Hedging Frequency, Transaction Costs, and Discrete Delta Replication
Summary
The document examines how often to rebalance a delta hedge for a European call under Black–Scholes assumptions and geometric Brownian motion. The questioner estimates the local hedging error from the option’s gamma and the squared price move, while proportional trading costs depend on the absolute change in delta and the underlying price. Summing the local error over a fixed horizon appears to make it independent of the hedge interval, prompting the question of how to choose a frequency.
The response derives that the number of shares traded per interval is approximately gamma times the underlying’s random price change. This makes expected proportional costs per hedge scale with the square root of the interval length; over a fixed horizon, total costs therefore rise as hedging becomes more frequent and diverge in the continuous-rebalancing limit. It then presents expressions for return, volatility, and a Sharpe ratio to optimize over frequency. These formulas are offered without a derivation or numerical validation, and their assumptions and notation require careful checking before practical use.
Key ideas
- For a small price move, the change in hedge shares is approximated by gamma times that move.
- With proportional transaction costs, expected cost per rebalance scales with the square root of the hedge interval.
- Across a fixed horizon, the stated cost scaling implies higher total costs as rebalancing becomes more frequent.
- The response proposes optimizing a Sharpe ratio over hedge frequency, using volatility and cost assumptions.
- The formulas are not derived or validated in the document, so their applicability should be checked before use.
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Full text
# Optimal Hedging Frequency using Black-Scholes and GBM (simplyfied)
# Optimal Hedging Frequency using Black-Scholes and GBM (simplyfied)
I am trying to derive an optimal hedging interval for a delta-replicated European call option from the Black-Scholes model. To do that, I would like to compare the resulting hedging error with my trading costs.
I am currently hedging (i.e. adjusting the delta of) the position every day and am trying to find out whether more frequent intraday hedging makes sense.
Unfortunately, I am making a mistake in my assumptions somewhere which I am hoping you can point out.
What I tried so far: Lets assume volatility and time to maturity stay the same and risk-free-rate = dividends = 0. I'll also use $\Delta \delta \approx \gamma \cdot \Delta S$ to describe the change in Delta for a small change in price.
I can roughly calculate the hedging error for a small price change using a Taylor expansion of the option price. $C(S) \approx C(S_0) + \delta(S_0) \cdot (S - S_0) + \frac{\gamma(S_0)}{2} \cdot (S - S_0)^2+... $ Since I hold $\delta(S_0)$ in my portfolio, the error for a small change in price is roughly $\frac{1}{2} \gamma (\Delta S)^²$
Additionally, I can calculate the trading costs as: $c \cdot \left| (\delta_t - \delta_0) \cdot S_{t} \right| \approx c \cdot \left| \gamma \cdot \Delta S \cdot (S_0 + \Delta S) \right|$ c is a constant representing my trading costs per trading volume.
Now, using geometric brownian motion $dS_t = \mu S_t \, dt + \sigma S_t \, dW_t$, I find the expected value for my hedging error and trading costs.
$\mathbb{E}[\frac{1}{2} \gamma (\Delta S)^²] \approx \frac{1}{2} \gamma S^2 \sigma^2 \Delta t$
$\mathbb{E}[c \left| \gamma \Delta S (S_0 + \Delta S) \right|] \approx c \gamma S^2 \sigma \sqrt{\Delta t} \sqrt{\frac{2}{\pi}}$
This seems to be correct in my simulations. However, when I try to calculate the total costs over a greater timespan T (one day for example), the hedging error seems to be the same no matter the hedging frequency. Let $n = \frac{T}{\Delta t}$ the number of intervals/hedges, then $\sum_{i=1}^{n} \frac{1}{2} \gamma S^2 \sigma^2 \Delta t = n \cdot \frac{1}{2} \gamma S^2 \sigma^2 \Delta t = \frac{1}{2} \gamma S^2 \sigma^2 T$
Adding this error to my costs (which decrease by increasing $\Delta t$) doesn't make much sense. Are my calculations wrong or my assumptions too simple? Do I have to calculate it completely differently? What am I doing wrong?
Please only use Black-Scholes and GBM in your answer and please also try to simplify it as much as possible. Thank you very much!
## Answer by Newquant (score 2)
https://quant.stackexchange.com/a/80986
Assuming proportional transaction costs (to the size of the trade):
The number of shares, N, to be hedged at each interval is given by $$ N = \Delta(S_t + dS, t + dt) - \Delta(S_t, t) \approx \Gamma_t * dS$$ which expands to: $$ N \approx \Gamma * \sigma * S * dW $$
The incremental cost is given by the number shares, N, multiplied by the share price, multiplied by the transaction cost fraction, $\alpha$: $$ Cost = |N*S| * \alpha $$ $$ Cost = |\Gamma * \sigma * S^2 * dW| * \alpha $$
$|dW| > 0$, so costs are, obviously, non-zero. Rewriting $dW$ as $z\sqrt{dt}$, we can see that for an option with a $\frac{T}{dt}$ number of hedges over the life, T, total costs will scale with $\frac{T}{dt} * \sqrt{dt} = \frac{T}{\sqrt{dt}}$. So as the hedging interval moves to 0, costs move to infinity.
The expected return of a hedged short option position is given by: $$\frac{\Gamma S^2}{2} * (\sigma^2_i - \sigma^2_r) * dt$$ and the volatility is given by: $$ \frac{\Gamma S^2 \sigma^2_r}{2} * \sqrt{\frac{\kappa -1}{N}} $$ where $\kappa$ is the raw kurtosis, under GBM $\kappa = 2$.
The expected return drag from increasing the hedging frequency is approximately: $$\Gamma * S^2 * \sigma_r \sqrt{\frac{2}{\pi * dt}}$$
Creating a Sharpe ratio gives: $$ \sqrt{\frac{N}{\kappa -1}} * (((\frac{\sigma^i}{\sigma_r})^2 -1) - \frac{\alpha\sqrt{\frac{8}{\pi}}}{\sigma_r * dt^\frac{3}{2}} )$$
You can then differentiate and maximise the sharpe ratio w.r.t hedging frequency.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.