Heston Stochastic Local Volatility Model

Klaus Spanderen1

R/Finance 2016University of Illinois, Chicago

May 20-21, 2016

1Joint work with Johannes Göttker-SchnetmannKlaus Spanderen Heston Stochastic Local Volatility Model 2016-05-20 1 / 19

Motivation

Combine two of the most popular option pricing models, the LocalVolatility model with x = ln St

dxt =

(rt − qt −

σ2LV (xt , t)

2

)dt + σLV (xt , t)dWt

and the Heston Stochastic Volatility model

dxt =(

rt − qt −νt

2

)dt +

√νtdW x

t

dνt = κ (θ − νt ) dt + σ√νtdW ν

t

ρdt = dW νt dW x

t

to control the forward volatility dynamics and the calibration error.

Klaus Spanderen Heston Stochastic Local Volatility Model 2016-05-20 2 / 19

Model Definition

Add leverage function L(St , t) and mixing factor η to the Heston model:

dxt =

(rt − qt −

L2(xt , t)2

νt

)dt + L(xt , t)

√νtdW x

t

dνt = κ (θ − νt ) dt + ησ√νtdW ν

t

ρdt = dW νt dW x

t

Leverage L(xt , t) is given by probability density p(xt , ν, t) and

L(xt , t) =σLV (xt , t)√E[νt |x = xt ]

= σLV (xt , t)

√ ∫R+ p(xt , ν, t)dν∫R+ νp(xt , ν, t)dν

Mixing factor η tunes between stochastic and local volatility

Klaus Spanderen Heston Stochastic Local Volatility Model 2016-05-20 3 / 19

Package RHestonSLV: Calibration and Pricing

Calibration:Calculate Heston parameters {κ, θ, σ, ρ, vt=0} and σLV (xt , t)Compute p(xt , ν, t) either by Monte-Carlo or PDE to get to theleverage function Lt (xt , t)Infer the mixing factor η from prices of exotic options

Package HestonSLV3 Monte-Carlo and PDE calibration3 Pricing of vanillas and exotic options like double-no-touch barriers3 Implementation is based on QuantLib, www.quantlib.org

Klaus Spanderen Heston Stochastic Local Volatility Model 2016-05-20 4 / 19

Cheat Sheet: Link between SDE and PDE

Starting point is a linear, multidimensional SDE of the form:

dx t = µ(x t , t)dt + σ(x t , t)dW t

Feynman-Kac: the price of a derivative u(x t , t) with boundarycondition u(xT ,T ) at maturity T is given by:

∂tu +n∑

k=1

µi∂xk u +12

n∑k ,l=1

(σσT

)kl∂xk∂xl u − ru = 0

Fokker-Planck: the time evolution of the probability density functionp(x t , t) with the initial condition p(x , t = 0) = δ(x − x0) is given by:

∂tp = −n∑

k=1

∂xk [µip] +12

n∑k ,l=1

∂xk∂xl

[(σσT

)kl

p]

Klaus Spanderen Heston Stochastic Local Volatility Model 2016-05-20 5 / 19

Fokker-Planck Forward Equation

The corresponding Fokker-Planck equation for the probability densityp : R× R≥0 × R≥0 → R≥0, (x , ν, t) 7→ p(x , ν, t) is:

∂tp =12∂2

x

[L2νp

]+

12η2σ2∂2

ν [νp] + ησρ∂x∂ν [Lνp]

−∂x

[(r − q − 1

2L2ν

)p]− ∂ν [κ (θ − ν) p]

Numerical solution of the PDE is cumbersome due to difficult boundaryconditions and the δ-distribution as the initial condition.

PDE can be efficiently solved by using operator splitting schemes,preferable the modified Craig-Sneyd scheme.

Klaus Spanderen Heston Stochastic Local Volatility Model 2016-05-20 6 / 19

Calibration: Fokker-Planck Forward Equation

3 Zero-Flux boundary condition for ν = {νmin, νmax}

3 Reformulate PDE in terms of q = ν1− 2κθ

σ2 or z = ln ν if the Fellerconstraint is violated

3 Prediction-Correction step for L(xt+∆t , t + ∆t)

3 Non-uniform grids are a key factor for success

3 Includes adaptive time step size and grid boundaries to allow forrapid changes of the shape of p(xt , ν, t) for small t

3 Semi-analytical approximations of initial δ-distribution for small t

Corresponding Feynman-Kac backward PDE is much easier to solve.

Klaus Spanderen Heston Stochastic Local Volatility Model 2016-05-20 7 / 19

Calibration: Monte-Carlo Simulation

The quadratic exponential discretization can be adapted to simulatethe Heston SLV model efficiently.

Reminder: L(xt , t) = σLV (xt ,t)√E[νt |x=xt ]

1 Simulate the next time step for all calibration paths2 Define set of n bins bi = {x i

t , xit + ∆x i

t } and assign paths to bins3 Calculate expectation value ei = E[νt |x ∈ bi ] over all paths in bi

4 Define L(xt ∈ bi , t) = σLV (xt ,t)ei

5 t ← t + ∆t and goto 1

Advice: Use Quasi-Monte-Carlo simulations with Brownian bridges.

Klaus Spanderen Heston Stochastic Local Volatility Model 2016-05-20 8 / 19

Calibration: Test Bed

Motivation: Set-up extreme test case for the SLV calibration

Local Volatility: σLV (x , t) ≡ 30%

Heston parameters:S0 = 100, ν0 = 0.09, κ = 1.0, θ = 0.06, σ = 0.4, ρ = −75%

Feller condition is violated with 2κθσ2 = 0.75

Implied volatility surface of the Heston model and the LocalVolatility model differ significantly.

Klaus Spanderen Heston Stochastic Local Volatility Model 2016-05-20 9 / 19

Calibration: Fokker-Planck PDE vs Monte-Carlo

Fokker-Planck Forward Equation, η=1.00

23

45

67

1

2

3

4

0.5

1.0

1.5

2.0

2.5

ln(S)

Time

Leve

rage

Fun

ctio

n L(

x,t)

0.5

1.0

1.5

2.0

2.5

Monte-Carlo Simulation, η=1.00

3

4

5

6

1

2

3

4

0.5

1.0

1.5

2.0

2.5

ln(S)

TimeLe

vera

ge F

unct

ion

L(x,

t)

0.5

1.0

1.5

2.0

2.5

Klaus Spanderen Heston Stochastic Local Volatility Model 2016-05-20 10 / 19

Calibration Sanity Check: Round-Trip Error for Vanillas

50 100 150 200 250

29.9

029

.95

30.0

030

.05

30.1

0

Round-Trip Error for 1Y Maturity

Strike

Impli

ed V

olatili

ty (in

%)

Monte-CarloFokker-Planck

Klaus Spanderen Heston Stochastic Local Volatility Model 2016-05-20 11 / 19

Case Study: Delta of Vanilla Option

Vanilla Put Option: 3y maturity, S0=100, strike=100

0.0 0.2 0.4 0.6 0.8 1.0

-0.4

0-0

.35

-0.3

0-0

.25

Delta of ATM Put Option

η

Del

ta

HestonBlack-ScholesHeston Minimum-VarianceLocal VolSLV

Klaus Spanderen Heston Stochastic Local Volatility Model 2016-05-20 12 / 19

Choose the Forward Volatility Skew Dynamics

Interpolate between the Local and the Heston skew dynamics by tuning η between 0 and 1.

0.5 1.0 1.5 2.0

2628

3032

3436

38Forward Starting Option: max(0, S2y - α*S1y)

Strike α

Impli

ed F

orwa

rd V

olatili

ty (in

%)

η=1.00η=0.50η=0.25η=0.00

Klaus Spanderen Heston Stochastic Local Volatility Model 2016-05-20 13 / 19

Case Study: Barrier Option Prices

DOP Barrier Option: 3y maturity, S0=100, strike=100

0 20 40 60 80 100

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

Barrier Option Pricing Local Vol vs SLV

Barrier

NPV

loca

l - N

PVS

LV

η = 1.0η = 0.5η = 0.2η = 0.1

Klaus Spanderen Heston Stochastic Local Volatility Model 2016-05-20 14 / 19

Case Study: Delta of Barrier Options

DOP Barrier Option: 3y maturity, S0=100, strike=100

0 20 40 60 80 100

-0.1

5-0

.10

-0.0

50.

00Barrier Option Δ local vs ΔSLV

Barrier

Δlo

cal -

ΔS

LV

η = 1.0η = 0.5η = 0.2η = 0.1

Klaus Spanderen Heston Stochastic Local Volatility Model 2016-05-20 15 / 19

Case Study: Double-No-Touch Options

Knock-Out Double-No-Touch Option: 1y maturity, S0=100

0.0 0.2 0.4 0.6 0.8 1.0

-0.0

50.

000.

050.

100.

15Double No Touch Option

NPVBS

NPV S

LV -

NPV B

S

Stochastic Local Volatility vs. Black-Scholes Prices

η=1.00η=0.75η=0.50η=0.25η=0.00

Klaus Spanderen Heston Stochastic Local Volatility Model 2016-05-20 16 / 19

Summary: Heston Stochastic Local Volatility

RHestonSLV: A package for the Heston Stochastic Local VolatilityModel

Monte-Carlo Calibration

Calibration via Fokker-Planck Forward Equation

Supports pricing of vanillas and exotic options

Implementation is based on QuantLib 1.8 and Rcpp

Package source code including all examples shown is on githubhttps://github.com/klausspanderen/RHestonSLV

Klaus Spanderen Heston Stochastic Local Volatility Model 2016-05-20 17 / 19

Literature

J. Göttker-Schnetmann and K. Spanderen.Calibration of the Heston Stochastic Local Volatility Model.http://hpc-quantlib.de/src/slv.pdf.

K.J. in t’Hout and S. Foulon.ADI Finite Difference Schemes for Option Pricing in the HestonModel with Correlation.International Journal of Numerical Analysis and Modeling,7(2):303–320, 2010.

Y. Tian, Z. Zhu, G. Lee, F. Klebaner and K. Hamza.Calibrating and Pricing with a Stochastic-Local Volatility Model.http://ssrn.com/abstract=2182411.

A. Stoep, L. Grzelak and C. Oosterlee,.The Heston Stochastic-Local Volatility Model: Efficient MonteCarlo Simulation.http://papers.ssrn.com/abstract_id=2278122.

Klaus Spanderen Heston Stochastic Local Volatility Model 2016-05-20 18 / 19