The payoff for basket option is max($w_1S_1+w_2S_2 -k,0)$. Using Ito's formula, I need to derive the PDE,
where $dS_1 = rS_1dt + \sigma_1 S_1dW_1$
$dS_2 = rS_2dt + \sigma_2 S_2dW_2$
I need some guidance in deriving the PDE for basket option similar to Black-Scholes PDE.
Things I have tried and know.
1) The PDE contains only terms in dt because of martingale property
2)$dg = g(0,x,y) + g_tdt +g_xdx + g_ydy + \frac{1}{2}g_{xx}dx^2 + \frac{1}{2}g_{yy}dy^2 + g_{xy}dxdy$ where g is the option and x = S1 and y = S2
3) Taking only the dt terms, $g_t+ g_x*r*S_1 + g_y*r*S_2 +\frac{1}{2}g_{xx}\sigma_1^2 S_1^2 + \frac{1}{2}g_{yy}\sigma_1^2 S_2^2 = rg$
Not sure how to incorporate the $w_1$ and $w_2$.
EDIT: Sorry I don't know why I was so sloppy with capital versus lowercase letters, I intended to make capitals random variables but I was sloppy and did it inconsistently in some places. I might edit it at some point.
This follows the structure of the derivation of the regular B-S equation by Thayer Watkins.
I guess the risk-free interest rate is $0$? Let the expiery time be $T$.
In principle the value of the call option is a function of the prices of both stocks, the strike price $k$, the time until expiery $t$, the volatility of both stocks and the expected rate of return. Of course we expect that like in the usual B-S formula $r$ will in the end play no role.
$$C = C(s_1,s_2,k,t,\sigma_1,\sigma_2,r)$$
Apply Ito's lemma
$$\mathrm{d}C = \left(C_t + C_{s_1}r + C_{s_2}r + \frac{\sigma_1^2}{2} S_1^2 C_{s_1, s_1} + \frac{\sigma_2^2}{2}S_2^2 C_{s_2,s_2} \right)\mathrm{d}t + C_{s_1}\sigma_1 S_1 \mathrm{d}W_1 + C_{s_1}\sigma_2 S_2 \mathrm{d}W_2$$
The term involving $C_{s_1,s_2}$ will go away since $W_1$ and $W_2$ are independent (I presume). Consider a portfolio short one call $-C$ and with $h_i$ shares of the underlying stock $S_i$
Then the change in value is: $$\mathrm{d}V = h_1 \mathrm{d} S_1 + h_2 \mathrm{d} S_2 - \mathrm{d}C$$
If $h_i = C_{s_i}$ then
$$\mathrm{d}V = C_{s_1}\mathrm{d}S_1 + C_{s_2}\mathrm{d}S_2 - \mathrm{d}C$$
So over a time interval $\mathrm{d}t$ the change in the value of the portfolio is
$$\mathrm{d}V = C_{s_1}\left(r S_1\mathrm{d}t + \sigma_1 S_1 \mathrm{d}W_1\right) + C_{s_2}\left(r S_2\mathrm{d}t + \sigma_2 S_2 \mathrm{d}W_2\right) - \left(C_t + C_{s_1}r + C_{s_2}r + \frac{\sigma_1^2}{2} C_{s_1, s_1} + \frac{\sigma_2^2}{2} C_{s_2,s_2} \right)\mathrm{d}t - C_{s_1}\sigma_1 S_1 \mathrm{d}W_1 - C_{s_1}\sigma_2 S_2 \mathrm{d}W_2$$
Cancel the terms involving $\mathrm{d}W_i$ and $r$
$$\mathrm{d}V = - \left(C_t + \frac{\sigma_1^2}{2} S_1^2 C_{s_1, s_1} + \frac{\sigma_2^2}{2} S_2^2 C_{s_2,s_2} \right)\mathrm{d}t$$
The portfolio is risk free, therefore it's value should increase at the risk free rate. You didn't say what that is, I'll assume $\mu = 0$ and you can generalize that if you need to.
$$\mathrm{d}V = \mu V \mathrm{d} t = \mu \left(C_{s_1}S_1 + C_{s_2}S_2 -C\right)\mathrm{d}t$$
For this to hold we require (again assuming $\mu = 0$)
$$ C_t + \frac{\sigma_1^2}{2} S_1^2 C_{s_1, s_1} + \frac{\sigma_2^2}{2} S_2^2 C_{s_2,s_2} = 0$$
Now you know the value of the option on the expiration day $C = \max\left(w_1 S_1 + w_2 S_2 - k,0\right)$; this gives you a final condition, I'll drop the explicit dependence of $C$ on the purely parametric variables.
$$C(s_1,s_2,T) = \max\left(w_1 s_1 + w_2 s_2 - k,0\right)$$