$$\begin{align} d\pi &= \theta dV + dS \\[3pt] & = (\theta \partial V/\partial t + \theta \mu S \partial V/\partial S + \theta S^2 \sigma^2 \partial^2 V/2\partial S^2 +\mu S ) dt + (\theta \sigma S\partial V/\partial t + \sigma S)dw \end{align} $$

In order for the portfolio to be riskless, they set $\theta = -(\partial V/\partial S) ^{-1}$. So essentially they are selling $1 / \Delta$ shares of the option and buying one stock. Why does this make the portfolio riskless?

  • 2
    $\begingroup$ Please see here and here. $\endgroup$ – Gordon Mar 28 '18 at 21:23

Your equation should read:

$$\begin{align} d\pi & = \theta\frac{\partial V}{\partial t}dt + \theta\frac{\partial V}{\partial S}dS + \frac{1}{2}\theta\frac{\partial^2 V}{\partial S^2}(dS)^2 +dS \\ & = \left(\theta\frac{\partial V}{\partial t} + \theta\frac{\partial V}{\partial S}\mu S + \frac{1}{2}\theta\frac{\partial^2 V}{\partial S^2}\sigma^2S^2+\mu S\right)dt +\left(\theta\color{red}{\frac{\partial V}{\partial S}}\sigma S + \sigma S\right)dw \end{align}$$

The only stochastic, i.e. risky term, in the equation above is:

$$ \left(\theta\frac{\partial V}{\partial S}\sigma S + \sigma S\right)dw $$

where $w$ is a Brownian Motion. Thus by setting:

$$\theta=-\frac{1}{\frac{\partial V}{\partial S}}$$

you cancel all stochastic terms and eliminate risk, therefore the portfolio must yield the risk free rate.

As an aside, note that the portfolio as defined here, $\pi = \theta V + S$ with $\theta=-(\partial V/\partial S)^{-1}$, is not strictly speaking self-financing $-$ check the comment by Gordon for more details.


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