Risk Latte - Black-Sholes PDE and Valuation of Options using Green’s Function

Black-Sholes PDE and Valuation of Options using Green’s Function

Danny Yap
CFE (July 2010) – Singapore

(Note by Team Latte: This is an abstract of Danny Yap’s brilliant CFE class presentation recently in Singapore. The class presentation was done using Excel spreadsheet and a full suite of option valuation using the Green’s Function was demonstrated and explained by Danny using Excel spreadsheet. Danny Yap is currently enrolled in our Certificate in Financial Engineering (CFE) course and attends classes in Singapore.)

If you believe the stock price process is a GBM,

Then Ito’s formula and a hedging argument, leads to the Black-Scholes Equation (BSE) for ,

To get the diffusion equation, make the change of variables,

Then the BSE becomes the Diffusion Equation (DE) for ,

In 1905, Einstein showed us that the DE arises from the Brownian motion of microscopic particles. Thus, both the BSE and the DE are based on the same underlying process. Solutions of the DE with known initial condition take the form,

or Green’s Function is called the Fundamental Solution of the DE.

Why use the Diffusion Equation?

Mathematical and numerical techniques developed for solving the DE can be applied.

1. Analytic solutions can be obtained:

  1. Green’s Function Methods - Calls, Puts, Binaries (We will do this on Excel spreadsheet!)
  2. Method of Images (Zero Boundary Conditions) –Barrier Options
  3. Impedance Boundary Conditions – Lookback Options
  4. Separation of Variables – Log and Power Contracts
  5. Duhame Integrals for Heat Equation - Rebates

2. Numerical methods can be used instead of Monte Carlo Simulations

  1. Finite-Difference Methods
  2. Fourier Transforms

Examples: Vanilla Call Option Price Using Green’s Function

For a call option, the boundary condition at is,

where is the Heaviside Step Function. Making the transformation of variables, the boundary condition on the DE at becomes,

We can put back into the integral, to get Black-Scholes Formula after “some” integration

Alternatively, we can solve for numerically and then transform back to get X (See Worksheet)

Initial Conditions for Other Options


Reference:
[1] Solution to the Black-Scholes Equation, S. Karim, MIT, May 2009
[2] Notes on Solving the Black-Scholes Equation, EOLA Investments, LLC, Oct 2009
[3] The Diffusion Equation – A Multidimensional Tutorial, T.S. Ursell, Caltech, 2007
[4] A. Einstein, Ann. d. Phys., 17, p. 549 (1905)

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