What makes a trading strategy self-financing?
A strategy buys and sells shares as prices move. When does it need no money beyond what it cost at the start?
When every purchase is paid from the strategy's own cash account and every sale is paid into it. Then nothing enters or leaves after the start, the value changes only through price moves and interest, and at every time the value equals the initial cost plus the gains so far.
Before you start · 3 quick checks on binomial prices, interest and trading gains, about 3 min
- Market. The n-period binomial tree of step 6: the stock starts at \(S_0 \gt 0\) and each period its price is multiplied by \(u\) or by \(d\), with \(0 \lt d \lt u\). A cash account grows by the factor \(1+r\) each period, with \(r \gt -1\) and \(d \lt 1+r \lt u\). Times are \(k = 0, 1, \dots, n\).
- Strategy. At each time \(k \lt n\), after seeing the prices so far, the strategy holds \(\Delta_k\) shares and \(B_k\) dollars in the cash account until time \(k+1\). Both may be negative: a short sale, a loan. At time \(n\) nothing is traded: \(\Delta_n = \Delta_{n-1}\).
- Value. \(V_k = \Delta_kS_k + B_k\) at every time \(k\). The initial cost is \(V_0\), the money put in at time 0.
- Outside money. At time \(k\) the cash account earns its interest, pays for shares bought at the price \(S_k\), receives the proceeds of shares sold, and changes by any money \(C_k\) put in from outside (\(C_k \gt 0\)) or taken out (\(C_k \lt 0\)): for \(k = 1, \dots, n\), \[ \begin{aligned} B_k = (1+r)B_{k-1} &- (\Delta_k - \Delta_{k-1})S_k \\ &+ C_k, \end{aligned} \] with \(C_n = 0\), since nothing is traded or taken out at the end.
- Self-financing. The strategy is self-financing if \(C_k = 0\) at every time \(k = 1, \dots, n-1\) on every path: after the initial cost, no money goes in or out.
- Gains. The gains up to time \(k\) are what the holdings earned: the shares from price changes, the cash from interest, \[ \begin{aligned} G_k = \sum_{j=1}^{k}\big(&\Delta_{j-1}(S_j - S_{j-1}) \\ &+ r(V_{j-1} - \Delta_{j-1}S_{j-1})\big), \end{aligned} \] since \(V_{j-1} - \Delta_{j-1}S_{j-1} = B_{j-1}\) is the cash held from \(j-1\) to \(j\). Each \(\Delta_{j-1}\) is chosen before the change \(S_j - S_{j-1}\) is known: the share gain is a left-endpoint sum, as on the Itô integral page (step 12).
Fine print: trading assumptions
- The market is frictionless: no transaction costs, no taxes, no bid-ask spread. Any real number of shares can be bought or sold short at the current price, and any amount can be lent or borrowed at the rate \(r\) per period.
- The stock pays no dividends, so a share earns only its price change. With dividends, the cash account would also receive them, and a short position would pay them.
- A strategy may depend on the path so far but not on later prices: the holdings are fixed when a node is reached, and two paths that reach the same price may hold different amounts.
- "Self-financing" is a statement about money flows only. It does not say the strategy is safe or that its value stays positive.
Keep the books along one path
At each time, set the shares with the slider and type the new cash balance. Then choose the next price move. The page marks every time at which the cash you typed does not match what the account holds after interest and the trade: that difference is outside money. At time 3 it lays out the books. Cash is typed to the cent at time 0; each period of interest can add one more decimal, so the box takes one more decimal place at each later time, enough for the exact balance a self-financed trade leaves.
"Start a new walk" clears the books and keeps your progress through the goals. "Restart the goals" also goes back to goal 1.
A fixed dollar amount, time-0 dollars and continuous time
The questions use the market of stage 1.
Write \(\tilde X_k = X_k/(1+r)^k\) for an amount at time \(k\) measured in time-0 dollars. For a self-financing strategy, \(V_k = \Delta_{k-1}S_k + (1+r)B_{k-1}\), and dividing by \((1+r)^k\) gives \(\tilde V_k = \Delta_{k-1}\tilde S_k + \tilde B_{k-1}\). Since \(\tilde V_{k-1} = \Delta_{k-1}\tilde S_{k-1} + \tilde B_{k-1}\),
\[ \tilde V_k - \tilde V_{k-1} = \Delta_{k-1}(\tilde S_k - \tilde S_{k-1}), \] \[ \tilde V_k = \tilde V_0 + \sum_{j=1}^{k}\Delta_{j-1}(\tilde S_j - \tilde S_{j-1}). \]In discounted dollars the cash account does not move between trades, so it contributes nothing: only the discounted stock price changes count. This is the form used in step 17, where discounted stock prices are martingales under the risk-neutral probabilities, and so is the discounted value of every self-financing strategy.
Let each period last \(\Delta t\) and let the cash account grow by \(1 + r\,\Delta t\) per period, for a rate \(r\) per unit of time. The one-period change of a self-financing strategy is then
\[ \begin{aligned} V_k - V_{k-1} &= \Delta_{k-1}(S_k - S_{k-1}) \\ &\quad + r(V_{k-1} - \Delta_{k-1}S_{k-1})\,\Delta t . \end{aligned} \]For continuous time, take these assumptions together: the information at time \(t\) is the price path up to \(t\); on a finite horizon \([0, T]\) the stock price follows a geometric Brownian motion under the real probabilities, \(dS = \mu S\,dt + \sigma S\,dW\) with constants \(\mu\) and \(\sigma \gt 0\) and \(S_0 \gt 0\), so \(S_t \gt 0\); the rate \(r\) is constant; and the holding \(\Delta_t\) never looks ahead (it is adapted, and progressively measurable) with \(\int_0^T \Delta_t^2S_t^2\,dt \lt \infty\) on every path. That is enough for the integrals below to exist; it is weaker than the mean-square condition of the Itô integral page, and on its own it does not make discounted values martingales. A strategy is called self-financing when, for every \(t\) in \([0, T]\), with probability 1,
\[ \begin{aligned} V_t = V_0 &+ \int_0^t \Delta_s\,dS_s \\ &+ \int_0^t r(V_s - \Delta_sS_s)\,ds, \end{aligned} \]written in short as
\[ dV = \Delta\,dS + r(V - \Delta S)\,dt . \]The first integral is the Itô integral: like the share gain on the tree, its sums use the holding at the start of each step (step 12). With a cash account that grows by \(1 + r\,\Delta t\) per period, as above, a strategy that resets its holding only at the times \(k\,\Delta t\) follows the one-period equation exactly, along any price path. A cash account that grows continuously, by \(e^{r\Delta t}\) per period, puts \((e^{r\Delta t} - 1)(V_{k-1} - \Delta_{k-1}S_{k-1})\) in place of the interest term, which agrees with it to first order in \(\Delta t\). Take an adapted \(\Delta\) with continuous paths on \([0, T]\), hold it at its value at the start of each step, and let \(S\) be a geometric Brownian motion; then, as stated here without proof, the share sums and the interest sums converge in probability to the two integrals as \(\Delta t \to 0\). The delta-hedging argument of step 16 starts from this equation.
There is no \(S\,d\Delta\) term because the cash account pays for every change in the holding. On the tree this is exact: the cash equation gives \(B_k - (1+r)B_{k-1} = -(\Delta_k - \Delta_{k-1})S_k\), and
\[ \begin{aligned} (\Delta_k - \Delta_{k-1})S_k &= (\Delta_k - \Delta_{k-1})S_{k-1} \\ &\quad + (\Delta_k - \Delta_{k-1})(S_k - S_{k-1}). \end{aligned} \]The two parts on the right are the tree's versions of \(S\,d\Delta\) and \(d[\Delta, S]\). When \(\Delta\) is itself an Itô process, Itô's product rule gives \(d(\Delta S) = \Delta\,dS + S\,d\Delta + d[\Delta, S]\), and self-financing means \(dB = rB\,dt - S\,d\Delta - d[\Delta, S]\), so the extra terms cancel.
The value is the cost plus the gains, at every time
Assumptions. The setup and its fine print: an n-period binomial tree with \(n \ge 2\), \(S_0 \gt 0\), \(0 \lt d \lt u\), \(r \gt -1\) and \(d \lt 1+r \lt u\), a frictionless market, no dividends, and any strategy \((\Delta_k, B_k)\) with outside money \(C_k\) defined by the cash equation, \(\Delta_n = \Delta_{n-1}\) and \(C_n = 0\).
Conclusion. (a) For every strategy, every path and every \(k\) from 1 to \(n\),
\[ \begin{aligned} V_k - V_{k-1} &= \Delta_{k-1}(S_k - S_{k-1}) \\ &\quad + r(V_{k-1} - \Delta_{k-1}S_{k-1}) + C_k, \end{aligned} \] \[ V_k = V_0 + G_k + \sum_{j=1}^{k} C_j . \](b) A strategy is self-financing if and only if \(V_k = V_0 + G_k\) for every path and every \(k\) from 1 to \(n\). In particular a self-financing strategy ends with \(V_n = V_0 + G_n\).
Each line of the proof is a claim about every strategy, every path and every \(k\) in the range it names. A candidate line that fails for one strategy is false, even if it holds for others.
The converse in line 5 needs the identity at every time. At the end alone it is not enough when there are two or more trading times: in goal 3 of stage 1, money put in at time 1 and taken out at time 2 cancel at time 3.
Problems
More problems
Further reading
- S. E. Shreve, Stochastic Calculus for Finance I: The Binomial Asset Pricing Model (Springer, 2004), ch. 1 and 2.
- S. E. Shreve, Stochastic Calculus for Finance II: Continuous-Time Models (Springer, 2004), ch. 4, "Stochastic Calculus."
- T. Björk, Arbitrage Theory in Continuous Time, 4th ed. (Oxford University Press, 2019), ch. 6, "Portfolio Dynamics."
- J. M. Harrison and S. R. Pliska, "Martingales and stochastic integrals in the theory of continuous trading," Stochastic Processes and their Applications 11 (1981), 215–260.