Building a Good Implied Volatility Surface and a Robust Local Volatility Model
Bruno Dupire Bloomberg L.P.
Sao Paulo, December 1, 2008
1) Building a Good Implied Volatility Surface
Uses of the Surface
Implied 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
Aug 07 Sept 07 Oct 07 Dec 07
Mar 08
Jun 08
Dec 08
Mar 09
S&P Strikes and Maturities
Jun 09
S&P 500
Wish List
A good implied volatility surface should be Accurate Smooth Arbitrage free Robust to missing data
What NOT to do
Interpolate prices Extrapolate volatilities with 0, flat or linear Create inter/intra maturity arbitrage
Price Interpolation
Listed Implied Volatilities
The first step is to get implied volatilities of listed options Synchronicity Implied forward/dividends DeAmericanization: EEP is model dependent
Issues with Stocks
Less option data American prices - DeAmericanization: 1 by 1 vs global Discrete dividends create discontinuities in implied volatilities Event calendar can be inserted
Event Calendar
What is Usually Done
Common market practice: 1) fit each market maturity with a few parameters (typically 4 or 5) 2) interpolate between maturities
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
The Plan
Chose a good fitting model Compute residuals (market model) Interpolate the residuals Add them back
The Logic
1) 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
Heston Model
Dynamics
dSt = (r q )dt + vt dWt St dvt = ( vt )dt + vt dBt dBt dWt = dt
Impact of VolVol
Impact of Mean Reversion
Impact of Correlation
Impact of Everything
Heston Computation
FFT: fractional/decay Integration Control variate Analytical asymptote
Heston Fit to S&P500
Residual Fitting
Non parametric interpolation Decaying extrapolation to inherit arbitrage free asymptotes
S&P 500 Residuals
S&P 500 Residuals
S&P 500 Fit
Cumulative variance as a function of strike. One curve per maturity. Dotted line: Heston, Red line: Heston + residuals, bubbles: market
RMS in bps BS: 305 Heston: 47 H+residuals: 7
S&P 500 implied volatility
2) Exploiting the Surface
S&P 500 RN Densities
S&P 500/Euro Stoxx 6m RN Densities
S&P 500/NASDAQ 6m RN Densities
Variance Swaps
Variance swap replicated by Log profile, which requires all strikes. Issues: Discrete strikes Impact of jumps Stochastic interest rates
Cheap/rich analysis
Residuals trading
3) Building LVM
Local Volatility Model
Simplest model to fit a full surface Forward volatilities that can be locked
dS = ( r d ) dt + ( S , t ) dW S C 2 (K , T ) 2 2C C = K (r d )K d C 2 T 2 K K
Summary of LVM
Simplest model that fits vanillas Second most used model (after Black-Scholes) in Equity Derivatives Local volatilities: forward vols that can be locked by a vanilla PF Stoch vol model calibrated
2 E[ t2 St = S ] = loc ( S , t )
If no jumps, deterministic implied vols => LVM
S&P500 implied and local vol
Exotic Pricing : OVME
Volatility Expansion
X : exotic option and : local variances functions
X ( + ) = X () + m( S , t ) ( S , t ) dS dt
S2 E tX | St = S + ( S , t ) is the sensitivity where m(S,t) = 2 of X to the local volatility at (S,t) (Frechet derivative).
One Touch Option - Price
Black-Scholes model S0=100, H=110, =0.25, T=0.25
One Touch Option -
1 O.T . : m( S , t ) = . .P.S 2 2
Up-Out Call - Price
Black-Scholes model S0=100, H=110, K=90, =0.25, T=0.25
Up-Out Call -
1 UOC : m( S , t ) = . .P.S 2 2
Vanilla hedging portfolios
S2 E tX | S t = S + ( S , t ) is the sensitivity of X Recall m( S , t ) = 2 to the local volatility at (S,t). Portfolio PF = ( K , T ) C K ,T dK dT of vanillas hedges all small volatility moves Solved by : 2 (K ,T ) 2 h( K , T ) 2 h( K , T ) + ( K , T ) = t S 2 (S,t), PF ( S , t ) = E tX | St = S = h( S , t )
Link /Vega
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 for exotic pricing and risk management
Quantitative Corner
