2
$\begingroup$

I was working on the definition of the self-financing portfolio.

Say $V=\phi_tS_t+\psi_t A_t$ where $S_t$ and $A_t$ are the stock price and the money market price at time $t$, resp, and $\phi_t$ and $\psi_t$ are the shares that are invested in stock and money market at time $t$, resp.

Then I came across the formula like this: $dV_t$=$\phi_t dS_t$+$\psi_t dA_t$+ $S_t d\phi_t$+$A_t d\psi_t$+$dS_t d\phi_t$+$dA_t d\psi_t$.

I have no idea what the last four terms mean. Precisely speaking, I don't know what does $d\phi_t$ or $d\psi_t$ mean. They are just some adapted random process and nothing more I know about them.

Moreover, most of reference book or lecture notes (like this, page 16), claim the above formula is based on the Ito lemma. Doesn't Ito Lemma/formula only apply to Ito process?

Many Thanks,


I should clarify my confusion a little bit. If I understand correctly, the differential notations don't have mathematical meaning. However, for $dS_t$ and $dA_t$, their integrals do. So I understand $dS_t$ and $dA_t$. But then $d\phi_t$ or $d\psi_t$ makes no sense to me. Because neither themselves or the integrals are defined.

Moreover, the approach of @Neeraj in the equation $3$ is more like a discrete approach, which is great and I can follow the logic. On the other hand, the note says the equation follows from Ito's lemma. In fact, I even saw some notes say $dA_t d\psi_t=0$ because of Ito's lemma. Those remarks really puzzle me a lot.

I hope I illustrate my confusion a little bit better. Thanks,

$\endgroup$
1
$\begingroup$

You have given the value of portfolio at time $t$ is \begin{equation} V_t=\phi_t S_t + \psi_t A_t \quad \cdots \cdots (1) \end{equation} where $\phi_t$and $\psi_t$ denote the number of units of the security and cash account respectively that is held in a portfolio at time t.

So, the value of portfolio at time $t+dt$ would be $$V_{t+dt}=(\phi_t + d\phi_t)(S_t + dS_t) + (\psi_t + d\psi_t)(A_t+dA_t)\quad \cdots \cdots (2)$$ where $d\phi_t$ and $d\psi_t$ are additional units bought or sold. Solved equation (2) and subtract equation (1) from equation (2), you will get $$V_{t+dt}-V_t = dVt = \phi_t dS_t+\psi_t dA_t+ S_t d\phi_t+A_t d\psi_t+dS_t d\phi_t+dA_t d\psi_t \quad\cdots (3)$$ Remember, in self-financing portfolio, there is no exogenous infusion or withdrawal of money; the purchase of a new asset must be financed by the sale of an old one. So, $d\phi_t$ and $d\psi_t$ must be zero. So last four term in equation will vanish and it lead to $$dV_t= \phi_t dS_t+\psi_t dA_t$$

This is exactly what given in your lecture notes.

$\endgroup$
  • $\begingroup$ Intuitively it is the change in $\phi_t$ betwwen $t$ and $t+dt$ $\endgroup$ – Alex C Jan 23 '16 at 21:39
  • $\begingroup$ Thank you a lot. But what confuses me is what exactly $d\phi_t$ means. If I understand correctly, the differential notations don't have mathematical meaning. However, for $dS_t$ and $dA_t$, their integrals do. So I understand $dS_t$ and $dA_t$. But I really don't get $d\phi_t$ or $\psi_t$. Moreover, your equation $3$ is more like a discrete approach, which is great. The notes on the other hand says it follows Ito's lemma. That puzzles me. In fact, I even saw some notes says $dA_t d\psi_t=0$ because of Ito's lemma. Does this clarify my confusion a little bit? Thanks a lot in advance. $\endgroup$ – Flowing Cloud Jan 23 '16 at 22:10
  • $\begingroup$ You say that $\phi_t $ "an arbitrary adapted random process". If I wrote it as $\phi(t,S)$ to make it clear that it is a function of time and the other random process in the problem, namely $S_t$ does that help? It is not arbitrary, it is yet to be defined more precisely but it will be a function of known things. $\endgroup$ – Alex C Jan 24 '16 at 14:38
  • $\begingroup$ @AlexC do you mean that $\phi_t$ must be a function of $S_t$? (if so, I now understand it.) Many thanks! $\endgroup$ – Flowing Cloud Jan 25 '16 at 5:39
  • $\begingroup$ @FlowingCloud The differentials must have a meaning, maybe it's a trivial result, but still. You could divide by dV to get 1 on the left, and relative rates of change on the right... $\endgroup$ – BAR Mar 24 '16 at 9:17
0
$\begingroup$

Let $[X,Y]_t$ be the quadratic covariation of two processes $X$ and $Y$ at time $t$. The actual precise meaning of $dXdY$ is $$dXdY= d[X,Y]_t$$ For example, the quadratic covariation of a Brownian motion $B_t$ with itself is $t$, so $$dB_t dB_t = d[B_t,B_t]_t=dt$$.

$\endgroup$
  • $\begingroup$ This sounds convincing. Will think about it, thanks! $\endgroup$ – Flowing Cloud Apr 20 '17 at 4:16

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.