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 options
Digital payoff
Forwards
Averaged 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 options
Range accrual
Target-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
K
t
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 options
Lookbacks
Asian average options
Trans-Atlantic options
Holder Extendible options
Knock-out and Knock-in barrier options
Multiple window barrier options
One-touch/No-touch options
Best/Worst options
Basket options
Beta Basket options.
Two-asset digitals
Two-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.
B
0)(
))()((PutVanilla
0)(
))()((CallVanilla
1
102
1
210
<−−=
∂
∂
=∆
−−−=
>=
∂
∂
=∆
−=
−
−
dN
S
Q
dNeSdKNeprice
dN
S
Q
dKNdNeSeprice
P
P
rTrT
C
C
rTrT

Currency prices follow stochastic processes:
i
dZ
i
St
i
dt
i
S
ii
dS )(σµ += i=1,2,…..N
j
dZ
i
dZ
ij
=ρ
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
stockprice,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
optionscall102Portfolio
2Portfolio2Portfolio1..25.0..8112...Portfolio2Portfolio1:wantWe
−∆==
===∆∆=−∆= andsoei
5.02
4
1
10priceoptionCall =−×=

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
SM
pm
pL
pu
S
Time
S
SdZtSdtqrdS )()( σ+−=

Using Monte-Carlo Simulations:
]|]0,[max[Pr 0SKSEeAmountemium T
Trd
−×= −
Simulated future carbon prices
1
10
100
1000
10000
2008 2012 2016 2020 2024 2028 2032 2036 2040 2044 2048
Year
Carbonprice
dtdZdZEtdZtdtttSd ijjiiii
ii
t ρσσµ =+−= ][);()()](5.0)([ln 2)()(
]ˆ)ˆˆexp[( 2
2
1)()(
iiii
i
t
i
tt ZttXX δσδσµδ +−=+

Finite Difference, Element, Volume Methods
0),()(
),(
])()([
),(
2
),(),(
2
222
=−
∂
∂
−+
∂
∂
+
∂
∂
tSVtr
S
tSV
Stqtr
S
tSVStS
t
tSV σ
S0
S

An Exotic Option: two-asset options
• 2 asset Black-Scholes equation:
• Payoff function
)max( 2,1 SSPayoff =
S2
∂
∂
σ
∂
∂
σ
∂
∂
ρσ σ
∂
∂ ∂
µ
∂
∂
µ
∂
∂
V
t
S
V
S
S
V
S
S S
V
S S
S
V
S
S
V
S
rV+ + + + + − =
1
2
1
2
01
2 2
1
2
1
2 2
2 2
2
2
2
2 1 2 1 2
2
1 2
1 1
1
2 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 Π−=∆•−∆Π−= T
loss
f
}),({min)()( αξξ ξαα
≥Ψ=≡ ∈
XXX R
VaR
][)( tail lossfECVaR −=≡ ααφ 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
∂
∂
=∆
.
2
0
Pr2
100
0
S
emiumS
∂
∂
=Γ
;
Pr
100
1
σ∂
∂
=
emium
v
);,,,,,
0
(Pr),,,
365
1,,
365/
0
(Pr µσµσµθ rTKSemiumrTKeSemium −−=
R
emium
∂
∂
=
Pr
100
1ρ

Example: Window No-Touch Option
FENICS FX Pricing Page:

Example: Window No-Touch Option

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
Frequency

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
Frequency

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
Frequency

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
functionyvolatilitlocal),(
).(),()]()([/
tS
tdWtSdttqtrSdS tt
σ
σ+−=

Stochastic volatility model
dtdZdWE
dZVdtVdV
dWSVdtSqrdS
tt
tttt
ttttt
ρ
ξθα
=
+−=
+−=
][
)(
)(

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.

Options for Managing Foreign Exchange Risk

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