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\textbf{Question}

Show that the Black-Scholes equation remains invariant under the
scaling $S'=\alpha S$ where $\alpha > 0$ is a constant.

A put option with strike $K$ is written on an asset which pays out on
a single, discrete yield $q$ at time $t_d < T$, where $T$ is the
expiry date of the put. Explain why the spot price jumps from $S$ to
$(1-q)S$ as the dividend date is crossed, but the option price remains
continuous. Denote the option price by $P(S,t;K,T)$.

Let $P_{BS}(S,t;K,T)$ denote the usual Black-Scholes value for a put
option on an asset which pays no dividends and has strike $K$, expiry
$T$. Show that
$$P(S,t;K,T) = \left \{ \begin{array}{ll} P_{BS}(S,t;K,T) &
\textrm{if} \ t_d < t < T,\\
(1-q)P_{BS}(S,t;K/(1-q), T) \ \ & \textrm{if} \ 0 \ge t < t_d.
\end{array} \right.$$

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\textbf{Answer}

Irrelevant whether they do this assuming $q=0$ or $q \ne 0$. 

For $q=0$, BS is $$\displaystyle \frac{\partial V}{\partial t}
+\frac{1}{2}\sigma^2 S^2V_{SS}+ (r-q)SV_S-rV=0$$
Now if $S'=\alpha S$ we have
$$ \displaystyle \frac{\partial}{\partial S} = \frac{\partial
S'}{\partial S}\frac{\partial}{\partial S'} =
\alpha\frac{\partial}{\partial S'},$$
$$\rm{so}\ \ \displaystyle S\frac{\partial}{\partial S}=
\frac{1}{\alpha}S'\alpha\frac{\partial}{\partial S'} =
S'\frac{\partial}{\partial S'}$$

Hence
\begin{eqnarray*}
\displaystyle S\frac{\partial}{\partial S} \left ( S\frac{\partial}{\partial
D} \right ) & = & \displaystyle S'\frac{\partial}{\partial S'} \left (
S' \frac{\partial}{\partial S'} \right )\\
\Rightarrow \displaystyle S^2\frac{\partial^2}{\partial S^2}
+S\frac{\partial}{\partial S} & = & \displaystyle
s'^2\frac{\partial^2}{\partial S'^2}+S'\frac{\partial}{\partial S'}\\
\Rightarrow \displaystyle S^2\frac{\partial^2}{\partial S^2} & = &
S'^2 \frac{\partial^2}{\partial S'^2}
\end{eqnarray*}

Thus $$V_t+\frac{1}{2}\sigma^2S'^2V_{S'S'}=(r-q)S'V_{S'}- RV =0$$
i.e. equation is invariant.


At time $t_d$ asset pays out a dividend of $qS$ (that is what a
dividend yield $q$ means), with certainty. If spot price immediately
after $t_d$ is not $(1-q)S$ we could arbitrage situation; eg if spot is
$\hat{S}>(1-q)S$ after $t_d$ - cost is $(1-q)S$, selling yields
$\hat{S}>(1-q)S$.

If spot is $\overline{S}<(1-q)S$ after $t_d$, buy asset before $t_d$,
collect dividend and then sell for $\overline{S}\ \Rightarrow$ risk
free profit.

Option doesn't pay any cash dividends, so must have
$V(t_d^-)=V(t_d^+)$.

Write this as
\begin{eqnarray*}
S & = & S^- \ \rm{at}\ t_d^-\\
S & = & (1-q)S^- \ \rm{at}\ t_d^+\\
V(S^-,t_d^-) & = & V(S^+, t_d^+)
\end{eqnarray*}

$\Rightarrow$ jump condition
\begin{eqnarray*}
v(S^-, t_d^-) & = & V(S^-(1-q), t_d^+) \ \ \rm{or\ just}\\
v(S, t_d^-) & = & V(S(1-q), t_d^+).
\end{eqnarray*}

for $t>t_d$ we have (since $q=0$ if there is only a DISCRETE dividend)
$$V_t+\frac{1}{2}\sigma^2S^2V_{SS}- rSV_S- rV=0,$$
$$V(S,T) = \rm{max}(K-S,0)$$
By definition, the solution of this problem is $P_{BS}(S,t;K,T)$ so
$$V=P_{BS}(S,t;K,T)\ \ \rm{for}\ t>t_d$$
Now write jump condition as $V(S,t_d^-)=V(S(1-q),t_d^+)$ so, at
$t_d^-$
$$V(S,t_d^-)=P_{BS}((1-q)S,t_d^-;K,T)$$
Now consider payoff for $P_{BS}((1-q)S,t;K,T)$,

i.e. $P_{BS}((1-q)S,T;K,T).$

It is
\begin{eqnarray*} P_{BS}((1-q)S,T;K,T) & = & \rm{max}(K-(1-q)S,0)\\
& = & (1-q)\rm{max}(\frac{K}{1-q}-S,0)
\end{eqnarray*}

Hence, since BS is invariant under $S\rightarrow(1-q)S$, is linear
problem for $V(S,t),\ t<t_d$ is equivalent to
$$V_t+\frac{1}{2}\sigma^2S^2V_{SS}+ rSV_S -rV =0,$$
$$V(S,T)=(1-q)\rm{max}(\frac{K}{1-q}-S,0)$$
i.e.
$$\displaystyle V=(1-q)P_{BS}(S,t;\frac{K}{1-q},T)$$
Hence the result.

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