Oil goes local

A TWO-FACTOR LOCAL VOLATILITY MODEL
FOR OIL AND OTHER COMMODITIES
15 // MAY // 2014
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OIL GOES
LOCAL
Introduction
// OIL GOES LOCAL – FRANCESCO CHIMINELLO
© Marie-Lan Nguyen / Wikimedia Commons
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 Cash+carry arbitrage not readily available
for many assets
 Need to model the dynamics of the whole
forward curve
 Options on the forwards




Expiry before the forward
Smile=>Local volatility needed
Not a shared volatility surface
Little or no early vol instruments
 Different behaviour by asset type
 Crude oil, Base/Precious metals, Softs...
 Volatile market
 Very high volatility common
 Very high skew/smile common
 High vol of vol
Introduction
// OIL GOES LOCAL – FRANCESCO CHIMINELLO
 Most commodities trade as
futures/forwards
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WTI forward curves
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Brent ATM volatility
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Brent Smile volatility
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 Simple: avoid overfitting
 Stable: avoid complex calibrations,
bootstraps if possible
 The (real life) hedge is the price, and the
hedge needs to be stable
 Must match liquid market
instruments
 Match Forwards by construction
 Match Vanillas by constructions
 Local volatility
 Exotics consistent with their hedges
 Capture the essential features of
the forward curve dynamics
 Needs to be investigated per asset
 Depends also on the intended trading
portfolio
 Build a usable, minimal model for
oil derivatives
Motivation
// OIL GOES LOCAL – FRANCESCO CHIMINELLO
 Commodity markets can be brutal
 Models need to be robust
Dynamics of the
forward curve:
historical analysis
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 Historical analysis => stylized facts
 Forwards are fixed date (not tenor)
 Analysis on prompt, second-prompt...
 Comparison with model needs exact tenors
 Quantities of interest
 Dynamics of individual forwards
 Instantaneous volatility curve
 Joint dynamics
 Covariance/correlation between
forwards
 Principal Components
Dynamics of the
forward curve:
historical analysis
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// OIL GOES LOCAL – FRANCESCO CHIMINELLO
WTI forward curves
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WTI: historical instantaneous vol term structure
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WTI: historical correlation term structure
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WTI: historical correlation term structure
Forward time to maturity (months)
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WTI: historical correlation term structure
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WTI: Eigenvalues
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WTI: first 6 Principal Components
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Copper: first 6 Principal Components
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Natural Gas: first 6 Principal Components
Dynamics of the
forward curve:
implied data
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 Short end vs long end of the curve
 Decorrelation between forwards
 Samuelson effect
 Historical instantaneous vol
 Average shape of implied ATM vol
 Volatility smile




Robust in high vol, high skew conditions
Avoid asymptotic arbitrage
Analytic derivatives
Smooth wrt input quotes
 Match market
 Match futures by construction
 Match options by construction
A model for oil:
minimal
requirements
// OIL GOES LOCAL – FRANCESCO CHIMINELLO
 2 factors
A minimal
model for oil:
lognormal
backbone
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 Decorrelated Brownians:
 Instantaneous variance:
A minimal
model for oil:
lognormal
backbone
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 Reminder: forwards are risk-neutral martingales
 Backbone dynamics:
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Reminder:
Forward
T forwards are martingales
Backbonein
dynamics:
observed
t
Correlated
 Decorrelated Brownians:
Brownians
 Instantaneous variance:
A minimal
model for oil:
lognormal
backbone
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

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 Decorrelated Brownians:
 Instantaneous variance:
A minimal
model for oil:
lognormal
backbone
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 Reminder: forwards are martingales
 Backbone dynamics:
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3 Model
Parameters
 Decorrelated Brownians:
 Instantaneous variance:
A minimal
model for oil:
lognormal
backbone
// OIL GOES LOCAL – FRANCESCO CHIMINELLO
 Reminder: forwards are martingales
 Backbone dynamics:
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 Decorrelated Brownians:
 Instantaneous variance:
A minimal
model for oil:
lognormal
backbone
// OIL GOES LOCAL – FRANCESCO CHIMINELLO
 Reminder: forwards are martingales
 Backbone dynamics:
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Shorthands:
 Total variance: arbitrary interval
 Total variance: market options
A minimal
model for oil:
lognormal
backbone
// OIL GOES LOCAL – FRANCESCO CHIMINELLO
 Compute total variance:
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A minimal
model for oil:
lognormal
backbone
Shorthands:
 Total variance: arbitrary interval
 Total variance: market options
Market total
variance
// OIL GOES LOCAL – FRANCESCO CHIMINELLO
 Compute total variance:
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Shorthands:
 Total variance: arbitrary interval
 Total variance: market options
A minimal
model for oil:
lognormal
backbone
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 Compute total variance:
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 Term structure of early implied ATM vol:
A minimal
model for oil:
lognormal
backbone
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 Normalisation to market ATM vol:
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 Term structure of early implied ATM vol:
A minimal
model for oil:
lognormal
backbone
// OIL GOES LOCAL – FRANCESCO CHIMINELLO
 Normalisation to market ATM vol:
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 Term structure of early implied ATM vol:
 Instantaneous covariance:
with shorthands:
A minimal
model for oil:
lognormal
backbone
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 Normalisation to market ATM vol:
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 Terminal correlation:
A minimal
model for oil:
lognormal
backbone
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 Terminal covariance:
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Once more
with a wilder
market
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-0.5
-1
1.5
1
0.5
0
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5/6/2013
10/18/2012
4/1/2012
9/14/2011
2/26/2011
8/10/2010
1/22/2010
7/6/2009
alpha beta rho
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2
-0.5
-1
1.5
1
0.5
0
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5/6/2013
Exciting market
10/18/2012
4/1/2012
9/14/2011
2/26/2011
8/10/2010
2
1/22/2010
7/6/2009
alpha beta rho
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Boring market
A minimal
model for oil:
smile and
local volatility
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46
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31
Apr-13
29
May-13
Jun-13
27
Jul-13
25
Aug-13
Sep-13
23
Oct-13
21
Nov-13
Dec-13
19
17
15
60
70
80
90
100
110
120
130
140
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33
31
Apr-13
29
May-13
Jun-13
27
Jul-13
25
Aug-13
Sep-13
23
Oct-13
21
Nov-13
Dec-13
19
17
15
0
0.2
0.4
0.6
0.8
1
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 Time extrapolation at ̴constant delta
 Early vol depends only on early ATM vol
and smile of standard options
 Simple
 Consistent time bucketing of vega
 Black-Scholes delta issues (as a smile
interpolator independent variable):
 Rootfinder needed to query volatility
 Slow
 Non-smooth
 ATM-Forward is not constant BS-delta
 Difficult to extrapolate in time
 Smile interpolator needs to be
swappable
 E.g.: splines, SVI...
 Examples here use spline interpolation
A minimal
model for oil:
smile and
local volatility
// OIL GOES LOCAL – FRANCESCO CHIMINELLO
 Parsimonious smile assumption:
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compare with Black-Scholes delta:
 Early skew rescaled to ATM vol
A minimal
model for oil:
smile and
local volatility
// OIL GOES LOCAL – FRANCESCO CHIMINELLO
 Smile interpolated in time along
isolines of reduced ATM delta:
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compare
ATM
vol with Black-Scholes delta:
No time term
Vol at strike
 Early skew rescaled to ATM vol
Time term
A minimal
model for oil:
smile and
local volatility
// OIL GOES LOCAL – FRANCESCO CHIMINELLO
 Smile interpolated in time along
isolines of reduced ATM delta:
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compare with Black-Scholes delta:
 Early
skew rescaled to ATM vol
Early
at the
Interpolator
strike vol
function
A minimal
model for oil:
smile and
local volatility
Rescaling
// OIL GOES LOCAL – FRANCESCO CHIMINELLO
 Smile interpolated in time along
isolines of reduced ATM delta:
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compare with Black-Scholes delta:
 Early skew rescaled to ATM vol
A minimal
model for oil:
smile and
local volatility
// OIL GOES LOCAL – FRANCESCO CHIMINELLO
 Smile interpolated in time along
isolines of reduced ATM delta:
 Implied vol known =>
Dupire local vol can be computed
 Apportion local variance to factors:
Proportionally to instantaneous
variance in the backbone
lognormal model
 Local volatility SDE:
A minimal
model for oil:
smile and
local volatility
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 Implied vol known =>
Dupire local vol can be computed
 Apportion local variance to factors:
Proportionally to instantaneous
variance in the backbone
lognormal model
Overall
local vol
 Local volatility SDE:
Factors
weights
A minimal
model for oil:
smile and
local volatility
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 Implied vol known =>
Dupire local vol can be computed
 Apportion local variance to factors:
Proportionally to instantaneous
variance in the backbone
lognormal model
 Local volatility SDE:
A minimal
model for oil:
smile and
local volatility
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 Historical: match the historical
covariance matrix
 Caveat: need to use exact time intervals
 Implied: in some markets,
information available:




Early options
Swaptions
Long-dated Asian options
Not recommended: calendar spreads
 Hybrid
 If only little implied info is available,
weight historical and implied data
Calibration
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 3 parameters: alpha, beta, rho
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 Need to simulate all the forwards
 High vol/skew require short steps
 Most of the trades are Asian anyway
 Analytic trades (exact and approximated)
 Any linear trade
 Vanilla Europeans
 By replication, any vanilla payout
 Asian options
 Swaptions
 Variance swaps
 Baskets (if correlation is high)
 PDE
 Trades on a single forward
 Most notably, Americans
Model usage
// OIL GOES LOCAL – FRANCESCO CHIMINELLO
 Exotic trades: Monte Carlo
 We presented a minimal but
robust 2-factors local volatility
model for oil
 Captures the essential stylized facts of the
forward curve dynamics
 Reproduces by construction forwards and
vanilla volatilities
 Calibration can be historical or implied
 Possible simple extensions:
 Time-dependent parameters
 e.g., handle very short end of the curve
 Different shapes of factors
 e.g., short factor for Agriculturals
 More complex extensions:
 Seasonality of correlation
 3 factors/effective option time
 Stochastic volatility
 with a single, shared vol process
 local vol component a must
 lack of calibration implied data
Final remarks
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Questions?
// OIL GOES LOCAL – FRANCESCO CHIMINELLO
© Marie-Lan Nguyen / Wikimedia Commons
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