How To Code the Newton-Raphson Method in Excel VBA Newton’s Algorithm

How To Code the Newton-Raphson Method in Excel VBA
Newton’s Algorithm
Newton's algorithm, alternately called the Newton-Raphson method, is
a numerical method for solving equations of the type f(x)=0. It has
numerous applications in finance, and we will use it to determine the
volatility surface for a call option using the Black Scholes VBA
function.
The above is the simpler method of interpretation, but the statement
f(x) = mx+c is actually derived from the more generalised Taylor's
series expansion where a continuous, differentiable function f may be
expanded about a point c as follows: f(x) =f(c)+f'(c)(x-c)+f''(c)/2!*(xc)^2+...
The method applies when we have a close approximation for a
solution to the equation. The method assumes that the tangent to the
graph at our approximation intersects the x-axis closer to the solution
of the equation, i.e. closer to where the line we are evaluating
intersects the x-axis. By solving for where the tangent intersects the xaxis, and repeating the process from that point, we get closer and
closer to the actual solution of the equation.
This is the univariate version of the Taylor expansion, but it will suffice
for finding, for example, volatility smiles since we assume that all other
variables remain constant.
Let's say we're at our approximate solution, x0. The value of y-axis at
this point is y = f(x0).
The equation of the intersection of the tangent with the y-axis is given
by the following:
0 = f'(x0)(x1-x0) + f(x0)
We know that the tangent has gradient f'(x) - the same gradient as the
line, from the definition of a tangent.
We also know that a univariate linear function takes the form f(x) = mx
+ c, where m is the gradient, x is the horizontal displacement from the
starting point, and c is the vertical displacement from the starting
point. As such, we have y = f'(x0)(x-x0) + f(x0) as the line of the
tangent. We solve this for y = 0 in order to get the next guesstimate of
the solution to f(x) = 0.
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Since the equation is linear, no second order derivatives exist so
these fall away, leaving
f(x) =f(c)+f'(c)(x-c). In our example, x0=c, giving y = f'(x0)(x-x0) + f(x0)
We then solve for x1
x1 = x0 - (f(x0)/f'(x0)
From this we can induce (though actual proof is out of scope here),
that
xn+1 = xn - f(xn)/f'(xn) , which is the formula for Newton's method.
The formula for Newton's method works in cases where the function is
differentiable, and where the gradient is not equal to zero.
How To Code the Newton-Raphson Method in Excel VBA
How To Code the Newton-Raphson Method in Excel VBA
VBA implementation of Newton's algorithm
Here is an implementation of Newton's algorithm for a simple function,
f(x) = x2 - 3.
For the volatility surface version, func(x_n) will be replaced by the
Black Scholes VBA pricing function, and func_dash(x_n) will be
replaced by the VBA function for calculating vega, the first derivative
of the option price with respect to volatility.
Function Newton(Seed As Double, Precision As Double) As Double
Dim x_next As Double, x_n As Double, error_val As Double
Dim ctr As Integer
x_n = Seed
ctr = 0
Do
x_next = x_n - func(x_n) / func_dash(x_n)
error_val = x_next - x_n
x_n = x_next
ctr = ctr + 1
Loop Until (Abs(error_val) <= Precision Or ctr = 1000)
Newton = x_next
End Function
Function func(x_n) As Double
func = x_n ^ 2 - 3
End Function
Function func_dash(x_n) As Double
func_dash = 2 * x_n
End Function
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How To Code the Newton-Raphson Method in Excel VBA