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I have got a Black-Scholes model with portoflio with two options which bond prices are $V_1$ and $V_2$ (with different maturities or strikes). The interest rate $r$ is stochastic and given by: $$ dr = u(r,t) dt + w(r,t)dW_t,$$ where $w$ and $u$ are some functions of $r$ and $t$.

The porfolio is given as follow: $$ \Pi = V_1 - \Delta V_2 $$

I need to prove that $$\boxed{\frac{\partial V}{\partial t} + \frac12 \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + (\mu -\lambda_S \sigma)S \frac{\partial V}{\partial S} - rV =0} \quad [1]$$

I found that problem in Paul Wilmott's book "On quantitative finance" on page 857, but I dont understand everything.

We know $$dV = \frac{\partial V}{\partial t}dt + \frac12 \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2}dt + \frac{\partial V}{\partial S}dS,\; \; \; [2]$$ and $$ d\Pi = dV_1 - \Delta d V_2, \; \; \; d\Pi = r\Pi dt. \; \; \; [3]$$

So after finding out form of $\Delta =\frac{\partial V_1/ \partial S}{\partial V_2/\partial S} $

and using $[2], \; [3]$ and $\Delta$ we can get

$$ \frac{1}{\partial V_1/ \partial S}\left[\frac{\partial V_1}{\partial t} + \frac12 \sigma^2 S^2 \frac{\partial^2 V_1}{\partial S^2} - rV_1\right] = \frac{1}{\partial V_2/ \partial S}\left[\frac{\partial V_2}{\partial t} + \frac12 \sigma^2 S^2 \frac{\partial^2 V_2}{\partial S^2} - rV_2\right]. \; \; \; [4]$$

And then the author in mentionet book writes it as

$$\frac{1}{\partial V_1/\partial S}\left[ \frac{\partial V_1}{\partial t} + \frac12 \sigma^2 S^2 \frac{\partial^2 V_1}{\partial S^2} - rV_1\right] = (\mu - \lambda_S \sigma)S \; \; \; [5]$$

Which I don't get - why $(\mu - \lambda_S \sigma)$ it's multiplyed by $S$? How does $\lambda$ or $\mu$ looks like?

Later I'm finding out if $V=S$ we have $\lambda_S = \frac{\mu - r}{\sigma}$ and $ \lambda_S$ is market price of risk for asset, so if we put that in $[5]$ we get

$$ \frac{\partial V}{\partial t} + \frac12 \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS \frac{\partial V}{\partial S} - rV =0. \; \; \; [6]$$

Is it a reason of why we wrote right side of $[4]$ that way? To get $[6]$ in that form?

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  • $\begingroup$ What’s $V$? It’s not defined. $\endgroup$ Commented May 13, 2021 at 20:55
  • $\begingroup$ We also don’t know what $w$ or $r$ represent. $\endgroup$ Commented May 13, 2021 at 21:02
  • $\begingroup$ I edited the question, I hope now it's more clear. $\endgroup$
    – Saguro
    Commented May 13, 2021 at 21:31
  • $\begingroup$ Still not clear to me what is the relationship of $V$ to the derivatives $V_1$ and $V_2$ or the portfolio $\Pi$. $\endgroup$ Commented May 14, 2021 at 11:50
  • $\begingroup$ in $[1]$ $V$ represents $V_1$ (after writing $V_2$ side in $[4]$ to the form of $[5]$ we don't need to use $_1$ in $V_1$). $dV$ in $[2]$ satisfy $V_1$ and $V_2$ $\endgroup$
    – Saguro
    Commented May 14, 2021 at 11:59

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