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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
- 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)

- 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

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

Introduction

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

FAST ACCURATE GENERAL

ING

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

Black-Scholes equation:

r: interest

q: dividend yield

σ: volatility

V: option price

S: underlying equity

ING

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)

- 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

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 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)

- 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

- 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

- 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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