Transcript FTCS Explicit Finite Difference Method for Evaluating

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
17 July 2015
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Outline
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Introduction and Motivation
Problem Statement
Solution Strategy
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Experimental Results
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FTCS Method
Assumptions
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
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We have applied the Forward Time Centered Space (FTCS)
method on the Black-Scholes equation to discretize the
Partial Differential Equation (PDE).
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Assumptions
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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…
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The complete non-linear Burger equation is of the form:
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From the Computational Fluid Dynamic (CFD) literature [2,
3] it is known that FTCS method works well for Burger’s
equation and hence it could be applied to solve the BlackScholes 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 Burger’s
equation is given by
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2-D Grid
pd
pm
pu
5
Maturity
Computation proceeds
4
3
2
N
1
0
-3
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-2
-1
0
Nj
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2
3
8
3-D Mesh for Evaluating Options
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Experimental Results
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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
Option Value
OV (T=1yr, vol=40%)
3.85
3.84
3.83
3.82
3.81
3.8
3.79
3.78
3.77
3.76
3.75
OV (T=1yr, vol=40%)
N = 10
N = 25
N = 50
N = 75
N = 100
Levels (N)
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Effect of  and T
Option Value vs. Number of Levels (N)
Option Value (OV)
5
OV (T=1yr, vol=40%)
4
OV (T=1yr, vol=25%)
3
OV (T=6Mts, vol=40%)
2
OV (T=6Mts, vol=25%)
OV (T=3Mts, vol=40%)
1
OV (T=3Mts, vol=25%)
0
N = 10
N = 25
N = 50
N = 75
N = 100
Number of Levels (N)
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Execution Time
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Variance of Option Value with K
Option Value (OV)
Option Value Vs. Strike Price
8
7
6
5
4
3
2
1
0
OV (T=1yr, vol=40%)
OV (T=1yr, vol=25%)
K=20
K=23
K=25
K=27
K=30
K=35
Strike Price (K)
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Effect of r
Option Value
5
4
Option Value
3
2
Option Value
1
0
r=
10%
r=
15%
r=
20%
r=
25%
r=
30%
Interest Rate
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Effect of l (Time steps)
Option Value Vs. Timesteps
Option Value (OV)
2.5
2
OV (N = 25, Nj = 100)
1.5
OV (N = 50, Nj = 100)
1
OV (N = 75, Nj = 100)
OV (N = 100, Nj = 100)
0.5
0
ts = 25
ts = 50
ts = 75 ts = 100 ts = 125
Timesteps
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Err Vs. Time steps
Err Vs. Timesteps
0.03
0.025
Err
0.02
0.015
Err
0.01
0.005
0
ts = 25
ts = 50
ts = 75
ts = 100
ts = 125
Timesteps
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Conclusion and Future Work

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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…
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