for accrued interest, the expected discounted payment stream of the buyer, also called expected discounted default leg (EDDL), is $$ {EDDL} = \sum_{j=1}^{n} d_{t_j} \Delta t_{j-1}\mathbb{E}{\left[ N_{t_j} + \frac{N_{t_{j-1}} - N_{t_j}}{2} \right]} $$ and the expected payment stream of the seller, also called expected discounted payment leg (EDPL), is $$ {EDPL}{s} = \sum{j=1}^{n} s d_{t_j} \mathbb{E}{[ \Delta L_{t_{j-1}} ]} , $$ where $\Delta t_{j-1} = t_{j} - t_{j-1}$, $\Delta L_{t_{j-1}} = L_{t_{j}} - L_{t_{j-1}}$, $L_{t} = (1 - R) C_t$, and $N_{t} = 1 - C_{t} = 1 - L_t / {(1 - R)} $.

The portfolio CDS is set up in the beginning such that $$ s = \frac{EDDL}{EDPL_{1}} . $$

CDO tranches

A (synthetic) CDO tranche is a financial contract between two parties in which the seller restructures a bond-portfolio into products of various credit quality. The buyer pays a periodic spread $s \Delta t N$ and receives a compensation for the recovered fraction of the portfolio loss between a lower attachment point $l$ and an upper attachment point $u$.

A (synthetic) CDO is a collection of (synthetic) CDO tranches with attachment points $$ 0 = l_1 < u_1 = l_2 < \ldots < u_{K-1} = l_{K} < u_{K} = 1 . $$

If accounting for accrued interest, the expected discounted payment stream of the buyer of tranche $k$, also called expected discounted default leg (EDDL), is $$ {EDDL}{k} = \sum{j=1}^{n} d_{t_j} \Delta t_{j-1} \mathbb{E}{[N_{k, t_j}]} + \frac{ \mathbb{E}{[N_{k, t_{j-1}}]} - \mathbb{E}{[N_{k, t_j}]} }{2} $$ and the expected payment stream of the seller of tranche $k$, also called expected discounted premium leg (EDPL), is $$ {EDPL}{k, s} = \sum{j=1}^{n} s d_{t_j} \mathbb{E}{[\Delta L_{k, t_{j-1}}]} , $$ where $\Delta t_{j-1} = t_j - t_{j-1}$, $\Delta L_{t_{j-1}} = L_{t_j} - L_{t_{j-1}}$, $$ L_{k, t} = \min{{ \max{{ {(1 - R)} C_t - l_k , 0 }} , u_k - l_k }} , $$ and $N_{k, t} = u_k - l_k - L_{k, t}$.

hsloot/cvalr documentation built on Sept. 24, 2022, 9:25 a.m.