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COHERENT RISKS

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Alexander S. Cherny

COHERENT RISKS

AND THEIR APPLICATIONS

- Why are coherent risks needed?
- How are coherent risks used?

COHERENT RISKS

Artzner, Delbaen, Eber, Heath (1997)

Definition.A coherent risk is a map r(X):

(i)r(X+Y) b r(X)+r(Y);

(ii) If X b Y, then r(X) r r(Y);

(iii)r(lX) = lr(X) for lr0;

(iv)r(X+m) = r(X)-m for m.

Theorem.ris a coherent risk r(X) = -minQDEQX.

Probabilistic scenarios

Terminal wealth of a portfolio

Scenario-based risk:r(X) = -min{X(w1),…,X(wN)},

where N and w1,…,wN are possible scenarios.

TV@R:r(X) = -E(X|Xbql),

where l(0,1) andql is the l-quantile of X.

XV@R:(27)r(X) = -Emin{X1,…,XN},

where N and X1,…,XNare independent copies of X.

Numbers in green are the numbers of papers on my website:

http://mech.math.msu.su/~cherny

Maximum:r1,…,rN are coherent risks

r(X)= max{r1(X),…,rN(X)}

is a coherent risk with D = D1…DN.

Conv. combination:r1,…,rN are coherent risks

- r(X)=l1r1(X)+…+lNrN(X)
is a coherent risk with D = l1D1+…+lNDN.

Convolution:r1,…,rN are coherent risks

r(X) = min{r1(X1)+…+rN(XN):X1+…+XN=X}

is a coherent risk with D = D1… DN.

(28)

X-P&L of a portfolio over the unit time period

F - increment of a market factor over this period

Problem:Risk of X driven by F = ?

Definition.(27)Factor risk of X driven by F:

rf(X;F) =r(j(F)), where j(z) =E(X | F=z).

This is a coherent risk with Df = {E(Z|F):ZD}.

FACTOR RISKS-II

X = X1+…+Xd

FACTOR RISKS-II

X = X1+…+Xd rf(X;F) = r(j(F)), where

j(z) = j1(z)+…+jd(z), ji(z) = E(Xi|F=z).

TV@R:rf(X;F) = -E(j(F)bql),

where qlis the l-quantile of j(F).

FACTOR RISKS-II

X = X1+…+Xd rf(X;F) = r(j(F)), where

j(z) = j1(z)+…+jd(z), ji(z) = E(Xi|F=z).

TV@R:rf(X;F) = -E(j(F)bql),

where qlis the l-quantile of j(F).

FACTOR RISKS-II

X = X1+…+Xd rf(X;F) = r(j(F)), where

j(z) = j1(z)+…+jd(z), ji(z) = E(Xi|F=z).

TV@R:rf(X;F) = -E(j(F)bql),

where qlis the l-quantile of j(F).

XV@R:rf(X;F) = -Emin{j(F1),…,j(FN)},

where F1,…,FNare independent copies of F.

COHERENT RISKS

Artzner, Delbaen, Eber, Heath (1997)

Definition.A coherent risk is a map r(X):

(i)r(X+Y) b r(X)+r(Y);

(ii) If X b Y, then r(X) r r(Y);

(iii)r(lX) = lr(X) for lr0;

(iv)r(X+m) = r(X)-m for m.

Theorem.ris a coherent risk r(X) = -minQDEQX.

Probabilistic scenarios

Terminal wealth of a portfolio

COHERENT RISKS

Artzner, Delbaen, Eber, Heath (1997)

Definition.A coherent risk is a map r(X):

(i)r(X+Y) b r(X)+r(Y);

(ii) If X b Y, then r(X) r r(Y);

(iii)r(lX) = lr(X) for lr0;

(iv)r(X+m) = r(X)-m for m.

Theorem.ris a coherent risk r(X) = -minQDEQX.

Probabilistic scenarios

Terminal wealth of a portfolio

X=+1 with P=0.96

X=-100 with P=0.04

l=0.05

V@Rl(X)=-1

-76 !

COHERENT RISKS

Artzner, Delbaen, Eber, Heath (1997)

Definition.A coherent risk is a map r(X):

(i)r(X+Y) b r(X)+r(Y);

(ii) If X b Y, then r(X) r r(Y);

(iii)r(lX) = lr(X) for lr0;

(iv)r(X+m) = r(X)-m for m.

Theorem.ris a coherent risk r(X) = -minQDEQX.

Probabilistic scenarios

Terminal wealth of a portfolio

COHERENT RISKS

Artzner, Delbaen, Eber, Heath (1997)

Definition.A coherent risk is a map r(X):

(i)r(X+Y) b r(X)+r(Y);

(ii) If X b Y, then r(X) r r(Y);

(iii)r(lX) = lr(X) for lr0;

(iv)r(X+m) = r(X)-m for m.

Theorem.ris a coherent risk r(X) = -minQDEQX.

Probabilistic scenarios

Terminal wealth of a portfolio

Do you agree that these two positions have the same risk?

Do you agree that the risk of any position coincides with the risk of the opposite position?

X=-1 with P=0.5 Y=-1 with P=0.5

X=+1 with P=0.5 Y=+0.5 with P=0.48

Y=+13 with P=0.02

EX=0 EY=0

QUADRATIC RISK-II

X=-1 with P=0.5 Y=-1 with P=0.5

X=+1 with P=0.5 Y=+0.5 with P=0.48

Y=+13 with P=0.02

EX=0, VarX=1EY=0, VarY=7.75

Do you agree that Y is 7 times riskier than X?

QUADRATIC RISK-III

r(X) = -EX+SvarX

is a coherent risk

But there exist better

coherent risks!

APPLICATIONS

Coherent risks provide a uniform basis for:

- risk measurement,
- capital allocation,
- risk management,
- pricing and hedging,
- assessing trades.

X – P&L earned by a company

X = (X1+…+Xd) – P&L earned by a company

Problem: How is the risk r(X) allocated

between the desks?

r(X1)+…+r(Xd)>r(X) – diversification!

Definition.Risk contribution of Y to X:

rc(Y;X) = -EQ*Y,

where Q*=argminQDEQX.

Capital allocation:rc(X1;X),…, rc(Xd;X).

P&L of a subportfolio

P&L of a portfolio

Scenario-based risk:r(X) = -min{X(w1),…,X(wN)},

where N and w1,…,wN are possible scenarios.

TV@R:r(X) = -E(X|Xbql),

where l(0,1) andql is the l-quantile of X.

XV@R:(27)r(X) = -Emin{X1,…,XN},

where N and X1,…,XNare independent copies of X.

Numbers in green are the numbers of papers on my website:

http://mech.math.msu.su/~cherny

Scenario-based risk:rc(Y;X) = -Y(wn*),

where n*=argminn=1,…,NX(wn).

TV@R:rc(Y;X) = -E(Y|Xbql),

where qlis the l-quantile of X.

XV@R:rc(Y;X) = -EYn*,

where n*=argminn=1,…,NXn,

(X1,Y1),…,(XN,YN) areindependent copies of (X,Y).

Properties:rc(X1;X)+…+rc(Xd;X) = r(X),

rc(Y;X) = lime0 e-1[r(X+eY)-r(X)],

YX r(X+Y) r(X)+ rc(Y;X).

Scenario-based risk:rc(Y;X) = -Y(wn*),

where n*=argminn=1,…,NX(wn).

TV@R:rc(Y;X) = -E(Y|Xbql),

where qlis the l-quantile of X.

XV@R:rc(Y;X) = -EYn*,

where n*=argminn=1,…,NXn,

(X1,Y1),…,(XN,YN) areindependent copies of (X,Y).

Properties:rc(X1;X)+…+rc(Xd;X) = r(X),

rc(Y;X) = lime0 e-1[r(X+eY)-r(X)],

YX r(X+Y) r(X)+ rc(Y;X).

Problem:E(X1+…+Xd) max,

XiAi – P&Ls available to the i-th desk,

r(X1+…+Xd)bC- firm’s capital.

Theorem. (25) If (X1,…,Xd) is optimal, then

EX1/rc(X1;X) =…= EXd/rc(Xd;X),

whereX = X1+…+Xd.

Problem:E(X1+…+Xd) max,

XiAi – P&Ls available to the i-th desk,

r(X1+…+Xd)bC- firm’s capital.

Theorem. (25) If (X1,…,Xd) is optimal, then

EX1/rc(X1;X) =…= EXd/rc(Xd;X),

RAROCc(X1 ; X) RAROCc(Xd ; X),

whereX = X1+…+Xd.

Question: Is it possible to decentralize the

procedure of imposing risk limits?

Yes!

Theorem. (27) If the limits are imposed on the

risk contributions and the desks are allowed

to trade these limits within the firm, then the

equilibrium is an optimal solution, and vice versa.

F - contingent claim

A – space of P&Ls of possible trading strategies

Problem:Find x and XA such that

r(X-F+x)b0 and x is as small as possible.

F - contingent claim

A – space of P&Ls of possible trading strategies

Problem:Find x and XA such that

r(X-F)bx and x is as small as possible.

Price: minXAr(X-F)

Hedge: argminXAr(X-F)

Quadratic risk:P – pricing measure

Price:EPF

Hedge: argminXAVar(X-F)

Risk-adjusted price:EPF+aVar(X*-F)

Which r to apply?

F - contingent claim

A – space of P&Ls of possible trading strategies

Problem:Find x and XA such that

r(X-F)bx and x is as small as possible.

Risk-adjusted price: minXAr(X-F)

Hedge: argminXAr(X-F)

Quadratic risk:P – pricing measure

Price:EPF

Hedge: argminXAVar(X-F)

Risk-adjusted price:EPF+aVar(X*-F)

Which r to apply?

Theorem.Ifr(Z)=-minQDEQZ, then

rm(Z) := minXAr(X+Z) = -minQDREQZ,

whereR={Q:EQX=0 XA}.

Risk-adjusted price of F equals

minXAr(X-F) = maxQDREQF = EQ*F

W – P&L of the firm’s overall portfolio

rm(W) = r(X*+W) = EQ**W

rm(W-F) -EQ**W+EQ**F if FW

Risk-adjusted price contribution of F to W

equalsEQ**F, where Q** = argminQDREQW.

Market-modified risk

Theorem.Ifr(Z)=-minQDEQZ, then

rm(Z) := minXAr(X+Z) = -minQDREQZ,

whereR={Q:EQX=0 XA}.

Risk-adjusted price of F equals

minXAr(X-F) = maxQDREQF = EQ*F

W – P&L of the firm’s overall portfolio

rm(W)= r(X*+W) = EQ**W

rm(W-F) -EQ**W+EQ**F if FW

Risk-adjusted price contribution of F to W

equalsEQ**F, where Q**=argminQDREQW.

Market-modified risk

Risk

Theorem.Ifr(Z)=-minQDEQZ, then

rm(Z) := minXAr(X+Z) = -minQDREQZ,

whereR={Q:EQX=0 XA}.

Risk-adjusted price of F equals

minXAr(X-F) = maxQDREQF = EQ*F

W – P&L of the firm’s overall portfolio

rm(W)= r(X*+W) = EQ**W

rm(W-F) -EQ**W+EQ**F if FW

Risk-adjusted price contribution of F to W

equalsEQ**F, where Q**=argminQDREQW.

Market-modified risk

Risk

Hedge

Theorem.Ifr(Z)=-minQDEQZ, then

rm(Z) := minXAr(X+Z) = -minQDREQZ,

whereR={Q:EQX=0 XA}.

Risk-adjusted price of F equals

minXAr(X-F) = maxQDREQF = EQ*F

W – P&L of the firm’s overall portfolio

rm(W)= r(X*+W) = EQ**W

rm(W-F) -EQ**W+EQ**F if FW

Risk-adjusted price contribution of F to W

equalsEQ**F, where Q**=argminQDREQW.

Market-modified risk

Risk

Hedge

Extreme measure

Theorem.Ifr(Z)=-minQDEQZ, then

rm(Z) := minXAr(X+Z) = -minQDREQZ,

whereR={Q:EQX=0 XA}.

Risk-adjusted price of F equals

minXAr(X-F) = maxQDREQF = EQ*F

W – P&L of the firm’s overall portfolio

rm(W)= r(X*+W) = EQ**W

rm(W-F) -EQ**W+EQ**F if FW

Risk-adjusted price contribution of F to W

equalsEQ**F, where Q**=argminQDREQW.

Market-modified risk

Risk

Hedge

Extreme measure

STATIC MODEL

Sn – price of the underlying at time n =0,1

W – P&L of a portfolio

r(h(S1-S0)+W) min, h

Pflug-Rockafellar-Uryasev method: r – TV@R

l-1E(q-h(S1-S0)-W)+- q min, h,q

m*, h*, q*

Risk:m*

Hedge:h*

Extreme measure:P( |h*(S1-S0)+W <q*)

W = f(S1), f is concave

Find a<b: P(S1(a,b)) = l,

E(S1 | S1(a,b)) =S0

W = f(S1), f is concave

Find a<b: P(S1(a,b)) = l,

E(S1 | S1(a,b)) =S0

Extreme measure:P( | S1(a,b))

W = f(S1), f is concave

Find a<b: P(S1(a,b)) = l,

E(S1 | S1(a,b)) =S0

Extreme measure:P( | S1(a,b))

Risk:-E(f(S1) | S1(a,b))

W = f(S1), f is concave

Find a<b: P(S1(a,b)) = l,

E(S1 | S1(a,b)) =S0

Extreme measure:P( | S1(a,b))

Risk:-E(f(S1) | S1(a,b))

Hedge:-(f(b)-f(a))/(b-a)

W=+1 with P=0.95

W=-19 with P=0.05

Quadratic hedging:

Coherent hedging:

Sn – price of the underlying at time n =0,…,N

sn – volatility at time n = 0,…,N

(Sn,sn) is a Markov process

Examples: GARCH, SV.

W– P&L produced by a portfolio

X– P&L produced by trading

r(X+W) min, XA

Sn – price of the underlying at time n =0,…,N

sn – volatility at time n = 0,…,N

(Sn,sn) is a Markov process

Examples: GARCH, SV.

W =(W1,..,WN)– stream of payments of a portfolio

X =(X1,..,XN)– stream of payments produced by trading

r(X+W) min, XA

Theorem.(32) If W corresponds to a portfolio of

European options, then the risk-adjusted price and

the hedge of W at time n are functions of n,Sn,Qn.

Sn = S0 exp{X1+…+Xn}, where Xnare i.i.d.

Wn=jn(Sn), where jnare concave

- – dynamic Tail V@R of order l
Find 0<a<b: P(exp(X)(a,b))=l,

E(exp(X)| exp(X)(a,b))=1.

Q** = P( |exp(X1)(a,b),…, exp(XN)(a,b))

X – P&L of a trade

Quality of X= Reward/Risk

Sharpe ratio:EX/VarX

RAROC: EX/V@R(X)

Gain-Loss ratio:EX+/EX-,

X+=max(X,0), X-=max(-X,0)

Coherent RAROC:EX/r(X),

r – coherent risk

ACCEPTABILITY INDICES

TV@R acceptability index:

a(X) = inf{l:E(X|Xbql)>0}-1.

ACCEPTABILITY INDICES

TV@R acceptability index:

a(X) = inf{l:E(X|Xbql)>0}-1.

ACCEPTABILITY INDICES

TV@R acceptability index:

a(X) = inf{l:E(X|Xbql)>0}-1.

TV@R acceptability index:

a(X) = inf{l:E(X|Xbql)>0}-1.

Definition.An acceptability indexis a map

a(X) = max{z[0,) :rz(X)<0},

where (rz)z[0,)is a family of coherent risks

increasing in z.

A(X) – performance measure

Convexity:A(X)rz, A(Y)rz A(X+Y)rz

Monotonicity:XbY A(X)b A(Y)

Arbitrage consistency:A(X)=+ X is an arbitrage

Measure Conv. Mon. Arb.

Sharpe ratio +

RAROC +

Gain-Loss ratio + + +

Coherent RAROC + +

Acceptability index + + +

A(X) – performance measure

Convexity:A(X)rz, A(Y)rz A(X+Y)rz

Monotonicity:XbY A(X)b A(Y)

Arbitrage consistency:A(X)=+ X is an arbitrage

Measure Conv. Mon. Arb.

Sharpe ratio +

RAROC +

Gain-Loss ratio + + +

Coherent RAROC + +

Acceptability index + + + (SP1)

A(X) – performance measure

Convexity:A(X)rz, A(Y)rz A(X+Y)rz

Monotonicity:XbY A(X)b A(Y)

Arbitrage consistency:A(X)=+ X is an arbitrage

Measure Conv. Mon. Arb.

Sharpe ratio +

RAROC +

Gain-Loss ratio + + +

Coherent RAROC + +

Acceptability index + + +(SP1)

(rz)z[0,) – family of coherent risks increasing in z

A – space of P&Ls of possible trading strategies

W – P&L of a portfolio

For any z[0,), find Xz = argminXArz(W+X)

Find z*such that rz*(W+Xz*)=0

a(W+X) max, XA

Hedge of W:Xz*

Risk measure:rz*

Pricing measure: Q**=argminQD*REQW

rz (W+Xz)

F – P&L of an additional trade

For any z[0,), find X’z =argminXArz(W+F+X)

Find z*’such that rz*’ (W+F+Xz*’)=0

Hedge of F:X’z*’- Xz*

F is profitable z*’ > z*

Theorem:(SP2)If FW, then F is profitable

EQ**F > 0.

Fair price of F:EQ**F

EQ**W= 0

= EQ**F

- Risk measurement Scenarios, TV@R, XV@R
Conv. comb., max, convolution

Factor risks

- Capital allocation Risk contribution
Extreme measure

- Risk management Trading risk limits
imposed on risk contributions

- Pricing and hedging Market-modified risk, Q*, Q**
Pflug-Rockafellar-Uryasev

- Assessing trades Acceptability indices
Appl. to pricing and hedging

THANK YOU FOR YOUR ATTENTION!