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Integrated cross asset front to back trading solutions
www.sungard.com/frontarena
FRONT ARENA
Jonas Persson, PhD
”A Finite Difference PDE solver in practise”
22 August 2007
Agenda
The FRONT ARENA Finite Difference PDE solver
Introduction The framework American options – an example Local volatility implementation Barrier options – challenging problems Performance/Accuracy Summary
Introduction
When analytical formulas are not enough
... numerical methods are necessary!
Example of such cases are
Dividends Local volatility Complex derivatives
Producing smooth Greeks with a numericalmethod is sometimes problematic.
The framework
Background and assumptions
Flat (or time-dependent) volatility framework (B&S ’73)
Local volatility for exotic options (Dupire ’93)
Absolute/Proportional dividend types supported
Term structure of interest rates
Handling dividends
Dividends must be handled as discrete,
approximating as dividend yield is not enough.
Example: U&O/D&O window barrier, Div at Td.
Another dividend example
A practical problem
The user wants to treat an Absolute dividend as
proportional (denoted AbsAsProp).
Reasons:
1. No non-volatile part in the stock price. (implied volatility for different maturities comparable)
2. Flatter implied vol surface from the market. (empirical)
We are handling this in the PDE solver.
The numerical framework
The B&S PDE is solved numerically using a
Finite Difference method with
Crank-Nicolson (normally)
- Second order accurate
- Centered differences
Euler backward (in some cases)
- Near barriers
- Used to dampen oscillations
The numerical framework
Implicit time-stepping gives:
Tri-diagonal linear systems of equations for each time-step
Solved using ”Thomas algorithm” with only O(8n) arithmetic operations per step
Example: American Put
For the American Put option we use an
”Operator splitting technique”
Introduced by Ikonen & Toivanen to handle the early
exercise feature.
”S. Ikonen, J. Toivanen, Operator Splitting
Methods for Pricing American Options,
Applied Mathematical Letters 17, 2004”
This has been extended in a number of papers to
stochastic volatility e.t.c.
American Put example - high Gamma peak
American Put option
Interesting case:
Strike at 100, T = 1y r = g = 5%, Flat volatility 26% One Absolute dividend of size 4.5 at t = 0.9y
High gamma peak for American Put (Abs div)
Value, Delta and Gamma for American Put, Abs dividend
Also this particular case with a Gamma jump is handled.
Exercise region for the American Put
Schematic early exercise region:
Without dividend With dividend
Local volatility
Dupire (’93) local volatility given an implied vol surface
2
2 222
1 12
2 2( )( , )
1loc
t r D StT SS t
Sd t S t d tS S S
20
1
log 1 2S S r D td
t
0
Local volatility
Implied volatility
Time to expiry
Interest rate
Carry cost
Strike price
Underlying price at calculation date
loc
t
r
D
S
S
Local volatility transformations
Transforming the formula yields
where we use
2 2
0
( , )
1 ln 1 12 2 2 2
loc
w tww x t
F w w wt w tF w w
0
2
Local volatility squared
Forward price of underlying at time t
Spot price
ln
( )
Derivative with respect to time
' and '' Derivative with respect to x
locw
F
F
x F
w t
w
Local volatility continued
Global parametric volatility surface built of time-skews (2nd degree polynomials)
Volatility is a function of forward prices and time
Properties of the volatility surface
Smooth representation with skew and smile parametrized
Works well with the Dupiré formula due to variance interpolation (explained shortly)
Linear interpolation on the variance
2 21 1 0 0 0 12
1 0
( ) ( )( , )
F t t t F t t tF t t
t t
0 0
1 1
0 1
The interpolated volatility value
Volatility at start skew at time
Volatility at end skew at time
t
t
t t t
Linear interpolation of the variance between the skewsalong constant forward price.
01
0201
21
22
tt
ttt
t
wtwtt
22
Linear vs. linear on variance interpolation
Linear interpolation of Global Volatility
0
0.05
0.1
0.15
0.2
0.25
0 0.5 1 1.5 2
Local Vol
Global Vol
Linear interpolation between variance
0
0.05
0.1
0.15
0.2
0.25
0 0.5 1 1.5 2
Expiry
Local VolGlobal Vol
Linear interpolation between values gives a non-smooth
volatility
where we use
Linear interpolation between variances gives a piece-wise constant local volatility
Local volatility continued
Calculating the local volatility:
The time derivative need not be calculated due to the transformation
The strike derivatives are approximated using central finite differences on the skews
Summary:
The transformation of the PDE and local volatility formulas together with the variance interpolation increases the speed of the calculations!
The barrier challenge
Challenges with barrier option:
Smooth Greeks near barriers difficult (e.g. for tree models).
Dividends introduce problems
Solution: Non-symmetric FD approximation near the
boundary. + Special tricks
A non-symmetric approximation
Lets take a first derivative as an example:
With the coefficients given by
Note that if the step sizes are equal we retrieve the standard
central approximation. Second derivatives are approximated
in a similar way.
11
iiiiiis
vcvbvas
u
i
)(,,
)(
iii
ii
ii
iii
iii
ii
hhh
hc
hh
hhb
hhh
ha
A non-symmetric approximation
The approximation closest to the barrier is
non-symmetric.
Example: Barrier option
Example: Up&Out European Call
Strike: 25 Barrier: 45 Expiry: 5y Interest rate and carry-cost: 3% Local volatility surface
Dividend structure
Time of dividend Size of dividend Type
0.5 years 1.0 Absolute
1.5 years 1.0 AbsAsProp
2.5 years 1.0 AbsAsProp
3.5 years 1.0 AbsAsProp
4.5 years 5% Proportional
Note: 1. AbsAsProp dividends require several grids to get Greeks.2. Dividends handled as dicrete.
Option Value and Delta
Value of the option
for some underlying
prices
Delta
Zoom of Delta and Gamma
A closer look at Delta
near the barrier.
Gamma
The barrier option again: An extreme case
Up&Out call,
Barrier at 60, strike at 40.AbsAsProp dividend of 2 at t=1y, T=2y, r=g=3%,
volatility=3%
Euler or Crank-Nicolson?
Delta using Crank-Nicolson for all steps
Delta Zoom of Delta
Euler or Crank-Nicolson?
Delta using the Euler method for all steps
Delta Zoom of Delta
Euler or Crank-Nicolson?
The choice of numerical method depends on ...
Accuracy considerations Non-smooth Greeks ? Discontinuities ? Speed
In this particular case:Low volatility often causes numerical problemsbecause of less damping inherent in the PDE.
The methods must cover also extreme cases!
Performance/Accuracy
Getting the numbers right involves many things:
Mathematical modelling The choosen model setup (incl. B&S, Divs, Vol) Calculation/Estimation of relevant parameters
Numerical considerations, such as Technical aspects (num. method e.t.c) Barrier treatment American feature treatment Richardson extrapolation Smoothing
Performance/Accuracy
Fast calculations is really important!
Or as a customer put it:
”- If I get really accurate prices too late they are completely useless!”
Performance/Accuracy
Example: American Put, Absolute/AbsAsProp dividend
Calculation of Price, Delta and Gamma
Average call time
in milliseconds
Absolute dividend
AbsAsProp dividend
30 time-steps 0.9277 2.9347
Tricks for smooth Greeks
Some tricks used to get smooth Greeks
Non-symmetric approx. (barriers) Fixed grid Smoothing of initial data Cubical spline approximations ... or non-polynomial approximations E.t.c.
Changing the settings
Many numerical parameters in the PDE solver
can be manipulated through the GUI.
E.g.: Number of time-steps Ikonen algorithm for American Put - On/Off Richardson Extrapolation - On/Off Smoothing of pay-off - On/Off Calculate Greeks - On/Off E.t.c.
Adjusting the number of time-steps
The number of time-steps can be adjusted per e.g. contract type (custom).
Summary – challenges
What makes the ”real-life” complicated
1. Dividends
2. Discontinuities - Non-smooth Greeks
3. ”Special cases” (really low vola. e.t.c.)
4. Fast calculations necessary!
5. Other features: rebates, quanto, window ...
Please consider all these things in your work!
The end
From www.Sloganizer.net
«Finite Differences, it's a kind of magic.»
Thank you for listening!
