IL GOES OCAL A TWO-FACTOR LOCAL VOLATILITY MODEL FOR OIL AND OTHER COMMODITIES 15 // MAY // 2014

Transcription

1 IL GOES OCAL A TWO-FACTOR LOCAL VOLATILITY MODEL FOR OIL AND OTHER COMMODITIES 15 MAY 2014

2 2 Marie-Lan Nguyen / Wikimedia Commons Introduction

3 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

4 WTI forward curves 4

5 Brent ATM volatility 5

6 Brent Smile volatility 6

7 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

8 8 Dynamics of the forward curve: historical analysis

9 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

10 WTI forward curves 10

11 WTI: historical instantaneous vol term structure 11

12 WTI: historical correlation term structure 12

13 WTI: historical correlation term structure 13 Forward time to maturity (months)

14 WTI: historical correlation term structure 14

15 WTI: Eigenvalues 15

16 WTI: first 6 Principal Components 16

17 Copper: first 6 Principal Components 17

18 Natural Gas: first 6 Principal Components 18

19 19 Dynamics of the forward curve: implied data

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22 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

23 23 A minimal model for oil: lognormal backbone

24 Reminder: forwards are risk-neutral martingales Backbone dynamics: 24 Decorrelated Brownians: A minimal model for oil: lognormal backbone Instantaneous variance:

25 Forward Reminder: T forwards are martingales observed Backbone in dynamics: t 25 Correlated Decorrelated Brownians: Brownians A minimal model for oil: lognormal backbone Instantaneous variance:

26 Reminder: forwards are martingales Backbone dynamics: 26 Decorrelated Brownians: A minimal model for oil: lognormal backbone Instantaneous variance:

27 Reminder: forwards are martingales Backbone dynamics: 27 Decorrelated Brownians: 3 Model Parameters A minimal model for oil: lognormal backbone Instantaneous variance:

28 Reminder: forwards are martingales Backbone dynamics: 28 Decorrelated Brownians: A minimal model for oil: lognormal backbone Instantaneous variance:

29 Compute total variance: 29 Shorthands: Total variance: arbitrary interval A minimal model for oil: lognormal backbone Total variance: market options

30 Compute total variance: 30 Shorthands: Total variance: arbitrary interval A minimal model for oil: lognormal backbone Total variance: market options Market total variance

31 Compute total variance: 31 Shorthands: Total variance: arbitrary interval A minimal model for oil: lognormal backbone Total variance: market options

32 Normalisation to market ATM vol: 32 Term structure of early implied ATM vol: A minimal model for oil: lognormal backbone

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34 Normalisation to market ATM vol: 34 Term structure of early implied ATM vol: A minimal model for oil: lognormal backbone

35 Normalisation to market ATM vol: 35 Term structure of early implied ATM vol: Instantaneous covariance: with shorthands: A minimal model for oil: lognormal backbone

36 36 Terminal covariance: Terminal correlation: A minimal model for oil: lognormal backbone

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39 39 Once more with a wilder market

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43 alpha beta rho /6/2009 1/22/2010 8/10/2010 2/26/2011 9/14/2011 4/1/ /18/2012 5/6/2013-1

44 alpha beta rho 44 2 Exciting market Boring market /6/2009 1/22/2010 8/10/2010 2/26/2011 9/14/2011 4/1/ /18/2012 5/6/2013-1

45 45 A minimal model for oil: smile and local volatility

46 Apr-13 May-13 Jun-13 Jul-13 Aug-13 Sep-13 Oct-13 Nov-13 Dec

47 Apr-13 May-13 Jun-13 Jul-13 Aug-13 Sep-13 Oct-13 Nov-13 Dec

48 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

49 Smile interpolated in time along isolines of reduced ATM delta: 49 compare with Black-Scholes delta: Early skew rescaled to ATM vol A minimal model for oil: smile and local volatility

50 Smile interpolated in time along isolines of reduced ATM delta: 50 ATM compare vol with Black-Scholes delta: No time term Vol at strike Time term Early skew rescaled to ATM vol A minimal model for oil: smile and local volatility

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

52 Smile interpolated in time along isolines of reduced ATM delta: 52 compare with Black-Scholes delta: Early skew rescaled to ATM vol A minimal model for oil: smile and local volatility

53 Implied vol known => Dupire local vol can be computed 53 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

54 Implied vol known => Dupire local vol can be computed 54 Apportion local variance to factors: Proportionally to instantaneous variance in the backbone lognormal model Overall Local volatility SDE: local vol Factors weights A minimal model for oil: smile and local volatility

55 Implied vol known => Dupire local vol can be computed 55 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

56 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

57 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

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 58

59 59 Marie-Lan Nguyen / Wikimedia Commons Questions?