Discounted Gains From Self-Financing Strategies

From Trading Positions to Discounted Gains

A trading strategy changes its holdings over time, but a self-financing strategy does not add or withdraw money when it rebalances. Its wealth changes only because the assets it already holds change in value. Discounting makes this bookkeeping especially direct: the change in discounted wealth is the position in each risky asset multiplied by that asset’s discounted price change.

Use a finite horizon with dates \(k=0,1,\ldots,N\). Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\geq0},\mathbb{P})\) carrying a Brownian motion \(B_t\). The market below has only finitely many trading dates, so the Brownian motion is not used. In particular, no Ito or Stratonovich integral is involved: gains are finite sums. The filtration at date \(k\), denoted \(\mathcal{F}_k\), records the information available when a position for the next period is chosen.

Suppose there are \(d\) risky assets, with price vector \(\mathbf{S}_k=(S_k^1,\ldots,S_k^d)\), and a strictly positive bank-account value \(A_k\), where \(A_0=1\). Assume the assets pay no dividends. For simplicity, take the bank account to be deterministic. Define the discounted price vector by \(\widetilde{\mathbf{S}}_k=\mathbf{S}_k/A_k\). A strategy chooses \(\boldsymbol{\theta}_k=(\theta_k^1,\ldots,\theta_k^d)\) shares of the risky assets and \(\phi_k\) units of the bank account for the period from \(k\) to \(k+1\). These holdings are \(\mathcal{F}_k\)-measurable: they may depend on information already observed, but not on the next price change.

Discounted wealth is \(\widetilde V_k=V_k/A_k\). The discounted gains accumulated by the risky holdings through date \(n\) are \(\widetilde G_n=\sum_{k=0}^{n-1}\boldsymbol{\theta}_k\cdot(\widetilde{\mathbf{S}}_{k+1}-\widetilde{\mathbf{S}}_k)\), with \(\widetilde G_0=0\). The bank account contributes no discounted price change: one unit of the bank account is worth \(A_k\) at date \(k\), or exactly one unit after discounting.

The Self-Financing Identity

Proof. Self-financing means that the holdings over period \(k\) value the wealth at both ends of that period. Divide the wealth at date \(k\) by \(A_k\), and the wealth at date \(k+1\) by \(A_{k+1}\). Since \(\phi_k\) is the number of bank-account units held throughout the period,

Subtracting cancels the bank-account holding and gives \(\widetilde V_{k+1}-\widetilde V_k=\boldsymbol{\theta}_k\cdot(\widetilde{\mathbf{S}}_{k+1}-\widetilde{\mathbf{S}}_k)\). Summing this equality for \(k=0,\ldots,n-1\) cancels the intermediate wealth terms: the left side is \(\widetilde V_n-\widetilde V_0\), and the right side is the stated gains sum. For \(n=0\), the sum is empty and equals zero, so the formula also holds at the initial date. \(\square\)

This identity also describes how to rebalance without injecting funds. Given wealth \(\widetilde V_k\) and a chosen risky position \(\boldsymbol{\theta}_k\), the discounted bank-account position must be \(\phi_k=\widetilde V_k-\boldsymbol{\theta}_k\cdot\widetilde{\mathbf{S}}_k\). It may be negative: that represents borrowing. The identity itself imposes no restriction excluding negative wealth or borrowing.

Rebalancing Along a Multi-Period Path

At a rebalancing date, the new risky position can depend on the prices observed so far. To remain self-financing, the bank-account holding must adjust so that the new portfolio has exactly the wealth already accumulated. The next example makes both the price-change calculation and that adjustment explicit.

Several Risky Assets and a Conditional Martingale Result

Discounting and self-financing alone do not make gains a martingale. A probability measure and a condition on discounted prices are needed. The following elementary result states precisely what is sufficient in this finite-date setting. It does not assert that such a measure exists.

Proof. Bounded holdings multiplied by integrable discounted prices have integrable products, so each gain increment is integrable. Since \(\widetilde G_k\) is a sum of integrable increments, it is integrable. For each \(k<N\), \(\boldsymbol{\theta}_k\) is \(\mathcal{F}_k\)-measurable, and the assumed conditional expectation of the discounted price change is zero. Therefore \(\mathbb{E}^{\mathbb{Q}}[\widetilde G_{k+1}-\widetilde G_k\mid\mathcal{F}_k]=\boldsymbol{\theta}_k\cdot\mathbb{E}^{\mathbb{Q}}[\widetilde{\mathbf{S}}_{k+1}-\widetilde{\mathbf{S}}_k\mid\mathcal{F}_k]=0\). Thus \(\mathbb{E}^{\mathbb{Q}}[\widetilde G_{k+1}\mid\mathcal{F}_k]=\widetilde G_k\), which is the martingale property. By the self-financing identity, \(\widetilde V_n=\widetilde V_0+\widetilde G_n\); adding an integrable initial value preserves integrability and the conditional-expectation identity. Hence discounted wealth is a martingale as claimed. \(\square\)

A Common Calculation Pitfall

When discount factors change between dates, compute each discounted price at its own date before subtracting. Do not subtract nominal prices and then treat that as a discounted increment. On the down-then-up path above, the first discounted change is \(900/11-100=-200/11\), not \(-100/11\). The second is \(10800/121-900/11=900/121\). With holdings \(2\) and then \(-1\), the gains are \(2(-200/11)-900/121=-5300/121\), not \(-3100/121\). Starting from \(1000\), the correct discounted terminal wealth is \(1000-5300/121=115700/121\). Direct portfolio valuation gives the same amount, as checked above.

The central bookkeeping distinction is between wealth and gains. Wealth includes the value of all current holdings, including the bank account. Discounted gains record only the changes in discounted risky-asset values multiplied by the positions held over each period. Self-financing is what makes their cumulative sum equal the change in discounted wealth; a martingale conclusion requires the additional measure and conditional-mean assumptions stated above.