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I am trying to figure out an intuitive explanation for the instantaneous change for the value of a portfolio (essentially I'm creating a self-financing portfolio to replicate a derivative payoff).

Suppose that at time t we hold the portfolio $\left(a_t,b_t,c_t\right)$ where $a_t,b_t$ and $c_t$ represent the number of units held at time t of securities with respective price processes $A_t,B_t$ and $C_t$. Assume $\left(a_t,b_t,c_t\right)$ are previsible. Letting $V_t$ be the value of this portfolio at time t.

The instantaneous change in the value of the portfolio, including cash inflows and outflows is therefore:

$dV_t=a_tdA_t+da_tA_t+da_tdA_t+b_tdB_t+db_tB_t+db_tdB_t+c_tdC_t+dc_tC_t+dc_tdC_t$

I understand that for $a_tdA_t$, it means the original holdings of $a_t$ multiplied by the change in value. For $da_tA_t$, it means changes in number of units held multiplied by the value at time t.

However, I'm struggling to understand intuitively why there is a $da_tdA_t$. Would greatly appreciate if anyone can help me understand!

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It's best to think of the sum of $da_t A_t$ and $da_t dA_t$: $$ da_t A_t + da_t dA_t = da_t(A_t + dA_t) $$ which is the cost of rebalancing at the new price $A_{t'} = A_t + dA_t$. You don't rebalance the portfolio at $t$ but at $t'$. And the self-financing condition means you need to finance this cost by rebalancing other assets (including possibly the money market account)

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    $\begingroup$ Take it 1 step further. If there are only two assets and you finance the cost of rebalancing A with the rebalancing of B then you will have $da_t(A_t+dA_t)=-db_t(B_t+dB_t)$. In that case (the self-financing case) the overall portfolio value change $dV_t=a_tdA_t+da_tA_t+da_tdA_t+b_tdB_t+db_tB_t+db_tdB_t$ simplifies to $dV_t=a_tdA_t+b_t dB_t$. A remarkable simplification. $\endgroup$
    – Alex C
    Oct 29, 2019 at 17:24
  • $\begingroup$ thanks yes that completes the answer. $\endgroup$
    – user34971
    Oct 29, 2019 at 18:21
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Say you start with 1000 units worth 100 equals value 100,000. You buy 100 more but the price falls to 90, giving you 1100*90 equals 99,000 value.

Change in Value =

dP * U = -10 * 1000 = -10,000

P * dU = 100 * 100 = 10,000

dP * dU = -10 * 100 = -1,000

The first two, the change in value of existing units, and the value brought in with new purchases of units, cancel out in this case. The change in aggregate value here is the change in value of the new units.

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