User:Fintor/Sandbox

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Fintor > talk > articles > sandbox | June 12 16:52 UTC

[edit] The Black Scholes PDE

1. The Black–Scholes PDE describes the evolution of the value of an option through time (t), as related to changes in the underlying stock price (S). The PDE is:

\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2\frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} = rV
V = V(S,t)\,

2. One solution to this PDE is the Black-Scholes formula for pricing a call option:

V(S,t) = S\Phi(d_1) - Ke^{-r(\tau)}\Phi(d_2) \,
Φ is the standard normal cumulative distribution function
τ is the time remaining until maturity

3. To check whether this result is a solution, substitute the Black-Scholes formula into the Black-Scholes PDE.

a. The partial derivatives are the Greeks:
  • \frac{\partial V}{\partial t} , Theta, = -\frac{S \phi(d_1) \sigma}{2 \sqrt{\tau}} - rKe^{-r \tau}\Phi(d_2)\
  • \frac{\partial V}{\partial S} , Delta, = \Phi(d_1) \
  • \frac{\partial^2 V}{\partial S^2} , Gamma, = \frac{\phi(d_1)}{S\sigma\sqrt{\tau}} \
b. Substituting:
LHS: = -\frac{S \phi(d_1) \sigma}{2 \sqrt{\tau}} - rKe^{-r \tau}\Phi(d_2)\ + \frac{1}{2}\sigma^2 S^2\frac{\phi(d_1)}{S\sigma\sqrt{\tau}} + rS\Phi(d_1)\
= -\frac{S \phi(d_1) \sigma}{2 \sqrt{\tau}} - rKe^{-r \tau}\Phi(d_2)\ + \frac{S \phi(d_1) \sigma}{2 \sqrt{\tau}} + rS\Phi(d_1)\
= rS\Phi(d_1)\ - rKe^{-r \tau}\Phi(d_2)\ .
RHS: = rV\,
= rS\Phi(d_1)\ - rKe^{-r \tau}\Phi(d_2)\ .
c. Conclusion: Since we have agreement, the Black-Scholes formula is a solution of the Black-Scholes PDE.

[edit] Bruces' Philosophers Song [1]

First heard on: Monty Python's Flying Circus
Composer: Eric Idle

Immanuel Kant was a real pissant
Who was very rarely stable.

Heidegger, Heidegger was a boozy beggar
Who could think you under the table.

David Hume could out-consume
Wilhelm Friedrich Hegel, [some versions have 'Schopenhauer and Hegel']

And Wittgenstein was a beery swine
Who was just as schloshed as Schlegel.

There's nothing Nietzsche couldn't teach ya
'Bout the raising of the wrist.
Socrates, himself, was permanently pissed.

John Stuart Mill, of his own free will,
On half a pint of shandy was particularly ill.

Plato, they say, could stick it away--
Half a crate of whisky every day.

Aristotle, Aristotle was a bugger for the bottle.
Hobbes was fond of his dram,

And René Descartes was a drunken fart.
'I drink, therefore I am.'

Yes, Socrates, himself, is particularly missed,
A lovely little thinker,
But a bugger when he's pissed.

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