Solutions to Black Scholes

Question

Consider the black scholes equation,

$$ \frac{\partial V}{\partial t } + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2 } + ( r-q )S\frac{\partial V}{\partial S }-rV =0 $$

How do I show that if $V( S, t)$ is a solution, then $S(\frac{\partial V}{\partial S })$ is also a solution?

I tried substituting $S(\frac{\partial V}{\partial S })$ to the equation and working through the calculations but it doesn't seem to work out.

On a related note, how do we also show that for $ \beta = 1-2(r-q)/\sigma^2$,

$$ W(S, t) = S^\beta V(\frac{1}{S}, t) $$

is also a solution?

The relevant partial derivatives are,

$$ \begin{align} \frac{\partial W}{\partial S} & = \beta S^{\beta -1 }V- S^{\beta -2}\frac{\partial V}{\partial S}\\ \frac{\partial^2 W}{\partial S^2} & = S^{\beta -4}\frac{\partial^2 V}{\partial S^2} -2S^{\beta -3}\frac{\partial V}{\partial S} + \beta(\beta -1 )S^{\beta-2}V \end{align} $$

So the various terms in the PDE are,

$$\begin{align} (r-q)S\frac{\partial W}{\partial S} & = (r-q) \left[ \beta S^{\beta }V- S^{\beta -1}\frac{\partial V}{\partial S} \right] \\ \frac{1}{2}\sigma^2S^2\frac{\partial^2W}{\partial S^2}& =\frac{1}{2}\sigma^2\left[ S^{\beta -2}\frac{\partial^2 V}{\partial S^2} -2S^{\beta -1}\frac{\partial V}{\partial S} + \beta(\beta -1 )S^{\beta}V \right] \end{align} $$

I can see that $ (r-q) \beta S^{\beta }V$ in the top term cancels with $\frac{1}{2}\sigma^2\beta(\beta -1 )S^{\beta}V $ in the bottom term.

So we end up with,

$$ (r-q)S\frac{\partial W}{\partial S}+\frac{1}{2}\sigma^2S^2\frac{\partial^2W}{\partial S^2}=-(r-q) S^{\beta -1}\frac{\partial V}{\partial S} + \frac{1}{2}\sigma^2\left[ S^{\beta -2}\frac{\partial^2 V}{\partial S^2} -2S^{\beta -1}\frac{\partial V}{\partial S} \right] $$

But beyond this I kind of stuck despite trying various manipulations.

Any help will be greatly appreciated!

Answer 1

First note that

$$\frac{\partial}{\partial S}\left(S\frac{\partial V}{\partial S}\right) = \frac{\partial V}{\partial S} + S \frac{\partial^2 V}{\partial S^2}$$

and

$$\frac{\partial^2}{\partial S^2}\left(S\frac{\partial V}{\partial S}\right) = 2\frac{\partial^2 V}{\partial S^2} + S \frac{\partial^3 V}{\partial S^3} \; .$$

Then, take the derivative of the Black-Scholes equation with respect to $S$.

$$\frac{\partial}{\partial T}\frac{\partial V}{\partial S } + \frac{1}{2}\sigma^2 \left(2S\frac{\partial^2 V}{\partial S^2} + S^2 \frac{\partial^3 V}{\partial S^3}\right) + ( r-q )\left(\frac{\partial V}{\partial S } + S\frac{\partial^2 V}{\partial S^2 }\right)-r\frac{\partial V}{\partial S} =0 \; .$$

Using our first two identities, we identify the second and third term

$$\frac{\partial}{\partial T}\frac{\partial V}{\partial S } + \frac{1}{2}\sigma^2 \left(S\frac{\partial^2}{\partial S^2}\left(S\frac{\partial V}{\partial S}\right)\right) + ( r-q )\left(\frac{\partial}{\partial S}\left(S\frac{\partial V}{\partial S}\right)\right)-r\frac{\partial V}{\partial S} =0 \; .$$

Multiplying everything by S, we get

$$\frac{\partial}{\partial T}\left(S\frac{\partial V}{\partial S }\right) + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2}{\partial S^2}\left(S\frac{\partial V}{\partial S}\right) + ( r-q )S\frac{\partial}{\partial S}\left(S\frac{\partial V}{\partial S}\right)-rS\frac{\partial V}{\partial S} =0 \; .$$

Answer 2

$$\frac{\partial V}{\partial t } + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2 } + ( r-q )S\frac{\partial V}{\partial S }-rV =0$$ We can regroup $$\frac{\partial V}{\partial t } + \frac{1}{2}\sigma^2 \left( S\frac{\partial }{\partial S}\right)^2 V + ( r-q - \sigma^2/2 )S\frac{\partial V }{\partial S } -rV =0.$$

Now observe that $S \frac{\partial}{\partial S}$ commutes with all the coefficients (i.e. terms in front) of $V$ since it commutes itself and doesn't interact with $\frac{\partial}{\partial t}.$

The result is now immediate.

(see my book Concepts etc section 5.8)

Answer 3

Ok for the second part on $W(S,t)=S^\beta V(1/S,t)$, I've figured it out. Just some carelessness on my part when tossing terms around. It goes as follows...

The partial derivatives are, $$\begin{align} \frac{\partial W}{\partial S} & = \beta S^{\beta -1 }V- S^{\beta -2}\frac{\partial V}{\partial S}\\ \frac{\partial^2 W}{\partial S^2} & = \beta \left[ -S^{\beta-3}\frac{\partial V}{\partial S} + (\beta-1)S^{\beta-2}V \right] - \left[ -S^{\beta-4} \frac{\partial^2 V}{\partial S^2} + (\beta-2)S^{\beta -3}\frac{\partial V}{\partial S} \right] \end{align}$$

Let $x=1/S$ and factor out $S^\beta$,

$$\begin{align} \frac{\partial W}{\partial S} & = S^{\beta } \left[ \beta xV -x^2\frac{\partial V}{\partial S} \right] \\ \frac{\partial^2 W}{\partial S^2}& = S^{\beta } \left[ \beta \left( -x^3\frac{\partial V}{\partial S} + (\beta-1)x^2V \right) -\left( -x^4\frac{\partial^2 V}{\partial S^2} +(\beta-2)x^3\frac{\partial V}{\partial S}\right)\right] \end{align}$$ We exclude $S^{\beta} $ since it's a common factor for all terms. The PDE terms are,

$$\begin{align} ( r-q )S\frac{\partial W}{\partial S} & = ( r-q )x^{-1}\frac{\partial W}{\partial S} \\ & = ( r-q )\left[ \beta V -x\frac{\partial V}{\partial S} \right]\\ \frac{1}{2}\sigma^2S^2\frac{\partial^2 W}{\partial S^2} &= \frac{1}{2}\sigma^2x^{-2}\frac{\partial^2 W}{\partial S^2}\\ &= \frac{1}{2}\sigma^2 \left[ \beta \left( -x\frac{\partial V}{\partial S} + (\beta-1)V \right) -\left( -x^2\frac{\partial^2 V}{\partial S^2} +(\beta-2)x\frac{\partial V}{\partial S}\right) \right] \end{align}$$

As mentioned earlier, the terms involving $V$ cancels, therefore $$\begin{align} ( r-q )S\frac{\partial W}{\partial S}+\frac{1}{2}\sigma^2S^2\frac{\partial^2 W}{\partial S^2} & = -( r-q )\left[x\frac{\partial V}{\partial S}\right] + \frac{1}{2}\sigma^2\left[(2-2\beta)x\frac{\partial V}{\partial S}+x^2\frac{\partial^2 V}{\partial S^2} \right] \\& =-( r-q )\left[x\frac{\partial V}{\partial S}\right] +2( r-q )\left[x\frac{\partial V}{\partial S}\right] +\frac{1}{2}\sigma^2x^2\frac{\partial^2 V}{\partial S^2}\\&=\frac{1}{2}\sigma^2x^2\frac{\partial^2 V}{\partial S^2}+( r-q )x\frac{\partial V}{\partial S} \end{align} $$

Combining with the other two terms we get,

$$ S^\beta \left[ \frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2x^2\frac{\partial^2 V}{\partial S^2}+( r-q )x\frac{\partial V}{\partial S} -rV \right]=S^\beta.0=0 $$ Therefore we've shown that if $ V(S,t)$ is a solution to the Black Scholes PDE, then $W(S,t)=S^\beta V(1/S,t)$ is also solution.