Portfolio Rebalancing Costs and the Cross-Term in Value Changes
Summary
The document explains why a discrete portfolio value change can contain a cross-term formed by a change in holdings multiplied by a change in an asset’s price. When holdings are adjusted during the same interval as prices move, the added units are not necessarily acquired at the interval’s starting price. Combining the holding-change and cross-term contributions values the trade at the updated price.
A numerical example shows a portfolio with existing units that buys more units while the asset price falls. The loss on the original holdings is offset in part by the value of the new purchase at the starting price; the cross-term accounts for the price change on those newly added units. For a self-financing strategy, the cost of rebalancing must be financed through adjustments to other assets, possibly including cash. The exposition uses discrete changes; continuous-time stochastic calculus may treat products of infinitesimal changes differently depending on the process assumptions.
Key ideas
- The cross-term captures the price movement applied to units added during the interval.
- Holding-change and cross-term contributions together value rebalancing at the updated price.
- A self-financing portfolio funds trades by adjusting other holdings, potentially including cash.
- The simple product-of-changes explanation concerns discrete changes and does not replace continuous-time process rules.
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Full text
# Instantaneous change in value of portfolio
# Instantaneous change in value of portfolio
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!
## Answer by user34971 (score 2, accepted)
https://quant.stackexchange.com/a/49426
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)
## Answer by demully (score 0)
https://quant.stackexchange.com/a/49404
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.