Processes matching given marginals A new fake Brownian motion Existence of Calibrated RSLV models Fake Brownian motion and calibration of a Regime Switching Local Volatility model Alexandre Zhou Joint work with Benjamin Jourdain Université Paris-Est Mathrisk PhD Seminar, CERMICS September 26 2016
Processes matching given marginals A new fake Brownian motion Existence of Calibrated RSLV models Plan Processes matching given marginals 1 Motivation Simulation of calibrated LSV models and theoretical results A new fake Brownian motion 2 The studied problem Main result Ideas of proof Existence of Calibrated RSLV models 3 The calibrated RSLV model Main result
Processes matching given marginals A new fake Brownian motion Existence of Calibrated RSLV models Outline Processes matching given marginals 1 Motivation Simulation of calibrated LSV models and theoretical results A new fake Brownian motion 2 The studied problem Main result Ideas of proof Existence of Calibrated RSLV models 3 The calibrated RSLV model Main result
Processes matching given marginals A new fake Brownian motion Existence of Calibrated RSLV models Motivation Fake Brownian motion A fake Brownian motion ( X t ) t ≥ 0 is a continuous martingale that has the same marginal distributions as the Brownian motion ( W t ) t ≥ 0 but is not a Brownian motion. Examples by Albin (2007) and Oleszkiewicz (2008) Hobson (2009): fake exponential Brownian motion and more general martingale diffusions. Stochastic processes matching given marginals is a question arising in mathematical finance.
Processes matching given marginals A new fake Brownian motion Existence of Calibrated RSLV models Motivation Trying to match marginals The market gives the prices of European Calls C ( T , K ) for T , K > 0 (idealized situation; in practice only ( C ( T i , K i )) 1 ≤ i ≤ I ). A model ( S t ) t ≥ 0 is calibrated to European options if � e − rT ( S T − K ) + � ∀ T , K ≥ 0 , C ( T , K ) = E . By Breeden and Litzenberger (1978), {prices of European Call options for all T , K > 0} ⇐ ⇒ {marginal distributions of ( S t ) t ≥ 0 }. Dupire Local Volatility model (1992), matching market marginals: dS t = rS t dt + σ Dup ( t , S t ) S t dW t � 2 ∂ T C ( T , K ) + rK ∂ K C ( T , K ) σ Dup ( T , K ) = K 2 ∂ 2 KK C ( T , K )
Processes matching given marginals A new fake Brownian motion Existence of Calibrated RSLV models Motivation LSV models Motivation: get processes with richer dynamics (e.g. take into account volatility risk) and satisfying marginal constraints. Alexander and Nogueira (2004) and Piterbarg (2006): Local and Stochastic Volatility (LSV) model dS t = rS t + f ( Y t ) σ ( t , S t ) S t dW t “Adding uncertainty” to LV models by a random multiplicative factor f ( Y t ) , ( Y t ) t ≥ 0 is a stochastic process.
Processes matching given marginals A new fake Brownian motion Existence of Calibrated RSLV models Motivation Calibration of LSV Models By Gyongy’s theorem (1988), the LSV model is calibrated to C ( T , K ) , ∀ T , K > 0 if � � f 2 ( Y t ) | S t σ 2 ( t , S t ) = σ 2 Dup ( t , S t ) E σ Dup ( t , x ) σ ( t , x ) = � E [ f 2 ( Y t ) | S t = x ] The obtained SDE is nonlinear in the sense of McKean: f ( Y t ) dS t = rS t dt + σ Dup ( t , S t ) S t dW t . � E [ f 2 ( Y t ) | S t ]
Processes matching given marginals A new fake Brownian motion Existence of Calibrated RSLV models Simulation of calibrated LSV models and theoretical results Simulation results Madan and Qian, Ren (2007): solve numerically the associated Fokker-Planck PDE, and get the joint-law of ( S t , Y t ) . Guyon and Henry-Labordère (2011): efficient calibration procedure based on kernel approximation of the conditional expectation. Subsequent extension to stochastic interest rates, stochastic dividends, multidimensional local correlation models,... However, calibration errors seem to appear when the range of f ( Y ) is too large.
Processes matching given marginals A new fake Brownian motion Existence of Calibrated RSLV models Simulation of calibrated LSV models and theoretical results Theoretical results Abergel and Tachet (2010): perturbation of the constant f case (Dupire) − → existence for the restriction to a compact spatial domain of the associated Fokker-Planck equation when sup f − inf f small. Global existence and uniquess to LSV models remain on open problem.
Processes matching given marginals A new fake Brownian motion Existence of Calibrated RSLV models Outline Processes matching given marginals 1 Motivation Simulation of calibrated LSV models and theoretical results A new fake Brownian motion 2 The studied problem Main result Ideas of proof Existence of Calibrated RSLV models 3 The calibrated RSLV model Main result
Processes matching given marginals A new fake Brownian motion Existence of Calibrated RSLV models The studied problem A simpler SDE Let Y be a r.v. with values in Y : = { y 1 , ..., y d } . We assume ∀ i ∈ { 1 , ..., d } , α i = P ( Y = y i ) > 0. We study the SDE (FBM), with f > 0: f ( Y ) dX t = dW t � E [ f 2 ( Y ) | X t ] X 0 ∼ µ . X 0 , Y , ( W t ) t ≥ 0 are independent.
Processes matching given marginals A new fake Brownian motion Existence of Calibrated RSLV models The studied problem Fake Brownian Motion Lemma If the positive function f is not constant on Y , then any solution to the SDE f ( Y ) = dW t , X 0 = 0 dX t � E [ f 2 ( Y ) | X t ] with Y and ( W t ) t ≥ 0 indep. is a fake Brownian motion. If ( X t ) t ≥ 0 is a Brownian motion then a.s. ∀ t ≥ 0, < X > t = t i.e. f 2 ( Y ) f ( Y ) √ ds a.e. E [ f 2 ( Y ) | X s ] = 1 = E [ f 2 ( Y ) | X s ] so that a.s. ∀ t ≥ 0, X t = W t . � = E � � � f 2 ( Y ) | X t f 2 ( Y ) Therefore X t ⊥ Y , E and � � f 2 ( Y ) = E f 2 ( Y ) is constant.
Processes matching given marginals A new fake Brownian motion Existence of Calibrated RSLV models Main result Existence to SDE (FBM) and fake Brownian motion We define for i ∈ { 1 , ..., d } , λ i : = f 2 ( y i ) , λ min : = min i λ i , λ max : = max i λ i . Theorem Under Condition (C): � λ i � λ i � � + λ max + λ min ( C ) : ∑ ∨ ∑ < 2 d + 4 . λ max λ i λ min λ i i i there exists a weak solution to the SDE (FBM) on [ 0 , T ] .
Processes matching given marginals A new fake Brownian motion Existence of Calibrated RSLV models Main result The associated Fokker Planck system For i ∈ { 1 , ..., d } , define p i s.t., for φ ≥ 0 and measurable, � = � � φ ( X t ) 1 { Y = y i } R φ ( x ) p i ( t , x ) dx . E The associated Fokker-Planck system is: � ∑ j p j � ∀ i ∈ { 1 , ..., d } , ∂ t p i = 1 2 ∂ 2 λ i p i xx ∑ j λ j p j p i ( 0 ) = α i µ ∑ j p j is solution to the heat equation.
Processes matching given marginals A new fake Brownian motion Existence of Calibrated RSLV models Ideas of proof Rewriting into divergence form The system can be rewritten in divergence form: ∂ t p 1 ∂ x p 1 · = 1 · 2 ∂ x ( I d + M ( p )) · · . ∂ t p d ∂ x p d ∑ j � = i λ j p j ∑ j ( λ i − λ j ) p j M ii ( p ) = � � 2 , ∑ j λ j p j λ i p i ∑ j ( λ j − λ k ) p j M ik ( p ) = , i � = k . � � 2 ∑ j λ j p j
Processes matching given marginals A new fake Brownian motion Existence of Calibrated RSLV models Ideas of proof Computing standard energy estimates (S.E.E) Multiply the system by ( p 1 , ..., p d ) , and integrate in x : ∂ x p 1 � � � d � 1 = − 1 · p 2 ∑ R ( ∂ x p 1 , ..., ∂ x p d ) ( I d + M ( p )) 2 ∂ t i dx dx . · 2 R i = 1 ∂ x p d Goal : S.E.E. in L 2 ([ 0 , T ] , H 1 ( R )) ∩ L ∞ ([ 0 , T ] , L 2 ( R )) . + ) ∗ = R d We want (coercivity property): for ( R d + \ { ( 0 , ..., 0 ) } + ) ∗ , ∀ y ∈ R d , y ∗ M ( ρ ) y ≥ ( ǫ − 1 ) | y | 2 . ∃ ǫ > 0 s . t . ∀ ρ ∈ ( R d
Processes matching given marginals A new fake Brownian motion Existence of Calibrated RSLV models Ideas of proof M ( ρ ) as a convex combination λ : = ∑ j λ j ρ j λ j ρ j ∑ k λ k ρ k , ∑ d ∑ j ρ j , w j : = j = 1 w j = 1. � � � � 1 − λ k , and if j � = k , M jk ( ρ ) = w j λ i . M ii ( ρ ) = ∑ j � = i w j λ − 1 λ Then M ( ρ ) = ∑ d j = 1 w j M j ( λ ) , where � � λ 1 λ − 1 · � λ j − 1 � − 1 λ � � � � � � � � 1 − λ j − 1 1 − λ j + 1 1 − λ 1 1 − λ d M j ( λ ) : = · · ← row j . 0 λ λ λ λ � λ j + 1 � − 1 λ · � � λ d λ − 1
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