Delta Hedging a Stock Option in the Black–Scholes Model
Summary
The document explains how to remove the random component from a portfolio made of an option and its underlying stock. Applying Itô’s lemma to the option value shows that the portfolio’s Brownian-motion exposure is proportional to the option’s delta and the stock exposure. Choosing the option position to cancel that combined exposure eliminates the stochastic term in the model, leaving a locally riskless portfolio that must earn the risk-free rate under the model’s assumptions.
The key correction is that the option’s differential includes its delta multiplying the stock-price change; omitting that factor leads to the wrong risk term. The note also cautions that the portfolio as defined may not be self-financing. Thus, the derivation illustrates delta hedging, but it does not by itself establish that a real trading position is continuously riskless: it relies on the Black–Scholes framework and leaves the self-financing details to further analysis.
Key ideas
- Itô’s lemma gives the option’s instantaneous exposure to movements in the underlying stock.
- The Brownian risk term combines the option delta exposure with the direct stock exposure.
- Setting the option position to offset the stock exposure removes the stochastic component locally.
- Under the model assumptions, a riskless portfolio earns the risk-free rate.
- The portfolio definition may not be self-financing, which limits the derivation.
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Full text
# Creating riskless portfolio in black scholes
# Creating riskless portfolio in black scholes
$$\begin{align} d\pi &= \theta dV + dS \\[3pt] & = (\theta \partial V/\partial t + \theta \mu S \partial V/\partial S + \theta S^2 \sigma^2 \partial^2 V/2\partial S^2 +\mu S ) dt + (\theta \sigma S\partial V/\partial t + \sigma S)dw \end{align} $$
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?
## Answer by Daneel Olivaw (score 2)
https://quant.stackexchange.com/a/39049
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.