Tranche Declension.

800px-CDO_-_FCIC_and_IMF_Diagram

With the CDO (collateralized debt obligation) market picking up, it is important to build a stronger understanding of pricing and risk management models. The role of the Gaussian copula model, has well-known deficiencies and has been criticized, but it continues to be fundamental as a starter. Here, we draw attention to the applicability of Gaussian inequalities in analyzing tranche loss sensitivity to correlation parameters for the Gaussian copula model.

We work with an RN-valued Gaussian random variable X = (X1, … , XN), where each Xj is normalized to mean 0 and variance 1, and study the equity tranche loss

L[0,a] = ∑m=1Nlm1[xm≤cm] – {∑m=1Nlm1[xm≤cm] – a}

where l1 ,…, lN > 0, a > 0, and c1,…, cN ∈ R are parameters. We thus establish an identity between the sensitivity of E[L[0,a]] to the correlation rjk = E[XjXk] and the parameters cj and ck, from where subsequently we come to the inequality

∂E[L[0,a]]/∂rjk ≤ 0

Applying this inequality to a CDO containing N names whose default behavior is governed by the Gaussian variables Xj shows that an increase in name-to-name correlation decreases expected loss in an equity tranche. This is a generalization of the well-known result for Gaussian copulas with uniform correlation.

Consider a CDO consisting of N names, with τj denoting the (random) default time of the jth name. Let

Xj = φj-1(Fjj))

where Fj is the distribution function of τj (relative to the market pricing measure), assumed to be continuous and strictly increasing, and φj is the standard Gaussian distribution function. Then for any x ∈ R we have

P[Xj ≤ x] = P[τj ≤ Fj-1j(x))] = Fj(Fj-1j(x))) = φj(x)

which means that Xj has standard Gaussian distribution. The Gaussian copula model posits that the joint distribution of the Xj is Gaussian; thus,

X = (X1, …., Xn)

is an RN-valued Gaussian variable whose marginals are all standard Gaussian. The correlation

τj = E[XjXk]

reflects the default correlation between the names j and k. Now let

pj = E[τj ≤ T] = P[Xj ≤ cj]

be the probability that the jth name defaults within a time horizon T, which is held constant, and

cj = φj−1(Fj(T))

is the default threshold of the jth name.

In schematics, when we explore the essential phenomenon, the default of name j, which happens if the default time τis within the time horizon T, results in a loss of amount lj > 0 in the CDO portfolio. Thus, the total loss during the time period [0, T] is

L = ∑m=1Nlm1[xm≤cm]

This is where we are essentially working with a one-period CDO, and ignoring discounting from the random time of actual default. A tranche is simply a range of loss for the portfolio; it is specified by a closed interval [a, b] with 0 ≤ a ≤ b. If the loss x is less than a, then this tranche is unaffected, whereas if x ≥ b then the entire tranche value b − a is eaten up by loss; in between, if a ≤ x ≤ b, the loss to the tranche is x − a. Thus, the tranche loss function t[a, b] is given by

t[a, b](x) = 0 if x < a; = x – a, if x ∈ [a, b]; = b – a; if x > b

or compactly,

t[a, b](x) = (x – a)+ – (x – b)+

From this, it is clear that t[a, b](x) is continuous in (a, b, x), and we see that it is a non-decreasing function of x. Thus, the loss in an equity tranche [0, a] is given by

t[0,a](L) = L − (L − a)+

with a > 0.

Optimal Hedging…..

hedging

Risk management is important in the practices of financial institutions and other corporations. Derivatives are popular instruments to hedge exposures due to currency, interest rate and other market risks. An important step of risk management is to use these derivatives in an optimal way. The most popular derivatives are forwards, options and swaps. They are basic blocks for all sorts of other more complicated derivatives, and should be used prudently. Several parameters need to be determined in the processes of risk management, and it is necessary to investigate the influence of these parameters on the aims of the hedging policies and the possibility of achieving these goals.

The problem of determining the optimal strike price and optimal hedging ratio is considered, where a put option is used to hedge market risk under a constraint of budget. The chosen option is supposed to finish in-the-money at maturity in the, such that the predicted loss of the hedged portfolio is different from the realized loss. The aim of hedging is to minimize the potential loss of investment under a specified level of confidence. In other words, the optimal hedging strategy is to minimize the Value-at-Risk (VaR) under a specified level of risk.

A stock is supposed to be bought at time zero with price S0, and to be sold at time T with uncertain price ST. In order to hedge the market risk of the stock, the company decides to choose one of the available put options written on the same stock with maturity at time τ, where τ is prior and close to T, and the n available put options are specified by their strike prices Ki (i = 1, 2,··· , n). As the prices of different put options are also different, the company needs to determine an optimal hedge ratio h (0 ≤ h ≤ 1) with respect to the chosen strike price. The cost of hedging should be less than or equal to the predetermined hedging budget C. In other words, the company needs to determine the optimal strike price and hedging ratio under the constraint of hedging budget.

Suppose the market price of the stock is S0 at time zero, the hedge ratio is h, the price of the put option is P0, and the riskless interest rate is r. At time T, the time value of the hedging portfolio is

S0erT + hP0erT —– (1)

and the market price of the portfolio is

ST + h(K − Sτ)+ er(T−τ) —– (2)

therefore the loss of the portfolio is

L = (S0erT + hP0erT) − (ST +h(K−Sτ)+ er(T−τ)) —– (3)

where x+ = max(x, 0), which is the payoff function of put option at maturity.

For a given threshold v, the probability that the amount of loss exceeds v is denoted as

α = Prob{L ≥ v} —– (4)

in other words, v is the Value-at-Risk (VaR) at α percentage level. There are several alternative measures of risk, such as CVaR (Conditional Value-at-Risk), ESF (Expected Shortfall), CTE (Conditional Tail Expectation), and other coherent risk measures. The criterion of optimality is to minimize the VaR of the hedging strategy.

The mathematical model of stock price is chosen to be a geometric Brownian motion, i.e.

dSt/St = μdt + σdBt —– (5)

where St is the stock price at time t (0 < t ≤ T), μ and σ are the drift and the volatility of stock price, and Bt is a standard Brownian motion. The solution of the stochastic differential equation is

St = S0 eσBt + (μ−1/2σ2)t —– (6)

where B0 = 0, and St is lognormally distributed.

Proposition:

For a given threshold of loss v, the probability that the loss exceeds v is

Prob {L ≥ v} = E [I{X ≤ c1} FY (g(X) − X)] + E [I{X ≥ c1} FY (c2 − X)] —– (7)

where E[X] is the expectation of random variable X. I{X < c} is the index function of X such that I{X < c} = 1 when {X < c} is true, otherwise I{X < c} = 0. FY (y) is the cumulative distribution function of random variable Y , and

c1 = 1/σ [ln(K/S0) − (μ−1/2σ2)τ] ,

g(X) = 1/σ [(ln (S0 + hP0)erT − h (K − f(X)) er(T−τ) −v)/S0 − (μ − 1/2σ2) T],

f(X) = S0 eσX + (μ−1/2σ2)τ,

c2 = 1/σ [(ln (S0 + hP0) erT − v)/S0 − (μ− 1/2σ2) T

X and Y are both normally distributed, where X ∼ N(0,√τ), Y ∼ N(0,√(T−τ).

For a specified hedging strategy, Q(v) = Prob {L ≥ v} is a decreasing function of v. The VaR under α level can be obtained from equation

Q(v) = α —– (8)

The expectations in Proposition can be calculated with Monte Carlo simulation methods, and the optimal hedging strategy which has the smallest VaR can be obtained from equation (8) by numerical searching methods….

Stationarity or Homogeneity of Random Fields

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Let (Ω, F, P) be a probability space on which all random objects will be defined. A filtration {Ft : t ≥ 0} of σ-algebras, is fixed and defines the information available at each time t.

Random field: A real-valued random field is a family of random variables Z(x) indexed by x ∈ Rd together with a collection of distribution functions of the form Fx1,…,xn which satisfy

Fx1,…,xn(b1,…,bn) = P[Z(x1) ≤ b1,…,Z(xn) ≤ bn], b1,…,bn ∈ R

The mean function of Z is m(x) = E[Z(x)] whereas the covariance function and the correlation function are respectively defined as

R(x, y) = E[Z(x)Z(y)] − m(x)m(y)

c(x, y) = R(x, x)/√(R(x, x)R(y, y))

Notice that the covariance function of a random field Z is a non-negative definite function on Rd × Rd, that is if x1, . . . , xk is any collection of points in Rd, and ξ1, . . . , ξk are arbitrary real constants, then

l=1kj=1k ξlξj R(xl, xj) = ∑l=1kj=1k ξlξj E(Z(xl) Z(xj)) = E (∑j=1k ξj Z(xj))2 ≥ 0

Without loss of generality, we assumed m = 0. The property of non-negative definiteness characterizes covariance functions. Hence, given any function m : Rd → R and a non-negative definite function R : Rd × Rd → R, it is always possible to construct a random field for which m and R are the mean and covariance function, respectively.

Bochner’s Theorem: A continuous function R from Rd to the complex plane is non-negative definite if and only if it is the Fourier-Stieltjes transform of a measure F on Rd, that is the representation

R(x) = ∫Rd eix.λ dF(λ)

holds for x ∈ Rd. Here, x.λ denotes the scalar product ∑k=1d xkλk and F is a bounded,  real-valued function satisfying ∫A dF(λ) ≥ 0 ∀ measurable A ⊂ Rd

The cross covariance function is defined as R12(x, y) = E[Z1(x)Z2(y)] − m1(x)m2(y)

, where m1 and m2 are the respective mean functions. Obviously, R12(x, y) = R21(y, x). A family of processes Zι with ι belonging to some index set I can be considered as a process in the product space (Rd, I).

A central concept in the study of random fields is that of homogeneity or stationarity. A random field is homogeneous or (second-order) stationary if E[Z(x)2] is finite ∀ x and

• m(x) ≡ m is independent of x ∈ Rd

• R(x, y) solely depends on the difference x − y

Thus we may consider R(h) = Cov(Z(x), Z(x+h)) = E[Z(x) Z(x+h)] − m2, h ∈ Rd,

and denote R the covariance function of Z. In this case, the following correspondence exists between the covariance and correlation function, respectively:

c(h) = R(h)/R(o)

i.e. c(h) ∝ R(h). For this reason, the attention is confined to either c or R. Two stationary random fields Z1, Z2 are stationarily correlated if their cross covariance function R12(x, y) depends on the difference x − y only. The two random fields are uncorrelated if R12 vanishes identically.

An interesting special class of homogeneous random fields that often arise in practice is the class of isotropic fields. These are characterized by the property that the covariance function R depends only on the length ∥h∥ of the vector h:

R(h) = R(∥h∥) .

In many applications, random fields are considered as functions of “time” and “space”. In this case, the parameter set is most conveniently written as (t,x) with t ∈ R+ and x ∈ Rd. Such processes are often homogeneous in (t, x) and isotropic in x in the sense that

E[Z(t, x)Z(t + h, x + y)] = R(h, ∥y∥) ,

where R is a function from R2 into R. In such a situation, the covariance function can be written as

R(t, ∥x∥) = ∫Rλ=0 eitu Hd (λ ∥x∥) dG(u, λ),

where

Hd(r) = (2/r)(d – 2)/2 Γ(d/2) J(d – 2)/2 (r)

and Jm is the Bessel function of the first kind of order m and G is a multiple of a distribution function on the half plane {(λ,u)|λ ≥ 0,u ∈ R}.