delhi client da-i
TRANSCRIPT
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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
FX Options : Market Mechanics
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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
FX Options : Market Mechanics
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
FX Options : Market Mechanics
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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
FX Options : Market Mechanics
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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
FX Options : Market Mechanics
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 !!