Consider a continuous time model for the price of stock $S_{t}, 0 \leq t \leq T$, and we make an assumption that $S_{T} / S_{0}$ does not depend on $S_{0}$.
Let $C$ be the price of a call option at time $t=0$ with the exercise price $K$ and expiration $T$. Denote $S_{0}=S$. Show that $C=C(S, K)$ as a function of $S$ and $K$ solves the partial differential equation $$ C(S, K)=S \frac{\partial C}{\partial S}+K \frac{\partial C}{\partial K} $$ You may assume all the necessary differentiability. Hint: write $C(S, K)=$ $S F(S / K)$ for some function $F$
Give an example of the model for the stock price $S_{t}$ that satisfies the above assumption and an example of the model that does not satisfy the assumption.
I am not sure how to go about this question. I will be grateful for any hits or help.
(StackExchange Quantitative Finance is a better forum for this type of question.)
Ignoring discounting, with $S_0=S$, under the pricing probability measure,
$$ C(S,K) = E\left[(S_T-K)^+\right] = S E\left[(S_T/S - K/S)^+\right]= S F(S/K)$$
for some deterministic function $F$. We then have:
$$S\partial C/\partial S + K\partial C/\partial K $$ $$= S\left(F(S/K) +S/KF'(S/K) \right) + KF'(S/K)(-S/K^2)$$ $$ = SF(S/K) = C(S,K) $$
Example of dynamics where $S_t/S_0$ is not dependent on $S_0$:
$$ S_t = S_0 \exp(\sigma W_t - \sigma^2/2 t) $$
Example of dynamics where $S_t/S_0$ is dependent on $S_0$:
$$ S_t = S_0 + \sigma W_t $$
($W_t$ being a standard Brownian motion and $\sigma$ a positive constant).