A Finite Differencing Solution for Evaluating European Prices

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A Finite Differencing Solution for Evaluating
European Prices

• Computational Finance cs 757
• Project CFWin03-33
• May 30, 2003
• Presented by Vishnu K Narayanasami
• vishnun_at_cs.umanitoba.ca

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Outline

• Introduction
• Problem Statement
• Background
• Methodology
• Experimental Analysis and Results
• Future Work

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Introduction

• European Call Options
• The buyer pays a one-time premium for buying a
  particular stock at a particular rate.
• The option can be exercised only at maturity.
• Finite Differencing
• The basic method of solving differential
  equations in a computer.
• Allows analysis of all kinds and shapes of
  objects.

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Problem Statement

• Evaluating prices of options is of practical
  importance and is a tedious task.
• Complicated finance problems lead to complex
  coupled equations.
• Closed form solutions of these equations are
  almost impossible.
• With finite differencing techniques, it is
  possible to model the problem and achieve better
  computational results.
• Implementing Crank-Nicholson scheme for
  evaluating European options

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Background

• Black-Scholes model
• Calculate theoretical call price based on five
  parameters strike price, stock price,
  volatility, time of expiration and short-term
  risk free interest.
• Black-Scholes equation is given as

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Background

• Crank-Nicholson Finite Differencing Technique
• Implicit method
• Uses Central-differencing
• System of linear equations
• Unconditionally stable
• Values of unknowns are assigned to the grid
  points.

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Major Difference

• Many finite differencing schemes Explicit
  methods, Classical Implicit, Implicit, etc.
• Crank-Nicholson method is fully implicit.
• Second order accurate in time whereas other
  schemes are first order accurate
• Unconditionally stable uses central
  differencing space derivative at time level
• n ½ (mean of other methods).

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Methodology
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Methodology (2)

• Discretized Black-Scholes equation

Where
is the timestep and
is the distance
between the nodes.
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Methodology - Pseudo Code (3)
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Methodology (4)
pm
pd
pu
Maturity values
N
Option Value
Nj
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Challenges

• To bridge finite differencing concepts to suit
  finance problems.
• I have already worked in finite differencing on a
  fluid mechanics problem, and it was complicated
  to let go my fixed mindset on that field and
  switch to finance.
• Implementing the boundary conditions for this
  project.
• Experienced memory problems during execution of
  the code.

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Experimental Analysis testbed

• The code was developed in Java and tested in the
  Linux machines in the Department of Computer
  Science with the following configuration

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Experimental Analysis Results (2)

• I could observe the general trend from the
  results that as the Strike Price (K) increases,
  the Option value decreases for European options,
  as in our assignment problem.
• I am still refining the code to achieve other
  results such as varying strike price over option
  values, increasing no. of levels, time steps,
  analyzing execution time.

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Continuing Work

• Making the code work completely to achieve
  satisfactory results
• Testing the code for differing parameters.

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Future Work

• Comparison with other finite differencing schemes
• Parallel implementation

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Thank you!