Self-financing portfolio

A portfolio is self-financing when every change in its asset holdings is paid for entirely by selling or buying assets within the portfolio, with no external cash added or removed.

For holdings with value $V_t={\varphi }_t{S}_t+{\psi }_t{B}_t$, the condition says that all gains and losses arise from the held assets, with no external cash transfer. The stochastic-integrability and admissibility requirements on the holdings are part of a valid trading strategy. Differentiating the value identity alone does not establish self-financing, since changes of holdings can contribute additional terms.

In the Black-Scholes model, take $B_t=e^{-\rho (T-t)}$ as a zero-coupon bond numéraire. Holding the fixed stock value $\theta B_t$ requires $g_t=\theta B_t/S_t$ shares. The self-financing conditions for smooth stock and bond holdings giveunits of the bond. Since $V_t=g_t{S}_t+h_t{B}_t$, this yields the displayed wealth. Under the risk-neutral measure, $V_t/B_t=w_0/B_0+\theta \sigma W_t^Q$: the discounted gain is a constant multiple of Brownian motion.

A constant-proportion portfolio holds a fixed fraction $\gamma$ of its portfolio wealth in the stock. In the Black-Scholes model with bank rate $\rho$, initial wealth $V_0$, and no consumption, its value isThe stock holding is $\gamma V_t/S_t$ and the remaining value $(1-\gamma )V_t$ is invested in the bank account. The Itô formula verifies $d{V}_t/V_t=[\rho +\gamma (\mu -\rho )]dt+\gamma \sigma d{W}_t$ when the physical stock drift is $\mu$.

For $p(x,t)=xg(x,t)+B_{t}h(x,t)$ and deterministic $d{B}_t=\rho B_{t}dt$, Itô formula shows that a self-financing portfolio must have $p_x=g$. This is equivalent to $x{g}_x+B_t{h}_x=0$. Differentiating it gives $p_{xx}=g_x$, so the remaining gain identity is $x{g}_t+B_t{h}_t+{\sigma }^2{x}^2{g}_x/2=0$. Conversely these two identities give $dp=gdS+hdB$. A value PDE alone does not verify arbitrary prescribed holdings.

If $B$ is a positive riskless numéraire of finite variation, $S$ the ex-dividend price vector and $D$ its cash-dividend rate, a self-financing portfolio with risky holdings $\pi$ satisfies $d(X/B)=\pi \cdot dY$. This follows by differentiating $X/B$, substituting $dX=\varphi dB+\pi \cdot (dS+Ddt)$ and using $X=\varphi B+\pi \cdot S$. If $Y$ is a martingale under an equivalent measure, discounted wealth is a local martingale. A deterministic lower bound on discounted wealth makes it a supermartingale, ruling out arbitrage. The cumulative dividends here are discounted at their own payment times.

For an integrable discrete trading strategy under an equivalent martingale measure, discounted portfolio wealth plus accumulated discounted strategy payments is a martingale. Conditioning its terminal value gives the displayed identity. A formula involving future payments alone requires terminal liquidation, $V_n=0$, or inclusion of that terminal value in the final payment. For an attainable claim paid only at maturity, liquidation yields its usual conditional discounted expected value.

A share paying a continuous dividend rate ${\delta }_t$ produces dollar gains $d{S}_t+{\delta }_{t}dt$. If its price follows a geometric Brownian motion with drift ${\mu }_{\mathrm{price}}$ and its dividend yield ${\delta }_t/S_t$ is constant, its drift in an investment-consumption problem is their sum. The self-financing portfolio equation uses this total return, even if the agent consumes the dividends rather than reinvesting them. Omitting the dividend yield changes both the optimal stock fraction and the equilibrium interest rate.

A self-financing portfolio is admissible when its wealth satisfies the prescribed lower-bound restriction, typically a deterministic bound below. Nonnegative wealth is sufficient. This condition permits supermartingale bounds on discounted gains and excludes doubling strategies.

Relative to a positive numéraire, a predictable self-financing strategy with risky holdings ${\theta }_t$ and discounted risky prices $X_t$ has discounted valueThus its gains are the discrete stochastic integral of its holdings against discounted price increments.

An integrable predictable strategy whose discounted gains ${\varphi }_n\Delta {\tilde{S}}_n$ are integrable preserves the martingale property of discounted prices in finite discrete time. Bounded holdings suffice. Without integrability the claim is false: let ${\tilde{S}}_0=3/2$, ${\tilde{S}}_1=1+U$, ${\tilde{S}}_2=1+U+\epsilon /2$, where $U$ is uniform on $(0,1)$ and $\epsilon$ is an independent fair sign. Holding zero stock first and $1/U$ shares in the second period, funded by the bank, gives zero-initial-cost terminal wealth $\epsilon /(2U)$, which is not integrable. Replication pricing requires the corresponding true-martingale condition, not merely formal self-financing.