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PLAN

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

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

EXAMPLES

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

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

[email protected]:r(X) = -E(X|Xbql),

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

[email protected]:(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

OPERATIONS

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)

FACTOR RISKS-I

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}.

X = X1+…+Xd

X = X1+…+Xd rf(X;F) = r(j(F)), where

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

[email protected]:rf(X;F) = -E(j(F)bql),

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

X = X1+…+Xd rf(X;F) = r(j(F)), where

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

[email protected]:rf(X;F) = -E(j(F)bql),

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

X = X1+…+Xd rf(X;F) = r(j(F)), where

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

[email protected]:rf(X;F) = -E(j(F)bql),

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

[email protected]:rf(X;F) = -Emin{j(F1),…,j(FN)},

where F1,…,FNare independent copies of F.

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

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

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

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

QUADRATIC RISK-I

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?

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 EY=0

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?

Coherent risks provide a uniform basis for:

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

CAPITAL ALLOCATION

X – P&L earned by a company

CAPITAL ALLOCATION

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

EXAMPLES

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

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

[email protected]:r(X) = -E(X|Xbql),

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

[email protected]:(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

RISK CONTRIBUTION

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

where n*=argminn=1,…,NX(wn).

[email protected]:rc(Y;X) = -E(Y|Xbql),

where qlis the l-quantile of X.

[email protected]: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).

RISK CONTRIBUTION

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

where n*=argminn=1,…,NX(wn).

[email protected]:rc(Y;X) = -E(Y|Xbql),

where qlis the l-quantile of X.

[email protected]: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).

RISK MANAGEMENT-I

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.

RISK MANAGEMENT-I

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.

RISK MANAGEMENT-II

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.

PRICING AND HEDGING

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.

PRICING AND HEDGING

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?

PRICING AND HEDGING

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

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 – [email protected]

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*)

EXAMPLE

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))

EXAMPLE

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))

EXAMPLE

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)

DYNAMIC MODEL

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

DYNAMIC MODEL

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.

EXAMPLE

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

Wn=jn(Sn), where jnare concave

- – dynamic Tail [email protected] 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))

ASSESSING TRADES

X – P&L of a trade

Quality of X= Reward/Risk

Sharpe ratio:EX/VarX

RAROC: EX/[email protected](X)

Gain-Loss ratio:EX+/EX-,

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

Coherent RAROC:EX/r(X),

r – coherent risk

ACCEPTABILITY INDICES

[email protected] 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.

COMPARISON OF PERFORMANCE MEASURES

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 + + +

COMPARISON OF PERFORMANCE MEASURES

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)

COMPARISON OF PERFORMANCE MEASURES

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)

APPLICATION TO PRICING AND HEDGING

(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

SUMMARY

- Risk measurement Scenarios, [email protected], [email protected]
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

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