# Definition of self-financing strategy

Consider a portfolio of two assets with prices $S_t$, $B_t$ and holdings $\Delta_t$ and $E_t$ respectively. So the portfolio value is

$$\Pi_t = \Delta_t S_t + E_t B_t$$

The portfolio is defined to be self-financing if we also have

$$d \Pi_t = \Delta_t d S_t + E_t d B_t$$

I'm trying to figure out an intuitive derivation if this definition. Here's my attempt:

Let $m_t$ be the amount of money moved from your $B_t$ holdings to $S_t$ over a short period of time. Then we have

$$d \Delta_t = \frac{m_t}{S_t} dt$$ $$d E_t = - \frac{m_t}{B_t} dt$$

A linear combination of these equations is

$$S_t d \Delta_t + B_t d E_t = 0$$

So we can now apply the Ito product rule and then cancel the first terms:

$$d \Pi_t = d(\Delta_t S_t) + d(E_t B_t) \\ = S_t d \Delta_t + \Delta_t dS_t + d \langle \Delta, S \rangle_t \\ + B_t d E_t + E_t dB_t + d \langle E, B \rangle_t \\ = \Delta_t d S_t + E_t d B_t + d \langle \Delta, S \rangle_t +d \langle E, B \rangle_t$$

So it remains only to show that the two cross variation terms cancel:

$$d \langle \Delta, S \rangle_t + d \langle E, B \rangle_t = 0$$

but I don't have a lot of intuition dealing with quadratic variation.

Intuition is easier in discrete time. Initially ($t=0$), your portfolio has value $$\Pi_0 = \Delta_0 S_0 + E_0 B_0.$$ If you don't do anything, at time $t=1$ its value becomes $$\Delta_0 S_1 + E_0 B_1.$$ If you want your strategy to be self-financing, you must ensure that your new portfolio at time $t=1$ can be bought with the asset you end up having from the previous period, namely: $$\Pi_1 := \Delta_1 S_1 + E_1 B_1 == \Delta_0 S_1 + E_0 B_1.$$ I.e. $$S_1 (\Delta_1-\Delta_0) + B_1 (E_1-E_0) = 0$$ or, for a continuum process, $$S_t d\Delta_t+B_t dE_t=0,$$ which is equivalent to $$d\Pi_t = d(\Delta_t S_t) + d(E_t B_t) = d\Delta_t S_t + \Delta_t dS_t + dE_t B_t + E_t dB_t = \Delta_t dS_t + E_t dB_t.$$ I recommend Baxter Rennie for a textbook with both good intuitive explanations and impeccable math.

• This is an amazingly clear answer! Thank you for taking out the time to write this. I have a short question: Baxter and Rennie—while defining a self-financing strategy—requires the portfolio process to be previsible. The way you have arrived at the definition of self-financing, which is also the way I derive it, doesn't seem to require previsiblilty. We are holding $(\Delta_t, E_t)$ over $t$ to $t+1$, which will be know to us at time $t$ even if the process is just adapted; previsiblilty is not required to know $(\Delta_t, E_t)$ at time $t$. Aug 26, 2019 at 4:51
• Really very intuitive explanation, thank you. But in general case the integration by part formula, which you use in your last series of equalities, should also include quadratic covariation terms. Can you please explain, why it gets canceled? I understand, that quadratic covariation of a FV process and continuous process is zero, but is it the case here? Mar 16, 2020 at 12:20

Roughly and intuitively speaking, a self financing strategy is (for me) a trading strategy which requires no extra cost during the trading except for the initial capital. Suppose that we are in a continuous time case. Let $(\Omega,\mathcal{F},P)$ be a prob space with filtration $\mathbb{F}$, $S$ (a semimartingle) be the discounted price process of the stock, $B$ the bank account ($B \equiv 1$), $\phi = (\theta, \eta)$ be a trading strategy, where $\eta$ is the number of units held in the bank account and $\theta$ (predictable process) in the stock $S$. Let $V$ be the value of the portfolio. The total cost of the strategy $\phi$ on $[0,t]$ is defined as $$C_t(\phi):= V_t(\phi) -(\theta\bullet S)_t,$$ where $(\theta\bullet S)_t$ (stochastic integral) denote the gain up to time $t$. In the lecture of mathematical finance that I followed, we defined $\phi$ to be self-financing if for all $t\ge 0$ $$C_{t}(\phi)=C_0(\phi) \, P\text{-a.s.}.$$ Note that $C_0(\phi) := V_0$ is the initial capital. Now it is possible to prove the following.

Lemma $\phi$ is self-financing iff $$V(\phi)=V_0(\phi) + (\theta\bullet S).$$ Indeed, there exists a bijection between self-financing strategies $\phi = (\theta,\eta)$ and the pairs $(V_0,\theta)$, where $V_0 ∈ L_0(\mathcal{F}_0)$ and $\theta$ is predictable and $S$-integrable. Explicitly, $V_0 = V_0(φ)$ and $$\eta = V_0 + (\theta\bullet S) − \theta \cdot S,$$ where $\theta \cdot S$ denotes a scalar product. Moreover, if $\phi = (\theta, \eta)$ is self-financing, then $\eta$ is also predictable.

The differential form of $V(\phi)=V_0(\phi) + (\theta\bullet S)$ is your definition of self financing strategy. I hope that it may be useful.

As requested in your comment, I add some details. It is not necessary to work with a discounted price process, but it make our life easier and it simplify the computations (and there is no loss of generality). Moreover, in order to understand the intuition behind the concept of self-financing strategy it is not relevant if we consider discounted or undiscounted price process. (In my opinion, in order to understand the intuition behind the concepts of mathematical finance it may be useful to study the discrete time case).

To say that a portfolio is self-financing means that when we rearrange the portfolio at time $t$ (e.g. from $\phi_{t-1}$ to $\phi_{t}$ in a discrete time model) there is no input/outflow of money. This means that you can rearrange your strategy using just the money which comes from the initial capital and the gain up to time $t$.

In order to motivate the above definition of self-financing strategy, we can look at the following (intuitive) argument. Suppose we keep a strategy $\phi=(\theta, \eta)$ constant between $t$ and $t+\Delta t$ and only change it from $\phi_t$ to $\phi_{t+\Delta t}$ at time $t$. Then in the interval $(t,\Delta t]$ the cost of this trading strategy is given by \begin{align} C_{t+\Delta t}-C_{t}&=(\phi_{t+\Delta t}-\phi_{t})\cdot(S_t,1) \\ &=\theta_{t+\Delta t}S_t +\eta_{t+\Delta t} -\theta_{t}S_t -\eta_t \\ &=\theta_{t+\Delta t}S_t +\eta_{t+\Delta t} -\theta_{t}S_t -\eta_t + [-\theta_{t+\Delta t}S_{t+\Delta t}+\theta_{t+\Delta t}S_{t+\Delta t}] \\ &=V_{t+\Delta t}-V_{t}-\theta_{t+\Delta t}(S_{t+\Delta t}-S_t), \end{align} recall that in discounted term the value of the portfolio is defined as $V_t := \theta_tS_t+\eta_t$. Summing up and taking $\Delta t$ small suggests the above (natural) definition of cumulative cost process, $$C_t(\phi):= V_t(\phi) - \int_{0}^t\theta_udS_u.$$ Hence, we call $\phi$ self-financing if $C_t(\phi)\equiv C_0(\phi)$ $P$-a.s. for all $t\ge0$, which means, according to our intuition, that the total cost of the strategy $\phi$ is known at time zero and we are sure that no extra money is required in the future.

In my opinion the same argumentation can be applied to the undiscounted case $\hat{B}\not\equiv 1$, which should give for the undiscounted pair $(\hat{S}_t,\hat{B}_t)$ the following definition of total cost \begin{align} \hat{C}_t := \hat{V}_t(\phi)-\int_{0}^t\theta_ud\hat{S}_u-\int_{0}^{t}\eta_ud\hat{B}_u. \end{align} Thus, we can conclude $\phi$ self-financing iff $$\hat{V}_t(\phi)= \hat{V}_0 +\int_{0}^t\theta_ud\hat{S}_u +\int_{0}^{t}\eta_ud\hat{B}_u,$$ whose differential form is your definition.

• Thanks for your answer. There are several things confusing me. First, why is $S$ the discounted stock price instead of the plain stock price? Is it necessary to lose symmetry by fixing a reference asset to discount with? And could you please elaborate on the intuition behind your alternative definition of a self-financing portfolio?
– user40167
Jun 19, 2016 at 4:26
• Edited. I hope that it can be useful.
– User
Jun 20, 2016 at 12:55

In fact, saying that the portfolio is self-financing means that the sum of the following terms equals zero : $S_t$ $d{\Lambda}_t+dE_tB_t+d< \Lambda ,S>_t+d< E ,B>_t =0$

The basic idea of self-financing portfolio is that the variations of the portfolio are only due to variations of asset prices and not from an injection of money. In most financial mathematics books, they just define the self-financing portfolio as you did without giving more details. However, vanishing the two first terms above means that the portfolio will not vary by changing the composition of the portfolio : we can rebalance between S and B but the total operation shoud yield a variation of zero. The two last terms express the stochastic covariation between the processes. If one of the two processes is continuous and deterministic, this will be zero. However, for general Ito processes, this could be potentially different from zero :

Imagine that at any moment you invest half of the stock prices in the asset prices, i.e. if $S_t$=40 then ${\Lambda_t}=20$. Hence, if the stock increases, the variation of the portfolio will be influenced by the variation of the stock and the related variation of $\Lambda$, which is not considered in the definition of a self-financing portfolio.

I hope this is more clear now :)

• Thank you, this is helpful. The mistake I made in the setup above was implicitly assuming that $\Delta_t$ and $E_t$ were differentiable. In your example where $\Delta_t = S_t/2$ this is clearly not the case and my argument doesn't go through. But the question remains: is it possible to give an intuitive definition in a similar spirit as in my original post?
– user40167
Jun 19, 2016 at 4:33

While Luigi Scorzato gives the best possible explanation of a self-financing portfolio, the issue of the residual quadratic terms in variation is missing there. I believe you can resolve it by claiming that $$dΔ_tSt+Δ_tdSt+dE_tBt+E_tdB_t$$ converges to $$d(Δ_tS_t)+d(E_tB_t)$$ in probability. In the case of Ito processes, $$dE_t$$ and $$dB_t$$ are distributed around their means (which are $$O(dt)$$ ) with the standard deviations $$O(\sqrt{dt})$$. The quadratic variation terns have the means and the standard deviations $$O(dt)$$. That is, the probability that the ratio of quadratic variation term to a linear term tends to zero, that probability tends to 1. For independent jump processes, the quadratic variation terms are non-zero only when both jumps are non-zero, and the probability of that compared to the probability of a non-zero jump of either process tends to zero with $$dt$$. Of course, this is nowhere near of a rigorous argument, but you asked for an intuitive answer. I hope this helps.