Building a Good Vol Surface

8/3/2019 Building a Good Vol Surface

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Building a Good Implied Volatility

Surface and a Robust LocalVolatility Model

Bruno Dupire

Bloomberg L.P.

Sao Paulo, December 1, 2008


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1) Building a Good ImpliedVolatility Surface


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Uses of the SurfaceImplied volatility surface is the central object for

derivatives, as yield curve is for fixed income

Gives prices of vanillas of all strikes and

maturities

Gives variance swaps term structure

Gives forward and local volatilities

Constrains the price of exotics

Identifies cheap/expensive strikes and maturities


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S&P Strikes and Maturities

T

K

Sept07

Oct07

Dec07

Mar08

Jun08

Dec08

Mar09

Jun09

Aug07


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S&P 500


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Wish List

A good implied volatility surface should be

Accurate

Smooth

Arbitrage free Robust to missing data


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What NOT to do

Interpolate prices

Extrapolate volatilities with 0, flat or linear

Create inter/intra maturity arbitrage


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Price Interpolation


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Listed Implied Volatilities

The first step is to get implied volatilities of

listed options

Synchronicity Implied forward/dividends

DeAmericanization: EEP is modeldependent


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Issues with Stocks

Less option data

American prices

- DeAmericanization: 1 by 1 vs global

Discrete dividends create discontinuities inimplied volatilities

Event calendar can be inserted


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Event Calendar


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What is Usually Done

Common market practice:

1) fit each market maturity with a few

parameters (typically 4 or 5)

2) interpolate between maturities


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The Meaning of Interpolation To interpolate is to guess

To extrapolate is even more so

Stabilize the known data

- easier with volatilities than prices

- still strong skew- other models may normalize further


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The Plan

Chose a good fitting model

Compute residuals (market model)

Interpolate the residuals

Add them back


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The Logic1) Fit of parametric model on all data

- Parametric => robust gap filling

- Model => arbitrage free

2) Non parametric fine tuning => accurate

Both 1) and 2) are infinitely smooth


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Heston ModelDynamics

dtdWdBdBvdtvdv

dWvdtqrS

dS

tt

tttt

tt

t

t

=

+=

+=

)(

)(


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Impact of VolVol


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Impact of Mean Reversion


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Impact of Correlation


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Impact of Everything


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Heston Computation FFT: fractional/decay

Integration

Control variate

Analytical asymptote


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Heston Fit to S&P500


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Residual Fitting Non parametric interpolation

Decaying extrapolation to inherit arbitragefree asymptotes


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S&P 500 Residuals


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S&P 500 FitCumulative variance as a function of strike. One curve per maturity.Dotted line: Heston, Red line: Heston + residuals, bubbles: market

RMS in bps

BS: 305Heston: 47H+residuals: 7


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S&P 500 implied volatility


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2) Exploiting the Surface


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S&P 500 RN Densities


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S&P 500/Euro Stoxx 6m RN Densities


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S&P 500/NASDAQ 6m RN Densities


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Variance SwapsVariance swap replicated by Log profile,

which requires all strikes. Issues: Discrete strikes

Impact of jumps Stochastic interest rates


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Cheap/rich analysisResiduals trading


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3) Building LVM


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Local Volatility Model Simplest model to fit a full surface

Forward volatilities that can be locked

( ) ( ) CdKCKdr

KCKTK

TC

dWtSdtdrS

dS

=

+=

2

2

2

2

2,

),()(


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Summary of LVM Simplest model that fits vanillas

Second most used model (after Black-Scholes) inEquity Derivatives

Local volatilities: forward vols that can be locked by avanilla PF

Stoch vol model calibrated

If no jumps, deterministic implied vols => LVM

),(][ 22 tSSSE loctt ==


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S&P500 implied and local vol


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Exotic Pricing : OVME


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Volatility Expansion

[ ]

).derivative(Frechetatyvolatilitlocalthetoof

ysensitivittheis),(|2

where

functionsvarianceslocal:and

optionexotic:

2

(S,t)X

tSSSES

m(S,t)

X

t

X

t

+==

dtdStStSmXX ),(),()()(

+=+


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One Touch Option - PriceBlack-Scholes model S0=100, H=110, =0.25, T=0.25


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One Touch Option -


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2...2

1),(:.. SPtSmTO =


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Up-Out Call - PriceBlack-Scholes model S0=100, H=110, K=90, =0.25, T=0.25


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Up-Out Call -


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2...2

1),(: SPtSmUOC =


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Vanilla hedging portfolios[ ]

[ ]

+

=

===

=

==

+

2

22

,

2

),(2

),(

),(),(

:bySolved

),(|),(,movesvolatility

smallallhedgesvanillasof),(Portfolio

.atyvolatilitlocaltheto

ofysensitivittheis),(|2

),(Recall

S

TKhTK

t

TKhTK

tShSSEtS(S,t)

dTdKCTKPF

(S,t)

XtSSSES

tSm

t

X

t

PF

TK

t

X

t


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/VegaLink


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Conclusion

Parametric fit for robustness

Non parametric interpolation for accuracy

Implied volatility surface is central to many uses

Local volatilities are the important first step forexotic pricing and risk management


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Quantitative Corner

Building a Good Vol Surface

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