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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

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fast solver three factor heston hull white model

Fast solver three-factor Heston / Hull-White model

Floris Naber

ING Amsterdam & TU Delft

Delft 22 March 15:30

www.ing.com

outline
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

introduction three factor model
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.

ING

introduction
Introduction

Simulation Heston processSimulation 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)

ING

introduction5
Introduction

Pricing equationfor the three-factor Heston / Hull-White model:

FAST ACCURATE GENERAL

ING

solving pde without boundary conditions
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

1 dimensional black scholes equation
1-dimensional Black-Scholes equation

Black-Scholes equation:

r: interest

q: dividend yield

σ: volatility

V: option price

S: underlying equity

ING

black scholes solved implicitly with pde9
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

black scholes solved explicitly on tree
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

1 dimensional hull white equation
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

ING

hull white solved implicitly with pde13
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

ING

hull white solved explicitly on tree
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

ING

conclusion
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

future goals
Future goals
  • More research on two methods to solve pdes
    • Explicit with RKC-methods
  • Investigating the Heston model
  • Implementing three-factor model solver

ING

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