Black-Scholes Equation - Riskless portfolio derivation

Question

The following is a summary of the derivation of the Black-Scholes equation as given on wikipedia (http://en.wikipedia.org/wiki/Black-Scholes_equation#Derivation) - I have a question regarding the assumption that the specified portfolio is self-financing.

We have a two asset market:

$dB_t = B_t r dt $

$dS_t = S_t (\mu dt + \sigma dW_t)$

We introduce a European option with price $v(t,S_t)$ at time $t$. We now consider a portfolio consisting of one option and -$\frac{\partial v}{ \partial S}$ stocks. Therefore if $X_t$ is our wealth at time $t$, we must have $X_t = v(t,S_t) - \frac{\partial v}{ \partial S} S_t$.

It is then claimed that we have $dX_t = dv(t,S_t) - \frac{\partial v}{ \partial S} dS_t$, as the portfolio is self-financing.

However, it seems to me that if we have a constant holding of 1 option in our portfolio, then the only way to make the overall portfolio self-financing is to have a constant holding of stock too (otherwise, if we increase/decrease our holding in stock, where do the extra funds for this come from?).

Typically, the way I have seen self-financing portfolios constructed is that the holding in 1 asset (for example, the risk-free asset) is not explicitly specified, and is determined by the self-financing condition (i.e. the condition that $X_t = \pi_t \cdot P_t$ and $dX_t = \pi_t \cdot dP_t$, where $\pi$ is the portfolio and $P_t$ is the price process - this gives a linear equation for the unspecified holding).

Based on the above, it seems that in order to have a self-financing portfolio where we hold a constant 1 option and $- \frac{\partial v}{ \partial S}$ shares, we must also have a dynamic holding in the risk-free asset which allows us to ensure that we can always have $- \frac{\partial v}{ \partial S}$ shares in our portfolio without injecting external funds (and thus breaking the self-financing condition). However, if we do have this holding of the risk-free asset in our portfolio as well, then our equation for the wealth process ($X_t = v(t,S_t) - \frac{\partial v}{ \partial S} S_t$) becomes incorrect, as we are not taking into account our holding in the risk-free asset.

In summary, I don't believe that the portfolio of 1 option and $-\frac{\partial v}{ \partial S}$ shares specified in the wikipedia derivation of the Black-Scholes equation is self-financing, but the derivation makes use of the fact that it [i]is[/i] self-financing. Am I missing something?

EDIT:

If the portfolio consisting of 1 option and $-\frac{\partial V}{\partial S}$ shares is self-financing, then we have the following:

$X_t = V(t,S_t) - \frac{\partial V}{\partial S}S_t$ (definition of wealth process)

$dX_t = dV(t,S_t) - \frac{\partial V}{\partial S}dS_t$ (as portfolio is assumed to be self-financing)

$dX_t = dV(t,S_t) - d(\frac{\partial V}{\partial S}S_t)$ (simply by definition of differentials)

Equating the RHS of the second and third equations above gives:

$dV(t,S_t) - \frac{\partial V}{\partial S}dS_t = dV(t,S_t) - d(\frac{\partial V}{\partial S}S_t)$

So $\frac{\partial V}{\partial S}dS_t = d(\frac{\partial V}{\partial S}S_t)$.

Using Ito's lemma on the RHS gives: $\frac{\partial V}{\partial S}dS_t = d(\frac{\partial V}{\partial S})S_t + \frac{\partial V}{\partial S}dS_t + d<\frac{\partial V}{\partial S},S>_t$.

And so $d(\frac{\partial V}{\partial S})S_t + d<\frac{\partial V}{\partial S},S>_t = 0$. (*)

Now, $d(\frac{\partial V}{\partial S}) = \frac{\partial^2 V}{\partial S \partial t} dt + \frac{\partial^2 V}{\partial S^2} dS_t + \frac{1}{2}\frac{\partial^3 V}{\partial S^3}d<S>_t$.

Therefore $d<\frac{\partial V}{\partial S},S>_t = \frac{\partial^2 V}{\partial S^2}S_t^2 \sigma^2 dt$.

Plugging these into (*) gives:

$\frac{\partial^2 V}{\partial S \partial t} dt + \frac{\partial^2 V}{\partial S^2} dS_t + \frac{1}{2}\frac{\partial^3 V}{\partial S^3}d<S>_t + \frac{\partial^2 V}{\partial S^2}S_t^2 \sigma^2 dt = 0$.

Therefore $\frac{\partial^2 V}{\partial S \partial t} dt + \frac{\partial^2 V}{\partial S^2} dS_t + \frac{1}{2}\frac{\partial^3 V}{\partial S^3}\sigma^2 S_t^2 dt + \frac{\partial^2 V}{\partial S^2}S_t^2 \sigma^2 dt = 0$.

The cofficient of $dS_t$ must be zero, so $\frac{\partial^2 V}{\partial S^2} = 0$, so $V(t, S) = f(t) + Sg(t)$. We could stop at this point, because we know that we can't satisfy the boundary condition $v(T,S) = \max(0,S-K)$, and therefore our assumption that we could hedge an option with a self-financing portfolio consisting of 1 option and $-\frac{\partial V}{\partial S}$ shares is wrong.

However, note that the coefficient of $dt$ must also be zero, and since we already have $\frac{\partial^2 V}{\partial S^2} = 0$, this gives $\frac{\partial^2 V}{\partial S \partial t} = 0$. Since $V(t, S) = f(t) + Sg(t)$, this implies that $g$ is constant. Therefore $\frac{\partial V}{\partial S}$ is constant, as I claimed in the comments below - i.e. in order for this to be a self-financing portfolio at all, the holding in the stock must be constant.

Answer 1

You are correct that showing the self-financing condition for the BS-portfolio is not as straightforward as one may think:

A portfolio $V_t(\alpha_t,\beta_t)$ (for stock $S_t$ and zerobond $B_t$) is self-financing iff:

$$V_t=\alpha_tS_t+\beta_t B_t$$

It further implies

$$dV_t=\alpha_tdS_t+\beta_tdB_t$$

To replicate a derivative $C(S_t,t)$ by a self-financing portfolio of stock and bond, set: $$dV_t=dC_t$$

The dynamics of $dC$ can be specified using Ito's Lemma on $C(S_t,t)$:

$$dC=\partial_tCdt+\partial_sCdS+\frac{1}{2}\sigma^2S_t^2\partial_{SS}Cdt=\partial_SCdS_t+(\partial_tC+\frac{1}{2}\sigma^2S_t^2\partial_{SS}C)dt$$

Next assume $C$ satisfies the BS-PDE:

$$\partial_tC+\frac{1}{2}\sigma^2S_t^2\partial_{SS}C=rC-rS_t\partial_S C$$

Inserting this into $dC$:

$$dC=\partial_SCdS_t+(C-S_t\partial_SC)rdt$$

Now we further have the bond-dynamics $dB_t=B_trdt$, so:

$$dC=\partial_SC\cdot dS_t+\left(\frac{C_t}{B_t}-\frac{S_t}{B_t}\partial_SC\right)\cdot dB_t$$

Finally, the coefficients before $dS_t$ and $dB_t$ are exactly the self-financing portfolio weights:

$$\left(\alpha_t=\partial_SC,\,\beta_t=\dfrac{C_t}{B_t}-\dfrac{S_t}{B_t}\partial_SC\right)$$

Answer 2

To show: $X:=(1,-\partial_SC)$ is a self-financing portfolio:

$$X\text{ self-financing}\leftrightarrow dX_t=adC_t+bdS_t\,\forall t\geq0$$

Let $C(S_t,t)\in C^2$, then by Ito formula:

$$dC=\partial_tCdt+\partial_sCdS+\frac{1}{2}\sigma^2S_t^2\partial_{SS}Cdt=\partial_SCdS_t+(\partial_tC+\frac{1}{2}\sigma^2S_t^2\partial_{SS}C)dt$$

Let $C(S_t,t)$ satisfy the BS-PDE in discounted form: $$\partial_tC+\frac{1}{2}\partial_{SS}C\sigma^2S^2=0$$

(Undiscounted form is $\partial_tC+\frac{1}{2}\sigma^2S^2\partial_{SS}C=rC-rS\partial_S C$).

Plugged in: $$dC=\partial_SCdS$$

So we get:

$$dX=adC+bdS=1dC-\partial_SCdS=\partial_SCdS-\partial_SCdS=0\,\,\forall t$$

So we have a riskless portfolio which satisfies the self-financing condition. (q.e.d.)