Merton {CreditRisk} R Documentation

## Merton's model

### Description

`Merton` calculates the survival probability Q(τ > T) for each maturity according to the structural Merton's model.

### Usage

```Merton(L, V0, sigma, r, t)
```

### Arguments

 `L` debt face value at maturity `t = T`; if the value of the firm V_T is below the debt face value to be paid in T the company default has occurred (it is a constant value). `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). `t` a vector of debt maturity structure. The last value of this vector rapresents the debt maturity T.

### Details

In Merton's model the default event can occur only at debt maturity T and not before. In this model the debt face value `L` represents the constant safety level. In this model the firm value is the sum of the firm equity value `St` and ad the firm debt value `Dt`. The debt value at time t < T is calculated by the formula:

D_t = L * \exp^{- r * (T - t)} - Put(t, T; V_t, L)

The equity value can be derived as a difference between the firm value and the debt:

S_t = V_t - D_t = V_t - L * \exp^{- r * (T - t)} + Put(t, T; V_t, L) = Call(t, T; V_t, L)

(by the put-call parity) so that in the Merton's model the equity can be interpreted as a Call option on the value of the firm.

### Value

`Merton` returns an object of class `data.frame` with:

• `Vt`: expected Firm value at time t < T calculated by the simple formula V_t = V_0 * \exp^{r * t}.

• `St`: firm equity value at each t < T. This value can be seen as a call option on the firm value `V_t`.

• `Dt`: firm debt value at each t < T.

• `Survival`: surviaval probability for each maturity.

### References

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

### Examples

```mod <- Merton(L = 10, V0 = 20, sigma = 0.2, r = 0.005,
t = c(0.50, 1.00, 2.00, 3.25, 5.00, 10.00, 15.00, 20.00))
mod

plot(c(0.50, 1.00, 2.00, 3.25, 5.00, 10.00, 15.00, 20.00), mod\$Surv,
main = 'Survival Probability for different Maturity \n (Merton model)',
xlab = 'Maturity', ylab = 'Survival Probability', type = 'b')
```

[Package CreditRisk version 0.1.3 Index]