Choosing Time Bumps and Conventions for Option Greeks

# Accurate calculation of time involving Greeks for practictioners


# Accurate calculation of time involving Greeks for practictioners












I am working as quantitative developer and currently I am reworking the numerical calculation of options greeks within our software. While doing so, I wondered the following: when calculating derivatives as a mathematician I would take very small steps sizes for a close approximation of the derivative by the difference quotient. When considering time derivatives, though, I think it might me reasonable from the point of view of a trader/hedger to also take into consideration what the next day(s) happens. For example, for Asian options on the next day there might be a change in the current realized average, which is used in the pricing model my company is using, when the next day is a settlement day for the Asian option.

Thus, my question is: is the mathematical point of view still valid or are bigger step sizes reasonable. And if so: is it still accurate to use closed formulas for time involving Greeks, where they exists, or should they always calculated numerically by bumping one day or so?

## Answer by Dimitri Vulis (score 2, accepted)

https://quant.stackexchange.com/a/80834

Suppose, for concreteness, that you have some value $V$ that depends, among other things, on time $t$ and on interest rate $r$ that has term structure.

For example, $V$'s sensitivity to changes in $r$s are traditionally reported in terms of 1 basis point $r$ move. If $r$ is high enough (tens of $\%$ are common in some emerging markets), a 1bp bump may be lost in numerical noise. Instead, you may need to perturb $r$ by a larger bump to calculate the sensitivity and then scale down the impact to 1 bp. Conversely, it is unlikely, but possible that $r$s are so close to 0, that bumping a single tenor by 1bp would be problematic, so you need to bump by less than 1bp and then scale up to 1bp for reporting.

The traditional time horizons for reporting P&L projected from passage of time / carry / roll-down are 1 calendar day and 6 months. I don't know if high-frequency trading folks look at less than 1 day time step.

As above, suppose you bump the date by 1 business day on a Friday. You're now re-pricing as of Monday. In order for the Friday results to be consistent with the Thursday results, you probably want to divide the impact by the number of calendar days, i.e. 3.

There are two basic ways to assume what happens to $r$ as time passes, and you may want to support both. One is, assume that the rates don't change: the 1 month rate today remains unchanged the next day. The other assumes that the forwards get realized.

The cross-gamma between 1-day $t$ and (parallel shift) $r$ may be useful, for example, you can include it in the P&L explain in order to reduce the unexplained P&L; you can use the cross-gammas to see how much the passage of $t$ affects $V$'s sensitivity to $r$.

Edit: Most people want to see their market risk figures in the form of P&L impact projected from standardized perturbations / risk scenarios - e.g. some rate moves 1bp or 2 standard deviation of a historical principal component or some other large move under a stress scenario. However some people like to see the same information presented in other equivalent ways - in addition, rather than instead of sensitivities - for example, key rate durations, or the number of commodity futures contracts needed to hedge the given dollar sensitivity to commodity price, or CDS notional needed to hedge a sensitivity to credit spread. While aggregating interest rate risk from multiple portfolios and finding optimal hedges using IR futures and swaps may be hard, showing a simplistic hedge ratio is less hard. It is nice to have such capabilities available in a risk library.

You want to allow your library's callers to calculate what they want, but be transparent about your methodology. You don't want the users of your library to get confused about what bump size was used, or how the impact was rescaled.

For example, the GIRR spec in Basel Committee on Banking Supervision - FRTB SA https://www.bis.org/basel_framework/chapter/MAR/21.htm $\S$21.19, asks to divide the impact of bumping $r$ by the bump size, so effectively it is scaled to a 100% bump, i.e. 10,000 bps. This isn't "wrong" - just less usual than showing a 1bp impact.

Likewise $\S$21.23--24 asks to divide the impact of bumping respectively commodity price and currency exchange rate by the bump size, this rescaling to 100% bump. For instruments having non-linear sensitivity, such as options, this figure may be very different from actually using a 100% bump or rescaling from a closed-form instantaneous derivative or from a bump other than 1%. It is helpful for a library to be very transparent what bump size was used, how the impact was rescaled, and to allow callers to calculate sensitivities to a variety of different bump sizes.