Self-Financing Portfolio

Why when we are using self-financing portfolios to replicate some external payoff we do not consider the quadratic variation of the portfolio weights? Say, in Black-Scholes world, when we are using $W_t = h_1(t)S_t + h_2(t)B_t$ where $S_t$ is the underlying stock which follows GBM and $B_t$ is the value of simple risk-free asset, we write the law of motion as follows?

$$dW = h_1(t)dS_t + h_2(t)dB_t$$

Why we have no $dh$ or $(dS)(dh)$ term here?

• Have you had a look at the definition of a self-financing portfolio? – Quantuple Mar 20 '17 at 8:06

As Quantuple implies in the comment, you do need to take all these terms into account. However, by the definition of a self-financing portfolio, they vanish. Using your notation, we generally have

$$\mathrm{d}W_t = h_1(t) \mathrm{d}S_t + S_t \mathrm{d}h_1(t) + \mathrm{d} \langle h_1, S \rangle_t + h_2(t) \mathrm{d}B_t + B_t \mathrm{d}h_2(t) + \mathrm{d} \langle h_2, B \rangle_t.$$

We require that

$$\mathrm{d}W_t = h_1(t) \mathrm{d}S_t + h_2 \mathrm{d}B_t.$$

The self-financing condition thus is

$$S_t \mathrm{d}h_1(t) + \mathrm{d} \langle h_1, S \rangle_t + B_t \mathrm{d}h_2(t) + \mathrm{d} \langle h_2, B \rangle_t = 0.$$

Note that in a Black-Scholes world, you'd always have $\mathrm{d} \langle h_2, B \rangle_t = 0$ anyways as $B$ is non-random.

Go gain some intuition for what this means economically, it is useful to re-write the self-financing condition as

$$\left( S_t + \mathrm{d}S_t \right) \mathrm{d}h_1(t) + \left( B_t + \mathrm{d}B_t \right) \mathrm{d}h_2(t) = 0.$$

I.e. we require that the value of the changes in the positions in $S$ and $B$ (at the new prices) cancel out.