Fast solver three-factor Heston / Hull-White model

Delft 22 March 15:30www.ing.com

Floris NaberING Amsterdam & TU Delft

ING 2

Outline

• Introduction to the problem (three-factor model) Equity underlying

Stochastic interest

Stochastic volatility

• Solving partial differential equations without boundary conditions

• 1-dimensional Black-Scholes equation

• 1-dimensional Hull-White equation

• Conclusion

• Future goals

ING 3

Introduction (Three-factor model)

• Underlying equity:

S: underlying equity, r: interest rate, q:dividend yield, v:variance

• Stochastic interest (Hull-White)

r: interest rate, θ:average direction in which r moves, a:mean reversion rate, :annual standard deviation of short rate

• Stochastic volatility (Heston)

v:variance, λ:speed of reversion, :long term mean, η:vol. of vol.

2( ( ) )t t rdr t ar dt dW

3( )t t tdv v v dt v dW v

r

1( )t t t t tdS r q S dt v S dW

ING 4

Introduction

Simulation Heston process Simulation Hull-White process

(λ:1, :0.35^2, η:0.5,v0:0.35^2,T:1) (θ:0.07, a:0.05, σ:0.01, r0:0.03)v

ING 5

Introduction

Pricing equation for the three-factor Heston / Hull-White model:

FAST ACCURATE GENERAL

2 2 2 2 22 2

12 13 232 2

22

2

V( ) ( ( ) ) ( )

1 1

2 2

10

2

r r r

V V Vr q S t ar v v rV

t S r v

V V V V VvS S v Sv v

r S r S v r r v

Vvv

ING 6

Solving pde without boundary conditions

Solving:

• Implicitly with pde-boundary conditions: whole equation as boundary condition using one-sided differences

• Explicitly on a tree-structured grid

ING 7

1-dimensional Black-Scholes equation

Black-Scholes equation:

r: interest

q: dividend yield

σ: volatility

V: option price

S: underlying equity

22 2

2

V 1( ) 0

2

V Vr q S S rV

t S S

ING 8

Black-Scholes(solved implicitly with pde)

ING 9

Black-Scholes(solved implicitly with pde)

• Inflow at right boundary, but one-sided differences wrong direction

• Non-legitimate discretization, due to pde-boundary conditions

(positive and negative eigenvalues)

• Actually adjusting extra diffusion and dispersion at boundary

ING 10

Black-Scholes (solved explicitly on tree)

• Upwind is used, so accuracy might be bad

• Strict restriction for stability of Euler forward Upperbound for spacestep with Gerschgorin

Example: r = 0.03, σ = 0.25, q = 0, S = [0,1000] gives N < 7

• Better time discretization methods needed, proposed RKC-methods.

ING 11

1-dimensional Hull-White equation

Hull-White equation:

r: interest rate

θ:average direction in which r moves

a:mean reversion rate

:annual standard deviation of short rate

22

2

V 1( ( ) ) 0

2 r

V Vt ar rV

t r r

r

ING 12

Hull-White (solved implicitly with pde)

Caplets:

ING 13

Hull-White (solved implicitly with pde)

• Flow direction same as one-sided differences as long as

• Discretization is not legitimate, but effects are hardly noticeable

max

( )tr

a

ING 14

Hull-White (solved explicitly on tree)

• Transformation applied to get rid of ‘-rV’

• Upwind is used

• Restriction on the time- and spacestep, but easier satisfied than Black-Scholes restriction

• Results look accurate

solV V V

ING 15

Conclusion

• Implicit methods with pde-boundary conditions: Give problems due to: non legitimate discretization and wrong

flow-direction

Put boundary far away to obtain accurate results

• Explicit methods: Very hard to satisfy stability conditions

Due to upwind less accurate

ING 16

Future goals

• More research on two methods to solve pdes Explicit with RKC-methods

• Investigating the Heston model

• Implementing three-factor model solver