The meaning of risk-neutral pricing?

Question

Assume that the underlying $S$ is some index, hence the risk-return $\mu=0$, where $S$ meets $$d S = \sigma S d W_t.$$

Let $V$ denote the price of the corresponding call option. To construct the related BS formula, I construct a portfolio $\Pi=V-\Delta S$, after setting a correct value of $\Delta$, I want the portfolio to be risk-free. That is $$d \Pi = r\Pi d t= r(V-\Delta S) d t,$$ where $r$ is risk-free rate.

hence by the Ito formula, I can get the BS-equation.

However, someone told me that the identity $d \Pi = r(V-\Delta S) d t$ should be $$d \Pi = (r*V-\mu*\Delta S) d t,$$ and then get another equation.

Since in my opinion, in the risk-neutral world, $\mu$ turns to be $r$ after applying the Girsonov transformation, and making the portfolio to be risk-free is under risk-neutral world. I agree with the first identity.

So my question is which one is correct? If is the latter one, what's meaning of risk-neutral pricing?

Thank you very much!

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Added 2016/10/26 10:39AM(+8)

Thanks for @MJ73550. I am sure the first one is right now.

However, if we distinguish the funding and lending rate for unsecurities (denote as $r_F$) and stock collateral(denote as $r_R$). Then maybe the identity $d \Pi = r(V-\Delta S) d t$ should be $$d \Pi = (r_F*V-r_R*\Delta S) d t,$$

Is this equation right?

Thanks again.

Answer 1

I'm not sure I really understand your question.

Am I right in thinking that it amounts to asking whether the BS formula should write: $$\frac{\partial V}{\partial t} + \alpha S \frac{\partial V}{\partial S} + \frac{1}{2} \frac{\partial^2 V}{\partial S^2} \sigma^2 S^2 - r V = 0$$ with $\alpha=r$ or $\alpha=\mu$?

If this is the case, it is self-financing portfolios whose $t$-value should emerge as $\Bbb{Q}$ martingales. Thus, if the stock pays dividend $\alpha = r-q \ne r$. If your model includes a more complex cost of carry/repo cost, it should transpire through $\alpha$.

When you introduce real world effects (collateral, lending/borrowing asymmetry etc.) it can of course become more complicated; see http://www.math.columbia.edu/~fts/What%20Rate%20to%20use%20v1.pdf (I did not check the validity of the equations but at least it will give you an idea of what effects can be included).

Answer 2

First equation is the right one.

But it won't help you to understand what happens behind risk-neutral pricing.

Indeed you just write that you invest in cash and stocks but you did not write self-financing conditions.

(1) Let $V_t$ be the value of the call,

(2) Let $\Delta_t$ be your delta of your delta hedge,

(3) Let $\Pi_t$ your cash remunerated at $r$.

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(3) $\Leftrightarrow d \Pi_t = r\Pi_t dt$

(1)+(2)+(3) $\Leftrightarrow V_t = \Pi_t + \Delta_t S_t$

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If your portfolio is split between cash and collateral.

You have

$V_t = \Pi^{cash}_t + \Pi^{collat}_t + \Delta_t S_t$

now you have $d\Pi^{cash}_t = r^{cash}_t\Pi^{cash}_t dt$ and $d\Pi^{collat}_t = r^{collat}_t\Pi^{collat}_t dt$