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Checking Self-Financing with Itô’s Formula

Article Quant Q&A · Author: piman314
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Summary

The document works through a two-asset portfolio consisting of a stock driven by Brownian motion and a constant savings account. Its purpose is to check whether time-varying holdings satisfy the self-financing condition: the change in portfolio value must equal the gains from holding the assets, with no external cash added or withdrawn.

The key step is to correctly substitute both holdings into the portfolio value. The resulting value is the square of Brownian motion minus time; applying Itô’s formula introduces a quadratic-variation term that cancels the explicit time change. The remaining differential equals the stock holding times the stock’s differential, while the savings account contributes nothing because its value is constant. This verifies self-financing for the stated setup. The illustration is a mathematical example rather than a trading strategy, and its conclusion depends on the specified asset processes and holdings.

Key ideas

  • A self-financing portfolio’s value change must equal gains from its asset holdings.
  • The portfolio value must include every term from both the stock and savings-account positions.
  • Itô’s formula adds a quadratic-variation term when differentiating the squared Brownian motion.
  • In this example, that term cancels the explicit time component, leaving the holdings-based gain.

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StatisticsRisk management
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# Stochastic Differentials - Ito's formula for a self-financing portfolio


# Stochastic Differentials - Ito's formula for a self-financing portfolio












Suppose I have a portfolio of stocks $(S)$ and savings account ($\beta_t$) then, the value is

$$V = a_t S_t + b_t \beta_t$$

and for this portfolio to be self replicating, we need by Ito's lemma $$dV = a_t dS_t + b d \beta_t$$

Now let $$a_t = 2B_t, b_t = -t - B_t^2 - 20B_t, S_t = 10 + B_t, \beta_t = 1$$ With $$B_t = \text{Brownian Motion at time t}$$

How can I show if this portfolio is self-financing?

I can write

$$V = a_t S_t + b_t \beta_t = 2B_t(10+B_t) - (t + B_t^2)$$ $$= 20B_t + 2B_t^2 - t - B_t^2 = 20B_t + B_t^2$$

Since $$S_t = 10 + B_t \to dS_t = dB_t ?$$ And $$\beta_t = 1 \to d \beta_t = 0 ?$$

Now I am having difficulty in evaluating $dV$ in these terms. Can someone help?

$$dV = \{....?\}$$

## Answer by AFK (score 2, accepted)

https://quant.stackexchange.com/a/17673

The portfolio is self-financing. You simply forgot a term in $b$ and a $-t$ term in $V$: \begin{eqnarray} V_t &=& a_t S_t + b_t \beta_t = (2B_t ) (10+ B_t) + (- t - B_t^2 - 20B_t)1 \\ &=& 20B_t + 2B_t^2 - t - B_t^2 - 20B_t \\ &=& B_t^2 - t \end{eqnarray} Applying Ito's lemma \begin{eqnarray} dV_t &=& (2B_t dB_t + \frac{1}{2}2d\langle B,B\rangle_t) - dt \\ &=& 2B_t dB_t \\ &=& a_t dS_t + b_t d\beta_t \end{eqnarray} Since $dS_t = dB_t$ and $d\beta_t = 0$, we have \begin{eqnarray} dV_t &=& a_t dS_t + b_t d\beta_t \end{eqnarray} which is a characterization of a self-financing portfolio.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.