slide1
Download
Skip this Video
Download Presentation
COHERENT RISKS

Loading in 2 Seconds...

play fullscreen
1 / 86

COHERENT RISKS - PowerPoint PPT Presentation


  • 201 Views
  • Uploaded on

Alexander S. Cherny. COHERENT RISKS. AND THEIR APPLICATIONS. PLAN. 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 );

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about ' COHERENT RISKS' - cybill


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
slide1

Alexander S. Cherny

COHERENT RISKS

AND THEIR APPLICATIONS

slide2
PLAN
  • Why are coherent risks needed?
  • How are coherent risks used?
slide3

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) = -minQDEQX.

Probabilistic scenarios

Terminal wealth of a portfolio

examples
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) = -Emin{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
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
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):ZD}.

slide7

FACTOR RISKS-II

X = X1+…+Xd

slide8

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).

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

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

slide9

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).

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

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

slide10

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).

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

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

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

where F1,…,FNare independent copies of F.

slide11

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) = -minQDEQX.

Probabilistic scenarios

Terminal wealth of a portfolio

slide12

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) = -minQDEQX.

Probabilistic scenarios

Terminal wealth of a portfolio

slide13
[email protected]

X=+1 with P=0.96

X=-100 with P=0.04

l=0.05

[email protected](X)=-1

-76 !

slide14

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) = -minQDEQX.

Probabilistic scenarios

Terminal wealth of a portfolio

slide15

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) = -minQDEQX.

Probabilistic scenarios

Terminal wealth of a portfolio

quadratic risk i1
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
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

EX=0 EY=0

slide20

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

EX=0, VarX=1EY=0, VarY=7.75

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

slide22

QUADRATIC RISK-III

r(X) = -EX+SvarX

is a coherent risk

But there exist better

coherent risks!

slide31

APPLICATIONS

Coherent risks provide a uniform basis for:

  • risk measurement,
  • capital allocation,
  • risk management,
  • pricing and hedging,
  • assessing trades.
capital allocation
CAPITAL ALLOCATION

X – P&L earned by a company

capital allocation1
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*=argminQDEQX.

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

P&L of a subportfolio

P&L of a portfolio

examples1
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) = -Emin{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
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) = lime0 e-1[r(X+eY)-r(X)],

YX r(X+Y)  r(X)+ rc(Y;X).

risk contribution1
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) = lime0 e-1[r(X+eY)-r(X)],

YX r(X+Y)  r(X)+ rc(Y;X).

risk management i
RISK MANAGEMENT-I

Problem:E(X1+…+Xd) max,

XiAi – P&Ls available to the i-th desk,

r(X1+…+Xd)bC- firm’s capital.

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

EX1/rc(X1;X) =…= EXd/rc(Xd;X),

whereX = X1+…+Xd.

risk management i1
RISK MANAGEMENT-I

Problem:E(X1+…+Xd) max,

XiAi – P&Ls available to the i-th desk,

r(X1+…+Xd)bC- firm’s capital.

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

EX1/rc(X1;X) =…= EXd/rc(Xd;X),

 

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

whereX = X1+…+Xd.

risk management ii
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
PRICING AND HEDGING

F - contingent claim

A – space of P&Ls of possible trading strategies

Problem:Find x and XA such that

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

pricing and hedging1
PRICING AND HEDGING

F - contingent claim

A – space of P&Ls of possible trading strategies

Problem:Find x and XA such that

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

Price: minXAr(X-F)

Hedge: argminXAr(X-F)

Quadratic risk:P – pricing measure

Price:EPF

Hedge: argminXAVar(X-F)

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

Which r to apply?

pricing and hedging2
PRICING AND HEDGING

F - contingent claim

A – space of P&Ls of possible trading strategies

Problem:Find x and XA such that

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

Risk-adjusted price: minXAr(X-F)

Hedge: argminXAr(X-F)

Quadratic risk:P – pricing measure

Price:EPF

Hedge: argminXAVar(X-F)

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

Which r to apply?

slide43
Theorem.Ifr(Z)=-minQDEQZ, then

rm(Z) := minXAr(X+Z) = -minQDREQZ,

whereR={Q:EQX=0 XA}.

Risk-adjusted price of F equals

minXAr(X-F) = maxQDREQF = 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 FW

 Risk-adjusted price contribution of F to W

equalsEQ**F, where Q** = argminQDREQW.

Market-modified risk

slide44
Theorem.Ifr(Z)=-minQDEQZ, then

rm(Z) := minXAr(X+Z) = -minQDREQZ,

whereR={Q:EQX=0 XA}.

Risk-adjusted price of F equals

minXAr(X-F) = maxQDREQF = 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 FW

 Risk-adjusted price contribution of F to W

equalsEQ**F, where Q**=argminQDREQW.

Market-modified risk

Risk

slide45
Theorem.Ifr(Z)=-minQDEQZ, then

rm(Z) := minXAr(X+Z) = -minQDREQZ,

whereR={Q:EQX=0 XA}.

Risk-adjusted price of F equals

minXAr(X-F) = maxQDREQF = 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 FW

 Risk-adjusted price contribution of F to W

equalsEQ**F, where Q**=argminQDREQW.

Market-modified risk

Risk

Hedge

slide46
Theorem.Ifr(Z)=-minQDEQZ, then

rm(Z) := minXAr(X+Z) = -minQDREQZ,

whereR={Q:EQX=0 XA}.

Risk-adjusted price of F equals

minXAr(X-F) = maxQDREQF = 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 FW

 Risk-adjusted price contribution of F to W

equalsEQ**F, where Q**=argminQDREQW.

Market-modified risk

Risk

Hedge

Extreme measure

slide47
Theorem.Ifr(Z)=-minQDEQZ, then

rm(Z) := minXAr(X+Z) = -minQDREQZ,

whereR={Q:EQX=0 XA}.

Risk-adjusted price of F equals

minXAr(X-F) = maxQDREQF = 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 FW

 Risk-adjusted price contribution of F to W

equalsEQ**F, where Q**=argminQDREQW.

Market-modified risk

Risk

Hedge

Extreme measure

slide55

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 – [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
EXAMPLE

W = f(S1), f is concave

Find a<b: P(S1(a,b)) = l,

E(S1 | S1(a,b)) =S0

example1
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))

example2
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))

example3
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)

example4
EXAMPLE

W=+1 with P=0.95

W=-19 with P=0.05

Quadratic hedging:

Coherent hedging:

dynamic model
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, XA

dynamic model1
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, XA

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.

example5
EXAMPLE

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

Wn=jn(Sn), where jnare concave

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
ASSESSING TRADES

X – P&L of a trade

Quality of X= Reward/Risk

Sharpe ratio:EX/VarX

RAROC: EX/[email protected](X)

Gain-Loss ratio:EX+/EX-,

X+=max(X,0), X-=max(-X,0)

Coherent RAROC:EX/r(X),

 r – coherent risk

slide71

ACCEPTABILITY INDICES

[email protected] acceptability index:

a(X) = inf{l:E(X|Xbql)>0}-1.

slide72

ACCEPTABILITY INDICES

[email protected] acceptability index:

a(X) = inf{l:E(X|Xbql)>0}-1.

slide73

ACCEPTABILITY INDICES

[email protected] acceptability index:

a(X) = inf{l:E(X|Xbql)>0}-1.

acceptability indices
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
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 measures1
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 measures2
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
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 = argminXArz(W+X)

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

 a(W+X)  max, XA

Hedge of W:Xz*

Risk measure:rz*

Pricing measure: Q**=argminQD*REQW

rz (W+Xz)

slide80
F – P&L of an additional trade

For any z[0,), find X’z =argminXArz(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 FW, then F is profitable 

EQ**F > 0.

Fair price of F:EQ**F

slide84

EQ**W= 0

= EQ**F

summary
SUMMARY

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

ad