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Black Scholes Differential Equation: Four Derivations
Replicating Portfolio Derivation
Given a Brownian motion process \( {\rm d}W \), retaining first order terms in \( {\rm d}t \), noting that \( {{\rm d}W}^2 = {\rm d}t \), and \( {\rm d}t {\rm d}W \) cross terms are higher than first order, an underlying stock, \( S \), undergoing a stochastic price process \( {\rm d}S \) with drift \( \mu \) and volatility \( \sigma \) takes the form:
$$ \eqalign{
{\rm d}S &= \mu S {\rm d}t + \sigma S {\rm d}W \\
( {\rm d}S )^2 &= \sigma^2 S^2 {\rm d}t \\
}
$$
Let the replicating portfolio \( \Pi \) equal the option value \( V \) plus an unknown Delta of the Stock, \( \Delta S \), acting as a Hedge. The unknown \( \Delta \) will be determined at the end.
$$ \eqalign{
\Pi &= V + \Delta S \\
{\rm d}\Pi &= {\rm d}V + {\rm d}(\Delta S) = {\rm d}V + \left\{ S {\rm d}\Delta + \Delta {\rm d}S \right\} = {\rm d}V + \Delta {\rm d}S \\[2mm]
}
$$
For now ignore and set \( {\rm d}\Delta = 0 \). Even though this should NOT be ignored, because \( {\rm d}\Delta(t,S) = \Delta_t {\rm d}t + \Delta_s {\rm d}S \) where \( \Delta_s \equiv \Gamma \), this term \( {\rm d}\Delta = 0 \) will be dropped because it is not needed for limited purposes of deriving the Black Scholes PDE, not a most general PDE. Completing this calculation for \( {\rm d}\Delta \ne 0 \) is an interesting direction of calculation which should be continued elsewhere just to see how far it can be taken and what can be done with it. Can all other Greeks be this way included? Does this lead to nothing but familiar equations or perhaps in a surprize to more accurate versions not yet explored?
Next are needed epressions for \( {\rm d}\Pi \) & \( {\rm d}V \). Two expressions for \( {\rm d}\Pi \): a first exploiting the risk neutral measure, under which \( \Pi \) earns the risk free rate (and so \( r \Pi {\rm d}t \) ) while the second arises from substitution in the first of \( \Pi = ( V + \Delta S ) \); that for \( {\rm d}V \) retains up to higher order terms \( V_{ss} \) since these are first order in time (as per that assumption):
$$ \eqalign{
{\rm d}\Pi &= r \Pi \ {\rm d}t = r V \ {\rm d}t + r \Delta S \ {\rm d}t \\
{\rm d}V &= V_t{\rm d}t + V_s{\rm d}S + \frac{1}{2} V_{ss} ({\rm d}S)^2 \\
&= \left( V_t + \frac{1}{2} V_{ss} \sigma^2 S^2 \right) {\rm d}t + V_s{\rm d}S
}
$$
Substituting then gives
$$ \eqalign{
r \Pi \ {\rm d}t &=
\left\{ \left( V_t + \frac{1}{2} V_{ss} \sigma^2 S^2 \right) {\rm d}t + V_s{\rm d}S \right\} + \Delta {\rm d}S \\
r V \ {\rm d}t + r \Delta S \ {\rm d}t &=
}
$$
Now subtracting \( r V \ {\rm d}t + r \Delta S \ {\rm d}t \) from both sides gives an expression which vanishes:
$$
\left\{ \left( V_t + \frac{1}{2} V_{ss} \sigma^2 S^2 \right) {\rm d}t + V_s{\rm d}S \right\} + \Delta {\rm d}S -
r V \ {\rm d}t - r \Delta S \ {\rm d}t = 0
$$
Collecting like terms for \( {\rm d}t \) and \( {\rm d}S \) then gives:
$$
\left( V_t + \frac{1}{2} V_{ss} \sigma^2 S^2 - rV - r \Delta S \right) {\rm d}t + ( V_s + \Delta ){\rm d}S = 0
$$
Finally, the condition to remove dependence upon the stochastic process \( {\rm d}S \), sets the definition for the Hedge \( \Delta = - V_s \). This ultimately defines the replicating portfolio and derives the Black Scholes partial differential equation:
$$ \eqalign{
\Pi &= V + \Delta S = V - V_s S \\
V_t + r V_s S + \frac{1}{2} \sigma^2 S^2 V_{ss} - r V &= 0 \hspace{1in} {\rm Black \ Scholes \ PDE} \\
\Theta - r \Delta S + \frac{1}{2} \sigma^2 S^2 \Gamma - r V &= 0 \hspace{1in} {\rm Black \ Scholes \ PDE \ in \ Greeks}
}
$$
Four different derivations of the Black Scholes partial differential equation are:
