12.3 Construction of the Black-Scholes PDE
12.3 Construction of the Black-Scholes PDE
Let S
be the stock, and C the call option on the stock.
From the Lemma the excess drift rate divided by the volatility is equal
for the stock and the call option. This
means that we have to simplify the equation
where
and
Substituting
for a
and q, we get
We now
take the following steps:
1.
Cancel the s in each denominator:
2.
Multiply the numerator and denominator of the right-hand side by C:
3.
Multiply each side by the remaining denominator:
4.
Cancel the term
on each side:
5.
Re-arrange to get
This is
the usual form of the Black-Scholes partial differential equation.
If the stock pays a
continuous dividend yield, q, then the
two volatility-adjusted drifts are still equal. This equality, however, is adjusted to account for the
continuous dividend yield (q) received by a stockholder:
Thus, by repeating the
above steps, an additional term is
created:
and the PDE is
A widely
used application of the continuous dividend yield model is the option pricing
model for currencies. Details of how the general method applies to currencies
are presented in the next topic Application:
Currency Options.
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