BlackCox {CreditRisk} R Documentation

## Black and Cox's model

### Description

`BlackCox` calculates the survival probability Q(τ > t) and default intensity for each maturity according to the structural Black and Cox's model.

### Usage

```BlackCox(L, K = L, V0, sigma, r, gamma, t)
```

### Arguments

 `L` debt face value at maturity `t = T` (it is a constant value). `K` positive parameter needed to calculate the safety level. `V0` firm value at time `t = 0` (it is a constant value). `sigma` volatility (constant for all t). `r` risk-free rate (constant for all t). `gamma` interest rate used to discount the safety level `Ht` (it is a constant value). `t` a vector of debt maturity structure (it is a numeric vector).

### Details

In Merton's model the default event can occurr only at debt maturity T while in Black and Cox's model the default event can occurr even before. In this model the safety level is given by the output `Ht`. Hitting this barrier is considered as an erlier default. Assuming a debt face value of `L` at the final maturity that coincides with the safety level in t = T, the safety level in t≤ T is the `K`, with K≤ L, value discounted at back at time t using the interest rate `gamma`, obtaining:

H(t | t≤ T) = K * \exp^{- γ * (T- t)}

The output parameter `Default.Intensity` represents the default intensity of Δ t. The firm's value `Vt` is calculated as in the `Merton` function.

### Value

This function returns an object of class `data.frame` containing firm value, safety level H(t) and the survival probability for each maturity. The last column is the default intensity calculated among each interval Δ t.

### References

David Lando (2004) Credit risk modeling.

Damiano Brigo, Massimo Morini, Andrea Pallavicini (2013) Counterparty Credit Risk, Collateral and Funding. With Pricing Cases for All Asset Classes.

### Examples

```mod <- BlackCox(L = 0.55, K = 0.40, V0 = 1, sigma = 0.3, r = 0.05, gamma = 0.04,
t = c(0.50, 1.00, 2.00, 5.00, 7.00, 10.00, 20.00, 30.00))
mod

plot(c(0.50, 1.00, 2.00, 5.00, 7.00, 10.00, 20.00, 30.00), mod\$Ht, type = 'b',
xlab = 'Maturity', ylab = 'Safety Level H(t)', main = 'Safety level for different
maturities', ylim = c(min(mod\$Ht), 1.5), col = 'red')
abline(h = 0.55, col = 'red')
lines(c(0.50, 1.00, 2.00, 5.00, 7.00, 10.00, 20.00, 30.00), mod\$Vt, xlab = 'Maturity',
ylab = 'V(t)', main = 'Value of the Firm \n at time t', type = 's')

plot(c(0.50, 1.00, 2.00, 5.00, 7.00, 10.00, 20.00, 30.00), mod\$Survival, type = 'b',
main = 'Survival Probability for different Maturity \n (Black & Cox model)',
xlab = 'Maturity', ylab = 'Survival Probability')

matplot(c(0.50, 1.00, 2.00, 5.00, 7.00, 10.00, 20.00, 30.00), mod\$Default.Intensity,
type = 'l', xlab = 'Maturity', ylab = 'Default Intensity')

```

[Package CreditRisk version 0.1.3 Index]