RootFinding Methods

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Root-Finding Methods

• The bisection method
• Newtons method
• Secant method

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1. The Bisection Method(1/2)

• Assume that f(x) is continuous on a given
  interval a, b and
• that it also satisfies
• f(a) f(b) lt 0 (1.1)
• Using the intermediate value theorem, the
  function f(x) must have at least one root in a,
  b.
• Algorithm Bisect (f, a, b, root, ?)
• Define c (a b) / 2
• If b-c ? ?, then accept root c, and exit.
• If sign (f(b))sign(f(c)) ? 0, then a c
  otherwise, b c.
• Return to step 1.

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1. The Bisection Method(2/2)

• There are several deficiencies in the algorithm
  Bisect. First, it
• does not take account of the limits of machine
  precision. The
• second major problem with Bisect is that it
  converges very
• slowly when compared with the methods defined in
  the
• following sections.
• The major advantages of the bisection method are
  
• (1) it is guaranteed to converge (provided f is
• continuous on a, b and equation (1.1) is
  satisfied), and
• (2) a reasonable error bound is available.

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2. Newtons Method(1/5)

• Assume that an initial estimate x0 is known for
  the desired
• root ? of f(x) 0. Newtons method will produce
  a sequence
• of iterates xn n ? 1, which we hope will
  converge to ?.
• The iteration formula
• The process is illustrated in Figure 2.2, for the
  iterates x1 and
• x2.

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2. Newtons Method(2/5)

• Theorem 2.1
• Assume f(x), f (x), and f (x) are continuous
  for all x in some
• neighborhood of ?, and assume f(?) 0, f (?) ?
  0. Then if x0
• is chosen sufficiently close to ?, the iterates
  xn, n ? 0, of (2.1)
• will converge to ?. Moreover,
• proving that the iterates have an order of
  convergence p 2.

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2. Newtons Method(3/5)

• Let x0 b and define the Newton iterates xn as
  in (2.1). Next,
• define a new sequence of iterates by
• with z0 a. The resulting iterates are
  illustrated in Figure 2.3.
• With the use of zn, we obtain excellent upper
  and lower
• bounds for ?. The use of (2.2) with Newtons
  method is
• called the Newton-Fourier method.

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2. Newtons Method(4/5)

• The Newton algorithm
• Algorithm Newton (f, df, x0, ?, root, itmax, ier)
• Remark df is the derivative function f (x),
  itmax is the maximum number of iterates to be
  computed, and ier is an error flag to the user.
• itnum 1
• denom df(x0).
• If denom 0, then ier 2 and exit.
• x1 x0-f(x0) / denom
• If , then set ier 0, root
  x1, and exit.
• If itnum itmax, set ier 1 and exit.
• Otherwise, itnum itnum 1, x0 x1, and go
  to step 3.

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2. Newtons Method(5/5)

• Discussions
• As with the earlier algorithm Bisect, no account
  is taken of
• the limits of the computer arithmetic, although a
  practical
• program would need to do such. Also, Newtons
  method is
• not guaranteed to converge, and thus a test on
  the number of
• iterates (step 7) is necessary.
• Another source of difficulty in some case is the
  necessity of
• knowing f (x) explicitly.

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3. The Secant Method(1/2)

• As with Newtons method, the graph of y f(x) is
  
• approximated by a straight line in the vicinity
  of the root ?. In
• this case, assume that x0 and x1 are two initial
  estimates of the
• root ?. Approximate the graph of y f(x) by the
  secant line
• determined by (x0, f(x0)) and (x1, f(x1)). Let
  its root be denoted
• by x2 we hope it will be an improved
  approximation of ?.
• This is illustrated in Figure 2.4.

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3. The Secant Method(2/2)

• The general formula based on this is

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Comparison of Newtons method and the secant
method

• Newtons method requires two function evaluations
  per
• iterate, that of f(xn) and f (xn), whereas the
  secant method
• requires only one function evaluation per
  iterate, that of f(xn)
• provided the needed function value f(xn-1) is
  retained from
• the last iteration. Newtons method is generally
  more
• expensive per iteration.
• On the other hand, Newtons method converges more
  rapidly order p 2 vs. the secant methods p
  1.62, and consequently it will require fewer
  iterations to attain a given desired accuracy.

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Initial Points

• Several approximate formulas for computing the
• implied volatility of an option exist. See
  Chance(1993), and
• Brenner and Subrahmanyam(1998). These approximate
  formulas serve as good initial points of the
  above root-finding methods.
• Reference
• Brenner, M., and M. Subrahmanyam, 1998, A Simple
  Solution to Compute Implied Standard Deviation,
  Financial Analysts Journal, Vol. 44, No. 5,
  80-83.
• Chance, D. M., 1993, Leap into the Unknown, Risk,
  Vol. 6, No. 5, 60-66.

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Example Solving implied volatility with the
Bisection Method
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Example Solving implied volatility with the The
Newton-Raphson Method

• Even starting with a bad initial point, the
  Newton-Raphson
• method converges to the true value very quickly.
• This is the case because the option price is a
  well behaved
• function (continuous and differentiable) of
  volatility.