Using Itô’s Lemma to Explain Delta-Neutral Risk
Summary
The note explains why a portfolio formed by shorting a derivative and holding a matching number of underlying shares has no instantaneous exposure to the underlying’s random movement. Applying Itô’s Lemma to the derivative value separates its change into a time component, a term proportional to the underlying’s change, and a second-order term involving quadratic variation. The share position cancels the term proportional to the underlying’s change, leaving drift terms that are known over the instant under the stated assumptions.
The explanation relies on the diffusion setting and treats the underlying’s quadratic variation as deterministic and of order dt. That assumption is appropriate for common diffusion models with specified volatility, but the note does not discuss stochastic volatility, jumps, transaction costs, or changes in the hedge ratio. Delta neutrality therefore describes local risk cancellation, not a guarantee that the portfolio remains risk-free over time.
Key ideas
- Itô’s Lemma decomposes derivative price changes into time, underlying movement, and quadratic variation terms.
- A matching underlying position cancels the derivative’s first-order exposure to the underlying.
- The remaining instantaneous change is deterministic when quadratic variation is known and proportional to dt.
- The hedge is local and depends on the diffusion assumptions.
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Full text
# How can Ito's Lemma be used to show that a delta-neutral portfolio is instantaneously risk-free?
# How can Ito's Lemma be used to show that a delta-neutral portfolio is instantaneously risk-free?
The lecture notes I am currently reading give the following example of a delta-neutral portfolio:
- minus one derivative (whose value at time $t$, when the value of the underlying is $S_t$, is denoted $f(t, S_t)$)
- $\Delta := \frac{\partial f}{\partial S_t}$ shares of the asset underlying the derivative
Following this example is a question which asks me to show that a delta-hedged portfolio with value $V(t, S_t)$ is instantaneously risk-free, if $S_t$ is a diffusion, by using Ito's Lemma. The first line of the solution of this questions states that:
> Ito's Lemma tells us that: $$dV(t, S_t) = \frac{\partial V}{\partial t} dt + \frac{\partial V}{\partial S_t} dS_t + \frac{1}{2}\frac{\partial^2 V}{\partial S_t^2} (dS_t)^2$$
Could anyone help me to understand how the above expression has been deduced?
## Answer by Arshdeep (score 1)
https://quant.stackexchange.com/a/57357
Once you subtract the delta term from your portfolio (thus canceling the middle term on the RHS), the two terms remaining have no uncertainty - they are deterministic. This is because square of $dS$ is the quadratic variation of the process S, presumably a deterministic (as in, known at the time of placing the hedge) quantity, and of the order $dt$.
Thus, the portfolio that remains has a deterministic drift with no uncertainty (No dependence on the brownian motion), so it is instantaneously risk free.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.