Estimating Delta-Hedged Option P&L from Theta and Volatility Moves

# Using Theta to Approximate the PNL of a Delta-Hedged Option Position


# Using Theta to Approximate the PNL of a Delta-Hedged Option Position












A long time ago, I was taught that there is a rule-of-thumb approximation of the 1-day PNL of a delta-hedged option position that uses theta. I searched the internet and couldn't find it, so I did some experimentation and found something that works.

In particular, let $n$ be the number of standard deviations that the underlying moves in a day (positive for gains, negative for losses). Let $\theta$ be the one-day theta of the option in question. Then the following equation seems to approximate the one-day PNL of a delta-hedged option position:

\begin{align*} \big(-\theta \cdot \,2^{(|n|)}\big) + 2\theta \end{align*}

I wrote some code to compare this approximation to the actual PNL as well as the canonical gamma-theta Taylor series approximation and it lines up well, especially for at-the-money options. Here is the graph comparing the three:

One of the reasons that this approximation is handy, is that it gives a succinct way of describing the PNL implications of your gamma position. I always found gamma hard to interpret from a PNL standpoint, and this approximation bridges that gap.

So, my question: is anyone else familiar with this approximation? And if so, is there any intuitive explanation of its theoretical underpinnings.

## Answer by Kermittfrog (score 2)

https://quant.stackexchange.com/a/82312

In the Black Scholes world, the PnL of a delta-hedged short call option is approximated by:

$$ \mathrm{PnL}\approx -\left(\frac{\partial C}{\partial t}\Delta t + \frac{1}{2}\frac{\partial^2 C}{\partial S^2}(\Delta S)^2\right)\tag{1} $$

Assuming no dividends, the call option greeks are

$$ \begin{align} \frac{\partial C}{\partial t}&\equiv \Theta=-S\sigma\frac{n(d_1)}{2\sqrt{T-t}}-rKe^{-r(T-t)}N(d_2)\tag{2}\\ \frac{\partial^2C}{\partial S^2}&\equiv\Gamma=\frac{n(d_1)}{S\sigma\sqrt{T-t}}\tag{3}\\ \end{align} $$ For interest rates close to zero, or OTM options, or options with short term to maturity, $\Theta$ is dominated by the first term,

$$ \Theta\approx -S\sigma\frac{n(d_1)}{2\sqrt{T-t}}\tag{4} $$

Substituting into $(3)$:

$$ \begin{align} \Gamma&\approx-2\frac{\Theta}{S^2\sigma^2}\tag{5} \end{align} $$

Finally, plug $(4)$ and $(5)$ into $(1)$ and rearrange:

$$ \begin{align} \mathrm{PnL}&\approx-\left(\Theta\Delta t -\Theta \frac{\Delta S^2/S^2}{\sigma^2}\right)\\ &=-\left(\Theta\Delta t -\Theta \frac{\Delta S^2/S^2}{\sigma^2\Delta t}\Delta t\right)\\ &=-\Theta\left(1 - k^2\right)\Delta t\tag{6} \end{align} $$

where we define $k\equiv \frac{\Delta S/S}{\sigma\sqrt{\Delta t}}$ as the relative return per unit of correspondingly time-scaled volatility in equation $(6)$.

From $(4)$ we find that that $\Theta<0$. Thus we have that as long as the underlying moves by less than a (time scaled) "standard deviation", $|k|\leq 1 \Leftrightarrow |dS/S| < \sigma\sqrt{\Delta t} $, the position will have a positive PnL.

Finally, we could also reformulate the pnl approximation in terms of $\Gamma$ instead of $\Theta$. From $(4)$, we find $\Theta\approx-\frac{1}{2}\Gamma S^2\sigma^2$, and thus:

$$ \begin{align} \mathrm{PnL}&\approx-\left(-\frac{1}{2}\Gamma S^2\sigma^2 \Delta t + \frac{1}{2}\Gamma \Delta S^2\right)\\ &=\frac{1}{2}\Gamma S^2\sigma^2\Delta t \left(1 - k^2\right) \end{align} $$

with $k$ as above.