# Black-Scholes Equation - Riskless portfolio derivation

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

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.

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I believe the question in the link actually refers to the case where the claim is replicated through a portfolio of bond and stock - my question is regarding the hedging portfolio of claim and stock. –  Mark Aug 14 '14 at 10:12
it's the same... –  athos Aug 14 '14 at 10:14
@Mark I added the Ito-proof on the self-fin. condition –  emcor Aug 14 '14 at 20:09
@emcor thanks for this. –  Mark Aug 14 '14 at 20:24

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:

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

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Thank you for your answer. I understand how a European call is replicated using a portfolio of stock and risk-free bond, but the wikipedia article I linked to has an argument involving a portfolio consisting of holdings in the claim and the stock. In your example, we specify that $\alpha_t = \frac{\partial V}{\partial S}$, and then $\beta_t$, the holding in the bond, is determined by the self-financing condition. However, in the wikipedia example, the holdings in the option and stock are fully specified, and I don't believe this leads to a self-financing portfolio. –  Mark Aug 13 '14 at 23:29
@Mark I see your point, the article uses Ansatz-technique which is not as straightforward. You can see that the portfolio $\pi$ is self-financing by calculating out the partial derivative of the black-scholes price, then you get: $dC=\partial_S CdS$ (where $C$ is the Black-Scholes-Formula), so $\Pi=0\forall t$, which is then obviously self-financing. –  emcor Aug 13 '14 at 23:49
Thanks again for your response. I don't see that the portfolio value being uniform shows that the portfolio is self-financing - consider a holding in the risk-free bond of $B_0^{-1}e^(-rt)$ at time $t$ - this portfolio always has value 1, but is not self-financing. For the portfolio suggested by wikipedia, the wealth process is $X_t = V(t, S_t) - \frac{\partial V}{\partial S} S_t$. The self-financing condition is $dX_t = dV(t, S_t) - \frac{\partial V}{\partial S} dS_t$... –  Mark Aug 14 '14 at 0:18
, and Ito's formula gives $dX_t = dV(t,S_t) - d(\frac{\partial V}{\partial S}) S_t - \frac{\partial V}{\partial S} dS_t - d<\frac{\partial V}{\partial S}, S_t>$. Equating these two gives $d(\frac{\partial V}{\partial S}) S_t = - d<\frac{\partial V}{\partial S}, S_t>$, which leads to a contradiction. –  Mark Aug 14 '14 at 0:22
@Mark $dX_t=dC-\partial_SCdS_t$ is just the definition of any self-financing portfolio with the given weights. You can define it so as an Ansatz, and then check if you get a solution. It must not be shown, because it is just a definition/condition which is imposed. It does not mean that there will be a solution under such conditions, but for European BS-Model there is. So the price $C(t,S_t)$ you get satisfies a self-fin. portfolio in itself as a solution under this assumption. –  emcor Aug 14 '14 at 0:37

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.)

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thanks again for your patience with my questions. If I have understood the above correctly, we're actually making two assertions about C: firstly, that the portfolio (1,-dC/dS) is self-financing [this is used to get the first equality in your final line of math], and we also assume that C satisfies a certain PDE, which implies that dC = dC/dS dS. However, it is not checked whether these two conditions on C are compatible, and I claim they are not (cont'd).. –  Mark Aug 14 '14 at 20:18
I have edited my OP to demonstrate what constraints on C are implied by forcing (1,-dC/dS) to be a self-financing portfolio, and it turns out that in order for this portfolio to be self-financing, we have the following SDE: $\frac{\partial^2 C}{\partial S \partial t} dt + \frac{\partial^2 C}{\partial S^2} dS_t + \frac{1}{2}\frac{\partial^3 C}{\partial S^3}\sigma^2 S_t^2 dt + \frac{\partial^2 C}{\partial S^2}S_t^2 \sigma^2 dt = 0$ And this implies that C must be of the form $C(t,S) = f(t) + kS$. Please let me know if you think there are any mistakes in my derivation of this in the OP. –  Mark Aug 14 '14 at 20:19
@Mark it looks like it makes sense but I'm too tired; obviously it can't be true because Black-Scholes model can also price e.g. inverted or squared options, I recommend to ask this as a separate new question on quant.se or math.se. Best, –  emcor Aug 14 '14 at 22:53