options for managing foreign exchange risk

Options for Managing Foreign Exchange

Dr Zili Zhu

Quantitative Risk Management

Mathematics, Informatics & Statistics

26th March 2010

CSIRO Mathematics, Informatics & Statistics www.cmis.csiro.au

Background of CSIRO

Organization:• Commonwealth Scientific and Industrial Research Organization

(7200 staff members)• Division of Mathematics, Informatics and Statistics (150 Scientists)• Quantitative Risk Management Group (25 scientists)

Commercial activities• CSIRO Exotic math for FX markets

• Consulting assignments for major banks

• Development of new options models for hedge funds.

• Development of major risk-management software.

• Rea-options valuation in energy industries.


Content

An introduction to common derivative products in FX

Understanding the key components of pricing derivatives.

How reliable are the pricing models given recent and excessive volatility

Other risk valuation methods


Financial Derivatives

Exchange markets: standardised Futures, swaps and options are actively traded on exchanges.

Over-the-counter (OTC) market: forwards, exotic options are traded directly among institutions and outside of exchanges.

Derivative – financial instrument whose value depends on other more basic variables (stocks, futures, FXs, interest rates), e.g. Vanilla call/put options on traded shares.

Simple standard derivatives:

Call/Put vanilla optionsDigital payoffForwardsAveraged rate.....


Multi-leg structure

• Zero cost• Tailor-made risk profile• Multiple expiries • Flexible



Some exotic options used in FX

Window barrier options (KO, KI, Touches, Digital)Basket optionsRange accrualTarget-redemption notes

B


Example: Reverse Knockout Call

Up and Out Call Payoff is:

V(S,T) = (S – K) if S < B V(S,T) = 0 if S B Barrier is: V(S,t) = 0 if S B

t

B

Kt

S

K

B

CSIRO Mathematical & Information Sciences www.cmis.csiro.au

A Typical Exotic Option: Two-Asset No-Touch

FENICS FX Pricing Page:


Other Exotic Options

Compound Call/Put Quanto optionsLookbacksAsian average optionsTrans-Atlantic options Holder Extendible optionsKnock-out and Knock-in barrier optionsMultiple window barrier optionsOne-touch/No-touch optionsBest/Worst optionsBasket optionsBeta Basket options.Two-asset digitalsTwo-asset Knock-out/in options…….


Exotic option: Beta Basket


How to Price Derivatives in FX

The price of a derivative should be the hedging cost of the derivative over its life cycle.

Financial mathematics is well established. Option-pricing formula and numerical methods are available.Industry conventions need to be considered.

B0)(

))()((Put Vanilla

0)(

))()(( Call Vanilla

1

102

1

210

dNS

Q

dNeSdKNeprice

dNS

Q

dKNdNeSeprice

PP

rTrT

CC

rTrT


Currency prices follow stochastic processes:

idZ

iSt

idt

iS

iidS )( i=1,2,…..N

jdZ

idZ

ij

Methodologies for Pricing Derivatives

$0.6

$0.7

$0.8

$0.9

$1.0

$1.1

$1.2

$1.3

$1.4

$1.5

0 0.2 0.4 0.6 0.8 1

time

stoc

k pr

ice,

S(t

)


Example of Pricing a Call Option – delta hedging

Portfolio= S0Δ – call option

Stock price, S0 = $10

Strike=$11

Stock price, ST = $12

Option price = $1

Portfolio1 = 12Δ - 1

Stock price, ST = $8

Option price = $0

Portfolio2 = 8Δ - 0Time period T

options call102Portfolio

2Portfolio2Portfolio1..25.0..8112 ...Portfolio2Portfolio1 : wantWe

andsoei

5.0241

10priceoption Call


Tree Methods

Trinomial Tree

2222

3

2

3 333

4

3

4 444

5

4

5 555

6

5

6 666

7

6

7 777

8

7

8 888

9

8

9 999

10

9

10 101010

11

10

11 111111

12

11

12 121212

3333

4

3

4 444

5

4

5 555

6

5

6 666

7

6

7 777

8

7

8 888

9

8

9 999

10

9

10 101010

11

10

11 111111

4444

5

4

5 555

6

5

6 666

7

6

7 777

8

7

8 888

9

8

9 999

10

9

10 101010

5555

6

5

6 666

7

6

7 777

8

7

8 888

9

8

9 999

6666

7

6

7 777

8

7

8 888

777

4.7050653

2.9506730

1.8504464

7.5025729

11.9634046

19.0765291

30.4189297

48.5052222

77.3451467

123.3325289

196.6627944

313.5932998

500.0475965

7.5018801

4.7046308

2.9504005

11.9622999

19.0747677

30.4161209

48.5007434

77.3380049

123.3211408

196.6446352

313.5643436

11.9611854

7.5011811

4.7041925

19.0729904

30.4132870

48.4962245

77.3307992

123.3096507

196.6263135

19.0711974

11.9600610

7.5004760

30.4104279

48.4916655

77.3235295

123.2980587

30.4075437

19.0693887

11.9589266

48.4870664

77.3161960

30.4017000

48.4824273

30.4046344

19.0675642

S

SL

SU

SMpm

pL

pu

S

Time

S

SdZtSdtqrdS )()(


Using Monte-Carlo Simulations:

]|]0,[max[Pr 0SKSEeAmountemium TTrd

1

10

100

1000

10000

2008 2012 2016 2020 2024 2028 2032 2036 2040 2044 2048

Year

Car

bo

n p

rice

dtdZdZEtdZtdtttSd ijjiiiiii

t ][);()()](5.0)([ln 2)()(

]ˆ)ˆˆexp[( 221)()(

iiiii

ti

tt ZttXX


Finite Difference, Element, Volume Methods

0),()(),(

])()([),(

2

),(),(2

222

tSVtr

S

tSVStqtr

S

tSVStS

t

tSV

S0

S


An Exotic Option: two-asset options

• 2 asset Black-Scholes equation:

• Payoff function

)max( 2,1 SSPayoff

S2

V

tS

V

SS

V

SS S

V

S SS

V

SS

V

SrV

1

2

1

201

2 21

2

12 2

2 22

2

22 1 2 1 2

2

1 21 1

12 2

2

S2

S2S1

)0,max( 21 11 KSwSwPayoff

)min( 2,1 SSPayoff


How Reliable are Pricing Models?

All models are constructed under certain assumptions.All models have their limitations.Model implementations can also have their own limitations.Computer code can often have bugs. Market data may not be arbitrage-free.Market data may be inconsistent.Models and pricing functions should have been tested for extreme

market conditions.On-going updates and maintenance are needed.Market is evolving, and models should too.


Practical Issues in Pricing Derivatives

Volatility is not constant, vol skew/smile exists.

Correlation is dependent on ATM price.

Correlation should be dependent on strike levels?

How to price basket options with skew.

How much correction is needed to get market price?

Compromise between speed and accuracy.


Volatility Smile/Skew

),()0(),( 0 VXVXVX T

lossf

}),({min)()(

XXXR

VaR

][)( tail lossfECVaR X


Hedging Principles

• Hedging to eliminate risk due to market movements in asset prices, volatility, interest-rates and correlations.

• The cost of hedging reflects the premium received from clients.

• Limit large down-side risk to P/L.

• Trading in derivatives without hedging is speculation.

• The objective of hedging is to protect business from unpredictable market movements on a daily basis.


Greek Hedging: Using Sensitivity Parameters

• Delta:

• Gamma:

• Vega

• Rho

• Time decay

;0

Pr

S

emium

.20

Pr2

1000

S

emiumS

;Pr

1001

emium

v

);,,,,,0

(Pr),,,3651,,365/

0(Pr rTKSemiumrTKeSemium

R

emium

Pr

1001


Example: Window No-Touch Option

FENICS FX Pricing Page:


Example: Greeks of Window No-Touch Option


Greeks: Window No-Touch Option


Delta hedging is automatically set for each individual option through the purchase/sell of underlying assets.

Other greek parameters such as gamma, vega, rho are balanced through the purchase/sell of vanilla and/or more liquid exotic options at portfolio level.

For options with discontinuous risk profiles or path-dependency (e.g. barrier options), hedging is difficult.

Portfolio Approach


Loss Distribution without Hedges

Target portfolio loss distribution

0

10

20

30

40

50

60

70

80

90

100

-6 -3.5 -1 1.5 4 6.5 9 11.5 14 16.5

Loss

Fre

qu

en

cy


A Greek Delta-Gamma Hedge To Reduce Risk

Delta-gamma hedge

0

100

200

300

400

500

600

-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2

Loss

Fre

qu

en

cy


Hedging Strategy

• Risk can only be reduced but not eliminated via hedging through greeks even if the Black-Scholes model is appropriate.

• Hedging through greeks is model dependent.• For commodities and energies (e.g. electricity), model

dependency can make hedging ineffective.


Hedging Through CVaR Minimisation

CVaR-minimising hedge

0

100

200

300

400

500

-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2

Loss

Fre

qu

en

cy

CSIRO Mathematics, Informatics &Statistics www.cmis.csiro.au

Other risk valuation methods

Implied volatility of Black-Scholes model is used for quoting FX options.

New valuation models are developed and implemented regularly.Every model has its drawbacks, and no model is perfect.Speed, accuracy and robustness need to be considered.


Local volatility surface model

functiony volatilitlocal ),(

).(),()]()([/

tS

tdWtSdttqtrSdS tt


Stochastic volatility model

dtdZdWE

dZVdtVdV

dWSVdtSqrdS

tt

tttt

ttttt

][

)(

)(


Summary

Introduction of derivatives in the FX market. A large number of options are available to accommodate

specific risk appetites and market views of end-users. The hedging of options can be implemented as part of a

structure. Full understanding of down-side risk of options is paramount

before trading. Introduced key concepts in pricing derivatives in the FX

market, and different pricing methods are available. All models have limitations. Implementation also has

limitations. Market data can be problematic. New and sophisticated models are created regularly. No

model is perfect.


Acknowledgments

• Thanks to FENICS FX, the global standard in FX options pricing and analysis, for the use of their trading system. The screenshots of pricing pages and market data pages in this presentation are from FENICS FX.

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