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Fintor > talk > articles > sandbox | December 19 • 10:46 UTC • 12:46 PM SAST

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The Black Scholes PDE

1. The Black–Scholes model is a PDE which describes the evolution of the value of a option, $v\,$, through time, $t\,$, as related to changes in the underlying stock price, $S\,$ with volatility $\sigma \,$, and for a risk free rate $r\,$. 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. For a call option, the solution to this PDE is the Black-Scholes formula; the option has strike price $K\,$, and time remaining until maturity $\tau \,$:

$V(S,\tau )=S\mathbb {N} (d_{1})-Ke^{-r(\tau )}\mathbb {N} (d_{2})\,$

where:

$d_{1}={\frac {\ln({\frac {S}{K}})+(r+{\frac {\sigma ^{2}}{2}})(\tau )}{\sigma {\sqrt {\tau }}}}$

$d_{2}=d_{1}-\sigma {\sqrt {\tau }}$

$\mathbb {N} (x)\,$ is the standard normal cumulative distribution function ${\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{x}e^{-z^{2}/2}\,dz$ (often written $\Phi$).

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\mathbb {N} '(d_{1})\sigma }{2{\sqrt {\tau }}}}-rKe^{-r\tau }\mathbb {N} (d_{2})\$

• ${\frac {\partial V}{\partial S}}$, Delta, $=\mathbb {N} (d_{1})\$

• ${\frac {\partial ^{2}V}{\partial S^{2}}}$, Gamma, $={\frac {\mathbb {N} '(d_{1})}{S\sigma {\sqrt {\tau }}}}\$

where: $\mathbb {N} '(x)\,$ is the standard normal Probability density function ${\frac {1}{\sqrt {2\pi }}}e^{-x^{2}/2}$ (often written $\phi$).

b. Substituting:

LHS: $=-{\frac {S\mathbb {N} '(d_{1})\sigma }{2{\sqrt {\tau }}}}-rKe^{-r\tau }\mathbb {N} (d_{2})\ +{\frac {1}{2}}\sigma ^{2}S^{2}{\frac {\mathbb {N} '(d_{1})}{S\sigma {\sqrt {\tau }}}}+rS\mathbb {N} (d_{1})\$

$=-{\frac {S\mathbb {N} '(d_{1})\sigma }{2{\sqrt {\tau }}}}-rKe^{-r\tau }\mathbb {N} (d_{2})\ +{\frac {S\mathbb {N} '(d_{1})\sigma }{2{\sqrt {\tau }}}}+rS\mathbb {N} (d_{1})\$

$=rS\mathbb {N} (d_{1})\ -rKe^{-r\tau }\mathbb {N} (d_{2})\$.

$=rV\,$.

RHS: $=rV\,$

c. Conclusion: Since we have agreement, the Black-Scholes formula is a solution of the Black-Scholes PDE.

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The Greeks

	Calls	Puts
value	$e^{-q\tau }S\mathbb {N} (d_{1})-e^{-r\tau }K\mathbb {N} (d_{2})\,$	$e^{-r\tau }K\mathbb {N} (-d_{2})-e^{-q\tau }S\mathbb {N} (-d_{1})\,$
delta	$e^{-q\tau }\mathbb {N} (d_{1})\,$	$-e^{-q\tau }\mathbb {N} (-d_{1})\,$
vega	$Se^{-q\tau }\mathbb {N} '(d_{1}){\sqrt {\tau }}=Ke^{-r\tau }\mathbb {N} '(d_{2}){\sqrt {\tau }}\,$
theta	$-e^{-q\tau }{\frac {S\mathbb {N} '(d_{1})\sigma }{2{\sqrt {\tau }}}}-rKe^{-r\tau }\mathbb {N} (d_{2})+qSe^{-q\tau }\mathbb {N} (d_{1})\,$	$-e^{-q\tau }{\frac {S\mathbb {N} '(d_{1})\sigma }{2{\sqrt {\tau }}}}+rKe^{-r\tau }\mathbb {N} (-d_{2})-qSe^{-q\tau }\mathbb {N} (-d_{1})\,$
rho	$K\tau e^{-r\tau }\mathbb {N} (d_{2})\,$	$-K\tau e^{-r\tau }\mathbb {N} (-d_{2})\,$
gamma	$e^{-q\tau }{\frac {\mathbb {N} '(d_{1})}{S\sigma {\sqrt {\tau }}}}\,$

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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.