Local Volatility Pricing Models for Long-Dated FX Derivatives

Transcription

1 Local Volatility Pricing Models for Long-Dated FX Derivatives G. Deelstra, G. Rayee Université Libre de Bruxelles Gregory Rayee (ULB) Bachelier Congress 2010 Toronto, june / 37

2 Outline of the talk Outline of the talk 1 Introduction 2 The Model 3 The local volatility function 4 Calibration 5 Extension Gregory Rayee (ULB) Bachelier Congress 2010 Toronto, june / 37

3 Introduction Introduction Recent years, the long-dated (maturity > 1 year) foreign exchange (FX) option s market has grown considerably Vanilla options (European Call and Put) Exotic options (barriers,...) Hybrid options (PRDC swaps) Gregory Rayee (ULB) Bachelier Congress 2010 Toronto, june / 37

4 Introduction Introduction A suitable pricing model for long-dated FX options has to take into account the risks linked to: domestic and foreign interest rates by using stochastic processes for both domestic and foreign interest rates dr d (t) = [θ d (t) α d (t)r d (t)]dt + σ d (t)dwd DRN (t), dr f (t) = [θ f (t) α f (t)r f (t)]dt + σ f (t)dwd FRN (t) the volatility of the spot FX rate (Smile/Skew effect) by using a local volatility σ(t, S(t)) for the FX spot ds(t) = (r d (t) r f (t))s(t)dt + σ(t, S(t))S(t)dWS DRN (t), by using a stochastic volatility ν(t) for the FX spot and/or jump ds(t) = (r d (t) r f (t))s(t)dt + ν(t)s(t)dws DRN (t), dν(t) = κ(θ ν(t))dt + ξ ν(t)dwν DRN (t) Gregory Rayee (ULB) Bachelier Congress 2010 Toronto, june / 37

5 Introduction Introduction Stochastic volatility models with stochastic interest rates: R. Ahlip, Foreign exchange options under stochastic volatility and stochastic interest rates, International Journal of Theoretical and Applied Finance (IJTAF), vol. 11, issue 03, pages , (2008). J. Andreasen, Closed form pricing of FX options under stochastic rates and volatility, Global Derivatives Conference, ICBI, (May 2006). A. Antonov, M. Arneguy, and N. Audet, Markovian projection to a displaced volatility Heston model, Available at (2008). A. van Haastrecht, R. Lord, A. Pelsser, and D. Schrager, Pricing long-maturity equity and fx derivatives with stochastic interest rates and stochastic volatility, Available at id= , (2008). A. van Haastrecht and A. Pelsser, Generic Pricing of FX, Inflation and Stock Options Under Stochastic Interest Rates and Stochastic Volatility, Available at id= , (February 2009). Local volatility models with stochastic interest rates: V. Piterbarg, Smiling hybrids, Risk, 66-71, (May 2006). Gregory Rayee (ULB) Bachelier Congress 2010 Toronto, june / 37

6 Introduction Introduction Advantages of working with a local volatility model: the local volatility σ(t, S(t)) is a deterministic function of both the FX spot and time. It avoids the problem of working in incomplete markets in comparison with stochastic volatility models and is therefore more appropriate for hedging strategies has the advantage to be calibrated on the complete implied volatility surface, local volatility models usually capture more precisely the surface of implied volatilities than stochastic volatility models Gregory Rayee (ULB) Bachelier Congress 2010 Toronto, june / 37

7 The model Introduction The calibration of a model is usually done on the vanilla options market local and stochastic volatility models (well calibrated) return the same price for these options. But calibrating a model to the vanilla market is by no mean a guarantee that all type of options will be priced correctly example: We have compared short-dated barrier option market prices with the corresponding prices derived from either a Dupire local volatility or a Heston stochastic volatility model both calibrated on the vanilla smile/skew. Gregory Rayee (ULB) Bachelier Congress 2010 Toronto, june / 37

8 Introduction Introduction A FX market characterized by a mild skew (USDCHF) exhibits mainly a stochastic volatility behavior, A FX market characterized by a dominantly skewed implied volatility (USDJPY) exhibit a stronger local volatility component. Gregory Rayee (ULB) Bachelier Congress 2010 Toronto, june / 37

9 Introduction Introduction The market dynamics could be better approximated by a hybrid volatility model that contains both stochastic volatility dynamics and local volatility ones. example: ds(t) = (r d (t) r f (t))s(t)dt + σ LOC2 (t, S(t)) ν(t)s(t)dws DRN (t), dr d (t) = [θ d (t) α d (t)r d (t)]dt + σ d (t)dwd DRN (t), dr f (t) = [θ f (t) α f (t)r f (t)]dt + σ f (t)dwf FRN (t), dν(t) = κ(θ ν(t))dt + ξ ν(t)dwν DRN (t). The local volatility function σ LOC2 (t, S(t)) can be calibrated from the local volatility that we have in a pure local volatility model! Gregory Rayee (ULB) Bachelier Congress 2010 Toronto, june / 37

10 The model The three-factor model with local volatility The spot FX rate S is governed by the following dynamics ds(t) = (r d (t) r f (t))s(t)dt + σ(t, S(t))S(t)dW DRN S (t), (1) domestic and foreign interest rates, r d and r f follow a Hull-White one factor Gaussian model defined by the Ornstein-Uhlenbeck processes { drd (t) = [θ d (t) α d (t)r d (t)]dt + σ d (t)dw DRN d (t), (2) dr f (t) = [θ f (t) α f (t)r f (t) ρ fs σ f (t)σ(t, S(t))]dt + σ f (t)dw DRN f (t), (3) θ d (t), α d (t), σ d (t), θ f (t), α f (t), σ f (t) are deterministic functions of time. Equations (1), (2) and (3) are expressed in the domestic risk-neutral measure (DRN). (WS DRN (t), Wd DRN (t), Wf DRN (t)) is a Brownian motion under the domestic risk-neutral measure Q d with the correlation matrix 1 ρ Sd ρ Sf ρ Sd 1 ρ df. ρ Sf ρ df 1 Gregory Rayee (ULB) Bachelier Congress 2010 Toronto, june / 37

11 The local volatility function first approach The local volatility derivation : first approach Gregory Rayee (ULB) Bachelier Congress 2010 Toronto, june / 37

12 The local volatility function first approach The local volatility derivation : first approach Consider the forward call price C(K, t) of strike K and maturity t, defined (under the t-forward measure Q t) by C(K, t) = C(K, t) P d (0, t) = EQt [(S(t) K) + ] = Differentiating it with respect to the maturity t leads to + (S(t) K)φ F (S, r d, r f, t)dsdr d dr f. K C(K, t) t = + K (S(t) K) φ F (S, r d, r f, t) t dsdr d dr f we have shown that the t-forward probability density φ F satisfies the following forward PDE: φ F t = (r d (t) f d (0, t)) φ F [(r d (t) r f (t))s(t)φ F ] x [(θ f (t) α f (t) r f (t))φ F ] z + 2 [σ(t, S(t))S(t)σ d (t)ρ Sd φ F ] x y [σ 2 (t, S(t))S 2 (t)φ F ] x [σ(t, S(t))S(t)σ f (t)ρ Sf φ F ] x z [(θ d (t) α d (t) r d (t))φ F ] y 2 [σ 2 d (t)φ F ] y [σ 2 f (t)φ F ] z [σ d (t)σ f (t)ρ df φ F ]. (4) y z Gregory Rayee (ULB) Bachelier Congress 2010 Toronto, june / 37

13 The local volatility function first approach The local volatility derivation : first approach Integrating by parts several times we get C(K, t) t + = f d (0, t) C(K, t) + [r d (t)k r f (t)s(t)]φ F (S, r d, r f, t)dsdr d dr f K (σ(t, K)K)2 φ F (K, r d, r f, t)dr d dr f = f d (0, t) C(K, t) + E Qt [(r d (t)k r f (t)s(t))1 {S(t)>K} ] (σ(t, K)K)2 2 C(K, t) K 2. This leads to the following expression for the local volatility surface in terms of the forward call prices C(K, t) σ 2 (t, K) = C(K,t) f t d (0, t) C(K, t) E Qt [(r d (t)k r f (t) S(t))1 {S(t)>K} ]. 1 K 2 2 C(K,t) 2 K 2 Gregory Rayee (ULB) Bachelier Congress 2010 Toronto, june / 37

14 The local volatility function first approach The local volatility derivation : first approach The (partial) derivatives of the forward call price with respect to the maturity can be rewritten as C(K, t) t = [ C(K,t) P d (0,t) ] t = C(K, t) t 2 C(t, K) K 2 = 2 C(K,t) [ P d (0,t) ] 1 2 C(t, K) K 2 = P d (0, t) K 2. 1 P d (0, t) + f d (0, t) C(t, K), Substituting these expressions into the last equation, we obtain the expression of the local volatility σ 2 (t, K) in terms of call prices C(K, t) σ 2 (t, K) = C(K,t) P t d (0, t)e Qt [(r d (t)k r f (t) S(t))1 {S(t)>K} ]. 1 2 K 2 2 C(K,t) K 2 Gregory Rayee (ULB) Bachelier Congress 2010 Toronto, june / 37

15 The local volatility function second approach The local volatility derivation : second approach Gregory Rayee (ULB) Bachelier Congress 2010 Toronto, june / 37

16 The local volatility function second approach The local volatility derivation : second approach Applying Tanaka s formula to the convex but non-differentiable function e t 0 r d (s)ds (S(t) K) + leads to e t t 0 r d (s)ds (S(t) K) + = (S(0) K) + r d (u)e u 0 r d (s)ds (S(u) K) + du 0 t + e u 0 r d (s)ds 1 {S(u)>K} ds u + 1 t e u 0 r d (s)ds dl K u (S) where L K u (S) is the local time of S defined by L K 1 t t (S) = lim 1 [K,K+ɛ] (S(s))d < S, S > s. ɛ 0 ɛ 0 Using ds(t) = (r d (t) r f (t))s(t)dt + σ(t, S(t))S(t)dWS DRN (t), taking the domestic risk neutral expectation of each side and finally differentiating, dc(k, t) = E Q d [e t 0 r d (s)ds (Kr d (t) r f (t)s(t))1 {S(t)>K} ]dt lim ɛ 0 EQ d [ 1 ɛ 1 [K,K+ɛ](S(t))e t0 r d (s)ds σ 2 (t, S(t)) S 2 (t)]dt. Gregory Rayee (ULB) Bachelier Congress 2010 Toronto, june / 37

17 The local volatility function second approach The local volatility derivation : second approach Using conditional expectation properties, the last term can be rewritten as follows 1 lim ɛ 0 ɛ EQ d [1 [K,K+ɛ] (S(t))e t0 r d (s)ds σ 2 (t, S(t))S 2 (t)] 1 = lim ɛ 0 ɛ EQ d [E Q d [e t0 r d (s)ds S(t)]1 [K,K+ɛ] (S(t))σ 2 (t, S(t))S 2 (t)] = E Q d [e t 0 r d (s)ds S(t) = K]p d (K, t) σ 2 (t, K) K 2 }{{} 2 C(K,t) K 2 where p d (K, t) = φ d (K, r d, r f, t) is the domestic risk neutral density of S(t) in K. This leads to the local volatility expression where the expectation is expressed under the domestic risk neutral measure Q d σ 2 (t, K) = C(K,t) E t Q d [e t 0 r d (s)ds (Kr d (t) r f (t)s(t))1 {S(t)>K} ] 1 2 K. (5) 2 2 C K 2 Gregory Rayee (ULB) Bachelier Congress 2010 Toronto, june / 37

18 The local volatility function second approach The local volatility derivation : second approach dq T dq d t0 = e r d (s)ds Pd (t,t ), you get the P d (0,T ) Making the well known change of measure : expression with the expectation expressed into the t-forward measure Q t σ 2 (t, K) = C(K,t) P t d (0, t)e Qt [(Kr d (t) r f (t)s(t))1 {S(t)>K} ] 1 2 K. 2 2 C K 2 Gregory Rayee (ULB) Bachelier Congress 2010 Toronto, june / 37

19 Calibration Calibration Gregory Rayee (ULB) Bachelier Congress 2010 Toronto, june / 37

20 Calibration Calibration Introduction Before pricing any derivatives with a model, it is usual to calibrate it on the vanilla market, determine all parameters present in the different stochastic processes which define the model in such a way that all European option prices derived in the model are as consistent as possible with the corresponding market ones. Gregory Rayee (ULB) Bachelier Congress 2010 Toronto, june / 37

21 Calibration Calibration Introduction The calibration procedure for the three-factor model with local volatility can be decomposed in three steps: 1 Parameters present in the Hull-White one-factor dynamics for the domestic and foreign interest rates, θ d (t), α d (t), σ d (t), θ f (t), α f (t), σ f (t), are chosen to match European swaption / cap-floors values in their respective currencies. 2 The three correlation coefficients of the model, ρ Sd, ρ Sf and ρ df are usually estimated from historical data. 3 After these two steps, the calibration problem consists in finding the local volatility function of the spot FX rate which is consistent with an implied volatility surface. Gregory Rayee (ULB) Bachelier Congress 2010 Toronto, june / 37

22 Calibration Calibration Introduction σ 2 (t, K) = C(K,t) P t d (0, t)e Qt [(Kr d (t) r f (t)s(t))1 {S(t)>K} ] 1 2 K. 2 2 C K 2 Difficult because of E Qt [(Kr d (t) r f (t)s(t))1 {S(t)>K} ] : there exists no closed form solution it is not directly related to European call prices or other liquid products. Its calculation can obviously be done by using numerical methods but you have to solve (numerically) a three-dimensional PDE: 0 = φ F t + (r d (t) f d (0, t)) φ F + [(r d (t) r f (t))s(t)φ F ] x + [(θ f (t) α f (t) r f (t))φ F ] z [σ(t, S(t))S(t)σ d (t)ρ Sd φ F ] x y 2 [σ 2 (t, S(t))S 2 (t)φ F ] x [σ(t, S(t))S(t)σ f (t)ρ Sf φ F ] x z + [(θ d (t) α d (t) r d (t))φ F ] y 2 [σ 2 d (t)φ F ] y [σ 2 f (t)φ F ] z 2 2 [σ d (t)σ f (t)ρ df φ F ]. (6) y z Gregory Rayee (ULB) Bachelier Congress 2010 Toronto, june / 37

23 Calibration Introduction First method : by adjusting the Dupire formula Gregory Rayee (ULB) Bachelier Congress 2010 Toronto, june / 37

24 Calibration first method Calibration : Comparison between local volatility with and without stochastic interest rates In a deterministic interest rates framework, the local volatility function σ 1f (t, K) is given by the well-known Dupire formula: C(K,t) σ1f 2 (t, K) = t + K(f d (0, t) f f (0, t)) C(K,t) + f K f (0, t)c(k, t). 1 2 K 2 2 C(K,t) K 2 If we consider the three-factor model with stochastic interest rates, the local volatility function is given by C(K,t) σ3f 2 (t, K) = P t d (0, t)e Qt [(Kr d (t) r f (t)s(t))1 {S(t)>K} ]. 1 2 K 2 2 C(K,t) K 2 We can derive the following interesting relation between the simple Dupire formula and its extension σ 2 3f (t, K) σ2 1f (t, K) = KP d (0, t){cov Q t [r f (t) r d (t), 1 {S(t)>K} ] + K 1 CovQ t [r f (t), (S(t) K) + ]}. 1 2 K 2 2 C K 2 (7) Gregory Rayee (ULB) Bachelier Congress 2010 Toronto, june / 37

25 Calibration first method Second method : by mimicking stochastic volatility models Gregory Rayee (ULB) Bachelier Congress 2010 Toronto, june / 37

26 Calibration By mimicking stochastic volatility models Calibrating the local volatility by mimicking stochastic volatility models Consider the following domestic risk neutral dynamics for the spot FX rate ds(t) = (r d (t) r f (t)) S(t) dt + γ(t, ν(t)) S(t) dws DRN (t) ν(t) is a stochastic variable which provides the stochastic perturbation for the spot FX rate volatility. Common choices: 1 γ(t, ν(t)) = ν(t) 2 γ(t, ν(t)) = exp( ν(t)) 3 γ(t, ν(t)) = ν(t) The stochastic variable ν(t) is generally modelled by a Cox-Ingersoll-Ross (CIR) process as for example the Heston stochastic volatility model: dν(t) = κ(θ ν(t))dt + ξ ν(t)dwν DRN (t) a Ornstein-Uhlenbeck process (OU) as for example the Schöbel and Zhu stochastic volatility model: dν(t) = k[λ ν(t)]dt + ξdwν DRN (t) Gregory Rayee (ULB) Bachelier Congress 2010 Toronto, june / 37

27 Calibration By mimicking stochastic volatility models Calibrating the local volatility by mimicking stochastic volatility models Applying Tanaka s formula to the non-differentiable function e t 0 r d (s)ds (S(t) K) +, where ds(t) = (r d (t) r f (t)) S(t) dt + γ(t, ν(t)) S(t) dws DRN (t) dc(k, t) = E Q d [e t 0 r d (s)ds (Kr d (t) r f (t)s(t))1 {S(t)>K} ]dt Here, the last term can be rewritten as lim ɛ 0 EQ d [ 1 ɛ 1 [K,K+ɛ](S(t))e t0 r d (s)ds γ 2 (t, ν(t))s 2 (t)]dt. 1 lim ɛ 0 ɛ EQ d [1 [K,K+ɛ] (S(t))e t0 r d (s)ds γ 2 (t, ν(t))s 2 (t)] 1 = lim ɛ 0 ɛ EQ d [E Q d [γ 2 (t, ν(t))e t0 r d (s)ds S(t)]1 [K,K+ɛ] (S(t))S 2 (t)] = E Q d [γ 2 (t, ν(t))e t 0 r d (s)ds S(t) = K]p d (K, t)k 2 = EQ d [γ 2 (t, ν(t))e t0 r d (s)ds S(t) = K] 2 C(K, t) E Q d [e t 0 r d (s)ds S(t) = K] K 2 K 2. (8) Gregory Rayee (ULB) Bachelier Congress 2010 Toronto, june / 37

28 Calibration By mimicking stochastic volatility models Calibrating the local volatility by mimicking stochastic volatility models E Q d [γ 2 (t, ν(t))e t 0 r d (s)ds C S(t) = K] E Q d [e t t = EQ d [e t 0 r d (s)ds (Kr d (t) r f (t)s(t))1 {S(t)>K} ] 0 r d (s)ds 1 S(t) = K] 2 K 2 2 C K }{{ 2 } σ 2 (t,k) Therefore, if there exists a local volatility such that the one-dimensional probability distribution of the spot FX rate with the diffusion ds(t) = (r d (t) r f (t)) S(t) dt + σ(t, S(t)) S(t) dws DRN (t), is the same as the one of the spot FX rate with dynamics ds(t) = (r d (t) r f (t)) S(t) dt + γ(t, ν(t)) S(t) dws DRN (t) for every time t, then this local volatility function has to satisfy σ 2 (t, K) = EQ d [γ 2 (t, ν(t))e t0 r d (s)ds S(t) = K] E Q d [e t 0 r d (s)ds S(t) = K] = E Qt [γ 2 (t, ν(t)) S(t) = K]. Gregory Rayee (ULB) Bachelier Congress 2010 Toronto, june / 37

29 Calibration By mimicking stochastic volatility models A particular case with closed form solution Consider the three-factor model with local volatility ds(t) = (r d (t) r f (t))s(t)dt + σ(t, S(t))S(t)dWS DRN (t), dr d (t) = [θ d (t) α d r d (t)]dt + σ d dwd DRN (t), dr f (t) = [θ f (t) α f r f (t) ρ fs σ f ν(t)]dt + σ f dwf DRN (t), Calibration by mimicking a Schöbel and Zhu-Hull and White stochastic volatility model ds(t) = (r d (t) r f (t))s(t)dt + ν(t)s(t)dws DRN (t), dr d (t) = [θ d (t) α d r d (t)]dt + σ d dwd DRN (t), dr f (t) = [θ f (t) α f r f (t) ρ fs σ f ν(t)]dt + σ f dwf DRN (t), dν(t) = k[λ ν(t)] dt + ξdwν DRN (t), The local volatility function is given by: σ 2 (T, K) = E Q T [ν 2 (T ) S(T ) = K] = E Q T [ν 2 (T )] if we assume independence between S and ν = (E Q T [ν(t )]) 2 + Var Q T [ν(t )] Gregory Rayee (ULB) Bachelier Congress 2010 Toronto, june / 37

30 Calibration By mimicking stochastic volatility models A particular case with closed form solution Under the T -Forward measure: dν(t) = [k(λ ν(t)) ρ dν σ d b d (t, T )ξ]dt + ξ dwν TF (t) T ν(t ) = ν(t)e k(t t) + t k(λ ρ dνσ d b d (u, T )ξ T )e k(t u) du + k t where b d (t, T ) = 1 α d (1 e α d (T t) ) ξe k(t t) dwν TF (u) so that ν(t ) conditional on F t is normally distributed with mean and variance given respectively by E Q T [ν(t ) F t] = ν(t)e k(t t) + (λ ρ dνσ d ξ α d k )(1 e k(t t) ) + ρ dνσ d ξ α d (α d + k)) (1 e (α d +k)(t t) ) Var Q T [ν(t ) F t] = ξ2 2k (1 e 2k(T t) ) Gregory Rayee (ULB) Bachelier Congress 2010 Toronto, june / 37

31 Calibration By mimicking stochastic volatility models A particular case with closed form solution σ 2 (T, K) = (E Q T [ν(t )]) 2 + Var Q T [ν(t )] ( = ν(t)e kt + (λ ρ dνσ d ξ α d k )(1 e kt ) + ρ ) dνσ d ξ 2 α d (α d + k)) (1 e (α d +k)t ) + ξ2 2k (1 e 2kT ) = σ 2 (T ) Figure: ξ = 20%, k = 50%, α d = 5%, ν(0) = 10%, σ d = 1%, λ = 20%, ρ dν = 1% Gregory Rayee (ULB) Bachelier Congress 2010 Toronto, june / 37

32 Extension Extension : Hybrid volatility model Gregory Rayee (ULB) Bachelier Congress 2010 Toronto, june / 37

33 Extension Hybrid volatility model Here we consider an extension of the three-factor model with local volatility that incorporates a stochastic component to the FX spot volatility by multiplying the local volatility with a stochastic volatility. Gregory Rayee (ULB) Bachelier Congress 2010 Toronto, june / 37

34 Extension Hybrid volatility model 1 Consider a hybrid volatility model where the volatility for the spot FX rate corresponds to a local volatility σ LOC2 (t, S(t)) multiplied by a stochastic volatility γ(t, ν(t)) where ν(t) is a stochastic variable, ds(t) = (r d (t) r f (t))s(t)dt + σ LOC2 (t, S(t))γ(t, ν(t))s(t)dws DRN (t), dr d (t) = [θ d (t) α d (t)r d (t)]dt + σ d (t)dwd DRN (t), dr f (t) = [θ f (t) α f (t)r f (t) ρ fs σ f (t)σ LOC2 (t, S(t))γ(t, ν(t))]dt + σ f (t)dwf DRN (t), dν(t) = α(t, ν(t))dt + ϑ(t, ν(t))dwν DRN (t). 2 Consider the three-factor model where the volatility of the spot FX rate is modelled by a local volatility denoted by σ LOC1 (t, S(t)), ds(t) = (r d (t) r f (t))s(t)dt + σ LOC1 (t, S(t))S(t)dWS DRN (t), dr d (t) = [θ d (t) α d (t)r d (t)]dt + σ d (t)dwd DRN (t), dr f (t) = [θ f (t) α f (t)r f (t) ρ fs σ f (t)σ LOC1 (t, S(t))]dt + σ f (t)dwf DRN (t). Gregory Rayee (ULB) Bachelier Congress 2010 Toronto, june / 37

35 Extension Hybrid volatility model Gyöngy s result Consider a general n-dimensional Itô process ξ t of the form: dξ t = δ(t, w)dw (t) + β(t, w)dt where W (t) is a k-dimensional Wiener process on a probability space (Ω, F, P), δ R n k and β R n are bounded F t-nonanticipative processes such that δδ T is uniformly positive definite. This process gives rise to marginal distributions of the random variables ξ t for each t. Gyöngy then shows that there is a Markov process x(t) with the same marginal distributions. The explicit construction is given by: dx t = σ(t, x t)dw (t) + b(t, x t)dt where: σ(t, x) = (E[δ(t, w)δ T (t, w) ξ t = x]) 1 2 b(t, x) = E[β(t, w) ξ t = x] Gregory Rayee (ULB) Bachelier Congress 2010 Toronto, june / 37

36 Extension Hybrid volatility model σ LOC2 (t, K) = σ LOC1 (t, K) E Q d [γ(t, ν(t)) r d (t) = x, r f (t) = y, S(t) = K] where the conditional expectation is by definition given by E Q d [γ(t, ν(t)) r d (t) = x, r f (t) = y, S(t) = K] 0 γ(t, ν(t))φ d (S(t) = K, r d (t) = x, r f (t) = y, ν(t), t)dν =. 0 φ d (S(t) = K, r d (t) = x, r f (t) = y, ν(t), t)dν Gregory Rayee (ULB) Bachelier Congress 2010 Toronto, june / 37

37 Thank you for your attention Gregory Rayee (ULB) Bachelier Congress 2010 Toronto, june / 37