1. Do not put content on the brand signature area Fast solver three-factor Heston / Hull-White model Delft 22 March 15:30 www.ing.com Floris Naber ING Amsterdam & TU Delft

2. Do not put content on the brand signature area Orange RGB= 255,102,000 Light blue RGB= 180,195,225 Dark blue RGB= 000,000,102 Grey RGB= 150,150,150 ING colour balance Guideline www.ing-presentations.intranet ING2 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

3. Do not put content on the brand signature area Orange RGB= 255,102,000 Light blue RGB= 180,195,225 Dark blue RGB= 000,000,102 Grey RGB= 150,150,150 ING colour balance Guideline www.ing-presentations.intranet ING3 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.

4. Do not put content on the brand signature area Orange RGB= 255,102,000 Light blue RGB= 180,195,225 Dark blue RGB= 000,000,102 Grey RGB= 150,150,150 ING colour balance Guideline www.ing-presentations.intranet ING4 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)

5. Do not put content on the brand signature area Orange RGB= 255,102,000 Light blue RGB= 180,195,225 Dark blue RGB= 000,000,102 Grey RGB= 150,150,150 ING colour balance Guideline www.ing-presentations.intranet ING5 Introduction Pricing equation for the three-factor Heston / Hull-White model: FAST ACCURATE GENERAL

6. Do not put content on the brand signature area Orange RGB= 255,102,000 Light blue RGB= 180,195,225 Dark blue RGB= 000,000,102 Grey RGB= 150,150,150 ING colour balance Guideline www.ing-presentations.intranet ING6 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

7. Do not put content on the brand signature area Orange RGB= 255,102,000 Light blue RGB= 180,195,225 Dark blue RGB= 000,000,102 Grey RGB= 150,150,150 ING colour balance Guideline www.ing-presentations.intranet ING7 1-dimensional Black-Scholes equation Black-Scholes equation: r: interest q: dividend yield σ: volatility V: option price S: underlying equity

8. Do not put content on the brand signature area Orange RGB= 255,102,000 Light blue RGB= 180,195,225 Dark blue RGB= 000,000,102 Grey RGB= 150,150,150 ING colour balance Guideline www.ing-presentations.intranet ING8 Black-Scholes(solved implicitly with pde)

9. Do not put content on the brand signature area Orange RGB= 255,102,000 Light blue RGB= 180,195,225 Dark blue RGB= 000,000,102 Grey RGB= 150,150,150 ING colour balance Guideline www.ing-presentations.intranet ING9 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

10. Do not put content on the brand signature area Orange RGB= 255,102,000 Light blue RGB= 180,195,225 Dark blue RGB= 000,000,102 Grey RGB= 150,150,150 ING colour balance Guideline www.ing-presentations.intranet ING10 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.

11. Do not put content on the brand signature area Orange RGB= 255,102,000 Light blue RGB= 180,195,225 Dark blue RGB= 000,000,102 Grey RGB= 150,150,150 ING colour balance Guideline www.ing-presentations.intranet ING11 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

12. Do not put content on the brand signature area Orange RGB= 255,102,000 Light blue RGB= 180,195,225 Dark blue RGB= 000,000,102 Grey RGB= 150,150,150 ING colour balance Guideline www.ing-presentations.intranet ING12 Hull-White (solved implicitly with pde) Caplets:

13. Do not put content on the brand signature area Orange RGB= 255,102,000 Light blue RGB= 180,195,225 Dark blue RGB= 000,000,102 Grey RGB= 150,150,150 ING colour balance Guideline www.ing-presentations.intranet ING13 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

14. Do not put content on the brand signature area Orange RGB= 255,102,000 Light blue RGB= 180,195,225 Dark blue RGB= 000,000,102 Grey RGB= 150,150,150 ING colour balance Guideline www.ing-presentations.intranet ING14 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

15. Do not put content on the brand signature area Orange RGB= 255,102,000 Light blue RGB= 180,195,225 Dark blue RGB= 000,000,102 Grey RGB= 150,150,150 ING colour balance Guideline www.ing-presentations.intranet ING15 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

16. Do not put content on the brand signature area Orange RGB= 255,102,000 Light blue RGB= 180,195,225 Dark blue RGB= 000,000,102 Grey RGB= 150,150,150 ING colour balance Guideline www.ing-presentations.intranet ING16 Future goals More research on two methods to solve pdes  Explicit with RKC-methods Investigating the Heston model Implementing three-factor model solver