delhi client da-i

7/31/2019 Delhi Client DA-I

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FX Options

Aashish Pitale

New Delhi

October 10, 2007

Global Markets, India


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Warm Up

In a Currency Option (say, EUR/USD), a Call is a Putand a Put is a Call?

True

False


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Warm Up

In a EUR/USD Option, if a customer is short EUR Put,at maturity, the customer would:

Buy USD

Sell USD

Buy JPY Depends

Depends, whether option expires in the money Sell USD


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Warm Up

Spot 1.4100, Fwd = 1.4100 (for all tenors) whichoption do you think will be more expensive?

1m 1.4150 EUR Call

3m 1.4150 USD Call

3m 1.4150 EUR Call 9m 1.4150 EUR Call

Cant Say


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Option Basics

Pricing

Volatility

Greeks

Risk Management

Agenda


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Option Basics


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FX Options : History

History & Development

Options became popular contracts in 1930s

Black & Scholes transformed the market by giving usa benchmark pricing theory in 1973

Interbank FX option market launched by 5 banks(including SCB) in 1982

by late 1980s traders had the computing power tolook at exotic options

1990s saw steady increase in liquidity combined withmore sophisticated pricing models: spreads tightenand more exotic products become feasible

FAS133 & IAS39 set back market growth, but nowrestarting again


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What is an Option ?

An option contract confers the right,but not the obligation,

to buy or sell a specific underlying asset


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Basic Option Terminology

OPTION TYPECALLOPTION PUT OPTION

EXERCISE TYPE

EUROPEANAMERICAN

BERMUDAN


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Basic Option Terminology

STRIKE PRICEIN THE MONEY OUT OF THE MONEY

AT THE MONEY

OPTION TYPE

VANILLA

BARRIER

DIGITAL


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Call Option P&L

Pay Off = Max( 0 , S X)

Pay Off

0

Spot, SStrike, X

P & L

0

Spot, S

Premium, C

Strike, X

OTM : S < XATMS : S = X

ITM : S > XATMF : F = X


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Option Vs Forward

Right to buy/sellObligation to buy/sellOptionForward

Upfront premiumNo premium

Payoff

0Spot

PREMIUM

Payoff

0Spot

S0


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Intrinsic and Time Value

OptionValue

FX RateStrike

Intrinsic value

Time value

Delta = slope of option

value line

Out of Money In the Money

ATM


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Boundary Condition

Pay Off

)0,( XSMaxCall T

)0,( TSXMaxPut


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Time

ValueIntrinsic

Value

+

Time till maturity

Interest Rate

Differential

Volatility

Difference between

strike and spot rate

Option Value


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Pricing Models

Option price

Binomial model

Monte Carlo Simulation

Black Scholes

Garman Kohlhagen


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Fundamental Pricing Principle

Price = Expected Discounted Cash Flows

Probability

Present Value

Inflows/Outflows


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Option Price

)0,max( XSMEc T

Expected Value Cash Flow: Pay Off

Multi-period StochasticDiscount Factor


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Option Price

)0,max( XSEec TrT

Present Value Cash Flow: Pay Off

Probability: Expected Value


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Option Price

C = N(d2)erT [ S*{N(d1)/N(d2)}e

rT X]

N(.) - Standard Normal Cumulative Distribution Function

C = SN(d1) Xe-rTN(d2)

Probability of

the option

ending in the

money

Present

Value Payoff from

exercise given

option ends up in

the money


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Basic pricing concepts

Option price

Put call parity

XFpc

SpXc

rT

TrrT f

e

ee

Boundary conditions

)ee,,0max(;0rTTr XSXScSc f


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Factors Affecting Option Value

Factor Call Value Put Value

Spot

Strike

Volatility

Domestic Interest Rates

Foreign Interest Rates


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Volatility


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Understanding volatility

Volatility

Historical

Implied

Forecasted

Actual


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Vols

Start, t=0 Start, t=T

S

Implied

Historical

Forecasted

Realized


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Understanding Volatility

HistoricalHow volatile was the spot in the past

This is a data analysis question

Standard deviation of continuous spot returns

Implied

Traders estimate of how volatile spot will be

The price of an option is quoted in implied vols

Black-Scholes translates premium into vols


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Understanding Volatility

ForecastedHow volatile the spot will be till options maturity

This a statistical estimation:

EWMA

GARCH

ARMA

Actual (Realized)

We cannot know this until it is too late!!


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Implied Volatility (Vol)

Implied Volatility is the one market parameter priced

exclusively by the options marketVol is a subjective expectation of the degree of future FXmovement

Vol changes over time

Vol is associated with strike Same Vol for same strike (irrespective of whether the option is call/put)

Different vols for different strikes

Vol is a traded instrument - and the market moves!

Live prices are quoted in premium terms & will change

depending on FX market; a vol price only moves with theoptions market

The benchmark vols are for ATM options (50 delta)


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FX Options : Market Mechanics

The Vol Smile

Normally OTM vols trade above ATM vol this is due to gap risk (e.g. devaluation), and the higher gearing

of OTM options (the lottery effect)

the butterfly measures this vol premium for OTM options over

ATM vol

(cash value of OTM options will always be less than ATM)

Most FX markets have a bias for calls over puts, orvice-versa (i.e. they are not symmetrical)

this is due to supply/demand factors, and if there is greateruncertainty on one side of the FX market

the risk-reversal measures this bias, in terms of the differencein vol for call and put options


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Implied Vols v/s Strike

Volat i l i ty

10

10.5

11

11.5

12

10p 25p ATMF 25c 10c

Simplistic Theoretical Assumption


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Fat Tails Volatility Smile

Greater Probability of Jumps

10

10.5

11

11.5

12

10p

ATMF

10c


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5.00%

5.50%

6.00%

6.50%

7.00%

7.50%

10D

15D

20D

25D

30D

35D

40D

45D

50D

45D

40D

35D

30D

25D

20D

15D

10D

Delta

Volatilit

25D USD Put

25D USD Call

The BS model is an idealisation of the behaviour of the underlying

asset price, to which the options market makes empirical adjustmentsvia the Smile.

Delta Smile Parameterization


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Interbank Volatility Quote


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Volatility Smile in Practice

25 Delta StrangleVols of a 25 Delta Call and a Putrelative to ATM vol

25 Delta RiskReversal

Difference between vols of 25 Deltacall and 25 delta put option

ATM VolatilityVolatility of an option with strikeequal to the forward rate


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Risk Reversal

R/R is collar / range forward Buy Call & Sell Put or Sell Call and Buy Put

Usually quoted for 25 Delta (both Call & Put)

R/R are Vega neutral hence vol spread is more important thanabsolute vol

1m 25d R/R

0.3/0.6 Fav USD Calls, or

0.3/0.6 Fav INR Puts (usually non-USD Ccy), or

0.3/0.6 INR Puts over

Buy USD Calls 0.30 vol higher than USD Put we sell

Sell USD Calls 0.60 vol higher than USD Put we buy

Bid/Offer w.r.t USD Calls

One needs the absolute vol for 25 Delta (either Call or Put) toknow actual vols


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Strangle/Butterfly/Fly

Straddle Buy Call, Buy Put at same strike K Strangle Buy Put at Strike A, Call at Strike B (A


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Strangle / Butterfly

1m 25d Fly in Black Scholes World?

Large positive value of Fly indicates smile with high

curvature

Negative value indicates sad face frown rarelyseen


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RR & Fly/Butterfly/Strangle - Exercise

Market Quotes:

25 Delta R/R 0.40% USD Call over25 Delta Strangle 0.7% over ATMF

ATMF 3.5%

Find the following:

25 D Strangle

25 Delta Call

25 Delta Put


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RR & Fly/Butterfly/Strangle

ATMF-2

252525

252525

dPutdCall

dFly

dPutdCalldRR

2

25

2525

2

252525

dRR

dFlyATMFdPut

dRRdFlyATMFdCall


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Volatility Surface

Combines volatility smile with volatility term structure Volatility term structure (Vols v/s Time to Maturity)

Helps to price option with any strike price and anymaturity

The effect of smile decreases as the option maturityincreases


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EUR/USD Volatility Surface

12.2811.3410.6310.4110.626m

1Y

3M

1M

1W

12.2611.2610.5510.3410.59

12.2611.3810.7010.4810.64

12.5711.7511.1010.9011.04

13.7413.0512.5012.4012.57

10C25CATM25P10 P


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Volatility Cone

Developed by Galen Burghardt Technique for visualizing current option implied

volatility relative to historic volatility at differentmaturity ranges

The maximum, average, minimum volatilities are fordifferent maturities are plotted for the sample horizonperiod


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Volatility cone : USD/INR

0.00%

1.00%

2.00%

3.00%

4.00%

5.00%

1W 1M 2M 3M 6M 1Y

Current

mean

max/min


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Greeks and Risk Management


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Greeks and Risk Management

Change in option value

Change in Volatility

Vega

Change in Interest rate

Rho

Change in Time

Theta

Change in spot

Delta

Gamma


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Greeks

Vega

Volga

Price

Gamma Vanna

Delta ThetaRho

dGamma/dS dVolga/dVdVolga/dSdGamma/dV

S V

S S

SSS

V V

V V V

R T



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FX Options : Greeks

Black Scholes and Greeks

Option Price

Option Definitionccy pair

tenorstrike

call / put

Market Variablesspot

forwardsinterest rates

volatility

BLACK -SCHOLESEQUATION The greeks:

delta, vega,rho, theta,

gamma, volga,vanna & other

market sensitivities


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Delta

Delta (D) is the rate of change of the option price with respect tothe underlying

Change in option price from infinitely smallchange in underlyingasset price

Option

price

A

BSlope = D

Spot price


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PriceYou own $10m of a

$Call Option, K=40

Time

decay

35 40 45

Delta:

+$1m

Delta:

+ $5m

Delta:

+$9m

TV

Delta

Delta


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Delta


Delta is the sensitivity of the option price to changes in the

underlying FX rate

Delta represents the proportion of FX that needs to bebought/sold in order to hedge the FX risk of the option

Delta (for a European vanilla option) also approximatelyrepresents the chance of the option being exercised (i.e.probability of ending ITM at maturity)

Strikes can be defined in terms of the delta

Delta =change in value of option

change in FX rate


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Delta

-0.

05

-0.

03

-0.

01

0.

01

0.

03

0.0

5

0.08

0.75

2.00

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Delta

Moneyness

Time (y)

Call Option Delta

0.9-1.0

0.8-0.9

0.7-0.8

0.6-0.70.5-0.6

0.4-0.5

0.3-0.4

0.2-0.3

0.1-0.2

0.0-0.1


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Vega is the sensitivity of the option price to changes in the

implied vol

If you own an option, you will normally be long vega- i.e. you make money if vols rise, and lose if they fall

Vega is greatest on longer-dated options

Vega =change in value of option

change in volatility

Vega



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Vega As a Risk Measure

Vega is usually expressed as change in value ofoption for 1% change in volatility

Volatility is quoted in % annualized terms

Vega hedging is done using another option

If V is the vega of the portfolio and Vt is the vega oftraded option, then the position ofV/ Vt in the tradedoption will make it vega neutral


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Vega

-0.

05

-

0.

03

-0.

01

0.

01

0.

03

0.

05

0.08

0.75

2.00

0

5

10

15

20

25

30

35

Vega

MoneynessTime

Call Option Vega

30-35

25-30

20-25

15-20

10-15

5-10

0-5

Change in option price due to change in volatility


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Gamma is the sensitivity of the option delta to changes in

the underlying FX rate

Gamma is the value that can be gained from owning anoption in order to trade the underlying FX rate

If you own an option, you will normally be long gamma- i.e. you get longer the underlying as spot goes up, andyou get shorter as spot goes down

Gamma is greatest on short-dated options close to expiry

Gamma =change in delta of option

change in FX rate

Gamma

FX Options : Gamma Scalping


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OptionValue

FX RateStrike

Delta is long - trader

can sell cash to rehedge

Delta is short - trader

can buy cash to rehedge

Gamma trading



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Gamma - Price Curvature

Call Option Price and PAY OFF

0 Strike

High

Curvature

LowerCurvature


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Gamma as a Risk Measure

Change of delta with price Large gamma - delta changes very fast

Portfolio has to be made delta neutral very frequently

Measures the curvature of the relationship between

option price and stock price Curvature leads to hedging error if portfolio is not

frequently rebalanced

G C t i k


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Gamma - Curvature risk

-0.

05

-0.

03

-0

.01

0.

01

0.0

3

0.

05

0.08

0.75

2.00

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Gamma

Moneyness

Time (y)

Call Option Gamma

0.9-1.0

0.8-0.9

0.7-0.8

0.6-0.70.5-0.6

0.4-0.5

0.3-0.4

0.2-0.3

0.1-0.2

0.0-0.1

V ill G k


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Vanilla Greeks

LONG CALL+ Delta

+ Gamma

+ Vega

SHORT CALL

- Delta

- Gamma

- Vega

LONG PUT- Delta

+ Gamma

+ Vega

SHORT PUT

+ Delta

- Gamma

- Vega


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Theta is the sensitivity of the option price to changes in time:

as time only goes in one direction, this is normally knownas time decay

Short theta is normally balanced against long gammapositions

Theta is the day-to-day cost of owning options: the price ofan option is determined by the perceived benefit ofowning the gamma versus the cost of the theta

Theta =change in value of option

change in time to expiry

Theta


Th t i k


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Theta as a risk measure

Theta is usually negative for an option As the time to maturity decreases option becomes

less valuable (Theta Decay)

Theta has greatest impact on short-term options

No uncertainty about the passage of time Theta indicates that the value of position will grow at

risk free rate if both delta and gamma are zero

If Theta is large in absolute terms, either delta or

gamma must be large

O ti P tf li


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Option Portfolio

Portfolio of Calls / Puts of different strikes, differentmaturities, different notional, different sides (buy / sell)

Portfolio Risk Management => Greeks !

Greeks are additive

Portfolio Delta = Sum of Deltas of all options (withappropriate signs)

Similarly, Portfolio Gamma, Portfolio Vega, etc


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Thanks, no questions please !!

delhi client da-i

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