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1. FRONT ARENA Jonas Persson, PhD ”A Finite Difference PDE solver in practise” 22 August 2007

2. 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

3. 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 numerical method is sometimes problematic.

4. 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

5. 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.

6. Another dividend example A practical problem The user wants to treat an Absolute dividend as proportional (denoted AbsAsProp). Reasons: • No non-volatile part in the stock price. (implied volatility for different maturities comparable) • Flatter implied vol surface from the market. (empirical) We are handling this in the PDE solver.

7. 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

8. 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

9. 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.

10. 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

11. 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.

12. Exercise region for the American Put Schematic early exercise region: Without dividend With dividend

13. Local volatility Dupire (’93) local volatility given an implied vol surface

14. Local volatility transformations Transforming the formula yields where we use

15. 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)

16. Linear interpolation on the variance Linear interpolation of the variance between the skews along constant forward price.

17. Linear vs. linear on variance interpolation Linear interpolation between variances gives a piece-wise constant local volatility Linear interpolation between values gives a non-smooth volatility where we use

18. 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!

19. 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

20. 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.

21. A non-symmetric approximation The approximation closest to the barrier is non-symmetric.

22. Example: Barrier option Example: Up&Out European Call • Strike: 25 • Barrier: 45 • Expiry: 5y • Interest rate and carry-cost: 3% • Local volatility surface

23. Dividend structure • Note: • AbsAsProp dividends require several grids to get Greeks. • Dividends handled as dicrete.

24. Option Value and Delta Value of the option for some underlying prices Delta

25. Zoom of Delta and Gamma A closer look at Delta near the barrier. Gamma

26. 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%

27. Euler or Crank-Nicolson? Delta using Crank-Nicolson for all steps Delta Zoom of Delta

28. Euler or Crank-Nicolson? Delta using the Euler method for all steps Delta Zoom of Delta

29. 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 problems because of less damping inherent in the PDE. The methods must cover also extreme cases!

30. 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

31. 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!”

32. Performance/Accuracy Example: American Put, Absolute/AbsAsProp dividend Calculation of Price, Delta and Gamma

33. 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.

34. 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.

35. Adjusting the number of time-steps The number of time-steps can be adjusted per e.g. contract type (custom).

36. Summary – challenges What makes the ”real-life” complicated • Dividends • Discontinuities - Non-smooth Greeks • ”Special cases” (really low vola. e.t.c.) • Fast calculations necessary! • Other features: rebates, quanto, window ... Please consider all these things in your work!

37. The end From www.Sloganizer.net «Finite Differences, it's a kind of magic.» Thank you for listening!