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My question concerns a topic in quantitative finance that I feel is often brushed under the table: is a given strategy self-financing.

We have two assets, one risky and one riskless, defined by the following SDEs:

\begin{align*} dS(t)&=\mu S(t)dt + \sigma S(t)dW(t)\\ S(0)&=S_0\\ dB(t)&=rB(t)dt\\ B(0)&=B_0 \end{align*}

with $\mu, \sigma, S_0, B_0>0$ constants, and $W(t)$ a Wiener-process.

Let's assume we have $V_0>0$ wealth we wish to invest in a portfolio of these assets. Our strategy is a special strategy: we wish to hold the ratio of the risky and the riskless asset constant at all times. Our strategy $\{\Delta(t), \beta(t)\}$ is thus defined as:

\begin{align*} V(t)&=\Delta(t)S(t)+\beta(t)B(t)\\ V(0)&=V_0\\ \frac{\Delta(t)S(t)}{V(t)}=x,&\quad\quad\frac{\beta(t)B(t)}{V(t)}=1-x \end{align*} with $x\in\mathbb{R}$ fixed. I know the self-financing condition is:

$$dV(t)=\Delta(t)dS(t)+\beta(t)dB(t)$$

During my studies I have checked the self-financing condition for the Black-Scholes-Merton formula for example, which did take a lot of calculations. What I feel the difference here might be is that in the BSM-formula the portfolio weights are given explicitly, whereas here they are given implicitly.

How do I check for the validity (self-financing) of this strategy? Intuitively I of course understand that I should be able to rebalance my portfolio in a way that the weights of the assets are constant, but I want to be able to show this in a rigorous way.

The closest I have come to an answer to this question is in P. Wilmott's Quantitative Finance (2nd Edition), Chapter 66: asset allocation in continuous time.

In the book Wilmott shows that the value process of such a portfolio is going to become: $$ dV(t) = \left(x(\mu-r)+r\right)V(t)dt+x\sigma V(t)dW(t) $$ which I of course understand intuitively, but he hasn't convinced me that this strategy is valid in the sense that it's self-financing.

I did apply Ito's lemma to the portfolio, and I arrived at: $$ dV(t)=\Delta(t)dS(t)+\beta(t)dB(t)+S(t)d\Delta(t)+B(t)d\beta(t)+d\langle\Delta,S\rangle(t)+d\langle\beta,B\rangle(t) $$ for which the sum of the last four terms should equal zero. I know that the last term $d\langle\beta,B\rangle(t)$ does equal zero because $B(t)$ is of bounded variation. But I'm having difficulty moving forward, since the strategy itself is defined implicitly, so I feel like I'm moving in circles.

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  • $\begingroup$ Remember that all these differentials are shorthand notations underpinning stochastic integration à la Ito. This means that for a differential $a(t) dX_t$, $dX_t = X_{t+dt}-X_t$ is a forward looking increment and $a(t)$ is adapted (+ regularity conditions), with $X_t$ asset $t$-value and $a(t)$ asset weight at $t$. Obviously the allocation strategy needs to be predictable. A common practical choice is to pick $a(t)$ constant in $[t,t+dt[$. This tells you that for constant weights between each rebalancing, the only thing you care about is if investing in the assets is a self-financing strategy. $\endgroup$ – Quantuple Jul 6 '17 at 7:31

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