ADI Finite Difference Schemes for the Calibration ofStochastic Local Volatility Models: An Adjoint Method

Maarten Wyns

Joint work with Karel in ’t Hout

WSC Spring Meeting 2017

Table of contents

I Introduction

I Difficulties

I Valuation of European options

I Adjoint calibration technique

I Numerical experiments

I Conclusions

2

Introduction: Why?

I European call option gives holder the right to buy a given assetat a prescribed date T for a prescribed price K

I Sτ foreign exchange rate at τ ≥ 0

I European call option: payoff u0(ST ) = max(ST − K , 0)

I Non-path-dependent European option:Fair value at τ = 0 is e−rdTE[u0(ST )] = e−rdTE[u0(S0e

XT )]

I rd , rf risk-free interest rates

3

Introduction: Why?

Modelling: Xτ = log(Sτ/S0)

Local volatility (LV) model

dXτ = (rd − rf − 12σ

2LV (Xτ , τ))dτ + σLV (Xτ , τ)dWτ

Stochastic local volatility (SLV) model

dXτ = (rd − rf − 12σ

2SLV (Xτ , τ)Vτ )dτ + σSLV (Xτ , τ)

√VτdW

(1)τ

dVτ = κ(η − Vτ )dτ + ξ√

VτdW(2)τ

Goal: Determine σSLV that reproduces market data

4

Difficulties

I For general σSLV European call values not known exactly

I Calibrate SLV model to underlying LV model

Gyongy ’86

LV and SLV model define same European call values if

σ2LV (x , τ) = σ2

SLV (x , τ)E[Vτ‖Xτ = x ] = σ2SLV (x , τ)

∫∞0 vp(x , v , τ)dv∫∞0 p(x , v , τ)dv

I E[Vτ‖Xτ = x ] dependent on σSLV in a non-trivial way

I For general σLV European call values not known exactly

5

Valuation of European options

Common: numerically solving the backward PDE (t = T − τ)

I LV model

ut = 12σ

2LV uxx + (rd − rf − 1

2σ2LV )ux

Maturity T → E[u0(S0eXT )] = u(X0,T )

I SLV model

ut = 12σ

2SLV vuxx + ρσSLV vuxv + 1

2ξ2vuvv

+ (rd − rf − 12σ

2SLV )ux + κ(η − v)uv

Maturity T → E[u0(S0eXT )] = u(X0,V0,T )

6

Theoretical alternative: forward PDE

I LV model

pτ = ∂2

∂x2

(12σ

2LV p)− ∂

∂x

((rd − rf − 1

2σ2LV )p

)E[u0(S0e

XT )] =∫∞−∞ u0(S0e

x)p(x ,T )dx

I SLV model

pτ = ∂2

∂x2

(12σ

2SLV vp

)+ ∂2

∂x∂v (ρσSLV vp) + ∂2

∂v2

(12ξ

2vp)

− ∂∂x

((rd − rf − 1

2σ2SLV )p

)− ∂

∂v (κ(η − v)p)

E[u0(S0eXT )] =

∫∞0

∫∞−∞ u0(S0e

x)p(x , v ,T )dxdv

7

Theoretical alternative: forward PDE

I LV model

pτ = ∂2

∂x2

(12σ

2LV p)− ∂

∂x

((rd − rf − 1

2σ2LV )p

)E[u0(S0e

XT )] =∫∞−∞ u(x ,T − τ)p(x , τ)dx

I SLV model

pτ = ∂2

∂x2

(12σ

2SLV vp

)+ ∂2

∂x∂v (ρξσSLV vp) + ∂2

∂v2

(12ξ

2vp)

− ∂∂x

((rd − rf − 1

2σ2SLV v)p

)− ∂

∂v (κ(η − v)p)

E[u0(S0eXT )] =

∫∞0

∫∞−∞ u(x , v ,T − τ)p(x , v , τ)dxdv

8

Adjoint calibration technique

FD DiscretizationBackward PDE, LV

↔ Adjoint DiscretizationForward PDE, LV

l (?)

Adjoint DiscretizationForward PDE, SLV

↔FD DiscretizationBackward PDE, SLV

(?) holds when similar discretizations for the backward PDEs

9

Adjoint discretization (LV model)

I FD discretization backward equation (t = T − τ):

U ′LV (t) = ALV (t)ULV (t), ULV,i (t) ≈ u(xi , t)

I Approximations PLV,i (τ) ≈∫ xi+0.5

xi−0.5p(x , τ)dx

ULV,i0(T ) = PTLV (τ)ULV (T−τ) = PT

LV (T )U0 =∑i

PLV,i (T )U0,i

I Adjoint discretization forward equation:

P ′LV (τ) = ATLV (T − τ)PLV (τ)

10

Adjoint discretization (SLV model)

I FD discretization backward equation: USLV,i ,j(t) ≈ u(xi , vj , t)

I Approximations PSLV,i ,j(τ) ≈∫ vj+0.5

vj−0.5

∫ xi+0.5

xi−0.5p(x , v , τ)dxdv

I Adjoint discretization forward equation:

USLV,i0,j0(T ) =∑i ,j

PSLV,i ,j(τ)USLV,i ,j(T − τ)

=∑i

∑j

PSLV,i ,j(T )

U0,i

11

Adjoint calibration

If

I Same discretization in x-direction of the backward equations

I Adjoint spatial discretization of the forward equations

I σ2LV (xi , τ) = σ2

SLV (xi , τ)∑

j vjPSLV,i,j (τ)∑j PSLV,i,j (τ)

≈ σ2SLV (xi , τ)

∫ ∞0

vp(xi ,v ,τ)dv∫ ∞0

p(xi ,v ,τ)dv

Then

I PLV,i (τ) =∑

j PSLV,i ,j(τ)

I ULV,i0(T ) = USLV,i0,j0(T )

12

Adjoint calibration

If

I Same discretization in x-direction of the backward equations

I Adjoint spatial discretization of the forward equations

I σ2LV (xi , τ) = σ2

SLV (xi , τ)∑

j vjPSLV,i,j (τ)∑j PSLV,i,j (τ) ≈ σ

2SLV (xi , τ)

∫ ∞0

vp(xi ,v ,τ)dv∫ ∞0

p(xi ,v ,τ)dv

Then

I PLV,i (τ) =∑

j PSLV,i ,j(τ)

I ULV,i0(T ) = USLV,i0,j0(T )

12

Non-linear system of ODEs

I Adjoint discretization yields system of ODEs for PSLV

I Corresponding discretization matrix depends on

σ2SLV (xi , τ) = σ2

LV (xi , τ)∑

j PSLV,i,j (τ)∑j vjPSLV,i,j (τ) (1)

I Non-linear system of ODEs

I Modified Craig–Sneyd implicit time stepping with inner iteration

13

Inner iteration

PSLV,n = PSLV,n−1 initial approximation to PSLV (τn);

for q is 1 to Q do

(a) approximate σ2SLV (xi , τn) by (1);

(b) update PSLV,n by performing a MCS time step;

end

I Q = 2 performs excellent

14

Numerical experiments, EUR/USD rate

S0 = 1.0764, rd = 0.03, rf = 0.01

Local volatility surface up to τ = 2 years

-20

x

Local vol surface

22τ

1

0

1

2

0

σLV

(x,τ

)

15

Stochastic parameters

Case 1 Case 2

κ 0.75 0.30η 0.015 0.04ξ 0.15 0.90ρ −0.14 −0.5T 2Y 2YS0 1.0764 1.0764V0 0.015 0.04

I Case 1: Actual EUR/USD parameters from Clarke (2011)

I Case 2: Challenging parameters from Andersen (2008)

16

Leverage surface, Case 1

-20

x

Leverage surface

22τ

1

0

20

10

0

σS

LV(x

,τ)

17

European call prices, Case 1

Strike K , maturity T → U0,i = max(S0exi − K , 0)

K/S0

0.6 0.8 1 1.2 1.4

C(K

,T)

0

0.1

0.2

0.3

0.4European call price, S

0= 1.0764, T = 2

ULV,i

0

(T)

USLV,i

0,j

0

(T)

18

Implied vols (in %), Case 1

ULV,i0(T )→ σimp,LV,B(K ,T ), USLV,i0,j0(T )→ σimp,SLV,B(K ,T )

K/S0

0.6 0.8 1 1.2 1.4

σB

S(K

,T)

8

10

12

14

Implied volatility, S0= 1.0764, T = 2

σimp,LV,B

(K,T)

σimp,SLV,B

(K,T)

19

Leverage surface, Case 2

-20

x

Leverage surface

22τ

1

5

10

00

σS

LV(x

,τ)

20

European call prices, Case 2

K/S0

0.6 0.8 1 1.2 1.4

C(K

,T)

0

0.1

0.2

0.3

0.4European call price, S

0= 1.0764, T = 2

ULV,i

0

(T)

USLV,i

0,j

0

(T)

21

Implied vols (in %), Case 2

K/S0

0.6 0.8 1 1.2 1.4

σB

S(K

,T)

8

10

12

14

Implied volatility, S0= 1.0764, T = 2

σimp,LV,B

(K,T)

σimp,SLV,B

(K,T)

22

Numerical results

Absolute implied vol error: |σimp,LV,B(K ,T )− σimp,SLV,B(K ,T )|

x-direction: 100 points, v -direction: 50 points, ∆τ = 1/100

K/S0 0.7 0.8 0.9 1.0 1.1 1.2 1.3

σimp,LV,B 10.23 9.19 8.99 9.61 10.70 11.68 12.49

Case 1 4e−3 3e−3 2e−3 1e−3 1e−3 2e−3 2e−3

Case 2 2e−3 4e−3 2e−4 5e−3 5e−3 4e−3 3e−4

23

Calibration time

I Calibration time ∼ desired accuracy LV discretization

I Calibration time ∼ desired accuracy v -direction

I x-direction: 100 points, v -direction: 50 points, time steps: 100→ Calibration time ≈ 1s

(Matlab code, Intel Core i7-3540M 3.00GHz, 8GB RAM)

24

Conclusions

I Adjoint calibration for exact match between semidiscrete LVand SLV model

I Time stepping and inner iteration for full discretization

I Fully discrete match upto temporal discretization error

I Spatial error � temporal error

I (1) Control discretization error within LV model(2) De facto exact calibration of the SLV model

25

Thank You!

26

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