pricing and risk managing libor exotics in...

Pricing And Risk Managing Libor Exotics in Practice

© Chinatrust 機密文件/Confidential/Draft

Abstract• A class of fixed income products called Libor Exotics has been actively

traded in financial markets. In this talk we first introduce these products in financial terms. We then explain how these products are priced, traded and risk managed in practice. Finally, we describe some of the difficulties we face, on the daily basis, in pricing and risk managing these products.

• To describe how financial engineering is applied in the industry through this specific example.


Summary

• Various products exist to take and manage yield curve risks.• To be able to identify risk and at the same time to manage risk are the

key to success.• Pricing and risk management are based on sound theory and uses

sophisticated mathematic methods.• Combined knowledge and skills of finance, mathematics and

programming are useful for working on the trading floors and at the risk management departments in Capital Market Groups.

• There are many unresolved practical issues for pricing.• Other areas such as Fx and credit are also actively using structured

products to transfer risks . Misuse of derivatives could do harm to economy. Subprime crisis is an example.


Summary

• Derivatives are cost effective tools to transfer risks from one area to another and among different asset classes.

• There are many derivatives actively traded in markets.• Pricing and risk management are based on sound theory and use

sophisticated mathematic methods.• Risk-management mechanism should be implemented to strictly control

and manage risks.• Combined knowledge and skills of finance, mathematics and

programming are useful for working on the trading floors and at the risk management departments in Capital Market Groups.


1. Yield Curve • Future cash flows often unknown and subject to various risks. They can

be estimated by yield curves. A yield curve is known today and will evolve in the future.

( ) ( ) ( ) ( )( ) ( ) 100$100$

.,10:100$0100$ ⎯⎯⎯⎯⎯⎯⎯⎯ ⎯←

<>=⇒> todayTDTatTD

stifSDtDDtD

( ) ( )( ) ( ) ( ) .,0,ln ttyetDtt

tDty −=>−=


1. Introduction

• USD Discount curve of March 3rd, 2008.

T y(T) D(T)0.00 2.85% 1.00000.26 2.85% 0.99270.35 2.77% 0.99050.42 2.69% 0.98860.51 2.73% 0.98620.76 2.56% 0.98081.01 2.46% 0.97552.01 2.47% 0.95163.01 2.76% 0.92034.01 3.07% 0.88415.01 3.36% 0.84536.01 3.61% 0.80497.01 3.83% 0.76448.02 4.02% 0.72479.01 4.18% 0.6861

10.01 4.32% 0.649212.01 4.54% 0.579615.02 4.77% 0.488520.02 4.96% 0.370225.03 5.02% 0.284330.02 5.04% 0.2199

Y ie ld V s. T e n o r

0.00%

1.00%

2.00%

3.00%

4.00%

5.00%

6.00%

0.0 0.3 0.3 0.4 0.5 0.8 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 12.0 15.0 20.0 25.0 30.0

Te nor


2. Callable Swap

• 5y semi-annual pay fixed swap: 000,000,100=N

[ ]{ }( )

( ) .

;

;,,

;:

;1,,2,1,0,,:;1,,2,1,0,5.0

;50

1,

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10

1

1

110

1+

Δ

+Δ

+

−=

+

+

−

⎯⎯⎯⎯ ⎯←

⎯⎯⎯ →⎯

==

−=−==−=Δ

=<<<<≤

+i

NTTTL

iNTK

ii

nii

ii

iii

nn

TatBPartyAParty

TatBPartyAParty

rateLibororfloatingTTLratefixedK

TDateSettlement

niTTPeriodniyTTT

yTTTT

iii

i

L

L

L

© Chinatrust 機密文件/Confidential/Draft 8

2. Callable Swap

Party A Party B

• Party A has the right but not the obligation to cancel the trade at every payment date after initial six-month lockout period.

TK*0.5*N

*0.5*N~

*0.5*N

5 year

N=$100,000,000

Party A Has the right to cancel.

( )1, +ii TTL( )1, +ii TTf


2. Callable Swap

• Discounted swap random value

• Non-arbitrage price

( ) ( ) ( )

( )[ ] ( ).,

,

1

1

01

1

01

1

011

+

−

=+

−

=+

−

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ii

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−

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Δ−=

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∑

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ii

n

iii

n

iii

n

iiiii

TNDTKTTf

TNDTKTNDTTTfV


2. Callable Swap• Forward rates are expectations of Libor rates.

• Par swap rate.

( ) ( )

( )

( ) ( ) ( )

( )( )( )

.1

;,1

1;0

;0

11,

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1,1

1

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iTD

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TTDTD

TTf

TTfTTDTD

T

TTi

iiii

Δ

−=

Δ+=

−−−−−−−−−−⎯⎯ ⎯←

⎯⎯⎯⎯ ⎯←⎯⎯ ⎯←

++

++

+

+Δ+

+

+

( ) ( )

( ).

,

1

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1

011

∑

∑−

=+

−

=++

Δ

Δ= n

iii

n

iiiii

TNDT

TNDTTTfS


2. Callable Swap• USD par swap rates for March 3rd, 2008. The data were used to

bootstrap the above yield curve.

Data Type Tenor RateDeposit Rate 3M 2.82%Deposit Rate 4M 2.75%Deposit Rate 5M 2.67%Deposit Rate 6M 2.71%Deposit Rate 9M 2.54%Swap Rate 1Y 2.46%Swap Rate 2Y 2.47%Swap Rate 3Y 2.75%Swap Rate 4Y 3.06%Swap Rate 5Y 3.33%Swap Rate 6Y 3.57%Swap Rate 7Y 3.78%Swap Rate 8Y 3.95%Swap Rate 9Y 4.09%Swap Rate 10Y 4.21%Swap Rate 12Y 4.41%Swap Rate 15Y 4.60%Swap Rate 20Y 4.76%Swap Rate 25Y 4.82%Swap Rate 30Y 4.85%

S wa p R a t e V s. T e n o r

0.00%

1.00%

2.00%

3.00%

4.00%

5.00%

6.00%

1Y 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y 10Y 12Y 15Y 20Y 25Y 30Y

Te nor


2. Callable Swap• Bermudan swaption: Can exercise at one of the time to obtain the

remaining swap from to .

• European swaption: Time payoff:

( ) { } 111

−==

⊂ nii

k

jn TTj

jnT nT

.1=k iT

( ) ( ) ( )

( ) ( ) ( ) .0,,max

0,,max

1

1

1

11

1

1

1

11

⎟⎟⎠

⎞⎜⎜⎝

⎛Δ−Δ≠

⎟⎟⎠

⎞⎜⎜⎝

⎛Δ−Δ

∑∑

∑∑−

=+

−

=++

−

=+

−

=++

n

ijjj

n

ijjjjj

n

ijjj

n

ijjjjj

TNDTKTNDTTTf

TNDTKTNDTTTL


2. Callable Swap

• At-the-Money European swaption and Cap market volatilities: Prices to volatilities via Black Formula.

• USD European swaption volatilities of March 3rd, 2008. The swaption which expires in 3y with an underlying of 7y swap has an annual volatility of 23.6%.TTE/Term 1Y 2Y 3Y 4Y 5Y 7Y 10Y1M 60.7% 62.2% 58.7% 54.5% 51.3% 43.4% 36.2%3M 59.7% 61.6% 56.5% 51.9% 48.5% 41.1% 34.5%6M 60.8% 56.4% 51.1% 46.8% 43.7% 37.4% 31.8%1Y 53.7% 47.4% 42.7% 39.4% 37.0% 32.4% 28.4%2Y 39.1% 35.7% 32.9% 30.9% 29.4% 26.8% 24.2%3Y 31.2% 29.1% 27.5% 26.1% 25.2% 23.6% 21.9%4Y 26.2% 25.0% 24.0% 23.2% 22.4% 21.3% 20.1%5Y 23.0% 22.1% 21.5% 20.9% 20.4% 19.5% 18.5%7Y 19.6% 19.1% 18.7% 18.3% 17.9% 17.4% 15.8%10Y 16.4% 16.1% 16.1% 15.8% 15.5% 15.3% 14.8%

TTE Volatility1Y 50.7%2Y 51.1%3Y 45.2%4Y 40.1%5Y 35.9%7Y 30.4%10Y 25.6%

( ) ( ) ( )( ) .,

;0,,,max

0

1

0101

rateswapparforwardTTS

TNDTTTSTTL

n

n

jjjnjj

=

Δ⎟⎠⎞⎜

⎝⎛ −∑

−

=++


2. Callable Swap

• Callable Swap: Swap + Bermudan swaption on opposite swap. If the Bermudan swaption is exercised, the newly entered swap can be used to cancel the original one.

• Risks from swap has been partially transferred to volatility. Options are cost effective tools for shifting risks, increasing liquidity.

• Care should be taken. CDOs have played a big role in subprime. Risk-management should be implemented.

Fixed Rate

Float Rate

Float Rate

Fixed Rate


2. Callable Swap• Application: Company A issues a 10y bond with coupon K = 4.2% to support

production activities.

• Hedge bond: A callable bond is the same as a callable swap on market risk part (not including liquidity risks and credit risks). Cash N today is the same as:

• Six month later, K = 3.8% may be good enough. So need callable.

N

N K

Float Rate N


3. Libor Exotic Swaps: Embed risks into fixed side.• Quanto Range Accrual Swap: A bank enters a swap receives TWD Libor

or Tibor. The swap pays a coupon depends on two predetermined levels L = 0% and U = 5% together with USD Libor as follows:

• Investors have views on the levels of USD Libor. They have risk to miss coupons for some periods.

( ) [ ]

( ) .

.;0

,, ;

1,

1

UTTLL

ii

iiK

Otherwise

ULTTLUSDKCoupon

≤≤

+

+=

⎪⎩

⎪⎨

⎧ ∈=

χ


3. Libor Exotic Swaps• Spread Range Accrual Swap: Form an index as the spread of USD 30y

swap rate and USD 10y swap rate.

• Investors have views on the shape of the long end of the yield curve. The correlation between the two is very hard to observe and to hedge.

( ) ( ) ( )

( )

( )

( ) ( ).

.0;0

0;

;

10300

1030

iyiy TSTS

i

i

iyiyi

K

TX

TXKCoupon

TSTSTX

−≤=

⎪⎩

⎪⎨

⎧

<

≥=

−=

χ


3. Libor Exotic Swaps• Dual Range Accrual Swap.

• Spread Option.

( ) ( ) ( )

( ) ( ) [ ]

( ) ( ) ( ) .

.;0

;,,0;

;

11030 ,0

1

1030

UTTLLTSTS

iii

iyiyi

iiiyiyK

Otherwise

ULTTLandTXKCoupon

TSTSTX

≤≤≥−

+

+=

⎪⎩

⎪⎨

⎧ ∈≥=

−=

χχ

( ) ( )( )( ).%0.1,*max 1030 iyiy TSTSMK −=


3. Libor Exotic Swaps• Snowball Swap (path-dependent)

• Many more: Target Redemption Swap, Knock-Out Swap, Snowball Range Accrual Swap, etc.

• The fixed rate in a vanilla swap is unchanged. Most of Libor Exotic Swaps have risky coupons built in to enhance yields. To hedge these risks is also expensive.

• Companies issue structured bonds. One can strip out coupons and avoid credit risks. Structured coupons are packed as Libor Exotic Swaps.

Denote by iK the coupon for the period [ ] 1,,2,1,0,, 1 −=+ niTT ii L . They are de-termined recursively,

( )( )⎩⎨⎧

−=−+==

+− .1,,2,1,0,,%5max%;5

11

0

niTTLKKK

iiii L


4. Pricing, trading and risk-managing• Structure Group: Design structure products to meet investors’ needs.

Some structured products have good market. Others are very client specific. (a) Financial knowledge – a must. (b) To write VBA – a plus. (c) Can explain prices and risks of sophisticated models – an asset.

• Marketing Group (Sales): Explore business opportunities from investorsand other banks. (a) Financial knowledge. (b) Communication skills.(c) Explain upsides and downsides of products and invest strategies to

customers. Understand model basics.• Trading Desks: Allow to hold risk limits. (a) Read markets to find

expectations. (b) Manage book risks through dynamic hedging. (c)Understand technical numbers, volatilities, etc. (d) Understand models.


4. Pricing, trading and risk-managing

• Pricing: Here are a few terms, all of 5y. Note that 5y TWD swap is traded at 2.46% on March 3rd, 2008.

(1) Quanto Range Accrual.

Tenor = 5y Amount = TWD 100 Mio Quarterly pay/fixing CTCB rcvs 3mCP Act/365, Modified Following CTCB pays K if 3mL < 5%, Act/365, Modified Following Upfront = 10 bps, K = ? Quote: K = 2.67%.



• Pricing

(1) Callable Quanto Range Accrual. If add callable feature: Non-call 3m. Upfront = 20 bps, K = ? Quote: K = 3.36%. (2) CMS Spread Range Accrual. CTCB pays K if 30y USD CMS – 10y USD CMS >= 0; Act/365, Modified Following Upfront = 25 bps, K = ? Quote: K = 3.06%.



• Trading: Once a price for a deal is acceptable for both parties, the deal will be done and booked into system until it expires.

• Hedging: The portfolio of above three deals with a 5y swap has a mark-to-market value of $2,332,871 TWD. Risks are:

USD Libor TWD LiborData Type Tenor DeltaCash 3m -116Cash 4m 0Cash 5m 0Cash 6m -150Cash 9m -503Swap 1Y -1,996Swap 2Y -3,722Swap 3Y -3,178Swap 4Y 22,290Swap 5Y 16,454Swap 6Y 297Swap 7Y 358Swap 8Y 408Swap 9Y 469Swap 10Y 11,345Swap 12Y 40,488Swap 15Y 23,617Swap 20Y -362Swap 25Y -511Swap 30Y -65,091

Data Type Tenor DeltaCash 3M 305Cash 6M 1,270Swap 1Y 1,580Swap 2Y 1,814Swap 3Y 1,175Swap 4Y 1,317Swap 5Y 155,054Swap 7Y 0Swap 10Y 0Swap 15Y 0



• Hedging: 1% Vega.USD Swaption VolatilityTTE/Term 1Y 2Y 3Y 4Y 5Y 7Y 10Y

1m 0 0 0 0 0 0 03m 0 0 0 0 0 0 2,7836m 0 0 0 0 0 0 5,8741Y 0 0 0 0 0 0 9,5912Y 0 0 0 0 0 0 4,4873Y 0 0 0 0 0 0 -4,9334Y 0 0 0 0 0 0 -12,6475Y 0 0 0 0 0 0 -6,3907Y 0 0 0 0 0 0 010Y 0 0 0 0 0 0 0TWD Swaption VolatilityTTE/Term 1Y 2Y 3Y 4Y 5Y 7Y 10Y

1m 0 0 0 0 0 0 03m 0 0 0 3,633 12,212 0 06m 0 0 0 9,610 8,135 0 01Y 0 0 1,355 8,312 770 0 02Y 0 514 4,199 1,436 0 0 03Y 207 1,449 544 0 0 0 04Y 631 219 0 0 0 0 05Y 103 0 0 0 0 0 07Y 0 0 0 0 0 0 010Y 0 0 0 0 0 0 0



• Risk managements in a bank exist in many different levels. • Desk level dynamic hedging is essential. Risk limits are the guidelines.

Risk management, back office, procedures, legal, accounting rules are followed. This is to control market risks and operation risks.

• Desk level P&L and risk are calculated every night after closing market data have entered into database. They are calculated by the riskmanagement group for official reports. Middle Office is responsible taking market data and issuing reports.

• Trading desks also do their own independent calculations. They also add various scenarios to reflect desks’ views and expectations of markets.



• Scenario and stress tests are also carried out for the trading floor. Scenarios are modified from time to time to reflect market conditions and the existing positions. Here is an example:

• Business. Analytic and IT.

Variables Scenarioes Shift Domestic Yield Curve -1.00% 0.01%Domestic Yield Curve -0.50% 0.01%Domestic Yield Curve -0.10% 0.01%Domestic Yield Curve -0.01% 0.01%Domestic Yield Curve 0.01% 0.01%Domestic Yield Curve 0.10% 0.01%Domestic Yield Curve 0.50% 0.01%All Risk factors Worst last 20y All Risk Factors Worst last 10y All Risk Factors Worst last 1y



• Value-At-Risk. Design, say, 365 scenarios all around today’s data. Calculate the P&L distribution of the bank portfolio. From the distribution1% worst possible loss will be reported. (a) Positions may come from different asset classes, sub-portfolios and systems. (b) Calculation efficiency should be resolved. For example, Gaussian Quadrature can help.

• Counterparty risks. Positive future cash flows might be lost due to counterparty default. It is preferred to calculate such risks and hedge such risks. Often, spreads are reserved for possible loss. (a)Calculations are difficult, especially one wants to hedge such risks. (b)There are many details such as netting agreement, etc. VaR project requires integrated efforts.


5. Modeling

• Prices of vanilla products are determined by supply and demand.• Models can be regarded as interpolation tolls.

• Validation Group is responsible for testing models of theoretical assumptions, inputs, model calibrations, and implementations. They are called Quantitative Analysts.

( )

.

lim,,

ValueSwapExoticLibor

discretizedPtSModel

Cap

SwaptionEuropean

RateSwap

datamarket→

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

⎟⎠⎞

⎜⎝⎛

∫∫∂∂

⎪⎪⎩

⎪⎪⎨

⎧

=

Ωξ

L


5. Modeling

• General Theory. Modeling is based on a sound theory. Roughly speaking, non-arbitrage in finance is equivalent to martingale in mathematics.

( )

( )( )

.

;

;

;0;,,,,

0 NeutralRiskPAccountMarketMoneyetN

PMeasureMartingaleEquivalentarbitrageNon

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DerivativeSpecialNNumerairePNEconomy

t

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THEORYt

t

−=→−−=∫

=

∃⎯⎯⎯ →←−

Σ→

>→Σ∑Ω→


5. Modeling

• Martingale: Let be the price for a derivative. Then

( )( )

( )( )

( ) ( ) ( )( )

( ) ( )( ) .0

00

;0,

0

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tNtR

P

P

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( )tR

( )( ) ( ) ( )

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,max

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Pt

=

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⎛⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡≥

τ

ττ

τ


5. Modeling

• Single rate modeling: Pricing of Spread Range Accrual Swap.Two swap rates to form the index:

• Numeraire: Money-Market-Account with deterministic discount.

• Rate distributions: Log-normal.

( ){ }21=jij TS

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etN

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.:

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10

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5.0 2

−=

+−

∈

=

==

nii

j

XTTjj

TT

NXXX

jeSTS jjj

ρ

σσ


5. Modeling

• In range probability:

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0

01

01

0

21

21

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TNK

EN

TSTSi

i

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∫∫ ≥−+

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( ) ( )

( )

( )

( )

( )( )

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( ) ( )( )( )[ ]

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;12

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5.05.000ln

,,,

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5.0

25.0

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2

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∫

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∫ ∫∫∫

∞−

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−

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=Φ

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=

=

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yd

TSTS

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yd

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π

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ρ

ρρ


5. Modeling• Term structure modeling. For callable products, one needs to know the

evolution of the entire yield curve. Even for Bermudan swaptions, there are several competing models which provide different estimations.

• Model choice. Randomness follows normal process is at most a very rough approximation. There is no hope to find THE interest rate process. Calibration is the key.

!ions!approximat needs BGM as simple as modelcorrect lTheoretica (5)money. losing hedging zeg-Zig stable. be Should (4)seconds). in swap a Bermudan (pricefast be Should (3)

prices. swaption uropeanrelevant E return Hencemarket. capnot andmarket swaption toCalibrate (2)

curve. yields today' Return)1(


5. Modeling

• Short rate models: Advantage: Relevant quantities can be recovered from short rates. Disadvantage: Not directly traded in markets and hence need complicated calibration numerical procedures.

• Hull-White Model: Popular due to its analytic tractability. BK or BDT model may be more correct in terms of positive rates. But it is very slow to calibrate them to today’s European swaption market.

( )( )

.,∫

=−

T

t

dssr

eTtD

( ) ( ) ( ) ( )( ) ( ) .tdWtdttrtattdr σθ +−=


5. Modeling

• Swap BGM Model. Taking as numeraire. Drift terms determined by non-arbitrage condition on bonds.

( )nTtB ,

( )( )( ) .:

;1,2,1,

nmm

tmm

m

TtoTrateswapofvaluettimetS

nmdWtStdS

−=+= LL σ

( )( )

( )( ) nmt

n

mP

n

m TTstTsBTsBE

TtBTtB

<<<⎥⎦

⎤⎢⎣

⎡Σ= ,

,,

,,

( )( )

( ) ( )( )

( )( ).1,,2,1,

,1

1

1

1

1

1

1,1

−=−=Δ

+Δ+∏

Δ+∏Δ−=

+

−

=+=

−

= +=≠+=

∑

∑ ∑

nmTTT

dWtST

tSTtST

tStdS

mmm

tmmn

mijj

imj

n

mi

i

mkjj

ikjmjkkk

m

m

L

σσσ


5. Modeling

• Libor BGM Model. Specifies Libor rates instead of swap rates. Inputted volatilities are cap volatilities. Approximations are needed to calibrate swaption volatilities.

( )( )

( )( ) .1,,2,1,

1

1

1

−=+Δ+

Δ−= ∑

−

+=

nmdWtFTtFT

tFtdF

tm

n

mi im

iFmii

m

m Lσσσ

( )( ) ( )

( ).

,

,,

1

1

1

11,

∑

∑−

=+

−

=++

Δ

Δ= n

miii

n

miiiiii

m

TtDT

TtDTTtLTtS


5. Modeling

• Early exercise. When tree can be used to approximate the processes, early exercise feature can be handled easily. In the Monte Carlo simulation setting, it becomes difficult. Here is a way to approximate:

• Two distributions are compared at the exercise times:

( ) ( ) ( )( ) ( )

( ) ( ) ( ).,:

,,,

,max:

1

1

1

1

∑−

=+

+

+

Δ−=

⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡Σ=

n

miimmmm

nmTnm

mPmm

TTDTKSTUvalueExercise

TTBTTB

TOETUTOvalueHoldingm

( ) ( ) ( )

( ) ( )⎭⎬⎫

⎩⎨⎧

=+∞−∞∈→ΣΩ

ΣΩ→ΣΩ⎥⎦⎤

⎢⎣⎡ Σ

∑=

M

ii

iit

tTtP

MiatUaPL

PLPLTfE

0

2

22

,,1,0),,(|,,

,,,,:|

L


5. Modeling

• Estimate holding value using Least-Square method. At the last exercise time point, the holding value equals

• Project holding value to a finite dimensional space:

( ) ( ) ( )( ) ( )

( )( ) ( ) .0,0,max

,,

,max

1

111 1

==

⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡Σ=

−

−−− −

nn

nnTnn

nPnn

TOTU

TTBTTB

TOETUTOn

( ) ( ) ( )( )

( ) ( )( ) ( ).,

;,,,

;,,,,

,

1

1

1 1

PTm

TTnm

mP

TTm

m

mm

mm

TU

PTTB

TOE

PPTO

ΣΩ∈

ΣΩ∈⎥⎦

⎤⎢⎣

⎡Σ

ΣΩ⊃ΣΩ∈

+

+

+ +


5. Modeling

• Project the holding value to a finite dimensional space by finding best approximation:

( ) ( ).,,0

PTUaHmT

M

jm

jjm ΣΩ⊂= ∑

=

( ) ( )( ) .

,~

11

1

0⎥⎦

⎤⎢⎣

⎡Σ=

++

+

=∑ mT

nm

mP

M

jm

jjm TTB

TOETUaO

A very important point here is that the exercise value ( )mTU is known at time

mT and hence can be used to form a subspace.


5. Modeling

• Can’t be extended to Libor Exotic Swaps directly: Exercise value is unknown at the exercise time. There is a way of iterating stopping times like finding a fixed point. For example, one starts by a trivial exercise policy of exercising at the first time. One then iterates the process to get a better policy.

( ) ( ) ( )( ) . timestopping:,

,max,00 τ

ττ

τ ⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡=

nPn TB

OETBTO


5. Modeling

• Black approximation: In the risk-neutral space, assume that all relevant swap rates follow log-normal distributions. One can use multi-layer binomial tree to price Libor Exotic Swaps in general and Bermudan swaption in particular.

• Another approximation is to freeze the stochastic drift terms in BGM to their time zero values.

• Not arbitrage-free: Rates are not martingales.

( )( ) ( ) ( ).0

;1,,2,1,25.0

0tWt

m

tmm

mmeStS

nmdWtdSσσ

σ+−=

−== L


6. Issues

• Need practical models to price Libor Exotic Swaps.

• With approximations, tree can be used to price Libor Exotic Swaps. But two factor trees are already slow and three factor trees are impractical.

• How to deal with barrier options in the multi-factor tree setting?

• Is there any adjustment orthogonal to the current pricing theory? Trading structured products has auction element. Can game theory play a role?

• Begin to see advanced mathematic tools. Some of functional analysis methods are coming into valuations.


7. Fx Options

• Options are used to hedge international cash flows. Again, they are instruments to transfer risks. For example, a Risk Reversal is a zero cost instrument which funds upside risks by downside risks. Taking the exchange rate between EUR and USD as an example. Its payoff at the expiry is listed below. The upside risk of stronger EUR and weaker USD is fully protected. On the other hand, there is a downside risk of falling exchange rate. The option holder will lose the call and the other party will exercise the put. Between the two strikes, nothing will happen.

( )( ) ( )( ) .,, lhlh KKUSDEURFTFKKTF >=−−− ++


7. Fx Options

• Other options. Fx options can be roughly divided into two generations. Furthermore, new structures have been added to the list in recent years. Options are cost effective tools for managing risks on the one hand, on the other hand, they may lead to pitfalls if misused.

• The representatives of the first generation exotics are barrier and Asian types. The second generation exotics contain spread, basket, best of and worst of options.

• There are many structured products. Basket linked note is one of them. Let us denote four currency pairs as follows:

( ) ( )( ) ( ) .,

,,

43

21

USDCAD

USDGBP

USDAUD

USDEUR

TXTXTXTX

==

==


7. Fx Options

• Baskets. Form a basket as

• There are many variations of basket forward and basket options. For example, one may choose weights carefully so that the basket of the four currencies matches USD as close as possible in terms of correlation. This is one way to use basket of currencies to hedge USD cash flows and at the same time to avoid a concentration of USD.

( ) ( ) .4,3,2,1,0,4

1

=<= ∑=

iwTXwTB ii

ii

( )( )( ).;;,max,min 21

notionalNstrikesKNKTBK

i ==


8. Credit Derivatives

• CDOs (Collateralized debt obligations). For the current subprime crisis, CDOs have been the main factor behind the massive asset writedownsin recent months.

• CDOs are a portfolio of other assets. The cash flows generated by the portfolio are divided into tranches. Here is an example.

• Equity tranche: Riskiest. AAA tranche: much safe.

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡−

⇒

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

trancheEquitymilliontranchemezzanineBB

trancheseniorratedAmilliontrancheseniorerAAAmillion

backedassetloans

bondsmortgages

million

2$4$

4$sup40$

securities

,

,,

50$


8. Credit derivative• Denote “default time”, “recovery rate”, “notional”, “conditional default

probability”, and “conditional survival probability” by the following notations. L( t ) will be the cumulative loss on the portfolio at time t. We have to find the distribution of L( t ) in order to calculate the expected loss.

{ } { } { } { } { }

( ) ( ) ( )ti

n

ii

ni

Vit

ni

Vit

nii

nii

nii

iNtL

pqN

≤=

=====

∑ −= τχδ

δτ

1

1|

1|

111

1

,,,,


8. Credit Derivative• One way to calculate the distribution of a random variable is to first

find its characteristic function. If we assume that the default times are independent conditional on some k dimensional factor V, then we have the following equations:

( )( ) ( )[ ] ( ) ( )

( )( ) ( ) ( )( )( )

( )( ) ( ) ( )( )( ) ( )dvvfupqu

upqVu

udPeeEu

kji

ji

R

n

jt

Vit

VittL

n

jt

Vit

VittL

tiuLtiuLPtL

∫∏

∏

∫

=≤−

=≤−

Ω

+=

+=

+∞∞−∈==

11

||

11

|||

,,

τδ

τδ

χϕϕ

χϕϕ

ϕ


8. Credit Derivative• CDOs are illiquid. • While marginal default probabilities can be hedged by CDSs which could

be illiquid themselves, correlations, the key element for the product and for pricing, are hard to hedge.

• Tranche levels are determined by rating agencies whose models are very complicated and historical dependent. The default probabilities of AAA tranche were under estimated. Agencies are busy modifying their rating models.

• Investors did not understand the risks built in CDOs, their did not trust rating agencies, they were not comfortable with the CDO prices. All these suspicious went to the next link in the chain. Finally, banks had to absorb losses.

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