# Strong Growth - TU Delft 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 2

Introduction (Three-factor model) Underlying equity: dSt (rt q ) St dt vt St dW1 S: underlying equity, r: interest rate, q:dividend yield, v:variance Stochastic interest (Hull-White) drt ( (t ) art )dt r dW2 r: interest rate, :average direction in which r moves, a:mean reversion rate, r:annual standard deviation of short rate Stochastic volatility (Heston) dvt (vt v)dt vt dW3 v:variance, :speed of reversion, v :long term mean, :vol. of vol.

ING 3 Introduction Simulation Heston process Simulation Hull-White process (:1, v :0.35^2, :0.5,v0:0.35^2,T:1) (:0.07, a:0.05, :0.01, r0:0.03) ING 4 Introduction Pricing equation for the three-factor Heston / Hull-White model:

V V V V (r q ) S ( (t ) ar ) (v v ) rV t S r v 1 2 2V 2V 2V 1 2 2V 2V vS 12 S v r 13Sv r

23 r v 2 2 2 r S r S v 2 r rv 1 2 2V v 2 0 2 v FAST ING ACCURATE 5

GENERAL Solving pde without boundary conditions Solving: Implicitly with pde-boundary conditions: whole equation as boundary condition using one-sided differences ING Explicitly on a tree-structured grid 6 1-dimensional Black-Scholes equation Black-Scholes equation:

2 V V 1 2 2 V (r q) S S rV 0 2 t S 2 S r: interest q: dividend yield : volatility V: option price S: underlying equity ING

7 Black-Scholes(solved implicitly with pde) ING 8 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)

ING Actually adjusting extra diffusion and dispersion at boundary 9 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, = 0.25, q = 0, S = [0,1000]

ING gives N < 7 Better time discretization methods needed, proposed RKCmethods. 10 1-dimensional Hull-White equation Hull-White equation: 2 V V 1 2 V ( (t ) ar ) r rV 0 2 t r 2

r r: interest rate :average direction in which r moves a:mean reversion rate r:annual standard deviation of short rate ING 11 Hull-White (solved implicitly with pde) Caplets: ING 12 Hull-White (solved implicitly with pde)

Flow direction same as one-sided differences as long as rmax ING (t ) a Discretization is not legitimate, but effects are hardly noticeable 13 Hull-White (solved explicitly on tree) Transformation V Vsol V

Upwind is used Restriction on the time- and spacestep, but easier satisfied than Black-Scholes restriction Results look accurate ING applied to get rid of -rV 14

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 15 Future goals

More research on two methods to solve pdes Explicit with RKC-methods Investigating the Heston model Implementing three-factor model solver ING 16