Black and Cox's model

Description

BlackCox calculates the survival probability Q(\tau > 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

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\le T is the K, with K\le L, value discounted at back at time t using the interest rate gamma, obtaining:

H(t | t\le T) = K * \exp^{- \gamma * (T- t)}

The output parameter Default.Intensity represents the default intensity of \Delta 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 \Delta 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')

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

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')

Firm value in Merton's model

Description

With this function we simulate n trajectories of firm value based on Merton's model.

Usage

Merton.sim(V0, r, sigma, t, n, seed = as.numeric(Sys.time()))

Arguments

Details

The trajectories are calculated according to the equation:

V_T = V_0 \exp{\int_0^T dln V_t}

Where we express dln V_t using Ito's lemma to derive the differential of the logarithm of the firm value as:

dln V_t =(\mu - \frac{\sigma^2}{2})dt + \sigma dW_t

Value

This function returns a matrix containing the simulated firm values.

References

Gergely Daròczi, Michael Puhle, Edina Berlinger, Péter Csòka, Dàniel Havran Màrton Michaletzky, Zsolt Tulasay, Kata Vàradi, Agnes Vidovics-Dancs (2013) Introduction to R for Quantitative Finance.

Examples

V <- Merton.sim(V0 = 20, r = 0.05, sigma = 0.2, t = seq(0, 30, by = 0.5), n = 5)
matplot(x = seq(0, 30, by = 0.5), y = V, type = 's', lty = 1, xlab = 'Time',
ylab = 'Firm value trajectories', main = "Trajectories of the firm values in the Merton's model")

Analytically - Tractable First Passage (AT1P) model

Description

at1p calculates the survival probability Q(\tau > t) and default intensity for each maturity according to the structural Analytically - Tractable First Passage model.

Usage

at1p(V0, H0, B, sigma, r, t)

Arguments

Details

In this function the safety level Ht is calculated using the formula:

H(t) = \frac{H0}{V0} * E_0[V_t] * \exp^{- B \int_0^t \sigma_u du}

The backbone of the default barrier at t is a proportion, controlled by the parameter H0, of the expected value of the company assets at t. H0 may depend on the level of the liabilities, on safety covenants, and in general on the characteristics of the capital structure of the company. Also, depending on the value of the parameter B, it is possible that this backbone is modified by accounting for the volatility of the company assets. For example, if B > 0 corresponds to the interpretation that when volatility increases - which can be independed of credit quality - the barrier is slightly lowered to cut some more slack to the company before going bankrupt. When B = 0 the barrier does not depend on the volatility and the "distance to default" is simply modelled through the barrier parameter H0.

Value

at1p returns an object of class data.frame containing the firm value, safety level H(t) and the survival probability for each maturity. The last column is the default intensity calculated among each interval \Delta t.

References

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

Examples

mod <- at1p(V0 = 1, H0 = 0.7, B = 0.4, sigma = rep(0.1, 10), r = cdsdata$ED.Zero.Curve,
t = cdsdata$Maturity)
mod

plot(cdsdata$Maturity, mod$Ht, type = 'b', xlab = 'Maturity', ylab = 'Safety Level H(t)',
main = 'Safety level for different maturities', ylim = c(min(mod$Ht), 1.5),
col = 'red')
lines(cdsdata$Maturity, mod$Vt, xlab = 'Maturity', ylab = 'V(t)',
main = 'Value of the Firm \n at time t', type = 's')

plot(cdsdata$Maturity, mod$Survival, type = 'b',
main = 'Survival Probability for different Maturity \n (AT1P model)',
xlab = 'Maturity', ylab = 'Survival Probability')

matplot(cdsdata$Maturity, mod$Default.Intensity, type = 'l', xlab = 'Maturity',
ylab = 'Default Intensity')

Black and Cox model calibration to market CDS data

Description

Compares CDS rates quoted on the market with theoric CDS rates calculeted by the function cds and looks for the parameters to be used into BlackCox for returning the default intensities corresponding to real market CDS rates performing the minimization of the objective function.

Usage

calibrate.BlackCox(V0, cdsrate, r, t, ...)

Arguments

Details

Inside calibrate.BlackCox, the function objfn takes the input a vector of parameters and returns the mean error occurred estimating CDS rates with cds function. The inputs used in cds are the default intensities calculated by the BlackCox function with the calibrated parameters. In particular the error is calculated as:

\frac{1}{n}\sum_{i=1}^n (c^{ds}-c^{ds}_{mkt})^2.

This quantity is a function of the default intensities and it is the objective function to be minimized in order to take optimal solutions for intensities.

Value

calibrate.BlackCox returns an object of class data.frame with calculated parameters of the BlackCox model and the error occurred in the minimization procedure.

References

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

Examples

calibrate.BlackCox(V0 = 1, cdsrate = cdsdata$Par.spread, r = 0.005, t = cdsdata$Maturity)

AT1P model calibration to market CDS data

Description

Compares CDS rates quoted on the market with theoric CDS rates calculeted by the function cds and looks for the parameters to be used into at1p for returning the default intensities corresponding to real market CDS rates performing the minimization of the objective function.

Usage

calibrate.at1p(V0, cdsrate, r, t, ...)

Arguments

Details

Inside calibrate.at1p, the function objfn takes the input a vector of parameters and returns the mean error occurred estimating CDS rates with cds function. The inputs used in cds are the default intensities calculated by the at1p function with the calibrated parameters. In particular the error is calculated as:

\frac{1}{n}\sum_{i=1}^n (c^{ds}-c^{ds}_{mkt})^2.

This quantity is a function of the default intensities and it is the objective function to be minimized in order to take optimal solutions for intensities.

Value

calibrate.at1p returns an object of class data.frame with calculated parameters of the at1p model and the error occurred in the minimization procedure.

References

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

Examples

calibrate.at1p(V0 = 1, cdsrate = cdsdata$Par.spread, r = cdsdata$ED.Zero.Curve,
t = cdsdata$Maturity)

Calibrate the default intensities to market CDS data

Description

Compares CDS rates quoted on market with theoric CDS rates and looks for default intensities that correspond to real market CDS rates trough a minimization problem of an objective function.

Usage

calibrate.cds(r, t, Tj, cdsrate, ...)

Arguments

Details

Inside calibrate.cds, the function err.cds takes the input a vector of intensities and return the mean error occurred estimating CDS rates with cds. In particular such error is calculated as:

\frac{1}{n}\sum_{i=1}^n (c^{ds}-c^{ds}_{mkt})^2.

This quantity is a function of default intensities and is the our objective function to be minimized in order to take optimal solutions for intensities.

Value

returns an object of class list with calculated intensities and the error occurred in the minimization procedure.

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

calibrate.cds( r = cdsdata$ED.Zero.Curve, t = seq(.5, 30, by = 0.5),
               Tj = c(1, 2, 3, 4, 5, 7, 10, 20, 30), cdsrate = cdsdata$Par.spread, RR = 0.4)

SBTV model calibration to market CDS data

Description

Compares CDS rates quoted on the market with theoric CDS rates calculeted by the function cds and looks for the parameters to be used into sbtv for returning the default intensities corresponding to real market CDS rates performing the minimization of the objective function.

Usage

calibrate.sbtv(V0, p, cdsrate, r, t, ...)

Arguments

Details

Inside calibrate.sbtv, the function objfn takes the input a vector of parameters and returns the mean error occurred estimating CDS rates with cds function. The inputs used in cds are the default intensities calculated by the sbtv function with the calibrated parameters. In particular the error is calculated as:

\frac{1}{n}\sum_{i=1}^n (c^{ds}-c^{ds}_{mkt})^2.

This quantity is a function of the default intensities and it is the objective function to be minimized in order to take optimal solutions for intensities.

Value

This function returns an object of class list with calculated parameters of sbtv model and the error occurred in the minimization procedure.

References

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

Examples

calibrate.sbtv(V0 = 1, p = c(0.95, 0.05), cdsrate = cdsdata$Par.spread,
r = cdsdata$ED.Zero.Curve, t = cdsdata$Maturity)

Calculates Credit Default Swap rates

Description

Calculates CDS rates starting form default intensities.

Usage

cds(t, int, r, R = 0.005, RR = 0.4, simplified = FALSE)

Arguments

Details

• Premium timetable is t_i; i=1,...,T. The vector starts from t_1\le 1, i.e. the first premium is payed at a year fraction in the possibility that the bond is not yet defaulted. Since premium are a postponed payment (unlike usual insurance contracts).

• Intensities timetable have domains \gamma_i; i=t_1,...,T.

• spot interest rates of bond have domain r_i; i=t_1,...,T. The function transforms spot rates in forward rates. If we specify that we want to calculate CDS rates with the simplified alghoritm, in each period, the amount of the constant premium payment is expressed by:

  \pi^{pb}=\sum_{i=1}^Tp(0,i)S(0,i)\alpha_i

  and the amount of protection, assuming a recovery rate \delta, is:

  \pi^{ps}=(1-\delta)\sum_{i=1}^Tp(0,i)\hat{Q}(\tau=i)\alpha_i

  If we want to calculate same quantities with the complete version, that evaluate premium in the continous, the value of the premium leg is calculated as:

  \pi^{pb}(0,1)=-\int_{T_a}^{T_b}P(0,t)\cdot(t-T_{\beta(t)-1}) d_t Q (\tau\geq t)+\sum_{i=a+1}^bP(0,T_i)\cdot\alpha_i * Q(\tau\geq T_i)

  and the protection leg as:

  \pi_{a,b}^{ps}(1):=-\int_{t=T_a}^{T_b}P(0,t)d*Q(\tau\geq t)

  In both versions the forward rates and intensities are supposed as costant stepwise functions with discontinuity in t_i

Value

cds returns an object of class data.frame with columns, for esch date t_i the value of survival probability, the premium and protection leg, CDS rate and CDS price.

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

cds(t = seq(0.5, 10, by = 0.5), int = seq(.01, 0.05, len = 20),
r = seq(0,0.02, len=20), R = 0.005, RR = 0.4, simplified = FALSE)

Calculate Credit Default Swap rates

Description

Calculate CDS rates starting from default intensities

Usage

cds2(t, Tj, tr, r, tint, int, R = 0.005, ...)

Arguments

Details

The function cds2 is based on cds but allows a more fine controll on maturities and on discretization of r and int. In particular input (t, tr, tint) can be of different length thanks to the function approx.

Value

An object of class data.frame that contains the quantities calculated by cds on Tj timetable.

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

cds2(t = c(1:20),Tj = c(1:20), tr = c(1:20), r = seq(0.01,0.06, len =20),
tint = c(1:20), int= seq(0.01,0.06, len =20))

CDS quotes from market

Description

• Maturity: Maturities of cds contracts expressed in years;

• Par.Spread: CDS rates quotes, spread that nullify the present value of the two legs;

• ED.Zero.Curve: EURIBOR interest rates (risk-free)

Usage

data(cdsdata)

Format

An object of class "data.frame".

Source

Thomson Reuters, CDS quotes of Unicredit on 2017-01-23

Cumulative Normal Distribution Function

Description

This function calculates the cumulative normal distribution function (CDF) for a given value x using the Hastings approximation method. This approximation is typically used in finance for the calculation of option pricing probabilities.

Usage

cum_normal_density(x)

Arguments

Details

The function uses a polynomial approximation as described by E.G. Haug in "The Complete Guide to Option Pricing Formulas" to estimate the CDF of a normal distribution. The coefficients used in the approximation are specifically chosen to minimize the error in the tail of the distribution, which is critical for financial applications like option pricing.

The polynomial approximation is applied to the normal density function:

N(x) = \frac{1}{\sqrt{2\pi}} e^{-x^2/2}

Then, the cumulative probability is adjusted based on the sign of x: - If x is non-negative, it returns \(1 - t\), where t is the polynomial approximation. - If x is negative, it returns \(t\).

The cumulative normal distribution function is important in statistics for hypothesis testing and in finance for the Black-Scholes option pricing formula.

Value

Returns the cumulative probability under the normal curve from \(-

\infty

\) to x.

References

Haug, E.G., The Complete Guide to Option Pricing Formulas. Hastings, C. Approximations for Digital Computers. Princeton Univ. Press, 1955.

Examples

cum_normal_density(1.96)  
cum_normal_density(-1.96) 

Generalized Black-Scholes Option Pricing Model

Description

This function calculates the price of a European call or put option using the generalized Black-Scholes formula, which extends the standard model to incorporate a continuous dividend yield.

Usage

generalized_black_scholes(TypeFlag = c("c", "p"), S, X, Time, r, b, sigma)

Arguments

Details

The generalized Black-Scholes formula considers both the risk-free rate and a cost of carry, making it suitable for a wider range of financial instruments, including commodities and currencies with continuous yields.

The pricing formula for call and put options is determined by:

C = S e^{(b-r)T} N(d_1) - X e^{-rT} N(d_2)

P = X e^{-rT} N(-d_2) - S e^{(b-r)T} N(-d_1)

where:

d_1 = \frac{\log(S / X) + (b + \sigma^2 / 2) T}{\sigma \sqrt{T}}

d_2 = d_1 - \sigma \sqrt{T}

and (N(\cdot)) is the cumulative normal distribution function, estimated by the 'cum_normal_density' function.

Value

Returns the price of the specified option (call or put).

References

Haug, E.G., The Complete Guide to Option Pricing Formulas.

Examples

# Calculate the price of a call option
generalized_black_scholes("c", S = 100, X = 100, Time = 1, r = 0.05, b = 0.05, sigma = 0.2)
# Calculate the price of a put option
generalized_black_scholes("p", S = 100, X = 100, Time = 1, r = 0.05, b = 0.05, sigma = 0.2)

Scenario Barrier Time-Varying Volatility AT1P model

Description

sbtv calculates the survival probability Q(\tau > t) and default intensity for each maturity according to the structural SBTV model.

Usage

sbtv(V0, H, p, B, sigma, r, t)

Arguments

Details

sbtv is an extension of the at1p model. In this model the parameter H0 used in the at1p model is replaced by a random variable assuming different values in different scenarios, each scenario with a different probability. The survival probability is calculated as a weighted avarage of the survival probability using the formula:

SBTV.Surv = \sum_{i = 1}^N p[i] * AT1P.Surv(H[i])

where AT1P.Surv(H[i]) is the survival probability computed according to the AT1P model when H_0 = H[i] and with weights equal to the probabilities of the different scenarios.

Value

sbtv returns an object of class data.frame containing the survival probability for each maturity. The last column is the default intensity calculated among each interval \Delta t.

References

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

Examples

mod <- sbtv(V0 = 1, H = c(0.4, 0.8), p = c(0.95, 0.05), B = 0, sigma = rep(0.20, 10),
            r = cdsdata$ED.Zero.Curve, t = cdsdata$Maturity)
mod

plot(cdsdata$Maturity, mod$Survival, type = 'b')