FXSkew2
Transcript of FXSkew2
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Modelling the FX Skewodelling the FX Skew
Dherminder Kainth and Nagulan SaravanamuttuDherminder Kainth and Nagulan Saravanamuttu
QuaRC, Royal Bank of ScotlandQuaRC, Royal Bank of Scotland
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Overview
o FX Markets
o Possible Models and Calibration
o Variance Swaps
o Extensions
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Spot
USDJPY Spot
USDJPY Spot
S p o t
99
102
105
108
111
114
117
120
123
126
129
132
135
99
102
105
108
111
114
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135
2 8 N ov 0 1
2 8 F e b 0 2
3 1 M a y 0 2
0 2 S e p 0 2
0 3 D e c 0 2
0 5 M ar 0 3
0 5 J un 0 3
0 5 S e p 0 3
0 8 D e c 0 3
0 9 M ar 0 4
0 9 J un 0 4
0 9 S e p 0 4
1 0 D e c 0 4
1 4 M ar 0 5
1 4 J un 0 5
1 4 S e p 0 5
1 5 D e c 0 5
1 7 M ar 0 6
1 9 J un 0 6
1 9 S e p 0 6
1 6 D e c 0 6
S p o t
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Volatility
USDJPY 1M Historic Volatility
4
5
6
7
8
9
10
11
12
13
14
15
16
17
4
5
6
7
8
9
10
11
12
13
14
15
16
17
2 8 N ov 0 1
2 8 F e b 0 2
3 1 M a y 0 2
0 2 S e p 0 2
0 3 D e c 0 2
0 5 M ar 0 3
0 5 J un 0 3
0 5 S e p 0 3
0 8 D e c 0 3
0 9 M ar 0 4
0 9 J un 0 4
0 9 S e p 0 4
1 0 D e c 0 4
1 4 M ar 0 5
1 4 J un 0 5
1 4 S e p 0 5
1 5 D e c 0 5
1 7 M ar 0 6
1 9 J un 0 6
1 9 S e p 0 6
1 4 D e c 0 6
V o l a t i l i t y
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European Implied Volatility Surface
• Implied volatility smile defined in terms of deltas
• Quotes available
‒ Delta-neutral straddle ⇒ Level
‒ Risk Reversal = (25-delta call ‒ 25-delta put) ⇒Skew
‒ Butterfly = (25-delta call + 25-delta put ‒ 2ATM) ⇒Kurtosis
• Also get 10-delta quotes
• Can infer five implied volatility points per expiry
‒ ATM
‒ 10 delta call and 10 delta put ‒ 25 delta call and 25 delta put
• Interpolate using, for example, SABR or Gatheral
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Implied Volatility Smiles
6
6.5
7
7.5
8
8.5
9
9.5
10
10.5
11
10C 25C ATM 25P 10P
delta
I m p l i e d V o l a t i
l i t y 1M
1Y
2Y
6
6.5
7
7.5
8
8.5
9
9.5
10
10.5
11
10C 25C ATM 25P 10P
delta
I m p l i e d V o l a t i l i t y 1M
1Y
2Y
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Liquid Barrier Products
• Some price visibility for certain barrier products in leading currency pairs (egUSDJPY, EURUSD)
• Three main types of products with barrier features
‒ Double-No-Touches
‒ Single Barrier Vanillas
‒ One-Touches
• Have analytic Black-Scholes prices (TVs) for these products
• High liquidity for certain combinations of strikes, barriers, TVs
• Barrier products give information on dynamics of implied volatility surface
• Calibrating to the barrier products means we are taking into account theforward implied volatility surface
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Double-No-Touches
• Pays one if barriers not breached through lifetime of product
• Upper and lower barriers determined by TV and UL=S2
• High liquidity for certain values of TV : 35%, 10%
time
0 T
F X
r a t e
U
L
S
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Double-No-Touches
• For constant TV, barrier levels are a function of expiry
80
90
100
110
120
130
140
0 0.5 1 1.5 2Expiry
B a r r i e r L e v e l
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Single Barrier Vanilla Payoffs
• Single barrier product which pays off a call or put depending on whether barrier is breached throughout life of product
• Three aspects
‒ Final payoff (Call or Put)
‒ Pay if barrier breached or pay if it is not breached (Knock-in or Knock-out)
‒ Barrier higher or lower than spot (Up or Down)
• Leads to eight different types of product
• Significant amount of value apportioned to final smile (depending on
strike/barrier combination)
• Not as liquid as DNTs
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One-Touches
• Single barrier product which pays one when barrier is breached
• Pay off can be in domestic or foreign currency
• There is some price visibility for one-touches in the leading currency markets
• Not as liquid as DNTs
• Price depends on forward skew
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Replicating Portfolio
60 70 80 90 100 110 12060 70 80 90 100 110 12060 70 80 90 100 110 12060 70 80 90 100 110 120
SpotKB
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Replicating Portfolio
60 70 80 90 100 110 12060 70 80 90 100 110 120
SpotKB
u < T
T
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60 70 80 90 100 110 120
Replicating Portfolio
60 70 80 90 100 110 12060 70 80 90 100 110 120
SpotKB
u < T
T
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One-Touches
• For Normal dynamics with zero interest rates
• Price of One-Touch is probability of breaching barrier
• Static replication of One-Touch with Digitals
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One-Touches
• Log-Normal dynamics
• Barrier is breached at time
• Can still statically replicate One-Touch
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One-Touches
• Introduce skew
• Using same static hedge
• Price of One-Touch depends on skew
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Model Skew
• Model Skew : (Model Price ‒ TV)
• Plotting model skew vs TV gives an indication of effect of model-implied smile
dynamics
• Can also consider market-implied skew which eliminates effect of particular
market conditions (eg interest rates)
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Possible Models and Calibration
o Local Volatility
o Heston
o Piecewise-Constant Heston
o Stochastic Correlation
o Double-Heston
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Local Volatility
• Local volatility process
• Ito-Tanaka implies
• Dupires formula
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Local Volatility Calibration to Europeans
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Local Volatility
• Gives exact calibration to the European volatility surface by construction
• Volatility is deterministic, not stochastic
• implies spot perfectly correlated to volatility
• Forward skew is rapidly time-decaying
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Local Volatility Smile Dynamics
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
75 85 95 105 115 125Strike
I m p l i e d V o l a t
i l i t y
Original
Shifted
ΔS
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Heston Model
• Heston process
• Five time-homogenous parameters
• Will not go to zero if
• Pseudo-analytic pricing of Europeans
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Heston Characteristic Function
• Pricing of European options
• Fourier inversion
• Characteristic function form
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Heston Smile Dynamics
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
75 85 95 105 115 125Strike
I m p l i e d V o l a
t i l i t y
Original
Shifted
ΔS
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Heston Implied Volatility Term-Structure
8.40%
8.50%
8.60%
8.70%
8.80%
8.90%
9.00%
9.10%
1W 1M 2M 3M 6M 1Y 2Y
Heston
Market
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Implied Volatility Term Structures
6
6.5
7
7.5
8
8.5
9
1W 1M 2M 3M 6M 1Y 2Y 3Y 4Y 5Y
USDJPY
EURUSD
AUDJPY
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Piecewise-Constant Heston Model
• Process
• Form of reversion level
• Calibrate reversion level to ATM volatility term-structure
time0 1W 1M 3M2M
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Piecewise-Constant Heston Characteristic Function
• Characteristic function
• Functions satisfy following ODEs (see Mikhailov and Nogel)
• and independent of
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Stochastic Volatility/Local Volatility
• Possible to combine the effects of stochastic volatility and local volatility
• Usually parameterise the local volatility multiplier, eg Blacher
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Stochastic Risk-Reversals
• USDJPY 6 month 25-delta risk-reversals
USDJPY (JPY call) 6M 25 Delta Risk Reversal
Ri s k R ev er s al
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
0 8 N ov 0 4
2 1 N ov 0 5
2 6 N ov 0 6
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Stochastic Correlation Model
• Introduce stochastic correlation explicitly but what process to use?
• Process has to have certain characteristics:
‒ Has to be bound between +1 and -1
‒ Should be mean-reverting
• Jacobi process
• Conditions for not breaching bounds
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Stochastic Correlation Model
• Transform Jacobi process using
• Leads to process for correlation
• Conditions
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Stochastic Correlation Model
• Use the stochastic correlation process with Heston volatility process
• Correlation structure
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Stochastic Correlation Calibration to Europeans and DNTs
Loss Function : 14.3
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Stochastic Correlation Calibration to Europeans and DNTs
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Multi-Scale Volatility Processes
• Market seems to display more than one volatility process in its underlying
dynamics
• In particular, two time-scales, one fast and one slow
• Models put forward where there exist multiple time-scales over which volatility
reverts
• For example, have volatility mean-revert quickly to a level which itself is
slowly mean-reverting (Balland)
• Can also have two independent mean-reverting volatility processes with
different reversion rates
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Double-Heston Model
• Double-Heston process
• Correlation structure
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Double-Heston Model
• Stochastic volatility-of-volatility
• Stochastic correlation
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Double-Heston Parameters
• Two distinct volatility processes
‒ One is slow mean-reverting to a high volatility ‒ Other is fast mean-reverting to a low volatility
‒ Critically, correlation parameters are both high in magnitude and of
opposite signs
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Double-Heston Calibration to Europeans and DNTs
Loss Function : 4.30
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Double-Heston Calibration to Europeans and DNTs
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Variance Swaps
o Product Definition
o Process Definitions
o Variance Swap Term-Structure
o Model Implied Term-Structures
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Variance Swap Definition
• Quadratic variation
• Variance swap price
• Price process
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Variance Process Definitions
• Define the forward variance
• Define the short variance process
• We already have models for describing
‒ Heston
‒ Double-Heston ‒ Double Mean-Reverting Heston (Buehler)
‒ Black-Scholes
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Variance Swap Term Structure
• Heston form for variance swap term structure
• Double-Heston
• Note the independence of the variance swap term-structure to the correlationand volatility-of-volatility parameters
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Volatility Swap Term Structure
0.09
0.095
0.1
0.105
0.11
0.115
0.12
0.125
0.13
1M 2M 3M 6M 9M 1Y 2Y
Double Heston
Heston
Local Volatility
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Extensions
o Stochastic Interest Rates
o Multi-Heston
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Stochastic Interest Rates
• Long-dated FX products are exposed to interest rate risk
• Need a dual-currency model which preserves smile features of FX vanillas
• Andreasens four-factor model
‒ Hull-White process for each short rate
‒ Heston stochastic volatility for FX rate ‒ Short rates uncorrelated to Heston volatility process
‒ Pseudo-analytic pricing of Europeans
‒ Can incorporate Double-Heston process for volatility and maintain rapid
calibration to vanillas
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Multi-Heston Process
• Can always extend Double-Heston to Multi-Heston with any number of
uncorrelated Heston processes
• Maintain pseudo-analytic European pricing
• In fact, using three Heston processes does not significantly improve on the
Double-Heston fits to Europeans and DNTs
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Summary
• FX markets exhibit certain properties such as stochastic risk-reversals and
multiple modes of volatility reversion
• Barrier products show liquidity - especially DNTs - and their prices are linked
to the forward smile
• The Double-Heston model captures the features of the market and recoversEuropeans and DNTs through calibration
• It also prices One-Touches to within bid/offer spread of SV/LV and exhibits
the required flexibility for modelling the variance swap curve
• Advantages are that it is relatively simple model with pseudo-analyticEuropean prices, and barrier products can be priced on a grid
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References
• D. Bates : Post-87 Crash Fears in S&P 500 Futures Options, National Bureau of
Economic Research, Working Paper 5894, 1997• S. Heston : A Closed-Form Solution for Options with Stochastic Volatility with
Applications to Bond and Currency Options, Review of Financial Studies, 1993
• H. Buehler : Volatility Markets ‒ Consistent Modelling, Hedging and PracticalImplementation, PhD Thesis, 2006
• M. Joshi : The Concepts and Practice of Mathematical Finance, Cambridge, 2003
• J. Andreasen : Closed Form Pricing of FX Options under Stochastic Rates andVolatility, ICBI, May 2006
• P. Balland : Forward Smile, ICBI, May 2006• S. Mikhailov and U. Nogel : Hestons Stochastic Volatility, Model Implementation,
Calibration and Some Extensions, Wilmott, 2005
• A. Chebanier : Skew Dynamics in FX, QuantCongress, 2006
• P. Carr and L. Wu : Stochastic Skew in Currency Options, 2004
• P. Hagan, D. Kumar, A. Lesniewski and D. Woodward : Managing Smile Risk,Wilmott, 2002
• J. Gatheral : A Parsimonious Arbitrage-Free Implied Volatility Parameterization with
Application the Valuation of Volatility Derivatives, Global Derivatives & RiskManagement, 2004
• [email protected], [email protected]
• www.quarchome.org