FX Options Fundamentals

FX Options FundamentalsJanuary 2006

Jean-Marc Servat

Director - FX Structuring - London

2CitiFX Structuring

Options Fundamentals: Agenda

Why Use Options?

Definition and Payout Profiles

Modelling Assets, Trees, Volatility

Value of Options: Time Value and Intrinsic Value, Drivers of Option Prices

Risk Management: Delta Hedging in Practice, Sensitivities, Greeks

Theory behind Delta Hedging

Call-Put parity

Fat tails, Smile and Volatility Surface

Classic Combinations of Vanilla Options

3CitiFX Structuring

Why Use Options?

Hedge a Specific Exposure: non constant, exposed over a range, currency clauses

Express a Precise Market View

Protect Uncertain Exposures from FX fluctuations

Volatile Forecasted Exposures: Business uncertainty

Contingent exposures: bid to award

Hedge Short Option Exposures: global product arbitrage

Minimize VaR

Trade Volatility as an uncorrelated asset class

What instruments are you using? Why?

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Definitions

and

Payout Profiles

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Definition

An option is a financial contract giving the buyer of the derivative, the right, but not the obligation to buy (for Call / Sell for a Put) a specified notional of currency against another one at a specific rate, on a specific date.

The seller of the option has the obligation to transact on the pre-agreed terms, if the owner exercise the option.

A rational buyer will only exercise the option if the action is beneficial

… compare with a Forward: both parties have the obligation to transact

VOCABULARY: Call, Put, Vanilla, Premium, Notional, Nominal, Strike,Underlying, Spot, Expiry, Tenor, Exercise,Deliver, Write

6CitiFX Structuring

Analogy: a Call Option on Equities

1-year 50.25 Call 1000 Shares of XYZ Inc.

The right to BUY: Call (vs: SELL = Put)

An Asset: the Underlying

In a pre-agreed amount: the notional

At a specific price: the strike (here is USD)

On (*) a pre-defined date: the expiry

on

(*) for European OptionsOn or Before = American optionsA Call on a Share is a Put on USD…

It is the right to exchange 2 Assets at a pre-defined rate, on a given date

7CitiFX Structuring

FX Call Option

1-year 1.2000 EUR Call 10 000 000 EUR

The right to BUY: Call (vs: SELL = Put)

An Asset: the Underlying

In a pre-agreed amount: the notional

At a specific price: the strike in USD per EUR

On (*) a pre-defined date: the expiry

on

(*) for European OptionsOn or Before = American options

It is the right to exchange 2 Assets at a pre-defined rate, on a given date

A Call on a EUR is a Put on USD…

(USD Put)

8CitiFX Structuring

Understanding the Definition

Strike

Spot

Premium(paid)

0

P&LAt expiryIn term

ccy

Long Option= the rightto transact

“The right, but not the obligation to buy, at a specific strike, on a specific date”Option holder will exercise at expiry if it is optimal.

OUT OFthe

Money

INthe

Money

ATthe

Money

9CitiFX Structuring

An Option is a “non linear instrument”

Call Option

Forw ard

ValueNow

0Spot

Strike

The Value (P&L) of a forward is a linear function of SpotThe Value of the option is a curve.

Value is expressed in term ccyEg: USD in EURUSD

What brings the Value UP when spot is below strike?

•the right to walk out of the forward (definition)

•the long gamma position…

10CitiFX Structuring

Payout Profiles of Elementary Options (“Vanillas”)

Long Put

Spot

P&LAt expiry

0

Strike

Long Call

SpotPremium

(paid)

0

P&LAt expiryIn term

ccyLong Option= the rightto transact

P&LAt expiry

Short Put

Spot0

Short Call

Spot

Premium(received)

0

P&LAt expiryIn term

ccy

Short Option= the obligation

to transactupon request

VOCABULARY: At-the-Money (F,S), In-the-Money, Out-of-the-Money


Premium of an Option

Premium in

Nominal in

% EUR (basis points) USD pips per EUR

EUR pips per USD % USD (basis points)USD

EUR USD

EUR USD

EUR

Multiply by Spot

Divideby Strike

ExampleSpot: 1.2024Strike: 1.2084 ATMF 3MEUR Nominal 10,000,000USD Nominal 12,084,000EUR Premium 181,154USD Premium 218,906

Premium in

Nominal in

1.8120% 0.0218

0.0150 1.803%

USD

EUR

USD

EUR


Modelling Asset Prices

Trees

Volatility


Random Walk

Ball can go left or right at each point with equal probability

Geometric Brownian Motion = movement of elementary particles

A particle position is unknown but expected…

Forward


Volatility Definition

Low VolatilityEUR CHF

Moderate VolatilityEUR USD

High VolatilityUSD ZAR

Volatility =Annualized Standard Deviation of Log-Returns

Realized Volatility = measure of past volatility of spot over a period

Implied Volatility = price quoted by market makers for options. Should be the expected volatility over the life of the option


1.00

1.05

1.10

1.15

1.20

1.25

1.30

1.35

1.40

Au

g-0

3

Se

p-0

3

Oct-

03

No

v-0

3

De

c-0

3

Ja

n-0

4

Fe

b-0

4

Ma

r-0

4

Ap

r-0

4

Ma

y-0

4

Ju

n-0

4

Ju

l-0

4

Au

g-0

4

Se

p-0

4

Oct-

04

No

v-0

4

De

c-0

4

Ja

n-0

5

Fe

b-0

5

Ma

r-0

5

Ap

r-0

5

Ma

y-0

5

Ju

n-0

5

Ju

l-0

5

Au

g-0

5

Se

p-0

5

Oct-

05

No

v-0

5

De

c-0

5

Ja

n-0

6

Fe

b-0

6

Spot

FWD

High Band 90%

Low Band 10%

Implied and Historical Volatility

6m HistoricalVolatility

6m ImpliedVolatility

Today


Properties of the Normal Distribution

+/- 1

Forward

68%

95%

+/- 2.6

99%

+/- 1.96

What is the expected trading range, given an implied volatility?

Note: 2-tail test vs VaR 1-tail test

Key concept for Value-At-Risk (VaR)


50% (*)

50%

Option Pricing – Binomial Trees

An intuitive view about how to price options

P&L of a 100 Strike Call

107.50

110.00

112.50

92.50

90.00

87.50

97.50

100.00

105.00

95.00

107.50 107.50

105.00 105.00

100.00100.00

97.50 97.50

92.5092.50

110.00110.00

102.50102.50

97.5095.00

90.0090.00

112.50

87.50

85.00 85.00

82.50

80.00

115.00115.00

117.50

5.00

0.00

0.00

0.00

0.00

0.00

15.00

10.00

8/256

28/256

8/256

1/256

28/256

56/256

56/256

70/256

102.50

20.00120.00 1/256

After 8 time steps,There are 256 possible paths

Call Strike 100

105.00

95.00

100.00

102.50

97.50

Start 100

Probabilities


Value of Options:

Time and Intrinsic Value

Drivers of Options price


Option Value = Intrinsic Value + Time Value

StrikeSpot

Value

Intrinsic Value

Time Value

Intrinsic Value is given by the Forward Rate and the Strike Price:Calls = Max {0, Forward – Strike}Put = Max {0, Strike – Forward}

Intrinsic value is the value of the option if you were to exercise it today at current forward rate

Option Value = Intrinsic Value + Time Value


Drivers of FX Option Prices

InterestRates

Spot

Time toExpiry

Strike Volatility

PricingModel

Theoretical Value

Delta Vega

ThetaRhoTho

21 dKedSeCall rTqT

T

TqrKS

d

)

2(ln

2

1

Tdd 12

K S

Tr, q

S = current Spot Rate (1 Foreign = S Domestic)K = Strike Rater = domestic continuously compounded risk free interest rateq = foreign continuously compounded risk free interest rateT = time in years till expiryN() = standard normal cumulative distribution function = implied volatility

Black –Scholes Formula for a Call:


Drivers of FX Option Prices

Driver Call Put

Spot (S)

Tenor (T)

Implied Volatility ()

IR Foreign (q)

IR Domestic (r)

Strike (K)

What happens to option price

if drivers increase in

value?

)()( 21 dKNdFNeCall rT T

T

K

F

d

2

)ln(2

1

Td

T

T

K

F

d

1

2

22

)ln(

Black and Scholes formula re-arranged with the Forward


Risk Management

Delta-Hedging in Practice

Sensitivities

Greeks


Definition of Greeks

Option positions give rise to various risks:

Delta : change in option price for changes in underlying spot

Gamma: rate of change of delta with spot

Theta: change in price of an option for a unit change in time (time decay)

Vega: change in option price given a one percent increase in implied volatility

These are called the Greeks.

Option trading is managing these risks.


Spot Sensitivity: P&L and Delta

3 months to Expiry

2 months to Expiry

1 month to Expiry

At-Expiry

3 months to Expiry

2 months to Expiry

1 month to Expiry

At-Expiry

P&L= Value relative to Spot

As time goes by, time value decays and curves gets close to “at expiry”

Delta = change of Value relative to Change in Spot = First derivative

Delta is positive between 0% and 100%(for a Call)

P&L

Delta

Spot

Spot


More about Delta

Delta represents the equivalent underlying position that would give the same P&L as the derivative for a small move in spot.

If you hold the derivative and take the opposite delta position, you isolate yourself from spot risk for short moves

Option Position Breakeven Graph Net Cash Position On Delta Hedge

Long Call Long Sell Outright

Long Put Short Buy Outright

Short Call Short Buy Outright

Short Put Long Sell Outright


Practicing Delta-Hedging

Let’s use Fox Online, mid spot, swaps and vol

Long a 1 month 50-delta GBP Call (OEC code), take note of premium, check Risk Graph - Plot P&L

Hedge with Forward transaction (FX code), Check Risk Graph, Plot P&L

Then move spot up, check the delta, delta hedge (sell GBP) at new forward

Move spot back to initial level, what is the delta? What is the new value of portfolio?, Why? Delta hedge again.

Move spot down, check delta, delta hedge (buy GBP)

Move spot back to initial level, what is the delta? What is the new value of portfolio?, Why?


Dealing Options and Delta

2 ways to deal options

Live price

Exchanging delta on a Spot reference or Vol Price

When fixing a Spot Ref, Price will be valid for a small change in spot, if vol does not change: If spot moves slightly, the slight change of value of the option will be compensated by the change of value of the spot exchange

Example: GBPUSD, spot ref 1.7680, Buying 10 Mio GBP a 1.7900 GBP Call 1-month

The delta is 30%, Price is 87 USD pips, Premium = 87 000 USD

When the trade is done, 5 minutes later , spot is at 1.7700

New price of option is USD 93 000

BUT Buyer still pays 87 000 USD AND sells 3 Mio GBP at 1.7680.

Buyer has lost 3 Mio GBP x 20 USD pips = 6 000 USD on the delta hedge

In practice, it is equivalent!


3 months to Expiry

2 months to Expiry

1 month to Expiry

At-Expiry

Spot Sensitivity: Delta and Gamma

3 months to Expiry

2 months to Expiry

1 month to Expiry

At-Expiry

Delta = change of Value relative to Change in Spot = First derivative

Delta is between 0% and 100% (for a Call)

Gamma = change of Delta relative to Change in Spot = Second derivative

Gamma options = short dated

Gamma is what makes an option a non linear instrument

Implied Vol is the price of an option = the price of gamma

Spot

Spot

Delta

Gamma


Other Risk Parameters: Vega and Theta

3 months to Expiry

2 months to Expiry

1 month to Expiry

At-Expiry

Vega = change of Value relative to Change in Implied Volatility

Vega is maximal at the money

Vega options = long dated

3 months to Expiry

2 months to Expiry

1 month to Expiry

At-Expiry

Theta = Time decay

Daily change of Value by tomorrow

Theta is maximum at the money

and for short dated options

Negative Theta (time decay) is the trade off of a long gamma position

One pays premium for the right to choose

SpotSpot

Vega

Theta


+ Call + Put - Call - Put

Delta + - - +Gamma + + - -Vega + + - -Theta - - + +

Greeks of Calls and Put, Long and Short

What is the best way to get long? Buying a Call or Selling a Put?Compare Gamma and Vega..Compare Gamma and Theta…


The Theory

Behind Delta-Hedging


Theory - Discounting with the Appropriate Rate

Theory : Investors expect higher returns when taking risk, ie expected return is higher when return is volatile

For a given, uncertain asset, at which rate should we discount cash flows ?

Example : Risk Free Rate of Return, Rf = 5%

Asset 1certain CF of 105 in 1 year

Asset 2 50% chance of 210, 50% = 0 in 1 year

T1Now 1 year

105PV = 100

Discountingat Rf=5%

T1Now 1 year

ExpectedValue105

PV < 100 Discountingat R = ? > 5%

210

0

P = 50%

P = 50%

A rational investor will require more than 5% return if there is volatility Asset 2 has a lower Present Value than Asset 1 1.

Volatility


Theory - Risk Neutral Valuation

Valuing the option :

estimating the Ra risk–adjusted rate,

or

creating a risk free portfolio (option with delta-hedge) and express the value of the option as the Rf discounted value of the expected value under Risk Neutral probabilities

Which World ? Investors’ attitude Discount Rate Probabilities

Risk Averse (real world)

Require compensation for

Risk

Ra

(adjusted by risk premium)

P1, P2...

True probabilities

Risk Neutral Indifferent to RiskRf

(Risk free)

P´1, P´2…

Risk adjusted

probabilities


Theory - Risk Neutral Valuation: Why is Delta hedging so important?

Now Now + T

S0 u

S0 d

S0

f

fu

fd

So = price of underlying at time 0f = price of derivative at time 0So u = price of underlying at expiry after an up move (u>1)So d = price of underlying at expiry after a down move (d<1)fu = price of derivative at expiry after an up movefd = price of derivative at expiry after a down mover = risk free rate of return

ud fuSfdS 00 dSuS

ff du

00

rTu efuSfS )( 00

rTu efuSSf )( 00

])1([ durT fppfef

du

dep

rT

This portfolio has the same value in the 2 states of the world : It is risk neutral This is the amount of underlying needed

to neutralize the derivative

Let’s create a portfolio made of the derivative and an amount of underlying, so that the portfolio future value has no uncertainty

Because there is no uncertainty,we can discount at the risk free rate of return

We deduct the current price of the derivative

We can simplify to where


Theory - Risk Neutral Valuation

f is the expected value under a probability p of the future pay-offs of the option, discounted at the risk free rate of return

p and (1-p) are called the risk neutral probabilities

p does not depend of the real world probabilities of a move up or down

Counter-intuitive!

Because these probabilities are already in the spot price!

])1([ durT fppfef

du

dep

rT

where

dSpupS 00 )1( The expected value of the underlying is

when we replace p by its value and simplify, we get rT

T eSSE 0)(

Provides the base of valuation methods

Analytical formulas (B&S)

Trees / Lattices / Finite Difference


Put-Call Parity


Put-Call Parity

With same strike: Call - Put = Forward

0

P&LAt expiryIn term

ccy

Long CallLong Call

Short Put

Example:

Long 1-Month EUR Call 1.20 on 10 Mio

=

Long 10 Mio EUR at 1-Month Forward rate 1.20

+

Long 1-Month EUR Put on 10 Mio EUR

Long Call

Short Put

Forward


Put-Call Parity in Practice

Spot is 1.2120. A 1-month EUR Call 1.2325 is a 30 delta. What is the delta of a 1.2325 Put?

The Call trades at 9.3% vol. What is the volatility price of the Put, same strike?

Spot is 1.2120. 1-month Forward is 1.2145. The 1.2325 is worth 63 USD pips. What is the value of the 1.2325 Put?

What are the Time and Intrinsic Value of these Calls and Puts?


Smile,Strangle,Risk Reversal and Volatility Curve


The Black & Scholes Pricing Model assumes

Constant Volatility in time

That Currency returns are normally distributed

However,

Volatility is not constant

Empirically, currency returns exhibit leptokurtosis (peaky distribution with fat tails)

Nevertheless the 1973 formula is still used today!

Black & Scholes Model Imperfections


When markets don’t follow a Normal Distribution

USD JPY October 1998

Normal distribution says that a once every 20 000 years, there will be a 5 standard deviation move…ie: 7% for a 48 hours trading period.

90

100

110

120

130

140

150

Oct

-95

Dec

-95

Feb

-96

Apr

-96

Jun-

96

Aug

-96

Oct

-96

Dec

-96

Feb

-97

Apr

-97

Jun-

97

Aug

-97

Oct

-97

Dec

-97

Feb

-98

Apr

-98

Jun-

98

Aug

-98

Oct

-98

Dec

-98

Feb

-99

Apr

-99

14% move in 48 hours

11% move in 2 weeks


The Option Market has adapted by

Keeping the same pricing formula for Vanilla Option

Each strike (delta) is priced with a specific volatility

Adjusting higher the price of OTM options

Differentiating the vol price of OTM Puts and Calls

This is known as “Volatility Smile”

Determinants of the Vol Smile are

Risk Reversals

Strangles

Market Adjusted…


Strangles

They measure the vol price difference between ATM options and OTM options

25 and 10 Delta have become the market reference

Per convention:

This corrects for the “fat tail” issue

ATM25Delta25Delta

25Delta Vol2

PutCallStrangle

ATM10Delta10Delta

10Delta Vol2

PutCallStrangle

Strangles: the Price of Tail Events


Effect of Strangles on Smile

Volatility

Call Delta

B&S Theo

Vo

lati

lity

Pri

ces

0% 100%

Put Delta100% 0%

ATM

Str

ang

le P

rice

sIn

crea

se

Effect of Strangles


Risk Reversals

They measure the vol price difference between OTM Calls and Puts having the same Delta

25 and 10 Delta have become the market reference

Per convention:

This corrects for the fact volatility changes as spot changes

25Delta25Delta25Delta PutCallReversalRisk

10Delta10Delta10Delta PutCallReversalRisk

Risk Reversals: Skewness


Effect of Risk Reversals on Smile

Volatility

Call Delta

B&S Theo

Vo

lati

lity

Pri

ces

0% 100%

Put Delta100% 0%

ATM

OTM Puts >OTM Calls

OTM Calls >OTM Puts

Effect of Risk Reversal


Because Risk Reversals measure the price difference between OTM Calls and Puts, they are often interpreted as the markets directional view.

This is wrong!

They reflect the expected move in implied vol given a move in spot.

The higher the RR’s absolute value, the more sensitive the implied vol to spot movements.

It has been empirically demonstrated that RR don’t have any directional predictive value on the underlying.

Interpreting Risk Reversals


Strangles, Risk Reversals and ATM prices may be used to determine the vol price of individual OTM options:

xDeltaxDeltaATMxDelta ReversalRisk 2

1StrangleVolCall

xDeltaxDeltaATMxDelta ReversalRisk 2

1StrangleVolPut

Pricing OTM Options

5

15

25

35

45

55

65

75

85

95

0.02

2.007.00%

8.00%

9.00%

10.00%

11.00%

12.00%

Delta

Tenor (yrs)

USD-JPY Volatility Surface

Example: ATM 10%, 25D RR +0.5%, 25D Strangle +0.25%

Call (25 Delta) = 10% + 0.25% + ½*0.5% = 10.5%

Put (25 Delta) = 10% + 0.25% - ½*0.5% = 10.0%


Classic Combination

of

Vanilla Options


Strike

StraddleLong Call and Put

same strike

Classic Combinations

StrangleLong Call and Put

different strikes

Collar(Risk Reversal)Long Call, Short Put

different strikes

Call SpreadLong Call lower Strike,Short Call higher strike

- Non directional- Volatile

- Non directional- Very Volatile

Short Straddle SeagullLong Call Spread

Short Put

ButterflyLong StraddleShort Strangle

- Bullish- Adapted to Stop Loss / take Profits

- Bullish with a target- Cheaper than Call

- Non directional- Expects low

volatility

- Bullish with target- Cheaper than Call

Spread

- Non Directional- Volatile within a

range

Ratio Call SpreadLong Call lower Strike,

Short Call higher strike double notional

- Bullish with a maximum target- Cheap, can even generate

premium


Thank you!

CitiFX Structuring

Mark Balascak New York 1-212-723-1113 [email protected] Michele Ghidoni London 44-20-7986-1807 [email protected] Godet London 44-20-7986-9381 [email protected] Stephane Knauf New York 1-212-723-1274 [email protected] Jean-Marc Servat London 44-20-7986-9379 [email protected] Wong Singapore 65-6328-2859 [email protected]

mailto:[email protected]



































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FX Options Fundamentals

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