Self-Financing Portfolios and Wealth Dynamics

Holdings, Asset Values, and Wealth

A portfolio’s wealth changes for two different reasons: its assets gain or lose value, or money is added to or removed from the portfolio. The distinction matters because a change in wealth is not automatically a trading gain. A deposit raises wealth but is not generated by the stock or bank account; a self-financing strategy has no such external cash flows.

Work on a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})$ carrying a Brownian motion $B_t$, and fix a finite horizon $T$. In the Black-Scholes model, suppose the stock price and bank-account value satisfy $dS_t=\mu S_t\,dt+\sigma S_t\,dB_t$ and $A_t=e^{rt}$, with $S_0>0$, $A_0=1$, and constant parameters $\mu,r\in\mathbb{R}$, $\sigma>0$. Thus $dA_t=rA_t\,dt$. Here $\mu$ is the stock’s drift and $\sigma$ its diffusion coefficient. This discussion uses $\mathbb{P}$; it does not assume the risk-neutral measure $\mathbb{Q}$ or that $\mu=r$.

Definition: A portfolio holds $\Delta_t$ shares of stock and $\beta_t$ units of the bank account at time $t$. Its wealth is $V_t=\Delta_tS_t+\beta_tA_t$. Holdings are predictable, so the position used in a stochastic integral is determined from information available just before the corresponding price increment. For the Itô integrals below, require the integrands to be progressively measurable and in $\mathcal{L}^2$: in particular, $\mathbb{E}\int_0^T(\Delta_t\sigma S_t)^2\,dt<\infty$. Also assume the ordinary drift integrals are finite, for example $\mathbb{E}\int_0^T|\Delta_t\mu S_t|\,dt<\infty$ and $\mathbb{E}\int_0^T|\beta_t rA_t|\,dt<\infty$.

The stock position is worth $\Delta_tS_t$. The remaining amount, $V_t-\Delta_tS_t$, is the dollar value held in the bank account; it equals $\beta_tA_t$. If this amount is negative, the portfolio borrows from the bank. The number of bank-account units is then $\beta_t=(V_t-\Delta_tS_t)/A_t$, since $A_t>0$.

What “Self-Financing” Means

For self-financing holdings, changes in wealth come only from the gains on the assets held. The gains over time are the stochastic integral of the stock position against stock-price changes and the ordinary integral of the bank-account position against bank-account changes. An adjustment to holdings is paid for by changing the other holding; it is not an external contribution. In particular, one does not apply the ordinary product rule to $\Delta_tS_t$ and count a term $S_t\,d\Delta_t$ as an additional gain: changing the number of shares is a trade, whose cost must be offset by the cash position.

Definition: A portfolio is self-financing on $[0,T]$ if, for every $t\in[0,T]$, $$ V_t=V_0+\int_0^t\Delta_s\,dS_s+\int_0^t\beta_s\,dA_s. $$ The Itô integral $\int_0^t\Delta_s\,dS_s$ is defined here through the drift integral $\int_0^t\Delta_s\mu S_s\,ds$ and the Itô integral $\int_0^t\Delta_s\sigma S_s\,dB_s$, whose progressively measurable integrand $\Delta_s\sigma S_s$ is required to be in $\mathcal{L}^2$. The bank-account integral is the ordinary time integral $\int_0^t\beta_s rA_s\,ds$, assumed finite. Equalities between processes are understood up to indistinguishability.

Theorem: Under these integrability conditions, a portfolio with $V_t=\Delta_tS_t+\beta_tA_t$ is self-financing if and only if it satisfies $$ dV_t=\Delta_t\,dS_t+r(V_t-\Delta_tS_t)\,dt. $$ Equivalently, in stock and Brownian terms, $$ dV_t=\big(\mu\Delta_tS_t+r(V_t-\Delta_tS_t)\big)\,dt+\sigma\Delta_tS_t\,dB_t. $$

Proof. Suppose first that the portfolio is self-financing. By its defining gains equation and $dA_t=rA_t\,dt$, $dV_t=\Delta_t\,dS_t+\beta_t rA_t\,dt$. The holdings identity gives $\beta_tA_t=V_t-\Delta_tS_t$, so substitution yields $dV_t=\Delta_t\,dS_t+r(V_t-\Delta_tS_t)\,dt$. Inserting the stock dynamics gives the displayed drift and diffusion form. Conversely, suppose the displayed equation holds and $V_t=\Delta_tS_t+\beta_tA_t$. Then $r(V_t-\Delta_tS_t)=r\beta_tA_t$, so integration gives $V_t=V_0+\int_0^t\Delta_s\,dS_s+\int_0^t\beta_s rA_s\,ds$. Since $dA_s=rA_s\,ds$, this is exactly the self-financing gains equation. Both implications hold, including when either holding is zero. $\square$

For clarity, the stochastic term in the expanded equation is an Itô integral with integrand $\sigma\Delta_tS_t$, assumed progressively measurable and square-integrable as stated above. In this case that integral is a true martingale: for simple predictable integrands, each increment has conditional expectation zero given the preceding information; approximation in $\mathcal{L}^2$ and the Itô isometry extend this property to the stated integrand class. In particular, its expectation is zero. No corresponding martingale claim is made for the full wealth process, which also contains drift terms.

Worked Portfolio Calculations

Worked Example: Fixed Holdings in Stock and Bank Account

Let $r=\tfrac14$, $\mu=\tfrac12$, and $\sigma=\tfrac12$. Suppose $S_0=4$, $V_0=10$, and the strategy holds one share and six bank-account units throughout: $\Delta_t=1$, $\beta_t=6$. Find wealth and verify its differential equation.

Since $A_t=e^{t/4}$, wealth is $V_t=S_t+6e^{t/4}$. At time zero this gives $V_0=4+6=10$, as specified. The stock dynamics are $dS_t=\tfrac12S_t\,dt+\tfrac12S_t\,dB_t$, and $d(6A_t)=\tfrac{6}{4}e^{t/4}\,dt=\tfrac32e^{t/4}\,dt$. Thus

dV_t=\left(\tfrac12S_t+\tfrac32e^{t/4}\right)dt+\tfrac12S_t\,dB_t.

The self-financing equation gives the same result: $\Delta_t=1$ and $V_t-\Delta_tS_t=6e^{t/4}$, so its drift is $\mu\Delta_tS_t+r(V_t-\Delta_tS_t)=\tfrac12S_t+\tfrac14(6e^{t/4})=\tfrac12S_t+\tfrac32e^{t/4}$; its diffusion coefficient is $\sigma\Delta_tS_t=\tfrac12S_t$. As an additional check, $\mathbb{E}[S_t]=4e^{t/2}$ for this geometric Brownian motion, so $\mathbb{E}[V_t]=4e^{t/2}+6e^{t/4}$, consistent with the explicit wealth expression.

Worked Example: Rebalancing Without an External Cash Flow

Let $r=\log 2$, so $A_1=2$. Initially the portfolio holds one share and six bank-account units. At time $1$, after observing $S_1$, it increases its stock holding to two shares. Determine the bank-account holding after the trade if the adjustment is self-financing, and check that wealth is unchanged at the trading instant.

Immediately before trading, wealth is $S_1+6A_1=S_1+12$. Buying one additional share costs $S_1$ dollars. Since one bank-account unit is worth $A_1=2$, the bank-account position must decrease by $S_1/2$ units. Therefore the new holding is $\beta_{1+}=6-\tfrac12S_1$. Immediately after the trade, the wealth is

2S_1+\left(6-\tfrac12S_1\right)A_1 =2S_1+\left(6-\tfrac12S_1\right)2 =2S_1+12-S_1 =S_1+12.

This equals the wealth immediately before trading. If $\beta_{1+}$ is negative, the portfolio borrows to finance the purchase; the same calculation still applies. The trade changes the holdings, not the wealth. After the trade, the chosen holdings are used for subsequent gains. Because the new position is selected using information at time $1$, it is held after that observation, as required for a predictable trading strategy.

Check: the increase in stock value is $S_1$. The decrease in bank-account value is $(S_1/2)A_1=(S_1/2)2=S_1$. The two changes cancel exactly, so no deposit or withdrawal is needed.

Worked Example: A Deposit Is Not a Trading Gain

Consider a bank-only portfolio with $r=\log 2$, initial wealth $V_0=10$, and a cash deposit of $4$ dollars at time $1$. Find its wealth at times just before and after the deposit, and at time $2$. Compare it with a self-financing bank-only portfolio.

Before the deposit, the account contains ten bank-account units, so $V_t=10A_t=10e^{t\log 2}$. Since $A_1=2$, wealth just before the deposit is $10\cdot2=20$. The deposit adds $4$ dollars, so wealth just after it is $24$. In bank-account units, the post-deposit holding is $24/A_1=24/2=12$.

That holding then grows from time $1$ to time $2$ by the factor $A_2/A_1=4/2=2$. Hence $V_2=24\cdot2=48$. Equivalently, the original $10$ dollars grows to $10A_2=10\cdot4=40$, while the deposit grows for one time unit to $4(A_2/A_1)=4\cdot2=8$; their sum is $40+8=48$.

Without the deposit, the self-financing bank-only portfolio would have wealth $10A_2=40$ at time $2$. The difference, $48-40=8$, is the grown value of the external deposit, not a trading gain. The wealth jump of $4$ dollars at time $1$ cannot be produced by the continuous bank-account dynamics alone.

External Cash Flows and the Wealth Ledger

External cash flows can be recorded separately. Let $C_t$ be a cumulative, adapted, finite-variation process representing net dollars added to the portfolio, with positive increments for deposits and negative increments for withdrawals. Use the same asset-integral conditions as above. The wealth ledger with external flows is $V_t=V_0+\int_0^t\Delta_s\,dS_s+\int_0^t\beta_s\,dA_s+C_t$, where $C_0=0$. Thus $dC_t$ records the external flow; it is not part of either asset’s return.

Proposition: Under the stated integrability conditions, the wealth ledger with cumulative external flow $C_t$ satisfies $$ dV_t=\Delta_t\,dS_t+r(V_t-\Delta_tS_t)\,dt+dC_t. $$ It is self-financing exactly when $C_t=0$ for all $t$ (or, equivalently, $C$ is constant when its initial value is absorbed into $V_0$).

Proof. From the ledger, $dV_t=\Delta_t\,dS_t+\beta_t\,dA_t+dC_t$. The bank-account dynamics give $\beta_t\,dA_t=\beta_t rA_t\,dt$, and the wealth identity gives $\beta_tA_t=V_t-\Delta_tS_t$. Substituting proves the displayed equation. If $C$ is identically zero, the ledger is the definition of self-financing. Conversely, if the portfolio’s wealth change consists only of the two asset gains, comparison with the ledger gives $C_t=0$ at every time, since both expressions have the same initial value and gains. If a constant $C$ is allowed, replacing $V_0$ by $V_0+C_0$ gives the equivalent zero-initial-flow convention. $\square$

Common Pitfalls

Counting a trade as a gain. A tempting but wrong calculation is to differentiate $V_t=\Delta_tS_t+\beta_tA_t$, keep only the asset-price terms, and then treat $S_t\,d\Delta_t$ as extra wealth. For a purchase of one share at time $1$, this would claim a wealth increase of $S_1$. The correction is that the purchase costs $S_1$: the bank-account value decreases by $S_1$, as the rebalancing example shows. Self-financing counts asset gains and losses, not the gross value of trades.

Confusing deposits with returns. If wealth jumps by $4$ dollars because of a deposit, the self-financing equation alone does not explain that jump. Record it as $dC_t$. Likewise, subtract external contributions when assessing investment performance; otherwise the measured wealth increase combines trading results with contributed capital.

The equation also separates the roles of the assets. The stock contributes drift $\mu\Delta_tS_t$ and diffusion $\sigma\Delta_tS_t$; the bank account contributes the interest $r(V_t-\Delta_tS_t)\,dt$ on the cash position. The multiplication rules $dB_t\,dB_t=dt$, $dB_t\,dt=0$, and $dt\,dt=0$ belong to Itô calculations; the wealth identity here follows directly from gains bookkeeping and does not require applying Itô’s formula.

Check Your Understanding

For each question, distinguish the asset gains from changes in the holdings or external cash flows.

Write the expanded drift and diffusion terms when a self-financing portfolio holds $\Delta_t$ shares and has wealth $V_t$.
A portfolio has wealth $V_t=15$, stock price $S_t=5$, and holds two shares. What is the dollar value in the bank account, and how many bank-account units does it hold if $A_t=3$?
At a rebalancing time, a portfolio buys $\delta$ additional shares at price $S_t$. What change in bank-account units finances this purchase?
Suppose a bank-only portfolio receives a withdrawal. How should the withdrawal enter the wealth ledger, and why is it not a self-financing gain?
State the square-integrability condition used for the Brownian integral in the stock gains term.