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Functional Ito Calculus
and PDE for Path-Dependent Options
Bruno DupireBloomberg L.P.
PDE and Mathematical Finance
KTH, Stockholm, August 19, 2009
Outline1) Functional Ito Calculus
• Functional Ito formula• Functional Feynman-Kac• PDE for path dependent options
2) Volatility Hedge
• Local Volatility Model• Volatility expansion• Vega decomposition• Robust hedge with Vanillas• Examples
1) Functional Ito Calculus
Why?
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Review of Ito Calculus
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• Functional Ito Calculus
current value
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Functional Feynman-Kac Formula
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dx t = a(X t ) dt + b(X t ) dwt
For g and r suitably integrable, g : ΛT →ℜ, r : Λ →ℜ we define f :Λ →ℜ by
f (Yt ) ≡ E[e− r(Z u )
t
T
∫ dug(ZT ) | Zt = Yt ]
where for u∈ [0, t],ZT (u) = Yt (u)
for u∈ [t,T],dzu = a(Zu) du + b(Zu) dwu
⎧ ⎨ ⎩
Then f satisfies
Δ t f (X t ) + a(X t )Δx f (X t ) − r(X t ) f (X t ) +b2(X t )
2Δxx f (X t ) = 0
(From functional Ito formula, using that e− r(Z u )
0
t
∫ duf (X t ) is a martingale)
Delta Hedge/Clark-Ocone
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Functional PDE for Exotics
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Recallthat
df (X t ) = Δx f (X t ) dx + Δ t f (X t ) dt +1
2b2Δxx f (X t ) dt
The portfolio PF of option f with a short position of Δx f stocksgives :
dPF(X t ) = Δ t f (X t ) dt +1
2b2Δxx f (X t ) dt
In the absence of arbitrage,
Δ t f (X t ) +1
2b2Δxx f (X t ) + r(X t )(Δx f (X t )x t − f (X t )) = 0
The ?/Θ trade - off for European options also holds for
path dependent options, even with an infinite number of state variables.
However, in general ? and Θ will be path dependent.
Classical PDE for Asian
term.convection bothering a is
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2) Robust Volatility Hedge
Local Volatility Model• Simplest model to fit a full surface• Forward volatilities that can be locked
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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
),(][ 22 tSSSE loctt
S&P500 implied and local vols
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 bpsBS: 305Heston: 47H+residuals: 7
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 to
local volatility
P&L from Delta hedging
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One Touch Option - Price
Black-Scholes model S0=100, H=110, σ=0.25, T=0.25
One Touch Option - Γ
PtSmTO ..2
1),(:..
Up-Out Call - Price
Black-Scholes model S0=100, H=110, K=90, σ=0.25, T=0.25
Up-Out Call - Γ
PtSmUOC ..2
1),(:
Black-Scholes/LVM comparison
price. LVM reach the toenables Scholes-Black theofinput y volatilitno case, In this
Vanilla hedging portfolio I
?),(function get the can we How
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Conclusion
• Ito calculus can be extended to functionals of price paths
• Local volatilities are forward values that can be locked
• LVM crudely states these volatilities will be realised
• It is possible to hedge against this assumption
• It leads to a strike/maturity decomposition of the volatility risk of the full portfolio