Why Delta Hedging Holds the Hedge Position Fixed During an Instant
Summary
This note addresses an apparent contradiction in Black–Scholes derivations: the delta-hedged portfolio is differentiated while its hedge quantity is treated as fixed, even though option delta changes with time and the underlying price. The accepted explanation distinguishes the number of underlying contracts held in the portfolio from the option’s Greek delta. For the infinitesimal portfolio change, the hedge position is set at its current value and treated as constant; the Greek determines that initial hedge quantity.
The note adds that delta changes as conditions evolve, so a hedger must rebalance over time. Rebalancing is separate from the instantaneous change calculation. A second answer disputes the accepted explanation and points to a published pricing-theory FAQ, so the document presents a contested issue rather than a complete derivation or consensus resolution. Readers should consult the cited treatment for the mathematical assumptions behind the portfolio argument.
Key ideas
- The symbol delta can refer either to a hedge quantity or to an option sensitivity.
- The infinitesimal portfolio change treats the already chosen hedge quantity as fixed.
- A delta hedge must be adjusted as the option’s Greek changes over time.
- The document records a disagreement about the derivation’s assumptions.
Tags
Full text
# Black-Scholes derivation assumption contradiction
# Black-Scholes derivation assumption contradiction
In many books and derivations of the Black-Scholes PDE one sees that
$$\Pi=V-\Delta F \Rightarrow d\Pi=dV-\Delta dF$$
which implicitly assumes that $d\Delta=0$. Somewhere down the road one then deduces that
$$\Delta=\frac{\partial V}{\partial F}$$
to simplify the equation. Doesn't this contradict the initial assumption that $d\Delta=0$? If one performs a full differentiation
$$\Pi=V-\Delta F \Rightarrow d\Pi=dV-\Delta dF - F d\Delta$$
the rest of the story goes wrong. Isn't it true that $\Delta = \Delta(t, S)$, i.e. is depending on time and the underlying stochastic process and hence has to be differentiated?
## Answer by pincopallino (score 2, accepted)
https://quant.stackexchange.com/a/10866
$\Pi$ is the value of a delta-hedged portfolio (option plus a short position in Δ underlying). The notation for $\Delta$ is overloaded. Here it represents the number of underlying contracts (f.ex shares) in your delta hedged portfolio, equal to the greek $\Delta$ when the portfolio is created. Therefore in the calculation of $d \Pi$, $\Delta$ (the number of shares in your portfolio) is treated as a constant.
Yes, the greek $\Delta$ evolves as the option approaches maturity and wrt $F$ and you will have to rebalance your portfolio. But this is not contemplated in the infinitesimal $d \Pi$.
## Answer by PedroCazorla (score 1)
https://quant.stackexchange.com/a/71224
The contradiction is true. See Question V in Peter Carr's FAQ's in Option Pricing Theory (1999).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.