fast solver three-factor heston / hull-white model
DESCRIPTION
Fast solver three-factor Heston / Hull-White model. Floris Naber ING Amsterdam & TU Delft. Delft 22 March 15:30 www.ing.com. Outline. Introduction to the problem (three-factor model) Equity underlying Stochastic interest Stochastic volatility - PowerPoint PPT PresentationTRANSCRIPT
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