Efficient Simulation of Generalized SABR andStochastic Local Volatility Models

Duy Nguyen1

Joint with Zhenyu Cui (Stevens) & J. Lars Kirkby (GeorgiaTech)

1Marist College

Overview

I SABR and generalized SABRI Motivation and OverviewI Probability distribution

I Simulation frameworkI Continuous time Markov chain (CTMC) approximationI Characteristic functions (ChF)I PROJ: simulation from ChF

I Numerical examples

SABR and generalized SABRThe standard SABR model introduced in Hagan et al. (2002)2{

dSt = VtSβt dW (1)

t ,

dVt = αVtdW (2)t ,

(1)

where α > 0 and β ∈ [0,1] are constants, andE[dW (1)

t dW (2)t ] = ρdt with −1 < ρ < 1.

I β = 1 : ST =

S0 exp{−1

2

∫ T0 V 2

s ds + ρα(VT − V0) +

√1− ρ2

∫ T0 VsdW (1)

s

}I log(ST )|(VT ,

∫ T0 Vsds) ∼

N (log(S0)− 12

∫ T0 V 2

s ds + ρα(VT − V0); (1− ρ2)

∫ T0 V 2

s ds)

I 0 ≤ β < 1 : It is difficult to derive the exact jointdistribution of ST and VT . To our best knowledge, there isno complete answer to this question.

2Hagan, P. S., D. Kumar, A. S. Lesniewski, and D. E. Woodward (2002).Managing smile risk. Wilmott Magazine pp. 84-108.

Generalized SABR

Consider the following stochastic volatility model:{dSt = m(Vt )S

βt dW (1)

t ,

dVt = α(Vt )dt + γ(Vt )dW (2)t ,

(2)

I α > 0 and β ∈ [0,1] are constants

I E[dW (1)t dW (2)

t ] = ρdt with −1 < ρ < 1.I Here it is assumed that the functions m(.), α(.), γ(.) satisfy

some technical conditions so the system has a unique (inlaw) solution.

GBM(SABR) m(v) = v , dvt = αvtdW (2)t

( Hagan et al. (2002)) f (v) = v/α, h(v) ≡ 0CIR (Heston) m(v) =

√v , dvt = κ(θ − vt )dt + σv

√vtdW (2)

t(Heston (1993)) f (v) = v/σv , h(v) = κ(θ − v)/σv

4/2 m(v) = a√

v + b/√

v , dvt = η(θ − vt )dt + σv√

vtdW (2)t

(Grasselli (2017)) f (v) = (av+b log(v))σv

, h(v) = η(aθ−b)σv

− aηvσv

+ (ηθbσv− bσv

2 ) 1v

Stein-Stein m(v) = v , dvt = η(θ − vt )dt + σv dW (2)t

(Stein and Stein (1991)) f (v) = v2

2σv, h(v) = σv

2 + ηθvσv− ηv2

σv

3/2 m(v) = 1/√

v , dvt = η[θ − vt ]dt + σv

√vtdW (2)

t

(Lewis (2000)) f (v) = log(v)σv

, h(v) =(ηθσv− σv

2

)1v −

ησv

Hull-White m(v) =√

v , dvt = av vtdt + σv vtdW (2)t

(Hull and White (1987)) f (v) = 2√

vσv, h(v) =

(avσv− σv

4

)√v

Scott m(v) = exp(v), dvt = η(θ − vt )dt + σv dW (2)t

(Scott (1987)) f (v) = ev

σv, h(v) = ev

(ηθσv

+ σv2 −

ηvσv

)α-Hyper m(v) = exp(v), dvt = (η − θ exp(av vt ))dt + σv dW (2)

t

(Da Fonseca and Martini (2016)) f (v) = ev

σv, h(v) = ev

(ησv

+ σv2

)− θ

σve(av +1)v

CEV-SV m(v) =√

v , dvt = η(θ − vt )dt + σv vαt dW (2)t

(Andersen and Piterbarg (2007)) f (v) = 1σv (3/2−α) v3/2−α, h(v) = η

σv(θ − v)v1/2−α + σv

2

( 12 − α

)vα−1/2

τ/2 models m(v) =√

v , dvt = ηvωt (θ − vt )dt + σv vτ/2t dW 2

t , τ 6= 3(Christoffersen et al. (2010)) f (v) = 2

σv (3−τ) v (3−τ)/2, h(v) = ησv

v (1−τ)/2+ω(θ − v) + σv4 (1− τ)v (τ−1)/2

Table: List of stochastic volatility models

Some selected previous works

1. Broadie, M. and O. Kaya (2006). Exact simulation ofstochastic volatility and other affine jump diffusionprocesses. Operations Research 54(2), 217-231.

2. Cai, N., Y. Song, and N. Chen (2017). Exact simulation ofthe SABR model. Operations Research 65(4), 931-951.

3. Kang, C., W. Kang, and J. M. Lee (2017). Exact simulationof the Wishart multidimensional stochastic volatility model.Operations Research 65(5), 1190-1206.

4. Rhee, C.-h. and P. W. Glynn (2015). Unbiased estimationwith square root convergence for SDE models. OperationsResearch 63(5), 1026-1043.

Our contributions

I We propose a unified efficient simulation framework for awide class of stochastic volatility models including Heston,3/2, 4/2, SABR, Heston-SABR,...

I Extensive numerical examples illustrate the accuracy andefficiency of our estimator, which compares favorably toexisting biased and unbiased simulation estimators in theliterature in terms of root mean square error (RMSE) andcomputational time.

The case β = 1

The model is reduced to{dSt = m(Vt )StdW (1)

t

dVt = α(Vt )dt + γ(Vt )dW (2)t .

I Cholesky decomposition: W (1)t := ρW (2)

t +√

1− ρ2W (3)t ,

where W (3)t |= W

(2)t .

I f (v) :=∫ v

0m(z)γ(z) dz

I h(v) := α(v)f ′(v) + 12γ

2(v)f ′′(v)

The case β = 1The model is reduced to{

dSt = m(Vt )StdW (1)t

dVt = α(Vt )dt + γ(Vt )dW (2)t

Under the assumption: h(v) = θ1 + θ2m2(v)

log(S∆)∣∣∣ (V∆,

∫ ∆

0m2(Vs)ds

)∼ N

(log(S0) + ζ1, (1− ρ2)

∫ ∆

0m2(Vs)ds

),

whereζ1 := −

(12 + ρθ2

) ∫ ∆0 m2(Vs)ds + ρ (f (V∆)− f (V0)− θ1∆) .

Hence, conditional on(

V∆,∫ ∆

0 m2(Vs)ds)

, the underlyingasset price can be simulated exactly with a single draw from anormal distribution

The case β = 0

The model is reduced to{dSt = m(Vt )[ρdW (2)

t +√

1− ρ2dW (3)t ],

dVt = α(Vt )dt + γ(Vt )dW (2)t .

Under the assumption: h(v) = θ1 + θ2m2(v) or ρ = 0 :

log(S∆)∣∣∣ (V∆,

∫ ∆

0m2(Vs)ds

)∼ N

(S0 + ζ2

(V0,V∆,

∫ ∆

0m2(Vs)ds

), (1− ρ2)

∫ ∆

0m2(Vs)ds

).

Interior Case: β ∈ (0,1)When β ∈ (0,1) and ρ = 0

P(

S∆ = 0 | S0 > 0,V∆,

∫ ∆

0m2(Vs)ds

)= 1− χ2

b(a; 0),

P(

S∆ ≤ ζ | S0 > 0,V∆,

∫ ∆

0m2(Vs)ds

)= 1− χ2

b(a; c),

where

a =1

v(∆)

(S1−β

01− β

)2

, b =1

1− β,

c =ζ2(1−β)

(1− β)2v(∆), v(∆) =

∫ ∆

0m2(Vs)ds,

χ2δ(x ;λ) is the non-central chi-square cumulative distribution

function with a non-centrality parameter λ and the degree offreedom δ.

Interior Case: β ∈ (0,1)When β ∈ (0,1) and ρ 6= 0

P(

S∆ = 0 | S0 > 0,V∆,

∫ ∆

0m2(Vs)ds,

∫ ∆

0h(Vs)ds

)≈ 1− χ2

b(a; 0),

P(

S∆ ≤ ζ | S0 > 0,V∆,

∫ ∆

0m2(Vs)ds,

∫ ∆

0h(Vs)ds

)≈ 1− χ2

b(a; c),

where

a =1

v(∆)

(S1−β

01− β

+ A∆

)2

, b = 2− (1− 2β − ρ2(1− β))

(1− β)(1− ρ2),

c =ζ2(1−β)

(1− β)2v(∆), v(∆) = (1− ρ2)

∫ ∆

0m2(Vs)ds.

A∆ = ρ

(f (V∆)− f (V0)−

∫ ∆

0h(Vs)ds

).

Continuous Time Markov Chain (CTMC)approximation

Consider the variance process

dVt = α(Vt )dt + γ(Vt )dW (2)t ,

I We approximate Vt by a continuous time Markov chainV m0

t taking m0 values {v0, . . . , vm0}I The generator Q = [qij ] satisfies the q-properties

qij :=

α−(vi)

ki−1+γ2(vi)− (ki−1α

−(vi) + kiα+(vi))

ki−1(ki−1 + ki), if j = i − 1,

α+(vi)

ki+γ2(vi)− (ki−1α

−(vi) + kiα+(vi))

ki(ki−1 + ki), if j = i + 1,

−qi,i−1 − qi,i+1, if j = i ,0, if j 6= i − 1, i , i + 1.

PropositionConsider a uniform variance grid {vj}m0

j=1, and let Ψ(·) be a

continuous function whose ∂4Ψ∂x4 is bounded. Assume that at

least one of the following conditions hold: i) Ψ(l∗) = Ψ(u∗) = 0,where [l∗,u∗] is the compact support of Ψ; ii) Vt has a compactstate space V with reflecting or absorbing boundaries. Then wehave weak convergence of the CTMC scheme with a quadraticconvergence order:∣∣∣E[Ψ(Vt )|V m0

0 = v ]− E[Ψ(V m0t )|V m0

0 = v ]∣∣∣ ≤ Cm−2

0 , t ∈ [0,T ],

for some C > 0 that depends on Ψ.

PropositionFor a time increment ∆ > 0, and function g : R→ R, thefollowing holds:The conditional ChF with (V0, V∆) = (vj , vk ) is given by

E[

exp

(iξ∫ ∆

0g(Vs)ds

) ∣∣vj , vk

]= Λj,k (ξ), j , k = 1, . . . ,m0,

where Λ(ξ) is the matrix exponential defined by

Λ(ξ) = exp(∆(Q + iξ · diag(g(v1),g(v2), . . . ,g(vm0))).

Biorthogonal projection approach

Goal: to simulate(∫ ∆

0 g(Vs)ds) ∣∣V0, V∆ from its ChF

I Given a function ϕ named the generator.I we construct a sequence on the support of f as

xk = x1 + (k − 1)h, k ∈ Z,

I Form the sequence{ϕa,k (x)}k∈Z := a1/2{ϕ(a(x − xk ))}k∈Z, which generatesan approximation spaceMa := span{ϕa,k}k∈Z

A‖g‖22 ≤∑k∈Z|〈g, ϕa,k 〉|2 ≤ B‖g‖22, ∀g ∈ L2(R),

B-spline generators

1. p = 0: Haar generator ϕ(y) ≡ ϕ[0](y) = 1[− 12 ,

12 ](y)

2. p = 1: Linear generator

ϕ(y) ≡ ϕ[1](y) =

{1 + y , y ∈ [−1,0],

1− y , y ∈ [0,1],

3. p ≥ 2 p-order B-spline generators are derived recursivelyby the convolution

ϕ(x) ≡ ϕ[p](x) = ϕ[0]?ϕ[p−1](x) =

∫ ∞−∞

ϕ[p−1](y−x)1[− 12 ,

12 ](y)dy .

I The orthogonal projection of any f ∈ L2(R) ontoMasatisfies

PMa f (y) =∑k∈Z

(∫ ∞−∞

f (x)ϕa,k (x)dx)ϕa,k (y) =

∑k∈Z

βa,k ·ϕa,k (y),

I βa,k := E[ϕa,k (X1)].I ϕ is the dual of ϕ. For example, for the case p-order

B-spine generators

ϕ ≡ ϕ[p](ξ) = ϕ[p](ξ)/Φ[p](ξ), ϕ[p](ξ) =

(sin(ξ/2)

(ξ/2)

)p+1

,

Φ[p](ξ) =

∫ p+12

− p+12

ϕ[p](x)2dx+2p+1∑k=1

cos(kξ)

∫ p+12

− p+12

ϕ[p](x)ϕ[p](x−k)dx .

I ||f − PMa f ||22 ≤ C||f (p+1)||22 · h2(p+1)

Projected density estimator

I The orthogonal projection of any f ∈ L2(R) ontoMasatisfies

PMa f (y) =∑k∈Z

βa,k · ϕa,k (y),

I βa,k := E[ϕa,k (X1)].I For the linear B-spline basis:

βa,k =a−1/2

π<{∫ ∞

0exp(−ixkξ) · φX (ξ) · ϕ(ξ/a)dξ

}, k ∈ Z,

where

φX (ξ) = E[eiXξ], ϕ(ξ) =12 sin2(ξ/2)

ξ2(2 + cos(ξ)), ξ ∈ R,

which are respectively the ChF of X , and the Fouriertransform of ϕ.

Single step algorithm

I Variance. To sample from V∆|V0 = vj0 = v03, and let

P∆j,k = e>j exp(Q∆)ek , j , k = 1, . . . ,m0.

We store a set of cumulative conditional probabilities forthe terminal state V∆ = vk ∈ Sv :

Fvj0 := {F v

j0,k}m0k=1, F v

j0,k =k∑

n=1

P∆j0,n, k = 1, . . . ,m0.

Simulating V∆|V0 = vj0 requires a single draw from astandard uniform random variable U ∼ U(0,1).

3Note that given any initial v0, it is simple to adjust the grid so that vj0 = v0

for some index j0 ∈ {1, . . . ,m0}.

I Integrated Variance I∆ =∫ ∆

0 g(Vs)ds|(V0 = vj0 , V∆ = vk ) :We draw a sample I∆|(vj0 , vk ) from its characteristicfunction using the projection density function.

I Underlying:β ∈ {0,1}

S∆ =

{S0 exp

(−(1

2 + ρθ2)I∆|(vj0 , vk ) + f ∆(vk , vj0) + Z

), β = 1,

S0 − ρθ2I∆|(vj0 , vk ) + f ∆(vk , vj0) + Z , β = 0.

Z ∼ N(0, (1− ρ2)I∆|(vj0 , vk )

),

f ∆(vk , vj0) := ρ(f (vk )− f (vj0)− θ1∆

).

β ∈ (0,1) :

S∆ =(

(1− β)2d(e + Z1)2 · (1− ρ2)I∆|(vj0 , vk )) 1

2(1−β).

Z1 ∼ N (0,1),d ,e are known.

Numerical example: Compare with Broadie and Kaya(2006)4 and Kang et al. (2017)5

Test Case (Heston) S0 T r κ θ v0 σv ρ

Case A 100 1 0.0319 6.21 0.019 0.010201 0.61 -0.70Case B 100 1 0.05 2.0 0.09 0.09 1.0 -0.3

Table: Baseline parameters for Heston experiments. q = 0, andK = S0.

4Broadie, M. and O. Kaya (2006). Exact simulation of stochastic volatilityand other affine jump diffusion processes. Operations Research 54(2),217-231

5Kang, C., W. Kang, and J. M. Lee (2017). Exact simulation of the Wishartmultidimensional stochastic volatility model. Operations Research 65(5),1190-1206

CTMC Euler KKL BK

Nsim m0 RMSE Time Steps RMSE Time RMSE Time RMSE Time

10000 20 3.70e-02 0.11 100 3.63e-01 0.03 7.80e-02 0.10 7.50e-02 3.840000 22 1.99e-02 0.16 200 2.27e-01 0.20 4.08e-02 0.42 3.73-02 15.2

160000 30 1.06e-02 0.33 400 1.46e-01 1.92 2.12e-02 1.63 1.86-02 60.0640000 34 4.24e-03 0.79 800 8.09e-02 16.11 1.11e-02 6.58 9.30e-03 239.4

Table: Heston: European call option on equity, case A parametersfrom Table 2. Ref. price: 6.80611.

10-2 10-1 100 101

Simulation time (seconds)

10-3

10-2

10-1

100

RM

S e

rror

CTMCEulerKKL

10-2 10-1 100 101

Simulation time (seconds)

10-3

10-2

10-1

100

RM

S e

rror

CTMCEulerKKL

Figure: Heston: Convergence of RMS error. The Left panelcorresponds to Case A, and the Right panel to Case B.

Numerical results for SABR model: Compare with Caiet al. (2017) 6

Test Case (SABR) S0 T ρ β α v0Case A (Interest rates) 0.05 1 0 0.55 0.03 0.20Case B (Foreign exchange) 1.10 1 0 0.70 0.10 0.20Case C (Equity derivatives) 100 1 0 0.60 0.20 0.30

Table: Baseline parameters for SABR experiments. We also setr = q = 0, and K = S0, unless stated otherwise.

6Cai, N., Y. Song, and N. Chen (2017). Exact simulation of the SABRmodel. Operations Research 65(4), 931-951

CTMC, Nsim = 50,000 Alternative Methods (Price)Case K Time (sec) Avg. Price Avg. Err. Std. Err. RMSE Exact Sim (Std.Err.) Expansion FDM

0.045 0.26 0.01732 -6.72e-05 6.86e-06 9.58e-05 0.01727 (9.19e-06) 0.01726 0.01725A 0.050 0.26 0.01513 -7.77e-05 7.09e-06 1.05e-04 0.01507 (8.69e-06) 0.01506 0.01505

0.055 0.26 0.01319 -9.79e-05 5.59e-06 1.13e-04 0.01311 (8.20e-06) 0.01311 0.013101.0 0.24 0.14197 -5.89e-06 2.63e-05 2.62e-04 0.14188 (5.21e-05) 0.14196 0.14197

B 1.1 0.24 0.08529 -4.48e-05 3.24e-05 3.25e-04 0.08518 (4.23e-05) 0.08524 0.085231.2 0.24 0.04680 4.52e-05 2.91e-05 2.93e-04 0.04679 (3.20e-05) 0.04683 0.0468390 0.23 10.03049 2.39e-04 1.47e-04 1.48e-03 10.02941 (1.48e-03) 10.03079 10.03078

C 100 0.23 1.90258 4.13e-04 6.54e-04 6.52e-03 1.90169 (8.94e-04) 1.90301 1.90294110 0.23 0.04480 -1.74e-04 1.26e-04 1.69e-03 0.04444 (1.30e-04) 0.04469 0.04468

Table: SABR: European call options. The expansion column isobtained using by expansion of Hagan et al. (2002). Exact simulationresults are obtained in Cai et al. (2017) using 10,240,000 samplepaths. For CTMC, we use 50,000 paths, and estimate the averageprice (CTMC) and error metrics and with respect to the finitedifference method (FDM) using 100 replications.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9β

0.080

0.081

0.082

0.083

0.084

0.085

0.086

0.087

price

FDM

CTMC

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9β

0.0

2.0

4.0

6.0

8.0

price

Figure: SABR: prices as function of β, for Case B (left) and Case C(right) from Table 4, with Nsim = 400,000.

0.1 0.2 0.3 0.4 0.5 0.6 0.7α

0.0850

0.0855

0.0860

0.0865

0.0870

0.0875

0.0880

0.0885

0.0890

price

FDM

CTMC

0.1 0.2 0.3 0.4 0.5 0.6 0.7α

1.88

1.90

1.92

1.94

1.96

1.98

price

Figure: SABR: prices as function of α, for Case B (left) and Case C(right) from Table 4, with Nsim = 400,000.

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5T

0.05

0.10

0.15

0.20

price

FDM

CTMC

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5T

1.00

1.50

2.00

2.50

3.00

3.50

4.00

4.50

price

Figure: SABR: prices as function of T , for Case B (left) and Case C(right) from Table 4, with Nsim = 400,000.

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8ρ

0.0849

0.0850

0.0851

0.0852

0.0853

0.0854

0.0855

0.0856

price

FDM

CTMC

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8ρ

1.894

1.896

1.898

1.900

1.902

1.904

1.906

price

Figure: SABR: prices as function of ρ, for Case B (left) and Case C(right) from Table 4, with Nsim = 400,000.

SABR model: Compare with Rhee and Glynn (2015)7

CTMC CTMC-MS Euler Unbiased

Nsim m0 RMSE Time m0 Steps RMSE Time Steps RMSE Time IRE RMSE Time

10000 20 1.46e-02 0.11 20 6 1.53e-02 0.02 100 1.58e-02 0.07 0.010 2.39e-02 0.1740000 22 7.67e-03 0.17 22 8 8.28e-03 0.10 200 8.32e-03 0.47 0.005 6.45e-03 0.66

160000 24 3.68e-03 0.35 24 10 3.21e-03 0.57 400 3.55e-03 4.33 0.002 2.47e-03 8.49640000 26 1.76e-03 1.01 26 12 1.45e-03 2.76 800 1.82e-03 34.41 0.001 1.15e-03 43.46

Table: SABR: European call option on equity, case C parameters.Ref. price: 1.90300.

7Rhee, C.-h. and P. W. Glynn (2015). Unbiased estimation with squareroot convergence for SDE models. Operations Research 63(5), 1026-1043.

SABR model: Compare with Rhee and Glynn (2015)8

10-2 10-1 100 101 102

Simulation time (seconds)

10-3

10-2

10-1

RM

S e

rror

CTMCCTMC-MSEulerUnbiased

Figure: SABR: European call option on equity, case C parameters.

8Rhee, C.-h. and P. W. Glynn (2015). Unbiased estimation with squareroot convergence for SDE models. Operations Research 63(5), 1026-1043.

References (incomplete list)I Broadie, M. and O. Kaya (2006). Exact simulation of stochastic volatility and other affine jump diffusion

processes. Operations Research 54(2), 217-231.

I Cai, N., Y. Song, and N. Chen (2017). Exact simulation of the SABR model. Operations Research 65(4),931-951.

I Cui, Z., J. Kirkby, and D. Nguyen (2017). A general framework for discretely sampled realized variancederivatives in stochastic volatility models with jumps. European Journal of Operational Research 262(1),381-400.

I Cui, Z., J. Kirkby, and D. Nguyen (2018a). A general framework for time-changed Markov processes andapplications. Working Paper.

I Cui, Z., J. Kirkby, and D. Nguyen (2018b). A general valuation framework for SABR and stochastic localvolatility models. SIAM Journal on Financial Mathematics 9(2), 520-563.

I Kang, C., W. Kang, and J. M. Lee (2017). Exact simulation of the Wishart multidimensional stochasticvolatility model. Operations Research 65(5), 1190-1206.

I Kirkby, J. (2015). Efficient option pricing by frame duality with the fast Fourier transform. SIAM Journal onFinancial Mathematics 6(1), 713-747.

I Kirkby, J. and S. Deng (2018). Static hedging and pricing of exotic options with payoff frames. MathematicalFinance, Forthcoming.

I Rhee, C.-h. and P. W. Glynn (2015). Unbiased estimation with square root convergence for SDE models.Operations Research 63(5), 1026-1043.

THANK YOU !!!