Financing and Overnight Profit in a Delta-Hedged Option Position
Summary
The document explains how interest enters the overnight profit of a delta-hedged option position. It represents the portfolio as a short option combined with a position in the underlying sized by the option’s delta. The net portfolio value determines whether the option premium leaves cash to invest or whether borrowing is needed to fund the hedge. If the position is delta-neutral and riskless under the assumed setup, its overnight financing return is based on the value of the entire portfolio, rather than treating the option premium’s interest in isolation.
The answer distinguishes this idealized hedged case from an overnight interval in which a large underlying move makes the hedge ineffective. In that case, the portfolio is no longer riskless, and the price-change terms in the textbook expression matter. The explanation is conceptual and assumes a financing account at the risk-free rate; it does not model transaction costs, changing delta, or other real-world frictions.
Key ideas
- A delta-hedged position combines a short option with an underlying position sized by delta.
- The net portfolio value determines the amount invested or borrowed overnight.
- In the idealized riskless case, the whole portfolio earns the risk-free financing return.
- Large overnight price moves can make the hedge ineffective and leave residual risk.
Tags
Full text
# Answer by emot (score 1, accepted)
# Why doesn't the overnight profit on a delta-hedged porfolio include interest on the initial selling/buying of the option?
I am self-studying and encountered the following passage from my textbook on the market maker's overnight profit on a delta-hedged portfolio:
I don't understand why their isn't a factor of $(e^{r/365} - 1)$ multiplied by the $C(S_0)$ term. My reasoning is that if a call/put is sold, the positive cash flow from the premium could be invested at the risk-free rate.
My understanding is that the overnight profit on a sold call would be the premium that the call sold for at day 0, which could be invested at the risk free rate overnight, minus the premium that the call could be sold for if held on to for another day, or $C(S_0)\left(e^{r/365} - 1\right) - C(S_1)$
Wouldn't it be more accurate to say that: $$\text{Profit} = -\left(C(S_1) - C(S_0)(e^{r/365} - 1)\right) +\Delta(S_1 - S_0) - (e^{r/365} - 1)\left(\Delta S_0 - C(S_0)\right)?$$
If we're factoring interest lost on self-financing the selling of an option and buying $\Delta$ shares of stock, why wouldn't we also factor interest gained by selling the option at $T = 0$?
## Answer by emot (score 1, accepted)
https://quant.stackexchange.com/a/31083
If you keep a delta-hedged portfolio $\pi$, your position looks like this:
$\pi=-C+\Delta S$
and it is worth $\pi$. If you get more money for selling call option that is needed for buying underlying to hedge, you invest it in the risk free money market account. If the money proceeds from selling the call option are not enough to buy the underlying, you borrow from the money market account at the risk free rate.
If it is hedged it has no risk, so it should earn the risk free rate, so overnight you earn what portfolio was worth the day before times risk free rate: $\pi (e^{r/365}-1)=(-C+\Delta S)(e^{r/365}-1)$
Equation from the reading (with extra change in the price of option and stock) assumes that your portfolio was not delta-neutral and the hedging was not effective - this happens when the moves in the underlying are large overnight. So effectively the equation from the reading is OK.
This is very well explained in Steven Shreve: Stochastic Calculus for Finance I, in the first chapters.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.