Transcript Strong Growth

Fast solver
three-factor Heston / Hull-White model
Floris Naber
ING Amsterdam & TU Delft
Delft 22 March 15:30
www.ing.com
Outline
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Introduction to the problem (three-factor model)
 Equity underlying
 Stochastic interest
 Stochastic volatility
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Solving partial differential equations without boundary conditions
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1-dimensional Black-Scholes equation
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1-dimensional Hull-White equation
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Conclusion
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Future goals
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Introduction (Three-factor model)
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Underlying equity:
dS t  ( rt  q ) S t dt 
v t S t dW1
S: underlying equity, r: interest rate, q:dividend yield, v:variance
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Stochastic interest (Hull-White)
drt  ( ( t )  art ) dt   r dW 2
r: interest rate, θ:average direction in which r moves, a:mean reversion
rate,  r:annual standard deviation of short rate
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Stochastic volatility (Heston)
dv t    ( v t  v ) dt   v t dW 3
v:variance, λ:speed of reversion, v :long term mean, η:vol. of vol.
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Introduction
Simulation Heston process
Simulation Hull-White process
(λ:1, v :0.35^2, η:0.5,v0:0.35^2,T:1)
(θ:0.07, a:0.05, σ:0.01, r0:0.03)
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Introduction
Pricing equation for the three-factor Heston / Hull-White model:
V
t


1
 (r  q )S
 V
vS
2
r
1
 V
2
 V
V
r
  (v  v )
2
  12 S
v r
S r
 V
V
v
2
  13 Sv
S v
 rV

1
2
 V
2
r
2
r
2
 v
2
v
2
  23 r
0
ACCURATE
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 V
2
2
FAST
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S
 ( ( t )  ar )
2
2
2
V
GENERAL
v
r v
Solving pde without boundary conditions
Solving:
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Implicitly with pde-boundary conditions:
 whole equation as boundary condition using one-sided differences
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Explicitly on a tree-structured grid
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1-dimensional Black-Scholes equation
Black-Scholes equation:
V
t
 (r  q )S
V
S

1
 S
2
2
r: interest
q: dividend yield
σ: volatility
V: option price
S: underlying equity
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 V
2
7
2
S
2
 rV  0
Black-Scholes(solved implicitly with pde)
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Black-Scholes(solved implicitly with pde)
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Inflow at right boundary, but one-sided differences wrong
direction
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Non-legitimate discretization, due to pde-boundary conditions
(positive and negative eigenvalues)
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Actually adjusting extra diffusion and dispersion at boundary
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Black-Scholes (solved explicitly on tree)
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Upwind is used, so accuracy might be bad
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Strict restriction for stability of Euler forward
 Upperbound for spacestep with Gerschgorin
Example: r = 0.03, σ = 0.25, q = 0, S = [0,1000]
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gives N < 7
Better time discretization methods needed, proposed RKCmethods.
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1-dimensional Hull-White equation
Hull-White equation:
V
t
 ( ( t )  ar )
V
r

1
2
r
2
r: interest rate
θ:average direction in which r moves
a:mean reversion rate
 r:annual standard deviation of short rate
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 V
2
r
2
 rV  0
Hull-White (solved implicitly with pde)
Caplets:
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Hull-White (solved implicitly with pde)
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Flow direction same as one-sided differences as long as
rm ax 
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 (t )
a
Discretization is not legitimate, but effects are hardly noticeable
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Hull-White (solved explicitly on tree)
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Transformation V  V so l V
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Upwind is used
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Restriction on the time- and spacestep, but easier satisfied than
Black-Scholes restriction
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Results look accurate
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applied to get rid of ‘-rV’
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Conclusion
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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
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Explicit methods:
 Very hard to satisfy stability conditions
 Due to upwind less accurate
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Future goals
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More research on two methods to solve pdes
 Explicit with RKC-methods
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Investigating the Heston model
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Implementing three-factor model solver
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