Understanding RisksIn Structured Products

Dong Qu

December 2011, Scuola Normale Superiore - Pisa

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Table of Contents

1. Common Denominator in Structured Products Business

2. Key Risk Characteristics of Structured Products

3. Correlation vs. Contagion

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1. Common Denominator in SP Business

Structured product business pillars:

Marketing/structuring;

Trading/risk managing;

Quantitative modeling;

Marketing/structuring: provide products to meet investors’ financial needs & risk appetites;

Trading/risk managing: hedge & mitigate risks;

Quantitative modeling: develop quantitative models to price the cost of hedging risks;

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Risk is the Common Denominator

Capital Return Risks

Growth Products Secured Linked to Markets Option Risks

(Impact Return)

Income Products

Not Secured

Much Enhanced Yield/Income

Option Risks (Impact Capital)

CPPI(Fund)

Protected But Not Secured Linked to Markets Gap Risks

(Impact Both)

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Quantitative Models

Quantitative models: mathematical tools to price the Risks and their hedging costs;

Good quantitative models:

Able to calibrate to the markets properly;

Able to capture key risk exposures;

Numerically stable for pricing;

Numerically stable for generating Greeks;

Computationally efficient.

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2. Key Risk Characteristics of Structured Products

Partial Differential Equation (PDE):

Risks Embedded in Options:

Time decay – Theta;

Spot – Delta;

Spot – Gamma;

Volatility – Vega;

Rate – Rho.

VrSVS

SVSqr

tV

2

222

21

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“Exotic” Risks Embedded in Options:

Dividend Risk;

Quanto Risk;

Volatility Skew/Smile Risk;

Gap Risk;

Hybrid Risk;

Correlation Risk.

“Exotic” Risks

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Risks That Supposed to be Trivial

Dividend Risk:

Most banks use deterministic sets of forecast dividends in pricing models;

During 2008 crises, banks had to mark down the dividends, resulted in substantial losses;

Quanto Risk:

Most banks use the simple quanto adjustment;

Again, during 2008 crises, banks’ equity positions lost lots of money on quanto (FX) exposures.

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Quanto Hedge Becomes Difficult …

Both equity and FX vol shoot up;

Correlation shoots up;

Stochastic vol effect becomes significant;

FX smile comes into play.

Risks that supposed to be insignificant became Massive!

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When market crashes, volatility shoots up!

DJ_STOXX50 Spot vs ImpVol

3500

3900

4300

4700

02/07/2007 26/08/2007 20/10/2007 14/12/2007 07/02/200810

20

30

40SpotImpVol

Volatility Skew/Smile Risk

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Delta in the Presence of Skew

Equity skew effectively illustrates the correlation between Spot and Volatility;

Option delta is a function of skew:

Skew dynamics:

Sticky delta (ATM vol moves with spot);

Sticky strike (Black-Scholes delta);

Sticky local vol (Static local vol, e.g. in PDE);

SV

SV

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Re-writing option delta:

Delta under skew dynamics:

L(t) = -1, sticky delta, real delta > BS delta;

L(t) = 0, sticky strike, real delta = BS delta;

L(t) = 1, sticky local vol, real delta < BS delta;

)(tLS

vega

SV

SV

SurfaceBSBS

Delta Hedge Under Skew Dynamics

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Delta Under Skew Dynamics (CALL)

Delta vs Maturity (European Call)

30%

40%

50%

60%

70%

80%

90%

0 1 2 3 4 5Maturity

L = -1L = -0.5L = 0L = 0.5L = 1

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Delta Under Skew Dynamics (PUT)

Delta vs Maturity (European Put)

-60%

-50%

-40%

-30%

-20%

-10%

0%

0 1 2 3 4 5Maturity

L = -1L = -0.5L = 0L = 0.5L = 1

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An Example of L(t) Term Structure

L(t) Term Structure

-100%

-75%

-50%

-25%

0%

0 2 4 6 8 10Years

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Skew Risks in Illiquid Markets

Skew risks are real:

Skew represents spot-vol correlation;

When market moves, volatility moves too!

In illiquid markets (e.g. emerging markets):

Option quotes are scarce, even for ATM;

Very difficult to obtain implied vols at various strikes;

Very little market information on volatility skew;

But the skew risks STILL exist and are REAL!

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Gap Risks in CPPI

CPPI uses GC measure to dynamically re-balance risky assets (e.g. shares) and risk free assets (e.g. cash);

Gap Condition (GC ):

Gap Condition is simply the gap between the current fund value and promised guarantee (pv_ed);

It is effectively an investment safety buffer.

TtTt DFGVGC

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CPPI - Dynamic Re-balance

Switching between risky and risk free assets:

If GC is large (e.g. > 25% Of Gt ), whole fund will be invested in risky assets, to fully benefit from market upside;

If GC is zero (i.e. Vt = Gt ), whole fund will be invested in cash, to fulfil the promised guarantee;

Between the two extreme cases, a portion will be invested in risky assets, with the remaining in risk free asset. The asset re-balance is conducted dynamically.

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An Example CPPI Path

CPPI Return vs Equity Path

0%

25%

50%

75%

100%

125%

150%

0 1 2 3 4 5

Equity PathCPPI Return

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Gap Risk

Risky Investment Proportion vs Gap Condition

0%

20%

40%

-10% -5% 0% 5% 10%

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Hybrid Risks

Equity Linked GAO: pension & insurance products, equity + interest rate + correlation risks:

Equity & Debt Products: convertible bonds, equity + credit + correlation risks:

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grNAKSNN

TTT

TT

BCBCBCBCB VScVr

SVS

SVSqr

tV

)(

21

2

222

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Stock markets correlation: global economy;

Credit markets correlation: global economy;

Stock/FX correlation: when markets are in distress;

Credit/FX correlation: there have always been strong links between emerging markets FX and sovereign credits;

Since credit crunch, many countries behave like emerging markets as far as credit is concerned.

3. Correlation vs. Contagion

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Stock Markets Correlation

Stock Markets (China, Europe, USA)

1000

2000

3000

4000

5000

6000

01/01/2008 08/12/2008 15/11/2009 23/10/2010 30/09/2011600

800

1000

1200

1400

1600SHCOMPSX5ESPX

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Credit Markets Correlation

Credit Default Swap

0

300

600

900

1200

01/01/2008 08/12/2008 15/11/2009 23/10/2010 30/09/2011

GERMANY

ITALY

Morgan Stanley

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Stock & FX Correlation

Stock & FX Correlation

1500

2000

2500

3000

3500

4000

4500

01/01/2008 08/12/2008 15/11/2009 23/10/2010 30/09/20111.1

1.2

1.3

1.4

1.5

1.6

1.7

SX5EEURUSD

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Credit & FX Correlation

Credit & FX Correlation

0

100

200

300

400

500

01/01/2008 08/12/2008 15/11/2009 23/10/2010 30/09/20111.1

1.3

1.5

1.7

ITALY

EURUSD

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UK - CDS vs. FX (Credit Crunch)

During Crunch: UK (FX vs CDS)

0

40

80

120

160

200

08/08/2008 03/11/2008 29/01/2009 26/04/2009 22/07/20091.2

1.4

1.6

1.8

2.0CDSGBPUSD

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What Happened In 2008?

Collateralized Debt Obligation (CDO): a pool of assets, with defined tranches of credit loss;

Each tranche can be sold to investors to suit their risk appetites;

The pool as a whole has no correlation risk, but pricing any single tranche CDO (STCDO) involves pricing in correlation risks;

One needs to model joint default risks (correlation);

Copula was used to model the joint default risks.

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Sub-Prime Distribution Mechanism

All elements were in place! Ready to go!

RMBS

(Sub-Prime)CDO

Tranches

Domestic Investors

Banks

(Pricing Model)

Foreign Investors

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CDO Lessons

There was a false sense of security – “we can now value CDOs”. The world had since sold far too much of them!

Simply because ones can value CDOs, it does NOT mean they can practically risk manage CDOs:

Hedging difficulties;

Over leveraging problems;

Correlation assumptions;

Liquidity issues;

Transparency & visibility issues;

Etc.

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What Is Happening Now (2011/2012)?

Instead of sub-prime mortgages in USA, it’s now sovereign debts in Europe;

Very high Debt to GDP Ratio: Greece (143%), Italy (119%), etc;

In addition, GDP does not grow but decline;

Faced with huge contagion risks, let’s hope politians find answers very very soon;

In fact, the contagion risks are surprisingly similar to basket correlation risks!

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Basket Options* Analogy

Basket correlation risks:

Cross Gamma: one underlying can affect the others;

Correlation risk: co-dependence can change;

Correlation skew risk: when markets fall, correlation shoots up;

Credit default risk: basket underlying can go bust;

* Qu, D. (2005), “Pricing Basket Options With Skew”, Wilmott Magazine, July

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Contagion Risks

Contagion Risks: versus basket correlation risks:

Cross Gamma: one country can affect the others;

Correlation risk: co-dependence can change;

Correlation skew risk: when in crisis, countries’ co-dependency becomes much stronger, hence increases chance of “getting worse together”;

Credit default risk: a country can default.

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Basket Correlation Contagion

The World Is Like A Correlated Basket !!

Questions?