FTCS Explicit Finite Difference Method for Evaluating European Options

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FTCS Explicit Finite Difference Method for
Evaluating European Options

• CS757 Computational Finance
• Project No. CS757.2003W-26

Amit Chhabra Department of Computer
Science University of Manitoba
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Outline

• Introduction and Motivation
• Problem Statement
• Solution Strategy
• FTCS Method
• Assumptions
• Experimental Results
• Effect of N
• Effect of Volatility and T
• Variance of Option Value with K
• Effect of r
• Effect of l (Time steps)
• Conclusion and Future Work

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Introduction and Motivation

• The pricing of financial instruments by numerical
  solutions of their pricing equation has become an
  important component in the arsenal of techniques
  available to practitioners of modern quantitative
  finance.
• The demand for complex financial instruments and
  the availability of powerful computers make the
  direct numerical solution of the governing
  pricing equation increasingly appealing approach
  to pricing

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Problem Statement

• The solution methods for the Black-Scholes model
  (used for the evaluation of option price) are
  computationally intensive.
• Moreover non-linearity of the BS model and
  real-time solution requirement makes it further
  difficult to solve the BS model.
• Thus, we developed an efficient and fast
  algorithm for evaluating European Option by
  Finite Differencing.

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Solution Strategy

• We have applied the Forward Time Centered Space
  (FTCS) method on the Black-Scholes equation to
  discretize the Partial Differential Equation
  (PDE).
• Assumptions
• The stock price follows the geometric Brownian
  motion with constant volatility ?
• There is a constant risk free interest rate r
• There are no arbitrage opportunities or
  transaction costs

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Solution Strategy cont

• The complete non-linear Burger equation is of the
  form
• From the Computational Fluid Dynamic (CFD)
  literature 2, 3 it is known that FTCS method
  works well for Burgers equation and hence it
  could be applied to solve the Black-Scholes
  equation given by

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FTCS Method

• In the explicit formulation of FTCS method, a
  first-order forward difference approximation and
  second-order central approximation for the time
  derivative and the spatial derivatives are used,
  resp.
• Hence, the Finite Difference Equation (FDE) for
  the Burgers equation is given by

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2-D Grid
pm
pu
pd
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Maturity
Computation proceeds
4
3
2
N
1
0
-3
-2
-1
0
1
2
3
Nj
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3-D Mesh for Evaluating Options
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Experimental Results

• To implement the pricing algorithm we used C
  language on Unix platform.
• We studied the effect of various parameters on
  the option value.
• It should be noted that the values of each
  parameter is varied only when its effect is
  studied on the option value.

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Effect of N
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Effect of ? and T
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Execution Time
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Variance of Option Value with K
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Effect of r
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Effect of l (Time steps)
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Err Vs. Time steps
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Conclusion and Future Work

• We developed an efficient algorithm fro pricing
  European options using finite differencing.
• We studied the effect of various parameters on
  the value of the option and concluded that
  increasing the number of time steps increases the
  accuracy of the option value. Also, after a
  certain value of time steps the option value
  stabilizes and err decreases.
• We considered a dividend paying asset.
• We notices that for bigger mesh, small machine
  size is a bottleneck. Hence we intend to
  parallelize the algorithm to be run on more than
  one processors and further decrease the execution
  time. Also, we intend to extend the algorithm for
  multiple assets.

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Thank You