Fast solver three-factor Heston / Hull-White model

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Fast solver three-factor Heston / Hull-White
model

• Floris Naber
• ING Amsterdam TU Delft

Delft 22 March 1530 www.ing.com
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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

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Introduction (Three-factor model)

• Underlying equity
• S underlying equity, r interest rate,
  qdividend yield, vvariance
• Stochastic interest (Hull-White)
• r interest rate, ?average direction in which r
  moves, amean reversion rate, annual standard
  deviation of short rate
• Stochastic volatility (Heston)
• vvariance, ?speed of reversion, long term
  mean, ?vol. of vol.

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Introduction

• Simulation Heston process
  Simulation Hull-White process
• (?1, 0.352, ?0.5,v00.352,T1) (?0.07,
  a0.05, s0.01, r00.03)

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Introduction

• Pricing equation for the three-factor Heston /
  Hull-White model
• FAST ACCURATE GENERAL

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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

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1-dimensional Black-Scholes equation

• Black-Scholes equation
• r interest
• q dividend yield
• s volatility
• V option price
• S underlying equity

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Black-Scholes(solved implicitly with pde)
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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

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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, s 0.25, q 0, S 0,1000
  gives N lt 7
• Better time discretization methods needed,
  proposed RKC-methods.

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1-dimensional Hull-White equation

• Hull-White equation
• r interest rate
• ?average direction in which r moves
• amean reversion rate
• annual standard deviation of short rate

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Hull-White (solved implicitly with pde)

• Caplets

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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

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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

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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

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Future goals

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