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: $\rho(X+ Y) \leq \rho(X) + \rho(Y)$

Delta-Gamma-Vega Approximations for nonlinear portfolios

Often in financial trading, instruments are not linear function of the underlyings. For instance, forward is just linear function of the underlying spot and can easily formulate linear approximation for the PnL. But for options, they are non-linear and clearly needed delta and/or delta-gamma based non-linear parametric approximations.

The formula for delta approximations is given by,

$PnL = n_S S_0 R + n_C \delta S_0 R$

For the delta-gamma approximations, fore note on the gamma, it is the second order derivative of delta from the change in the underlying. It essentially captures the non-linearity from the taylor series expansion.

$PnL = (n_s + \sum_{k} n_k \delta_k ) S_0 R + (\sum_{k} n_k \gamma_k) \frac{1}{2} S_0^2 R^2$

The Value-at-Risk for the delta portfolio can be calculated from,

$VaR_{\alpha} (PL) = |\delta| S_0 \sigma VaR_{\alpha}(X)$

Given the increase volatility contracts traded, its imperative to include the vega approximation to the taylor series, so the formula we see will be given by,

$PL_{\alpha}(X) = (n_S + \sum_{k} n_k \delta_k) S_0 R + (\sum_k n_k \gamma_k) \frac{1}{2} S_0^2 R^2 + (\sum_k n_k vega_k) \Delta \sigma*$

So far we have seen the PnL approximations and VaR calculations based on delta, gamma and vega sensitivities. Note that we haven’t include the rho and theta primarily they underlying is not too sensitive on very short term. We will now see the formula for the delta, gamma and vega to impute in the above equations,

$\delta_{call} = \frac{ln \frac{Se^{rT}}{K}}{\sigma \sqrt{T}} + \frac{1}{2} \sigma \sqrt{T}$
$\delta_{put} = \delta_{call} - 1$
$\gamma_{call|put} = \frac{1}{S_0 \sigma \sqrt{T}} \frac{1}{\sqrt{2 \Pi}} e^{-\frac{d1^2}{2}}$
$vega_{call|put} = S_0 \sqrt{T} \frac{1}{\sqrt{2 \Pi}} e^{- \frac{d1^2}{2}}$

Note in the Gamma equation, ensure to substitute $\delta_1$ and NOT $N(d1)$

Black hole Theory & Quantum Physics

In the universe, there are many number of black holes which form high density, gravitational pull that even light cannot pass through it. So, when we trace back to the formation of our universe way back to Hubbler’s modified estimation 13.8 billion years ago, universe was in-fact black hole, a very small condensed with huge amount of four forces, gravity, electromagnetic fields, strong and weak nuclear forces which were so packed within tiny tiny particle that when gravity got exploded, the whole new universe inflated and hence formation of different gas from base hydrogen & helium atoms formed. So far back tracing using the light emitted from back from the formation of universe we can guess the age of our universe but what it really takes to have in first place that tiny tiny dense state of huge gravitational and electromagnetic force, how it happened ? Does it means, there’s multiverse meaning multiple universe we are one among them or previous to the formation of universe whether there was existing one which was absorbed by one large giant black hole. We will see more about the theory and what we have theoretically observed so far.

Alright, let’s see what happens when star like Sun started to run out of the fuel. Well, we all know the core of the sun needs $H_2$ i.e., hydrogen to burn and this is also one of the core gas along with helium which then forms class of other heavy gas elements in the universe. Now, when star dies, hydrogen runs out and it uses helium to produce other heavy gas elements such as lithium and beryllium. During this time, it inflates in huge size and carbon it generates will form iron. At one stage when all the helium gases are finished, it will either form huge red dwarf or inflate and form supernova and form itself neutron star. So, a neutron star is like black hole such that its mass is so dense that even tiniest size mass is as huge as mass of Earth.

So, big bang happened predicted earlier by 13.77 billion years and our solar system formed roughly 4.6 billion years, so roughly 9.17 years it took for our solar system to form. Earlier, we could imagine universe was expanding with other galaxies, solar systems within each ones, exoplanets, asteroids, comets, dwarf, neutron star etc.,. So, a picture explains more beautifully as below,

https://en.wikipedia.org/wiki/Big_Bang

A new study in Nature finds that the Milky Way is part of a broader supercluster of 100,000 galaxies known as Laniakea.

The above supercluster named ‘Laniakea’ is one recently formed by group of scientist after observing movements of close spatial galaxies and our milky way galaxy as you see is one corner of this super massive cluster. Like this, there are many many superclusters in universe, where each galaxies boots millions, billions of stars and each star having its own solar system planets orbiting around them.

We will see next extra terrestrial life forms most importantly Titan methane lakes that orbiting around Saturn and read some interesting news on this.