Modelling Volatility Skews

1
Modelling Volatility Skews

• Bruno Dupire
• Bloomberg
• bdupire_at_bloomberg.net
• London, November 17, 2006

2
OUTLINE

• Generalities
• Leverage and jumps
• Break-even volatilities
• Volatility models
• Forward Skew
• Smile arbitrage

3
Generalities
4
Market Skews

• Dominating fact since 1987 crash strong negative
  skew on
• Equity Markets
• Not a general phenomenon
• Gold FX
• We focus on Equity Markets

5
Skews

• Volatility Skew slope of implied volatility as a
  function of Strike
• Link with Skewness (asymmetry) of the Risk
  Neutral density function ?

6
Why Volatility Skews?

• Market prices governed by
• a) Anticipated dynamics (future behavior of
  volatility or jumps)
• b) Supply and Demand
• To arbitrage European options, estimate a) to
  capture risk premium b)
• To arbitrage (or correctly price) exotics, find
  Risk Neutral dynamics calibrated to the market

7
Modeling Uncertainty

• Main ingredients for spot modeling
• Many small shocks Brownian Motion (continuous
  prices)
• A few big shocks Poisson process (jumps)

8
2 mechanisms to produce Skews (1)

• To obtain downward sloping implied volatilities
• a) Negative link between prices and volatility
• Deterministic dependency (Local Volatility Model)
• Or negative correlation (Stochastic volatility
  Model)
• b) Downward jumps

9
2 mechanisms to produce Skews (2)

• a) Negative link between prices and volatility
• b) Downward jumps

10
Leverage and Jumps
11
Dissociating Jump Leverage effects

• Variance
• Skewness

12
Dissociating Jump Leverage effects

• Define a time window to calculate effects from
  jumps and
• Leverage. For example, take close prices for 3
  months
• Jump
• Leverage

13
Dissociating Jump Leverage effects
14
Dissociating Jump Leverage effects
15
Break Even Volatilities
16
Theoretical Skew from Prices
? gt

• Problem How to compute option prices on an
  underlying without options?
• For instance compute 3 month 5 OTM Call from
  price history only.
• Discounted average of the historical Intrinsic
  Values.
• Bad depends on bull/bear, no call/put parity.
• Generate paths by sampling 1 day return
  recentered histogram.
• Problem CLT gt converges quickly to same
  volatility for all strike/maturity breaks
  autocorrelation and vol/spot dependency.

17
Theoretical Skew from Prices (2)

• Discounted average of the Intrinsic Value from
  recentered 3 month histogram.
• ?-Hedging compute the implied volatility
  which makes the ?-hedging a fair game.

18
Theoretical Skewfrom historical prices (3)

• How to get a theoretical Skew just from spot
  price history?
• Example
• 3 month daily data
• 1 strike
• a) price and delta hedge for a given within
  Black-Scholes model
• b) compute the associated final Profit Loss
• c) solve for
• d) repeat a) b) c) for general time period and
  average
• e) repeat a) b) c) and d) to get the theoretical
  Skew

19
Strike dependency

• BE volatility is an average of returns, weighted
  by the Gammas, which depend on the strike

20
Theoretical Skewfrom historical prices (4)

21
Theoretical Skewfrom historical prices (4)

22
Theoretical Skewfrom historical prices (4)

23
Theoretical Skewfrom historical prices (4)

24
Local Volatility Model
25
One Single Model

• We know that a model with dS s(S,t)dW would
  generate smiles.
• Can we find s(S,t) which fits market smiles?
• Are there several solutions?
• ANSWER One and only one way to do it.

26
The Risk-Neutral Solution
But if drift imposed (by risk-neutrality),
uniqueness of the solution
27
Forward Equation

• BWD Equation price of one option for
  different
• FWD Equation price of all options
  for current
• Advantage of FWD equation
• If local volatilities known, fast computation of
  implied volatility surface,
• If current implied volatility surface known,
  extraction of local volatilities

28
Forward Equations (2)

• Several ways to obtain them
• Fokker-Planck equation
• Integrate twice Kolmogorov Forward Equation
• Tanaka formula
• Expectation of local time
• Replication
• Replication portfolio gives a much more financial
  insight

29
Volatility Expansion

• K,T fixed. C0 price with LVM
• Real dynamics
• Ito
• Taking expectation
• Equality for all (K,T) ?

30
Summary of LVM Properties

• is the initial volatility surface
• compatible with local vol
• compatible with
• (calibrated SVM are noisy versions of LVM)
• deterministic function of (S,t) (if no
  jumps)
• future smile FWD smile from local vol
• Extracts the notion of FWD vol (Conditional
  Instantaneous Forward Variance)

31
Stochastic Volatility Models
32
Heston Model
Solved by Fourier transform
33
Role of parameters

• Correlation gives the short term skew
• Mean reversion level determines the long term
  value of volatility
• Mean reversion strength
• Determine the term structure of volatility
• Dampens the skew for longer maturities
• Volvol gives convexity to implied vol
• Functional dependency on S has a similar effect
  to correlation

34
(No Transcript)
35
(No Transcript)
36
(No Transcript)
37
(No Transcript)
38
(No Transcript)
39
Spot dependency

• 2 ways to generate skew in a stochastic vol model
• Mostly equivalent similar (St,st ) patterns,
  similar future
• evolutions
• 1) more flexible (and arbitrary!) than 2)
• For short horizons stoch vol model ? local vol
  model independent noise on vol.

40
SABR model

• F Forward price
• with correlation r

41
(No Transcript)
42
(No Transcript)
43
(No Transcript)
44
(No Transcript)
45
(No Transcript)
46
The SABR Claim
47
Market behavior
48
SABR fallacy

• SABR claims to dissociate vol dynamics from Skew
  fitting
• Many banks manage their vol risk with SABR
• BUT if 2 SABR models fit the same skew, they
  generate essentially the same vol dynamics, which
  coincide in average with LVM dynamics!

49
Calibration according to SABR

• Backbone as a function of F with
  frozen
• depends only
  on
• Many banks calibrate by
• 1) Estimating from historical backbone
• 2) Then adjusting to fit the skew

50
Problems

• Freezing ignores which actually impacts
  the average backbone
• SABR can be rewritten with the lognormal
  volatility
• In this parameterization instead of
  , freezing gives
  the backbone is then constant!
• The backbone depends on the
  parameterization , it is not intrinsic to
• the model it is a flawed concept

51
Message from LVM

• In particular for short
  maturities
• so average backbone local vols!
• In the absence of jumps, the skew is due to
  average levels
• of vols higher on the downside
• If you were in that region
• increases

52
2 fitting models

• SABR A
• SABR B
• calibrated to A
• Same skew
  Different backbones

53
Comparison
Scattered plot Average
backbone
in average
Same skew similar vol dynamics
LVM vol dynamics
54
SABR Conclusion

• To dissociate vol dynamics from skew fitting,
  jumps are needed
• In a smile (as opposed to skew) dominated market,
  it is less clear as and cancel their
  first order impact
• Banks may be unaware of the inconsistency of
  their risk management

55
Forward Skew
56
Forward Skews
In the absence of jump model fits market
This constrains a) the sensitivity of the ATM
short term volatility wrt S b) the average
level of the volatility conditioned to STK. a)
tells that the sensitivity and the hedge ratio of
vanillas depend on the calibration to the
vanilla, not on local volatility/ stochastic
volatility. To change them, jumps are
needed. But b) does not say anything on the
conditional forward skews.
57
Controlling Fwd Skew

• Many products depend on short term skew in the
  future
• Example Napoleon, globally/locally
  capped/floored cliquet
• What does current vol surface tell us on future
  short term skew?

Local cap/floor
Global cap/floor
58
Toy model
A)
freeze vol at beginning of month
flat 1 month skew but if decreases generate
initial skew
but flat future skew
59
Toy model
B)
(or slightly increasing in )
but not flat future skew
60
Limitations

• Behavior of future 1 month skew very different if
  start date is mid month instead of beginning of
  month
• The amplitude of jumps is a measure of the
  wrongness of the model
• LVM corresponds to the jumpless case

61
Sensitivity of ATM volatility / S
At t, short term ATM implied volatility
st. As st is random, the sensitivity is defined
only in average
In average, follows . Optimal
hedge of vanilla under calibrated stochastic
volatility corresponds to perfect hedge
ratio under LVM.
62
Market Model of Implied Volatility

• Implied volatilities are directly observable
• Can we model directly their dynamics?
• where is the implied volatility of a given
• Condition on dynamics?

63
Drift Condition

• Apply Itos lemma to
• Cancel the drift term
• Rewrite derivatives of
• gives the condition that the drift of
  must satisfy.
• For short T, we get the Short Skew Condition
  (SSC)
• close to the money
• ? Skew determines u1

64
Optimal hedge ratio ?H

• BS Price at t of Call option
  with strike K, maturity T, implied vol
• Ito
• Optimal hedge minimizes PL variance

Implied Vol sensitivity
BS Vega
BS Delta
65
Optimal hedge ratio ?H II

• With
• ? Skew determines u1, which determines ?H

66
Smile Arbitrage
67
Deterministic future smiles

• It is not possible to prescribe just any future
  smile
• If deterministic, one must have
• Not satisfied in general

68
Det. Fut. smiles no jumps gt FWD smile

• If
• stripped from Smile S.t
• Then, there exists a 2 step arbitrage
• Define
• At t0 Sell
• At t
• gives a premium PLt at t, no loss at T
• Conclusion independent of
  
• from initial smile

S
69
Consequence of det. future smiles

• Sticky Strike assumption Each (K,T) has a fixed
  independent of (S,t)
• Sticky Delta assumption depends only on
  moneyness and residual maturity
• In the absence of jumps,
• Sticky Strike is arbitrageable
• Sticky ? is (even more) arbitrageable

70
Example of arbitrage with Sticky Strike

• Each CK,T lives in its Black-Scholes (
  )world
• PL of Delta hedge position over dt
• If no
  jump

71
Arbitrage with Sticky Delta

• In the absence of jumps, Sticky-K is
  arbitrageable and Sticky-? even more so.
• However, it seems that quiet trending market (no
  jumps!) are Sticky-?.
• In trending markets, buy Calls, sell Puts and
  ?-hedge.
• Example

S
PF
?-hedged PF gains from S induced volatility moves.
Vega gt Vega
S
PF
Vega lt Vega
72
Conclusion

• Both leverage and asymmetric jumps may generate
  skew but they generate different dynamics
• The Break Even Vols are a good guideline to
  identify risk premia
• The market skew contains a wealth of information
  and in the absence of jumps,
• The spot correlated component of volatility
• The average behavior of the ATM implied when the
  spot moves
• The optimal hedge ratio of short dated vanilla
• The price of options on RV
• If market vol dynamics differ from what current
  skew implies, statistical arbitrage

73

• BLOOMBERG, BLOOMBERG PROFESSIONAL, BLOOMBERG
  MARKETS, BLOOMBERG NEWS, BLOOMBERG ANYWHERE,
  BLOOMBERG TRADEBOOK, BLOOMBERG BONDTRADER,
  BLOOMBERG TELEVISION, BLOOMBERG RADIO, BLOOMBERG
  PRESS and BLOOMBERG.COM are trademarks and
  service marks of Bloomberg Finance L.P., a
  Delaware limited partnership, or its
  subsidiaries. The BLOOMBERG PROFESSIONAL service
  (the "BPS") is owned and distributed locally by
  Bloomberg Finance L.P. (BFLP) and its
  subsidiaries in all jurisdictions other than
  Argentina, Bermuda, China, India, Japan and Korea
  (the "BLP Countries"). BFLP is a wholly-owned
  subsidiary of Bloomberg L.P. ("BLP"). BLP
  provides BFLP with all global marketing and
  operational support and service for these
  products and distributes the BPS either directly
  or through a non-BFLP subsidiary in the BLP
  Countries.