° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° OIL GOES LOCAL A TWO-FACTOR LOCAL VOLATILITY MODEL FOR OIL AND OTHER COMMODITIES Do not move or delete this text box. For cropping purposes only, and does not print. 18 // FEB // 2015

OIL GOES LOCAL – FRANCESCO CHIMINELLO 2 // Introduction © Marie-Lan Nguyen / Wikimedia Commons

OIL GOES LOCAL – FRANCESCO CHIMINELLO 3 //  Most commodities trade as futures/forwards  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 4 // WTI forward curves

OIL GOES LOCAL – FRANCESCO CHIMINELLO 5 // Brent ATM volatility

OIL GOES LOCAL – FRANCESCO CHIMINELLO 6 // Brent Smile volatility

OIL GOES LOCAL – FRANCESCO CHIMINELLO 7 //  Commodity markets can be brutal  Models need to be robust  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 8 // Dynamics of the forward curve: historical analysis

OIL GOES LOCAL – FRANCESCO CHIMINELLO 9 //  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

OIL GOES LOCAL – FRANCESCO CHIMINELLO 10 // WTI forward curves

OIL GOES LOCAL – FRANCESCO CHIMINELLO 11 // WTI: historical instantaneous vol term structure

OIL GOES LOCAL – FRANCESCO CHIMINELLO 12 // WTI: historical correlation term structure

OIL GOES LOCAL – FRANCESCO CHIMINELLO 13 // WTI: historical correlation term structure Forward time to maturity (months) 1 curve / 3 months

OIL GOES LOCAL – FRANCESCO CHIMINELLO 14 // WTI: historical correlation term structure

OIL GOES LOCAL – FRANCESCO CHIMINELLO 15 // WTI: Eigenvalues

OIL GOES LOCAL – FRANCESCO CHIMINELLO 16 // WTI: first 6 Principal Components

OIL GOES LOCAL – FRANCESCO CHIMINELLO 17 // Copper: first 6 Principal Components

OIL GOES LOCAL – FRANCESCO CHIMINELLO 18 // Natural Gas: first 6 Principal Components

OIL GOES LOCAL – FRANCESCO CHIMINELLO 19 // Dynamics of the forward curve: implied data

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OIL GOES LOCAL – FRANCESCO CHIMINELLO 22 //  2 factors  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 23 // A minimal model for oil: lognormal backbone

OIL GOES LOCAL – FRANCESCO CHIMINELLO 24 //  Reminder: forwards are risk-neutral martingales  Backbone dynamics:  Decorrelated Brownians:  Instantaneous variance: A minimal model for oil: lognormal backbone

OIL GOES LOCAL – FRANCESCO CHIMINELLO 25 //  Reminder: forwards are martingales  Backbone dynamics:  Decorrelated Brownians:  Instantaneous variance: A minimal model for oil: lognormal backbone Forward T observed in t Correlated Brownians

OIL GOES LOCAL – FRANCESCO CHIMINELLO 26 //  Reminder: forwards are martingales  Backbone dynamics:  Decorrelated Brownians:  Instantaneous variance: A minimal model for oil: lognormal backbone Factor 2 Decaying Factor 2 Decaying Factor 1 Parallel Factor 1 Parallel Normalisation (to mkt vol)

OIL GOES LOCAL – FRANCESCO CHIMINELLO 27 //  Reminder: forwards are martingales  Backbone dynamics:  Decorrelated Brownians:  Instantaneous variance: A minimal model for oil: lognormal backbone 3 Model Parameters

OIL GOES LOCAL – FRANCESCO CHIMINELLO 28 //  Reminder: forwards are martingales  Backbone dynamics:  Decorrelated Brownians:  Instantaneous variance: A minimal model for oil: lognormal backbone

OIL GOES LOCAL – FRANCESCO CHIMINELLO 29 //  Compute total variance: Shorthands:  Total variance: arbitrary interval  Total variance: market options A minimal model for oil: lognormal backbone

OIL GOES LOCAL – FRANCESCO CHIMINELLO 30 //  Compute total variance: Shorthands:  Total variance: arbitrary interval  Total variance: market options A minimal model for oil: lognormal backbone Market total variance

OIL GOES LOCAL – FRANCESCO CHIMINELLO 31 //  Compute total variance: Shorthands:  Total variance: arbitrary interval  Total variance: market options A minimal model for oil: lognormal backbone

OIL GOES LOCAL – FRANCESCO CHIMINELLO 32 //  Normalisation to market ATM vol:  Term structure of early implied ATM vol: A minimal model for oil: lognormal backbone

OIL GOES LOCAL – FRANCESCO CHIMINELLO 33 //

OIL GOES LOCAL – FRANCESCO CHIMINELLO 34 //  Normalisation to market ATM vol:  Term structure of early implied ATM vol: A minimal model for oil: lognormal backbone

OIL GOES LOCAL – FRANCESCO CHIMINELLO 35 //  Normalisation to market ATM vol:  Term structure of early implied ATM vol:  Instantaneous covariance: with shorthands: A minimal model for oil: lognormal backbone

OIL GOES LOCAL – FRANCESCO CHIMINELLO 36 //  Terminal covariance:  Terminal correlation: A minimal model for oil: lognormal backbone

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OIL GOES LOCAL – FRANCESCO CHIMINELLO 39 // Once more with a wilder market

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OIL GOES LOCAL – FRANCESCO CHIMINELLO 42 //

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OIL GOES LOCAL – FRANCESCO CHIMINELLO 44 // Exciting marketBoring market

OIL GOES LOCAL – FRANCESCO CHIMINELLO 45 // A minimal model for oil: smile and local volatility

OIL GOES LOCAL – FRANCESCO CHIMINELLO 46 //

OIL GOES LOCAL – FRANCESCO CHIMINELLO 47 //

OIL GOES LOCAL – FRANCESCO CHIMINELLO 48 //  Parsimonious smile assumption:  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 49 //  Smile interpolated in time along isolines of reduced ATM delta: 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 50 //  Smile interpolated in time along isolines of reduced ATM delta: compare with Black-Scholes delta:  Early skew rescaled to ATM vol A minimal model for oil: smile and local volatility ATM vol No time term Vol at strike Time term

OIL GOES LOCAL – FRANCESCO CHIMINELLO 51 //  Smile interpolated in time along isolines of reduced ATM delta: compare with Black-Scholes delta:  Early skew rescaled to ATM vol A minimal model for oil: smile and local volatility Early at the strike vol Interpolator function Interpolator function Rescaling

OIL GOES LOCAL – FRANCESCO CHIMINELLO 52 //  Smile interpolated in time along isolines of reduced ATM delta: 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 53 //  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

OIL GOES LOCAL – FRANCESCO CHIMINELLO 54 //  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 Overall local vol Factors weights Factors weights

OIL GOES LOCAL – FRANCESCO CHIMINELLO 55 //  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

OIL GOES LOCAL – FRANCESCO CHIMINELLO 56 //  3 parameters: alpha, beta, rho  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

OIL GOES LOCAL – FRANCESCO CHIMINELLO 57 //  Exotic trades: Monte Carlo  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 58 //  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

OIL GOES LOCAL – FRANCESCO CHIMINELLO 59 // Questions? © Marie-Lan Nguyen / Wikimedia Commons