The Black Scholes PDE
edit
1. The Black–Scholes model is a PDE which describes the evolution of the value of a option ,
v
{\displaystyle v\,}
, through time,
t
{\displaystyle t\,}
, as related to changes in the underlying stock price,
S
{\displaystyle S\,}
with volatility
σ
{\displaystyle \sigma \,}
, and for a risk free rate
r
{\displaystyle r\,}
. The PDE is:
∂
V
∂
t
+
1
2
σ
2
S
2
∂
2
V
∂
S
2
+
r
S
∂
V
∂
S
=
r
V
{\displaystyle {\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
)
{\displaystyle 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
{\displaystyle K\,}
, and time remaining until maturity
τ
{\displaystyle \tau \,}
:
V
(
S
,
τ
)
=
S
N
(
d
1
)
−
K
e
−
r
(
τ
)
N
(
d
2
)
{\displaystyle V(S,\tau )=S\mathbb {N} (d_{1})-Ke^{-r(\tau )}\mathbb {N} (d_{2})\,}
where:
d
1
=
ln
(
S
K
)
+
(
r
+
σ
2
2
)
(
τ
)
σ
τ
{\displaystyle d_{1}={\frac {\ln({\frac {S}{K}})+(r+{\frac {\sigma ^{2}}{2}})(\tau )}{\sigma {\sqrt {\tau }}}}}
d
2
=
d
1
−
σ
τ
{\displaystyle d_{2}=d_{1}-\sigma {\sqrt {\tau }}}
N
(
x
)
{\displaystyle \mathbb {N} (x)\,}
is the standard normal cumulative distribution function
1
2
π
∫
−
∞
x
e
−
z
2
/
2
d
z
{\displaystyle {\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{x}e^{-z^{2}/2}\,dz}
(often written
Φ
{\displaystyle \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 :
∂
V
∂
t
{\displaystyle {\frac {\partial V}{\partial t}}}
, Theta ,
=
−
S
N
′
(
d
1
)
σ
2
τ
−
r
K
e
−
r
τ
N
(
d
2
)
{\displaystyle =-{\frac {S\mathbb {N} '(d_{1})\sigma }{2{\sqrt {\tau }}}}-rKe^{-r\tau }\mathbb {N} (d_{2})\ }
∂
V
∂
S
{\displaystyle {\frac {\partial V}{\partial S}}}
, Delta ,
=
N
(
d
1
)
{\displaystyle =\mathbb {N} (d_{1})\ }
∂
2
V
∂
S
2
{\displaystyle {\frac {\partial ^{2}V}{\partial S^{2}}}}
, Gamma ,
=
N
′
(
d
1
)
S
σ
τ
{\displaystyle ={\frac {\mathbb {N} '(d_{1})}{S\sigma {\sqrt {\tau }}}}\ }
where:
N
′
(
x
)
{\displaystyle \mathbb {N} '(x)\,}
is the standard normal Probability density function
1
2
π
e
−
x
2
/
2
{\displaystyle {\frac {1}{\sqrt {2\pi }}}e^{-x^{2}/2}}
(often written
ϕ
{\displaystyle \phi }
). b. Substituting:LHS :
=
−
S
N
′
(
d
1
)
σ
2
τ
−
r
K
e
−
r
τ
N
(
d
2
)
+
1
2
σ
2
S
2
N
′
(
d
1
)
S
σ
τ
+
r
S
N
(
d
1
)
{\displaystyle =-{\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})\ }
=
−
S
N
′
(
d
1
)
σ
2
τ
−
r
K
e
−
r
τ
N
(
d
2
)
+
S
N
′
(
d
1
)
σ
2
τ
+
r
S
N
(
d
1
)
{\displaystyle =-{\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})\ }
=
r
S
N
(
d
1
)
−
r
K
e
−
r
τ
N
(
d
2
)
{\displaystyle =rS\mathbb {N} (d_{1})\ -rKe^{-r\tau }\mathbb {N} (d_{2})\ }
.
=
r
V
{\displaystyle =rV\,}
.RHS :
=
r
V
{\displaystyle =rV\,}
c. Conclusion: Since we have agreement, the Black-Scholes formula is a solution of the Black-Scholes PDE.
The Greeks
edit
Calls
Puts
value
e
−
q
τ
S
N
(
d
1
)
−
e
−
r
τ
K
N
(
d
2
)
{\displaystyle e^{-q\tau }S\mathbb {N} (d_{1})-e^{-r\tau }K\mathbb {N} (d_{2})\,}
e
−
r
τ
K
N
(
−
d
2
)
−
e
−
q
τ
S
N
(
−
d
1
)
{\displaystyle e^{-r\tau }K\mathbb {N} (-d_{2})-e^{-q\tau }S\mathbb {N} (-d_{1})\,}
delta
e
−
q
τ
N
(
d
1
)
{\displaystyle e^{-q\tau }\mathbb {N} (d_{1})\,}
−
e
−
q
τ
N
(
−
d
1
)
{\displaystyle -e^{-q\tau }\mathbb {N} (-d_{1})\,}
vega
S
e
−
q
τ
N
′
(
d
1
)
τ
=
K
e
−
r
τ
N
′
(
d
2
)
τ
{\displaystyle Se^{-q\tau }\mathbb {N} '(d_{1}){\sqrt {\tau }}=Ke^{-r\tau }\mathbb {N} '(d_{2}){\sqrt {\tau }}\,}
theta
−
e
−
q
τ
S
N
′
(
d
1
)
σ
2
τ
−
r
K
e
−
r
τ
N
(
d
2
)
+
q
S
e
−
q
τ
N
(
d
1
)
{\displaystyle -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
τ
S
N
′
(
d
1
)
σ
2
τ
+
r
K
e
−
r
τ
N
(
−
d
2
)
−
q
S
e
−
q
τ
N
(
−
d
1
)
{\displaystyle -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
τ
e
−
r
τ
N
(
d
2
)
{\displaystyle K\tau e^{-r\tau }\mathbb {N} (d_{2})\,}
−
K
τ
e
−
r
τ
N
(
−
d
2
)
{\displaystyle -K\tau e^{-r\tau }\mathbb {N} (-d_{2})\,}
gamma
e
−
q
τ
N
′
(
d
1
)
S
σ
τ
{\displaystyle e^{-q\tau }{\frac {\mathbb {N} '(d_{1})}{S\sigma {\sqrt {\tau }}}}\,}
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.
This is a Wiki pedia user page . This is not an encyclopedia article or the talk page for an encyclopedia article. If you find this page on any site other than Wiki pedia , you are viewing a mirror site . Be aware that the page may be outdated and that the user in whose space this page is located may have no personal affiliation with any site other than Wiki pedia . The original page is located at https://en.wiki pedia.org/wi ki/User:Fintor/Sandbox .