Structural Credit Merton Model

We will see how to derive the structural credit merton model, another way from reduced form model, where the default is modelled from the underlying market traded debt.

$E_T = max( A_T - D, 0)$
$E_t = A_t N(d_1) - D e-^{rT} N(d_2)$
$d1 = \frac {ln (\frac{Ae^{rT}}{D})}{\sigma_A \sqrt{T}} + \frac{1}{2} \sigma_A \sqrt{T}$

Now, the leverage can be written as, $L = \frac{\hat{D}}{A}$ , where $\hat{D} = D_0 e^{(r - y)T}$

Now, equity call on the underlying debt can be written in terms of leverage, asset vol and time to maturity as,

$E_t = A_t [ N(d_1) + L N(d_2) ]$
$d1 = \frac {-ln(L) + (r + \frac{\sigma_A^2}{T} ) T}{\sigma_A \sqrt{T}}$

Credit Value Adjustments (CVA)

CVA’s can be priced unilaterally or bilaterally. We will see some of the terms in CVA,

Future Exposure (FE):
Underlying positive (simulated) future exposure.
$E(t_j, \omega_s) := (V(t_j, \omega_s)^{+}) = max(V(t_j, \omega_s), 0)$
Worst Potential Future Exposure (WPFE):
Maximum positive future exposure
$EE(t_j) = max(E(t_j, \omega_s))$

Let’s now focus on the steps involved in CVA pricing, note that, CVA pricing given below

$CVA = LGD (PD_{(0, T)}) D_{(0, T)} E[max(V_j,0)^+]$

1. Price the forward contract
2. Compute the max exposure on each scenario
3. Price for the CVA
4. Subtract from the forward price determined from #1 to get the overall exposure

Let’s see how to compute CVA from 2-step binomial forward price process.

Compute the forward price
Compute the up/down movement, $d = \frac{1}{u}$
Compute the up/down probability, $p_u = \frac{e^{rT} -d}{u -d} ; p_d = 1 - p_u$
Create 4 panels
1st panel – Stock price with up/down
2nd panel – Forward price $V_t = S_t - F e^{-rT}$
3rd panel – $Max(V^+, 0)$
4th panel – Prob with up/down, $D(0,T)$
Calculate EE, $EE = \sum_{i=1}^n D(0,T) max(V^+, 0)$
Calculate $CVA = LGD PD_{(0,T)} D_{(0, T)} E_Q(max(V^+, 0))$

Credit Default Swaps (CDS)

Formal Definition:

Credit Default Swaps are insurance contracts that protects the buyer from credit events of underlying reference entity in exchange for series of premium payments.

We will see the derivation of credit default swaps below.

$PremL_{a,b,R} = \frac{R}{2} \sum_{i=a+1}^b D_{0, T_i} \alpha_i (Q_{\tau > T_{i-1}} - Q_{\tau > T_i}) + \sum_{i=a+1}^b R D_{0, T_i} \alpha_i Q_{\tau > T_i}$
$ProtL_{a,b,R} = LGD \sum_{i=a+1}^b D(0, T_i) (Q_{\tau > T_{i-1}} - Q_{\tau > T_i} )$

BootStrap CDS rates from quoted SPC, given LGD / Rec. Rate,

$R_{Tn} (CDS) = \frac{LGD (\sum D Q_{t-i} - \sum D Q_{t} ) }{ \frac{1}{2} (\sum D \alpha_i Q_{t-i} + \sum D \alpha_i Q_{t} ) }$

Default Prob on Risky ZCB

With No Recovery Rate

$V_D = E_Q [ N D(0, T) Q_{\tau > T} + 0 D(0, T) Q_{\tau \leq T} ]$
$V_D = N D(0, T) Q_{\tau \leq T}$
$V_D = N D(0, T) (1 - Q_{\tau \leq T})$
$Q_{\tau \leq T} = 1 - \frac{V_D}{V_{RF}}$
$Q_{\tau \leq T} = 1 - e^{-(y_D - y_{RF})}$

Now, the survival prob and credit spread can be calculated as below,

$Q_{\tau > T} = - e^{-(y_D - y_{RF})}$
$y_{cs} = - \frac{1}{T} ln (Q(\tau > T))$

With Recovery Rate

$V_D = E_Q [ N D(0, T) \textbf{I}_{\tau > T} + N Rec D(0, T) \textbf{I}_{\tau \leq T}]$
$V_D = N D(0, T) Q_{\tau > T} + N Rec D(0, T) Q_{\tau \leq T}$

From this we can derive for the credit spread for ZCB with recovery rate,

$y_cs = - \frac{1}{T} ln ( Q_{\tau > T} + Rec Q_{\tau \leq T} )$
$y_cs = - \frac{1}{T} ln ( Q_{tau >T} (1 - Rec) + Rec )$

Fixed Effects Estimator

$Cov(X_i, \alpha_i) \neq 0$ , meaning that $\alpha_i$ can be arbitarirly correlated to $X_i$ .
Estimate,
$\hat{\beta_{DV}} = [X_{DV}' X_{DV}]^{-1} X_{DV}' y_{DV}$

Random Effects Estimator

$Cov(X_i, \alpha_i) = 0$
$\hat{\beta_{GLS}} = [ X_{DV}' X_{DV} + T \omega^2 X_B' X_B]^{-1} X_{DV}' y_{DV} + T \omega^2 X_B' y_B'$
Where, $\omega = \frac{\sigma_{\epsilon}}{\sqrt{T \sigma_{\alpha}^2 + \sigma_{\epsilon}^2}}$

AutoRegressive Conditional Heteroskedasticity (ARCH)

We will see some of the conditional and unconditional moments for ARCH,

$y_t = \theta y_{t-1} + \epsilon_t, |\theta| < 1$
$\epsilon_t = u_t^2 (\omega + \alpha \epsilon_t^2)$

Let’s see the expectation and variances,

$E[\epsilon_t|I_{t-1}] = 0$
$var[\epsilon_t|I_{t-1}] = (\omega + \alpha \epsilon_t^2)$
$E[y_t I_{t-1}] = \theta y_{t-1}$

Linear Dependent Variables

We will see some of the features of linear dependent variables and what best models can be adopted to estimate the parameters and regressors.

Linear Probability Model, this is the simplest choice in the class of the linear dependent variables class fo models and they simply estimate from OLS regression. The main disadvantage being that the probability can exceed 0 or 1, but we know that, $P \in (0,1)$ . Also the error term,

$\epsilon_t = \{ -x_i \beta, when : y_i = 0 \} or$
$\epsilon_t = \{ y_i - x_i \beta, when : y_i = 1 \}$

Due to the heteroskedasticity nature of the error, we need heteroskedasticity-robust standard errors and also as the dependent variable is always evaluated at probability of 1, ie., $P[y_i = 1]$ , for values exceeding 0 or 1, they are truncated and many true values cannot be directly estimated at the extreme ends.

We will another class of model, namely Tobit and we can use this in class of censored and truncated models.

Duration-Convexity Measures

The duration and convexity are bond sensitivity measures used to approximate the PnL without resorting to full revaluation. Note that, they perform good with short time duration, but as time duration gets long, the approximation gets worse.

Bond Valuation: $V_0 = \sum_{k} F_k e^{ - Y_k T_k }$
Duration: $D = \frac { \sum_k T_k F_k e^{- Y_k T_k}}{V_0}$
Convexity: $C = \frac { \sum_k T_k^2 F_k e^{ -Y_k T_k}}{V_0}$
Sensitivity PnL: $PnL_{DC} = -D V_0 \Delta_y + \frac{1}{2} C^2 V_0 \Delta^2_y$
Exact PnL = $V_{T+n} - V_{T}$

Now, when the time horizon is modified, lets say to recalculate the portfolio with 1 days horizon, we will then use for full revaluation,

PnL: $\sum_k F_k e^{ -Y_k (T_k - \frac{1}{365})}$

Value-At-Risk, Expected Shortfall Measures & Properties

The risk quantification was first introduced by Markowiz from his famous Capital Asset Pricing Measure (CAPM) portfolio theory which essentially states,

$E[R_t] = R_f + \beta (E[R_m] - R_f)$

Meaning that, expected return should be measured by the systematic risk priced by the market beta. Hence variance is captured by the second order moment,

$\sigma^2 = E[X^2] - E[X]^2$

But this doesn’t capture the correlation and as option derivatives grew by mid 70’s and 80’s, clearly needed a measure which captures effectively a single measure of risk across portfolio, risk measures and this is proposed by JP Morgan around c.1992.

Value-At-Risk parametric formula is given by,

$VaR_q(X) = -E[X] + \sigma Z_{X}[1-\alpha]$
$Left Quantile: q_{\alpha}^-[X] = inf \{ X \in R : P[x \leq X] \geq \alpha \}$
$Right Quantile: q_{\alpha}^+[X] = inf \{ X \in R : P[x \leq X] > \alpha \}$

Now for the expected shortfall, we shall see the formula given below,

$ES_q(X) = -\frac{1}{\alpha} \int_{0}^{\infty} VaR(u) du$
$ES_q(X) = - \frac{1}{\alpha} \frac{ \int_{-\infty}^{-VaR_{\alpha}} x f(x) dx } { P[X \leq -VaR_{\alpha}}]$

Now, let’s look at the coherent properties of VaR & Expected Shortfall risk measures given below, note that from below properties VaR satisfies first three whereas ES satisfies all of them, hence ES is coherent risk measure. As side note, there’s been proposal for convexity (CO) measure which is slightly modified version of the sub additivity which better suggests for the coherent risk measures.

Monotonicity: $X \geq Y, then, \rho(X) \leq \rho(Y)$
Cash Additivity: $\rho(Y + m) = \rho(Y) - m$
Positive Homogeneity: $\rho(\lambda Y) = \lambda \rho(Y)$
SubAdditivity: $\rho(X+ Y) \leq \rho(X) + \rho(Y)$