Understanding the portfolio used in the derivation of the Black-Scholes PDE
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In the derivation of the Black-Scholes equation, I see that a portfolio $Pi=V-Delta S$ is used, where $Delta$ turns out to be $frac{partial{V}}{partial S}$. But to determine $Delta$, it is assumed to be constant. So my confusion comes from the reasoning why $Delta$ is constant as when I look at the graphed solutions for a call option, $frac{partial{C}}{partial S}$ is clearly not constant when S is less than the price of the strike price.
finance
asked Dec 27 '18 at 17:41
DLBDLB
738
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add a comment |
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In the derivation of the Black-Scholes equation, I see that a portfolio $Pi=V-Delta S$ is used, where $Delta$ turns out to be $frac{partial{V}}{partial S}$. But to determine $Delta$, it is assumed to be constant. So my confusion comes from the reasoning why $Delta$ is constant as when I look at the graphed solutions for a call option, $frac{partial{C}}{partial S}$ is clearly not constant when S is less than the price of the strike price.
finance
asked Dec 27 '18 at 17:41
DLBDLB
738
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$begingroup$
Is your $y$-axis $C$, $V$ or $Pi$? It may be worth editing in a proof that $partial_S V$ being constant would imply $partial_S C$ is too.
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– J.G.
Dec 27 '18 at 17:51
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@J.G. the y-axis is C- the call option value and in derivations, people state $dPi=dV-Delta dS$ implying |delta is a constant.
$endgroup$
– DLB
Dec 27 '18 at 19:37
add a comment |
$begingroup$
In the derivation of the Black-Scholes equation, I see that a portfolio $Pi=V-Delta S$ is used, where $Delta$ turns out to be $frac{partial{V}}{partial S}$. But to determine $Delta$, it is assumed to be constant. So my confusion comes from the reasoning why $Delta$ is constant as when I look at the graphed solutions for a call option, $frac{partial{C}}{partial S}$ is clearly not constant when S is less than the price of the strike price.
finance
asked Dec 27 '18 at 17:41
DLBDLB
738
$endgroup$
In the derivation of the Black-Scholes equation, I see that a portfolio $Pi=V-Delta S$ is used, where $Delta$ turns out to be $frac{partial{V}}{partial S}$. But to determine $Delta$, it is assumed to be constant. So my confusion comes from the reasoning why $Delta$ is constant as when I look at the graphed solutions for a call option, $frac{partial{C}}{partial S}$ is clearly not constant when S is less than the price of the strike price.
finance
finance
asked Dec 27 '18 at 17:41
DLBDLB
738
asked Dec 27 '18 at 17:41
DLBDLB
738
asked Dec 27 '18 at 17:41
DLBDLB
738
asked Dec 27 '18 at 17:41
DLBDLB
738
asked Dec 27 '18 at 17:41
DLBDLB
738
738
$begingroup$
Is your $y$-axis $C$, $V$ or $Pi$? It may be worth editing in a proof that $partial_S V$ being constant would imply $partial_S C$ is too.
$endgroup$
– J.G.
Dec 27 '18 at 17:51
$begingroup$
@J.G. the y-axis is C- the call option value and in derivations, people state $dPi=dV-Delta dS$ implying |delta is a constant.
$endgroup$
– DLB
Dec 27 '18 at 19:37
add a comment |
$begingroup$
Is your $y$-axis $C$, $V$ or $Pi$? It may be worth editing in a proof that $partial_S V$ being constant would imply $partial_S C$ is too.
$endgroup$
– J.G.
Dec 27 '18 at 17:51
$begingroup$
@J.G. the y-axis is C- the call option value and in derivations, people state $dPi=dV-Delta dS$ implying |delta is a constant.
$endgroup$
– DLB
Dec 27 '18 at 19:37
$begingroup$
Is your $y$-axis $C$, $V$ or $Pi$? It may be worth editing in a proof that $partial_S V$ being constant would imply $partial_S C$ is too.
$endgroup$
– J.G.
Dec 27 '18 at 17:51
$begingroup$
Is your $y$-axis $C$, $V$ or $Pi$? It may be worth editing in a proof that $partial_S V$ being constant would imply $partial_S C$ is too.
$endgroup$
– J.G.
Dec 27 '18 at 17:51
$begingroup$
@J.G. the y-axis is C- the call option value and in derivations, people state $dPi=dV-Delta dS$ implying |delta is a constant.
$endgroup$
– DLB
Dec 27 '18 at 19:37
$begingroup$
@J.G. the y-axis is C- the call option value and in derivations, people state $dPi=dV-Delta dS$ implying |delta is a constant.
$endgroup$
– DLB
Dec 27 '18 at 19:37
add a comment |
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$begingroup$
Is your $y$-axis $C$, $V$ or $Pi$? It may be worth editing in a proof that $partial_S V$ being constant would imply $partial_S C$ is too.
$endgroup$
– J.G.
Dec 27 '18 at 17:51
$begingroup$
@J.G. the y-axis is C- the call option value and in derivations, people state $dPi=dV-Delta dS$ implying |delta is a constant.
$endgroup$
– DLB
Dec 27 '18 at 19:37