fx options fundamentals
TRANSCRIPT
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
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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
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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
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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)
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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
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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…
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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
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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
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Modelling Asset Prices
Trees
Volatility
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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
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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
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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
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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)
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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
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Value of Options:
Time and Intrinsic Value
Drivers of Options price
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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
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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:
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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
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Risk Management
Delta-Hedging in Practice
Sensitivities
Greeks
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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.
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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
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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
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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?
27CitiFX Structuring
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!
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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
29CitiFX Structuring
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
30CitiFX Structuring
+ 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…
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The Theory
Behind Delta-Hedging
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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
33CitiFX Structuring
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
34CitiFX Structuring
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
35CitiFX Structuring
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
36CitiFX Structuring
Put-Call Parity
37CitiFX Structuring
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
38CitiFX Structuring
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?
39CitiFX Structuring
Smile,Strangle,Risk Reversal and Volatility Curve
40CitiFX Structuring
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
41CitiFX Structuring
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
42CitiFX Structuring
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…
43CitiFX Structuring
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
44CitiFX Structuring
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
45CitiFX Structuring
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
46CitiFX Structuring
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
47CitiFX Structuring
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
48CitiFX Structuring
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%
49CitiFX Structuring
Classic Combination
of
Vanilla Options
50CitiFX Structuring
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
51CitiFX Structuring
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]
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