The PDE approach to American Options

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The PDE approach to American Options

• Dustin Lennon
• Nigel Thavasi

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Contents

• The Canonical Free Boundary Problem
• Financial Intuition of Black-Scholes
• Why are American Options different?
• American Options as FBPs
• Linear Complementarity
• Onto Numerics
• Finite Difference Formulation
• Poisson/Laplace Equation
• Obstacle Problem
• Results
• Wilmotts Example
• Numerical PDE vs. Binomial Tree
• Summary
• Future research
• Additional References

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The Canonical FBP
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The Canonical FBP (contd)

• An elastic string held at A and B is stretched by
  a smooth object
• However, a priori, we dont know the points of
  contact P, Q and R
• What is known?
• Either the string is in contact with the
  obstacle, in which case, its position is known
• Or, it must satisfy an equation of motion, which
  in this case means it must be straight

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The Canonical FBP (contd)

• Under the following conditions, the solution to
  the obstacle problem can be shown to be unique
• String must be above or on the obstacle
• String must have negative or zero curvature
• String must be continuous
• String slope must be continuous

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Financial Intuition of BS

• Backward heat equation

Financial interpretation
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American vs. European

• Right to early exercise converts the BS PDE to an
  inequality
• Early exercise imposes the additional constraint
  of
• At each time, one must determine the option value
  and also for each value of S, if the option
  should be exercised ? Free Boundary Problem

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American Options as FBPs

• Analogue constraints
• Option value must be greater than or equal to the
  payoff function (a)
• BS equation is replaced by an inequality (ß)
• Option value must be continuous (?)
• Option delta must be continuous (d)

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American Options as FBPS (contd)
Q
Q
The example of the American put suffices for
conceptual illustration.
Constraints (a), (?) and (d)
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American Options as FBPs (contd)

• We now establish constraint (ß), albeit
  heuristically.
• Since the option can be exercised at any time,
  the arbitrage argument used for the European
  option no longer leads to a unique value for the
  return on the ?-hedged portfolio.

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American Options as FBPs (contd)
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American Options as FBPs (contd)

• In the first instance, while the BS equation is
  not satisfied, the inequality is and in the
  second instance, the equality of BS is attained.
  The financial intuition is obtained by consulting
  the BS operator, and one sees the relationship
  quite clearly

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American Options as FBPs (contd)

• We now establish the necessary boundary
  conditions.
• Consider the American Put.

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American Options as FBPs (contd)
Region of slope matching
American Put
European Put
Payoff Function
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American Options as FBPs (contd)
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American Options as FBPs (contd)
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American Options as FBPs (contd)
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American Options as FBPs (contd)
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American Options as FBPs (contd)
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American Options as FBPs (contd)
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American Options as FBPs (contd)

• Therefore our boundary conditions are
• It is important to note that the second condition
  is NOT derived from the functional form of the
  payoff function.
• The reason is that the value of the stock that
  makes exercise optimal is NOT known a priori.

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Linear Complementarity

• The FBP formulation is plagued with the explicit
  dependence on the free boundary which translates
  into problems with attempts at numerical
  solutions.
• Hence, the linear complementarity (LC)
  formulation is used so that a well posed
  numerical problem emerges.

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Linear Complementarity (contd)
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Linear Complementarity (contd)

• Intuition While we cannot say when contact with
  the string occurs, we can deduce the
  contemporaneous nature of the elements at contact
  
• Close inspection of the previous graph reveals
  the following

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Linear Complementarity (contd)
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Linear Complementarity (contd)

• In general, a problem of the form below is known
  as a complementarity problem linear simply
  indicates that each of the elements of the
  product are linear
• This formulation is similar to what the
  economists affectionately refer to as the
  Envelope Theorem.

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Linear Complementarity (contd)

• Therefore, the linear complimentarity formulation
  is

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Linear Complementarity (contd)
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Linear Complementarity (contd)
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Linear Complementarity (contd)
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Onto Numerics

• The additional right to exercise early
  complicates the mathematics significantly by
  making it a free boundary problem
• All the difficulty arises from not knowing a
  priori what value of the stock makes early
  exercise optimal
• In order to have a well defined FBP, care is
  taken to ensure that the boundary conditions are
  derived INDEPENDENT of the free boundary and in
  this case, we used arbitrage

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Onto Numerics (contd)

• However, the stark limitation of the FBP
  formulation is its explicit dependence on the
  free boundary, which leads to complication in
  posing a well defined numerical problem.
• So the move to linear complementarity is made.
  We see that in this formulation, the explicit
  dependence on the free boundary is removed, and
  we can now pose a well defined numerical problem.
• The linear complementarity also leads into the
  idea of a variational inequality (finite element).