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Functional Itô Calculus and PDEs for Path-Dependent Options. Bruno Dupire Bloomberg L.P. Rutgers University Conference New Brunswick, December 4, 2009. Outline. Functional It ô Calculus Functional Itô formula Functional Feynman-Kac PDE for path dependent options 2) Volatility Hedge

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Functional Itô Calculus and PDEs for Path-Dependent Options


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functional it calculus and pdes for path dependent options

Functional Itô Calculus and PDEs for Path-Dependent Options

Bruno Dupire

Bloomberg L.P.

Rutgers University Conference

New Brunswick, December 4, 2009

outline
Outline
  • Functional Itô Calculus
  • Functional Itô formula
  • Functional Feynman-Kac
  • PDE for path dependent options

2) Volatility Hedge

  • Local Volatility Model
  • Volatility expansion
  • Vega decomposition
  • Robust hedge with Vanillas
  • Examples
review of it calculus
Review of Itô Calculus

current value

  • 1D
  • nD
  • infiniteD
  • Malliavin Calculus
  • Functional Itô Calculus

possible evolutions

p l of a delta hedged vanilla

Option Value

P&L

Break-even

points

Delta hedge

P&L of a delta hedged Vanilla
local volatility model
Local Volatility Model
  • Simplest model to fit a full surface
  • Forward volatilities that can be locked
summary of lvm
Summary of LVM
  • Simplest model that fits vanillas
  • In Europe, second most used model (after Black-Scholes) in Equity Derivatives
  • Local volatilities: fwd vols that can be locked by a vanilla PF
  • Stoch vol model calibrated 
  • If no jumps, deterministic implied vols => LVM
s p 500 fit
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

hedge within outside lvm
Hedge within/outside LVM
  • 1 Brownian driver => complete model
  • Within the model, perfect replication by Delta hedge
  • Hedge outside of (or against) the model: hedge against volatility perturbations
  • Leads to a decomposition of Vega across strikes and maturities
implied and local volatility bumps
Implied and Local Volatility Bumps

implied to

local volatility

one touch option price
One Touch Option - Price

Black-Scholes model S0=100, H=110, σ=0.25, T=0.25

up out call price
Up-Out Call - Price

Black-Scholes model S0=100, H=110, K=90, σ=0.25, T=0.25

conclusion
Conclusion
  • Itô calculus can be extended to functionals of price paths
  • Price difference between 2 models can be computed
  • We get a variational calculus on volatility surfaces
  • It leads to a strike/maturity decomposition of the volatility risk of the full portfolio