Merton {CreditRisk}

Merton's model

Description

Merton calculates the survival probability Q(\tau > 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 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 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: survival 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')