Continuous Delta Hedging P&L from Implied and Realized Volatility
Summary
The document derives the instantaneous profit and loss of a delta-hedged option when the hedge and option valuation use implied volatility but the underlying realizes a different volatility. Applying the Black–Scholes pricing equation reduces the P&L rate to one-half the realized-minus-implied variance, multiplied by the underlying price squared and the option’s implied-volatility gamma. This gives a direct way to interpret the hedge: positive gamma earns when realized variance exceeds implied variance, while the reverse difference creates losses, all else equal.
The derivation concerns continuous rebalancing and a risk-free financing adjustment. Although each instantaneous increment follows the stated relation, cumulative P&L depends on the underlying’s path because gamma changes with price and time; paths near the strike tend to expose the position to higher gamma. The note does not provide a numerical lifetime profit formula for the example, and its result abstracts from discrete hedging, transaction costs, jumps, and model misspecification.
Key ideas
- Under continuous delta hedging, instantaneous P&L depends on the gap between realized and implied variance.
- The variance gap is scaled by the underlying price squared and option gamma.
- Gamma in the derivation is calculated using the implied volatility used to price the option.
- Cumulative hedging P&L is path-dependent because gamma varies with the underlying price and time.
- Price paths near the strike can generate larger absolute P&L through higher gamma exposure.
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Full text
# Continuous delta hedge formula
# Continuous delta hedge formula
When we buy a call and continuously delta hedge using some implied volatility $\sigma_i$, what is the formula for our aggregate profit given that the actual realized volatility is $\sigma_r$?
Say $S_0 = 1000, \sigma_i = 0.25, \mu = 0.10$, and the call has expiry a year from now.
How does the formula look like in terms of $\sigma_r$? What happens if $\sigma_r =0$? $> \sigma_i$? $< \sigma_i$?
## Answer by LocalVolatility (score 8)
https://quant.stackexchange.com/a/33452
This is a slightly extended version of my comment that summarizes the main result of the reference that I provided.
This problem is discussed in detail in Chapter 12 of Wilmott (2006), which is based on the paper Ahmad and Wilmott (2005). See also the related Question 9 in Carr (2005).
In your case, you are selling and delta hedging the option using the implied volatility $\sigma_{(i)}$ while the actual volatility of the underlying asset is $\sigma_{(r)}$. Your portfolio profit and loss is given by the sum of value changes of (i) the derivative and (ii) the hedging position, both using the implied volatility. We have
\begin{equation} \mathrm{d} \Pi_t = \mathrm{d}V_t^{(i)} - \Delta_t^{(i)} \mathrm{d}S_t - r \left( V_t^{(i)} - \Delta_t^{(i)} S_t \right) \mathrm{d}t. \end{equation}
Since
\begin{equation} \mathrm{d}V_t^{(i)} = \frac{\partial V^{(i)}}{\partial t} \mathrm{d}t + \underbrace{\frac{\partial V^{(i)}}{\partial S}}_{= \Delta^{(i)}} \mathrm{d}S_t + \frac{1}{2} \underbrace{\frac{\partial^2 V^{(i)}}{\partial S}}_{=\Gamma^{(i)}} \mathrm{d} \langle S \rangle_t, \end{equation}
we get
\begin{equation} \mathrm{d}\Pi_t = \left( \frac{\partial V^{(i)}}{\partial t} + \frac{1}{2} \sigma_{(r)}^2 S_t^2 \frac{\partial^2 V^{(i)}}{\partial S^2} - r \left( V_t^{(i)} - \Delta_t^{(i)} S_t \right) \right) \mathrm{d}t. \end{equation}
Now we use that $V^{(i)}$ satisfies the Black-Scholes PDE
\begin{equation} \frac{\partial V^{(i)}}{\partial t} + r S_t \underbrace{\frac{\partial V^{(i)}}{\partial S}}_{=\Delta^{(i)}} + \frac{1}{2} \sigma_{(i)}^2 S_t^2 \underbrace{\frac{\partial^2 V^{(i)}}{\partial S^2}}_{=\Gamma^{(i)}} - r V^{(i)} = 0 \end{equation}
to obtain
\begin{equation} \mathrm{d}\Pi_t = \frac{1}{2} \left( \sigma_{(r)}^2 - \sigma_{(i)}^2 \right) S_t^2 \Gamma^{(i)} \mathrm{d}t. \end{equation}
I.e. over each short interval, your profit and loss is proportional to the difference in realized to implied variance times the current gamma (computed using the implied volatility). While $\mathrm{d}\Pi_t$ is deterministic, the overall hedging profit and loss over the lifetime of the option is path-dependent. Its absolute value is higher for paths that fluctuate around the strike (where the gamma is higher).
References
Ahmad, Riaz and Paul Wilmott (2005) "Which Free Lunch Would You Like Today, Sir? Delta Hedging, Volatility Arbitrage and Optimal Portfolios," Wilmott Magazine, available here
Carr, Peter (2005) "FAQs in Option Pricing Theory", Working Paper, available here
Wilmott, Paul (2006) Paul Wilmott on Quantitative Finance, Vol. 1: Wiley, 2nd EditionShown 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.